Resolution over Linear Equations and Multilinear Proofs

RESOLUTION O VER LINEAR EQUA TIONS AND MUL TILINEAR PR OOFS RAN RAZ AND IDDO TZAMERET Abstra ct. W e develop and study the complexit y of prop ositional proof systems of v arying strength extending resolution by a llo wing it to operate with disjunctions of linear equations instead of clauses. W e demonstrate p olynomial-size refutations for hard t autologi es like the pigeonhole prin- ciple, Tseitin graph tautologies and the clique-coloring tautologies in these pro of systems. Using the (mo notone) in terp olatio n by a comm unication game technique we establish an exponential-size lo w er b ound on refutations in a certain, considerably strong, frag ment o f resolution o ver linear equations, as w ell as a general p olynomial upper bound on (non-monotone) in terpolants in th is fragmen t. W e then apply t hese results to extend and impro ve previous results on multili near pro ofs (o ver ﬁelds of characteri stic 0), as studied in [ R T06 ]. Sp eciﬁcally , w e sho w the follo wing: • Proofs op erating with depth-3 multilinear form ulas p olynomially simulate a certain, consid- erably strong, fragmen t of resolution o ver l inear equ ations. • Proofs op erating with depth-3 multilinear form ulas admit p olynomial-size refutations of t he pigeonhole principle and Tseitin graph tautolog ies. The former imp ro v e o ver a p revious result that established small multilinear p roofs only for the functional pigeonhole principle. The latter are diﬀerent than p revious pro ofs, and apply to multilinear pro ofs of Tseitin mo d p graph tautologies ov er an y ﬁeld of characteristi c 0. W e conclude by co nnecting resolution o ver linear equations with extensions of the cutting planes proof system. Contents 1. In tro duction 2 1.1. Comparison to Earlier W ork 4 1.2. Summary of Resu lts 5 2. Notatio n and Ba c kground on Prop ositional Pro of Systems 7 3. Resolution o v er L in ear E quations and its Subsys tems 8 3.1. Disjunctions of Linear Equ ations 8 3.2. Resolution o ver Linear Equations – R(lin) 9 3.3. F ragment of Resolution o v er Linear Equations – R 0 (lin) 10 4. Reasoning and C oun ting in side R(lin) and its Subsystems 11 4.1. Basic Reasoning inside R(lin) and its Subsystems 12 4.2. Basic Coun ting inside R(lin) and R 0 (lin) 13 5. Implicational Completeness of R(lin) and its Subsystems 16 6. Short Pro ofs for Hard T autologies 17 6.1. The Pigeonhole Principle T autologie s in R 0 (lin) 17 6.2. Tseitin mo d p T au tologies in R 0 (lin) 19 2000 Mathematics Subje ct Classiﬁc ation. 03F20, 68Q17, 68Q15. Key wor ds and phr ases. proof complexity , resolution, algebraic pro of systems, multilinear pro ofs, cu tting planes, feasible monotone interpolation. The ﬁrst author was supported by The Israel Science F oundation and The Minerva F ound ation. The second author was supp orted b y The Israel Science F oundation (grant no. 250/0 5). 1 6.3. The Clique-Coloring Principle in R(lin) 23 7. In terp olation Results f or R 0 (lin) 26 7.1. Int erp olation for Semantic Refutatio ns 26 7.2. P olynomial Upp er Bound s on Inte rp olan ts for R 0 (lin) 28 8. Size Lo wer Bounds 30 9. Applications to Multilinear Pr oofs 32 9.1. Bac kground on Algebraic and Multilinear Pro ofs 32 9.2. F rom R(lin) Pro ofs to PC R Pro ofs 34 9.3. F rom PCR Pro ofs to Multilinear Pro ofs 35 9.4. Small Depth-3 Multilinear Pr oofs 38 10. Relations with Extensions of Cutting Planes 38 App endix A. F easible Monotone Int erp olation 41 Ac kno w ledgmen ts 42 References 42 1. Introduction This pap er considers t w o kind s of pr oof systems. The ﬁ r st kind are extensions of resolution that op erate with d isjunctions of linear equations with in tegral co eﬃcien ts in stead of clauses. The second kind are algebraic pro of systems op er ating w ith multilinea r arithmetic form ulas. Pro ofs in b oth kind s of systems establish the u nsatisﬁabilit y of form ulas in conju nctiv e norm al f orm (CNF). W e are primarily concerned w ith connections b et w ee n these t w o f amilies of pro of systems and with extending and imp ro v in g p revious r esults on m ultilinear pro ofs. The resolution system is a p opu lar prop ositional pro of sys tem that establishes the un satisﬁabilit y of CNF formula s (or equiv alently , the truth of tautologies in disj unctiv e normal f orm) by op erating with clauses (a clause is a d isjunction of pr op ositional v ariables and their negati ons). It is well kno wn that resolution cannot p ro vide small (that is, p olynomial-size ) p ro ofs for many basic co unt- ing argument s. The most notable example of this are the s trong exp onen tial lo wer b ound s on the resolution refutation size of the pigeonhole principle and its diﬀeren t v arian ts (Hak en [ Hak85 ] w as the ﬁ rst to establish such a low er b ound; see also [ Razb02 ] f or a su r v ey on th e pro of complexit y of the pigeonhole p rinciple). Due to th e p opularity of resolution b oth in practice, as the core of many automated theorem p ro v ers, and as a theoretical case-study in prop ositional pr oof complexit y , it is natur al to consider w eak extensions of resolution that can o v ercome its ineﬃciency in pro vid- ing pr oofs of coun ting argument s. The p ro of sys tems w e present in this pap er are extensions of resolution, of v arious s trength, that are suited for this pu rp ose. Prop ositional pro of systems of a diﬀeren t nature that also attracted muc h atten tion in p ro of complexit y theory are algebr aic pr o of systems , which are pr oof systems op erating with (multiv ariate) p olynomials o v er a ﬁeld. In this pap er, w e are particularly in terested in algebraic pro of systems that op erate with m ultilinear p olynomials represent ed as m ultilinear arithmetic form ulas, called by the generic name multiline ar pr o ofs (a p olynomial is multiline ar if the p o w er of eac h v ariable in its monomials is at most one). The in vestiga tion in to su c h pro of systems w as initiated in [ R T06 ], and here we contin u e this line of r esearc h. This researc h is motiv ated on the one hand b y the apparen t considerable strength of such systems; and on the other hand, by the kno wn s u p er-p olynomial size lo w er b ounds on m ultilinear formulas computing certain imp ortan t functions [ Raz04 , Raz06 ], com bined with the general working assum ption that establishing lo w er b ound s on the size of obje cts a pro of sy s tem manipulates (in this case, multil inear formulas) is close to establishing lo wer b ound s on the size of the pr o ofs themselv es. 2 The basic pro of system w e sh all study is denoted R(lin). The pr oof-lines 1 in R(lin) pro ofs are disjunctions of lin ear equations with integral co eﬃcien ts ov er the v ariables ~ x = x 1 , . . . , x n . It turns out that (already p rop er su bsystems of ) R(lin) can hand le very elegan tly basic coun ting argumen ts. The follo wing deﬁnes the R(lin) pro of system. Giv en an initial CNF, we translate ev ery clause W i ∈ I x i ∨ W j ∈ J ¬ x j (where I are the ind ices of v ariables with p ositiv e p olarities an d J are the indices of v ariables with negativ e p olarities) p ertaining to the CNF, in to the disju nction W i ∈ I ( x i = 1) ∨ W j ∈ J ( x j = 0). Let A and B b e t w o disjunctions of linear equations, and let ~ a · ~ x = a 0 and ~ b · ~ x = b 0 b e tw o linear equatio ns (where ~ a, ~ b are t w o v ecto rs of n in tegral coeﬃcien ts, and ~ a · ~ x is the scalar pro du ct P n i =1 a i x i ; and similarly for ~ b · ~ x ). The ru les of inf er en ce b elonging to R(lin) allo w to deriv e A ∨ B ∨ (( ~ a + ~ b ) · ~ x = a 0 + b 0 ) from A ∨ ( ~ a · ~ x = a 0 ) and B ∨ ( ~ b · ~ x = b 0 ) (or similarly , to derive A ∨ B ∨ (( ~ a − ~ b ) · ~ x = a 0 − b 0 ) from A ∨ ( ~ a · ~ x = a 0 ) and B ∨ ( ~ b · ~ x = b 0 )). W e can also simplify d isj unctions by d iscarding (un satisﬁable) equations of the form (0 = k ), for k 6 = 0. In addition, for ev ery v ariable x i , w e shall add an axiom ( x i = 0) ∨ ( x i = 1), whic h forces x i to tak e on only Bo olean v alues. A deriv ation of the emp ty disju nction (whic h stands for f alse ) from the (trans late d) clauses of a CNF is called an R(lin) r efutation of the giv en CNF. T h is wa y , ev ery unsatisﬁable CNF has an R(lin) refutation (this can b e pr o ved b y a straigh tforw ard simulati on of resolution b y R(lin)). The basic idea connecting r esolution op erating w ith d isjunctions of lin ear equations and m ultilin- ear pro ofs is this: Whenev er a disjunction of linear equations is simple enough — and sp eciﬁcally , when it is close to a symmetric f unction, in a mann er made precise — then it can b e repr esen ted b y a small size and small d epth multilinear arith m etic form ula ov er ﬁ elds of c haracteristic 0. This idea was already u sed (somewhat implicitly) in [ R T06 ] to obtain p olynomial-size multili near pro ofs op erating w ith depth-3 m ultilinear form ulas of the functional p igeo nhole p r inciple (this principle is wea k er than the p igeonhole principle). In the curren t pap er we generalize previous results on m ultilinear pro ofs by fu lly using this idea: W e sh o w ho w to p olynomially simulate with multilinea r pro ofs, oper ating with small depth multilinear formulas, certa in short pro ofs carr ied in s ide resolu- tion o v er linear equations. This enables us to pro vide n ew p olynomial-size multi linear pro ofs for certain hard tautol ogies, improving results from [ R T06 ]. More sp eciﬁcally , we introd u ce a certain fragmen t of R(lin), whic h can b e p olynomially s im u- lated b y depth-3 multi linear p ro ofs (that is, multilinear pro ofs op erating with depth-3 multil inear form ulas). On the one hand this fragment of resolution o v er linear equ atio ns already is suﬃcien t to formalize in a transparent w a y basic counting arguments, and so it ad m its sm all pro ofs of the pigeonhole principle and the Tseitin mo d p formulas (whic h yields some new upp er b ounds on m ultilinear pro ofs); and on the other hand w e can use the (monotone) in terp olation tec hnique to establish an exp onentia l-size lo wer b ound on refutations in this fragmen t as we ll as d emonstrating a general (non-monotone) p olynomial upp er b ound on in terp olan ts for this fragmen t. The p ossibilit y that multilinear p ro ofs (p ossib ly , op erating with dep th -3 multilinear form ulas) p ossess the feasible monotone in terp olat ion prop erty (and hence, adm it exp on ential- size lo w er b ounds) remains op en. Another f amily of p rop ositional p ro of systems we discu s s in relation to the systems menti oned ab o v e are the cutting planes system and its extensions. The cutting planes pro of system op erates with linear ine qu alities with in tegral co eﬃcien ts, and this system is very close to the extensions of resolution we presen t in th is pap er. In particular, the follo wing simple observ atio n can b e used to p olynomially simula te cutting planes pro ofs with p olynomially b ound ed coeﬃcien ts (and some of its extensions) inside resolution o v er lin ear equations: The truth v alue of a linear inequ alit y ~ a · ~ x ≥ a 0 (where ~ a is a v ector of n in tegral co eﬃcien ts and ~ x is a v ect or of n Bo ole an v ariables) is 1 Eac h element (usually a form ula) of a pro of-sequence is referred to as a pr o of-line . 3 equiv alen t to the tru th v alue of the follo w in g disjunction of linear equalities: ( ~ a · ~ x = a 0 ) ∨ ( ~ a · ~ x = a 0 + 1) ∨ · · · ∨ ( ~ a · ~ x = a 0 + k ) , where a 0 + k equ als the sum of all p ositive co eﬃcien ts in ~ a (that is, a 0 + k = max ~ x ∈{ 0 , 1 } n ( ~ a · ~ x )). Note on terminology . All the pro of systems considered in this pap er in tend to pr ov e the unsatis- ﬁability o ver 0 , 1 v alues of collections of clauses (p ossib ly , of translation of the clauses to disj u nctions of linear equ atio ns). In other wo rds, pro ofs in such pro of systems intend to r efute the collections of clauses, whic h is to v alidate their n egati on. Therefore, thr oughout this pap er we shall sometime sp eak ab out r efutations and p ro ofs interc hangeably , alw a ys intending refutations, u nless otherwise stated. 1.1. Comparison t o Earlier W ork. T o th e b est of our kno wledge th is pap er is the ﬁrst that considers resolution pro ofs op erating with disjunctions of linear e quations . Previous w orks consid- ered extensions of resolution ov er lin ear ine qualities augmente d with the cutting planes inference rules (the resulting p r oof system denoted R (C P )). In fu ll generalit y , we sho w that resolution o v er linear equations can p olynomially sim ulate R(CP ) wh en the co eﬃcien ts in all the inequalities are p olynomially b ounded (ho w ev er, th e con v erse is not known to h old). On the other h an d , w e shall consider a certain fr agmen t of resolution ov er linear equations, in which we do n ot eve n kno w ho w to p olynomially sim ulate cutting planes p ro ofs with p olynomially b ound ed coeﬃcients in inequ aliti es (let alone R(CP) w ith p olynomially b ounded co eﬃcien ts in inequalities). W e no w shortly discuss the previous work on R(CP) and relate d pr oof systems. Extensions of resolution to d isjunctions of linear ine qualities w er e ﬁ rst considered by Kra j ´ ı ˇ cek [ Kra98 ] who deve lop ed the pr oof sy s tems LK(C P ) and R(CP). The L K (CP) sy s tem is a ﬁrst-order (Gen tzen-st yle) sequ en t calculus that op erates with linear in equ alitie s in s tead of atomic f orm ulas and augments the standard ﬁrst-order sequ ent calculus inference rules with the cu tting planes inference r ules. The R(CP) pr o of s y s tem is essent ially r esolution o v er linear inequalities, th at is, resolution that operates with disjunctions of linear inequalities instead of clauses. The main motiv ation of [ Kra98 ] is to extend the feasible in terp olation tec hnique and consequently the lo w er b ounds results, fr om cutting planes and resolution to stronger p ro of systems. That pap er establishes an exp onent ial-size lo wer b ound on a restricted v ersion of R(CP) p ro ofs, namely , when the n umber of inequalities in eac h p r oof-line is O ( n ε ), w here n is the num b er of v ariables of the initial form ulas, ε is a small enough constan t and the coeﬃcient s in the cutting planes in equ alitie s are p olynomially b ounded. Other pap ers considering extensions of resolution o v er linear inequalities are the more recen t pap ers by Hirsch & Ko jevniko v [ HK06 ] and Ko jevnik o v [ Ko j07 ]. The ﬁ rst pap er [ HK06 ] considers a com bination of resolution with LP (an in complete subs ystem of cutting planes based on s imple linear programming reasoning), with the ‘lift and p ro ject’ p ro of system (L&P), and w ith th e cutting planes p ro of system. The second pap er [ Ko j 07 ] deals with imp ro v in g the parameters of the tree-lik e R(CP) lo wer-boun ds obtained in [ Kra98 ]. Whereas pr evious results concerned pr imarily with extendin g the cutting planes pr oof system, our foremost motiv ation is to extend and impro v e p revious r esu lts on alge braic pro of systems op erating w ith multilinear formulas ob tained in [ R T06 ]. In th at pap er the concept of multili near pro ofs was in tro duced and sev eral basic results concerning multilinear p r oofs w ere prov ed. In particular, p olynomial-size pro ofs of t w o imp ortan t com binatorial principles w ere demonstrated: the functional pigeonhole prin ciple and the Ts eitin (mod p ) graph tautologies. I n the current pap er w e impro v e b oth these results. As ment ioned ab o v e, motiv ated by relations with multilinear pro ofs op erating with d epth-3 m ul- tilinear form ulas, w e s h all consid er a certa in su bsystem of resolution o v er linear equations. F or this su bsystem we apply t w ice the in terp olation by a comm unicatio n game tec hniqu e. Th e ﬁrst 4 application is of the non -monotone v ersion of the tec hnique, and the second application is of the monotone ve rsion. Namely , the ﬁrst application pro vides a general (non-mon otone) in terp olation theorem that demonstrates a p olynomial (in the size of refutations) upp er b ound on in terp olan ts; The pro of uses the general metho d of transform ing a refutation in to a Karc hmer-Wigderson com- m unication game for t w o play ers, from wh ic h a Bo olean circuit is then attainable. In particular, w e shall app ly the in terp olation theorem of Kra j ´ ı ˇ cek from [ Kra97 ]. The second application of the (monotone) in terpolation b y a communicatio n game tec hniqu e is imp licit and p ro ceeds b y using the lo w er b ound criterion of Bonet, Pitassi & Raz in [ BPR97 ]. Th is criterion states that (semant ic) pro of sy s tems (of a certain natural and standard kind) whose p ro of-lines (considered as Bo olean functions) h a ve lo w communicatio n complexit y cannot prov e eﬃcien tly a certain tautology (namely , the clique-col oring tautologies). 1.2. Summary of Results. T his pap er introd uces and connects sev eral new concepts and ideas with some known ones. I t identiﬁes new extensions of resolution op erating with linear equations, and relates (a certain) such extension to m ultilinear pro ofs. The up p er b ounds for the pigeonhole principle and Tseitin mo d p form ulas in fragments of resolution o ver linear equations are new. By generalizing the mac hinery deve lop ed in [ R T06 ], these upp er b ounds yield new and impro v ed re- sults concerning m ultilinear p ro ofs. Th e lo wer b ound for the clique-coloring formulas in a fr agmen t of r esolution ov er linear equ atio ns emplo ys the s tand ard m onotone interp olation b y a communica- tion game tec hnique, an d sp eciﬁcally utilizes the theorem of Bonet, Pitassi & Raz from [ BPR97 ]. The general (non-monotone) in terp olatio n result for a fragmen t of resolution ov er linear equations emplo ys the theorem of Kra j ´ ı ˇ cek f r om [ Kra97 ]. The upp er b ound in (the stronger v arian t of – as d escrib ed in the in trod uction) resolution o v er linear equ atio ns of the clique-c oloring formula s follo w s that of A tserias, Bonet & Esteban [ ABE02 ]. W e no w giv e a detailed outline of th e results in this pap er . The pro of systems. In Section 3 w e formally deﬁne t wo extensions of resolution of decreasing strength allo wing r esolution to op erate w ith disjun ctions of linear equations. The size of a linear equation a 1 x 1 + . . . + a n x n = a 0 is the sum of all a 0 , . . . , a n written in unary notation . The size of a disjun ction of linear equations is the tota l size of all linear equatio ns in the disjun ction. The size of a pr oof op erating with d isjunctions of linear equations is the total size of all the disjunctions in it. R(lin): This is the stronger pr oof system (describ ed in the in tro duction) that op erates with disjunctions of lin ear equ ations with in teger coeﬃcients. R 0 (lin): T h is is a (prov ably p r op er) fragment of R(lin). It op erates with disju nctions of (arbi- trarily many) linear equations whose v ariables ha v e constan t co eﬃcien ts, und er the restriction th at ev ery d isjunction can b e partitioned into a constan t num b er of sub-d isjunctions, where eac h sub - disjunction either consists of lin ear equ ations that diﬀer only in their free-terms or is a (trans latio n of a) clause. Note that any single linear ine quality with Bo olean v ariables can b e represented by a disj u nction of linear equations that diﬀer only in their f ree-terms (see the example in the in tro duction section). So the R 0 (lin) pro of system is close to a pro of system op erating with disjun ctions of constant n um b er of lin ear inequ alitie s (with constan t in tegral co eﬃcients). In f act, disju nctions of linear equations v arying only in their free-terms, ha v e more (expr essiv e) strength than a s in gle inequalit y . F or instance, th e p ar ity f unction can b e easily repr esen ted b y a d isjunction of lin ear equ atio ns, while it cannot b e represent ed b y a single linear inequalit y (or eve n by a d isj unction of linear inequalities). As already men tioned, the motiv ation to consider the restricted pro of system R 0 (lin) comes from its relation to m ultilinear pro ofs op erating with depth-3 multil inear formulas (in short, depth-3 5 m ultilinear pro ofs): R 0 (lin) corresp onds roughly to the sub system of R(lin) th at w e kno w how to simulate by depth -3 m ultilinear pro ofs via the tec hnique in [ R T06 ] (the tec hnique is based on con v erting d isjunctions of linear forms int o symmetric p olynomials, w hic h are kno wn to ha v e small depth-3 multilinea r form ulas). This sim ulation is then applied in order to imp ro v e o v er known upp er b ounds for depth-3 multilinear p r oofs, as R 0 (lin) is already suﬃcien t to eﬃcien tly prov e certain “hard tautologies”. Moreo v er, we are able to establish an exp onentia l lo w er b ou n d on R 0 (lin) refutations size (see b elow for b oth upp er and lo w er b ounds on R 0 (lin) pro ofs). W e also establish a sup er-p olynomial separation of R(lin) from R 0 (lin) (via th e clique-coloring principle, for a certain choic e of parameters; see b elo w). Short refut ations. W e demonstrate the follo wing short refutations in R 0 (lin) and R(lin): (1) P olynomial-siz e r efu tatio ns of the pigeo nhole principle in R 0 (lin); (2) P olynomial-siz e r efu tatio ns of Tseitin mod p graph formulas in R 0 (lin); (3) P olynomial-siz e refutations of the clique-coloring form ulas in R(lin) (for certain parameters). The refutations h ere follo w by direct simulat ion of the Res(2) r efu tatio ns of clique-coloring form ulas from [ ABE02 ]. All th e thr ee f amilies of f orm ulas ab ov e are pr omin en t “hard tautologies” in pro of complexit y literature, w h ic h means that strong size lo w er b ound s on pr oofs in v arious pro of systems are kno wn for them (for the exact form ulatio n of these families of formulas see Section 6 ). In terp olation results. W e provide a p olynomial upp er-b oun d on (n on-monotone) in terp olan ts corresp onding to R 0 (lin) refu tatio ns; Namely , we sho w that an y R 0 (lin)-refutation of a give n form ula can b e tr ansformed into a (non-monotone) Bo olean circuit compu ting the corresp ondin g int erp olan t function of the formula (if there exists su ch a f unction), with at most a p olynomial increase in size. W e employ the general in terp olatio n theorem of Kra j ´ ı ˇ cek [ Kra97 ] for semantic p ro of systems. Lo w er b ounds. W e pro vide the follo wing exp onential lo wer b oun d: Theorem 1. R 0 (lin) do es not have sub-exp onential r efutations for the cliqu e -c oloring formulas. This result is p ro v ed b y app lyin g a result of Bonet, Pitassi & Raz [ BPR97 ], that (implicitly) us e the monotone interpolation by a comm unication game tec hnique f or establishing an exp onentia l- size low er b ound on refutations of general seman tic pro of sys tems op erating with pro of-lines of low comm unication complexit y . Applications to m ultilinear pro ofs. Multilinear pro of sys tems are (seman tic) refutation sys- tems op erating with multilinear p olynomials ov er a ﬁ xed ﬁeld, wher e ev er y multil inear p olynomial is represent ed b y a m ultilinear arithmetic form ula. In this pap er w e sh all consider multilinear form ulas o ver ﬁelds of charact eristic 0 only . The size of a multilinear pro of (that is, a pr o of in a multilinear pro of system) is the tota l size of all multilinea r form ulas in th e pro of (for formal deﬁnitions concerning m ultilinear p ro ofs see Section 9 ). W e shall ﬁrst connect m ultilinear pro ofs with resolution o v er linear equations by the follo wing result: Theorem 2. M ultiline ar pr o ofs op er ating with depth- 3 multiline ar formulas over char acteristic 0 p olynomial ly-simulate R 0 (lin) . An immediate coroll ary of this theorem and the upp er b ounds in R 0 (lin) d escrib ed ab o v e are p olynomial-size m ultilinear p ro ofs for the pigeo nhole pr in ciple and the Tseitin m od p form ulas. (1) P olynomial-siz e d epth-3 m ultilinear refutations for the pigeonhole principle o ver ﬁelds of c haracteristic 0. This impr o ves o v er [ R T06 ] that s h o w s a similar upp er b ound for a we ak er principle, namely , the functional pigeonhole principle. (2) P olynomial-siz e d ep th-3 m ultilinear refutations for the Ts eitin mo d p graph formulas o v er ﬁelds of c haracte ristic 0. T hese refutations are diﬀerent than those demonstrated in [ R T06 ], 6 and further they establish short m ultilinear refu tatio ns of the Tseitin mo d p graph formulas o v er any ﬁeld of char acteristic 0 (the pro of in [ R T06 ] show ed ho w to refute the Tseitin m o d p f ormulas by m ultilinear refutations only o v er ﬁelds that con tain a pr imitiv e p th ro ot of unit y). Relations with cutting planes pro ofs. As men tioned in the introdu ctio n, a pro of system com- bining resolution with cutting planes w as pr esen ted b y Kra j ´ ı ˇ cek in [ Kra98 ]. Th e r esulting system is denoted R(CP) (see Section 10 for a deﬁnition). When the co eﬃcien ts in the linear inequalities inside R(CP) pro ofs are p olynomially b ounded, the resulting p r oof system is denoted R(CP*). W e establish the follo wing simulati on result: Theorem 3. R(lin) p olynomial ly simulates r esolution over cutting planes ine qualities with p olyno- mial ly b ounde d c o eﬃcients R(CP*) . W e do not kn o w if the con verse also holds. 2. Not a tion and Backgr ound o n Propositional Proof S ystems F or a natural num b er n , w e u se [ n ] to denote { 1 , . . . , n } . F or a v ector of n (inte gral) coeﬃcien ts ~ a and a v ector of n v ariables ~ x , w e den ote by ~ a · ~ x the scalar pro du ct P n i =1 a i x i . If ~ b is another v ector (of length n ), then ~ a + ~ b denotes the add ition of ~ a and ~ b as ve ctors, and c ~ a (for an integ er c ) denotes the pro du ct of the scalar c with ~ a (where, − ~ a denotes − 1 ~ a ). F or t w o linear equations L 1 : ~ a · ~ x = a 0 and L 2 : ~ b · ~ x = b 0 , th eir addition ( ~ a + ~ b ) · ~ x = a 0 + b 0 is denoted L 1 + L 2 (and their subtraction ( ~ a − ~ b ) · ~ x = a 0 − b 0 is d enoted L 1 − L 2 ). F or t w o Bo olean assignments (iden tiﬁed as 0 , 1 strin gs) α, α ′ ∈ { 0 , 1 } n w e write α ′ ≥ α if α ′ i ≥ α i , for all i ∈ [ n ] (where α i , α ′ i are the i th b its of α and α ′ , r esp ectiv ely). W e n ow recall some basic concepts on prop ositional pro of systems. F or bac kground on algebraic pro of systems (and sp eciﬁcally m ultilinear pro ofs) see Section 9 . Resolution. In order to p ut our work in context, w e need to deﬁne the resolution refutation system. A CNF f orm ula o v er the v ariables x 1 , . . . , x n is d eﬁ ned as follo ws. A liter al is a v ariable x i or its n egat ion ¬ x i . A clause is a disjunction of literals. A CNF formula is a conjunction of clauses. The size of a clause is the num b er of literals in it. Resolution is a complete and sound pro of system for un satisﬁable C NF formulas. Let C and D b e t w o clauses con taining n either x i nor ¬ x i , the r esolution rule allo ws one to d eriv e C ∨ D from C ∨ x i and D ∨ ¬ x i . T he clause C ∨ D is called the r esolvent of the clauses C ∨ x i and D ∨ ¬ x i on the v ariable x i , and we also sa y that C ∨ x i and D ∨ ¬ x i w ere r esolve d over x i . The we akening rule allo ws to deriv e the clause C ∨ D fr om the clause C , for an y tw o clauses C, D . Deﬁnition 2.1 ( Resolution ) . A r esolution pr o of of the clause D fr om a CN F formula K is a sequence of clauses D 1 , D 2 , . . . , D ℓ , su c h that: (1) eac h clause D j is either a clause of K or a resolv en t of t w o previous clauses in the sequence or derived b y the w eak ening ru le from a previous clause in the sequence; (2) the last clause D ℓ = D . The si ze of a resolution p ro of is the sum of all the sizes of the clauses in it. A r esolution r efutation of a CNF formula K is a resolution pro of of the empty clause ✷ from K (the empt y clause s tand s for f al se ; that is, the empty clause has no satisfying assignmen ts). A pro of in resolution (or any of its extensions) is called also a derivation or a pr o of-se quenc e . Eac h sequence-ele men t in a pr oof-sequence is called also a pr o of-line . A pro of-sequence con taining the pro of-lines D 1 , . . . , D ℓ is also s aid to b e a derivation of D 1 , . . . , D ℓ . 7 Co ok-Rec kho w pro of systems. F ollo w ing [ CR79 ], a Co ok- R e ckhow pr o of system is a p olynomial- time algorithm A that receiv es a Bo olean form ula F (for in s tance, a CNF) and a string π o v er some ﬁnite alphab et (“the (prop osed) refutation” of F ), su c h that there exists a π with A ( F , π ) = 1 if and only if F is un satisﬁable. Th e c ompleteness of a (Co ok-Rec khow) pro of system (with r esp ect to th e set of all unsatisﬁable Bo olean formulas; or for a subset of it, e.g. the s et of unsatisﬁable CNF form ulas) stand s for the f act that ev ery unsatisﬁable form ula F has a s tr ing π (“the refutation of F ”) so that A ( F , π ) = 1. The soundness of a (Co ok-Rec kho w ) p r oof system stands for the fact that every form ula F so th at A ( F , π ) = 1 for some string π is unsatisﬁable (in other w ords, no satisﬁable form ula has a refutation). F or instance, resolution is a C ook-Rec kho w pro of system, since it is complete and sound for the set of unsatisﬁable CNF formulas, and giv en a CNF formula F and a strin g π it is easy to c hec k in p olynomial-time (in b oth F and π ) whether π constitutes a r esolution r efu tatio n of F . W e shall also consider pro of systems th at are not necessarily (that is, n ot known to b e) Co ok- Rec kho w p ro of systems. Sp eciﬁcally , m ultilinear p ro of systems (o ver large enough ﬁelds) meet the requirement s in the deﬁnition of C ook-Rec kho w pro of systems, exc ept that the condition on A ab o v e is r elaxed: w e allo w A to b e in pr ob abilistic p olynomial-time BPP (wh ich is not known to b e equal to deterministic p olynomial-time). P olynomial sim ulations of pro of systems. When comparing the strength of d iﬀeren t pro of systems w e shall conﬁne ourselves to CNF form ulas only . That is, w e consider p rop ositional pr oof systems as p ro of systems for the set of u n satisﬁable CNF formulas. F or that pu rp ose, if a pr oof system do es not op er ate with clauses dir ectly , then w e ﬁ x a (direct) tr anslatio n from clauses to the ob jects op er ated by the pro of system. This is d one for b oth r esolution o v er linear equations (whic h op erate with disju nctions of lin ear equations) and its fragmen ts, and also f or m ultilinear pro ofs (w h ic h op erate with multil inear p olynomials, represente d as m ultilinear formulas); see for example Subsection 3.1 for s uc h a direct translation. Deﬁnition 2.2. Let P 1 , P 2 b e tw o pro of systems for the set of unsatisﬁable CNF form ulas (we iden tify a CNF formula with its corresp onding translation, as discus s ed ab ov e). W e sa y th at P 2 p olynomial ly simulates P 1 if giv en a P 1 refutation π of a CNF formula F , then there exists a refutation of F in P 2 of size p olynomial in the size of π . In case P 2 p olynomially sim ulates P 1 while P 1 do es n ot p olynomially s imulates P 2 w e sa y that P 2 is strictly str onger than P 1 . 3. Resolution over Linear Equa tions and its Sub systems The pro of systems we consider in this section are extensions of r esolution. Pro of-lines in res- olution are clauses. Instead of this, th e extensions of resolution w e consider here oper ate with disjunctions of linear equ atio ns with in teg ral coeﬃcien ts. F or this section we use the conv ention that all th e form al v ariables in th e prop ositional pro of systems considered are tak en from the set X := { x 1 , . . . , x n } . 3.1. Disjunctions of L ine a r Equations. F or L a linear equation a 1 x 1 + . . . + a n x n = a 0 , the righ t hand side a 0 is called the fr e e-term of L and the left h and sid e a 1 x 1 + . . . + a n x n is called the line ar form of L (the linear form can b e 0). A disjunction of line ar e quations is of the follo wing general form:  a (1) 1 x 1 + . . . + a (1) n x n = a (1) 0  ∨ · · · ∨  a ( t ) 1 x 1 + . . . + a ( t ) n x n = a ( t ) 0  , (1) where t ≥ 0 and the co eﬃcien ts a ( j ) i are inte gers (for all 0 ≤ i ≤ n, 1 ≤ j ≤ t ). W e discard du plicate linear equations from a disjun ctio n of linear equations. The semanti cs of su ch a d isj unction is the natural one: W e sa y th at an assignment of in teg ral v alues to the v ariables x 1 , ..., x n satisﬁes ( 1 ) 8 if and only if there exists j ∈ [ t ] so that the equ atio n a ( j ) 1 x 1 + . . . + a ( j ) n x n = a ( j ) 0 holds un der the giv en assignmen t. The s ym b ol | = d enotes th e semantic implic ation relation, that is, for ev ery collec tion D 1 , . . . , D m of disjunctions of linear equations, D 1 , . . . , D m | = D 0 means that ev ery assignment of 0 , 1 v alues that s atisﬁes all D 1 , . . . , D m also satisﬁes D 0 . 2 In this case w e also sa y that D 1 , . . . , D m semantic al ly imply D 0 . The size of a line ar e q u ation a 1 x 1 + . . . + a n x n = a 0 is P n i =0 | a i | , i.e., the sum of the b it sizes of all a i written in unary n otat ion. Accordingly , the size of the line ar form a 1 x 1 + . . . + a n x n is P n i =1 | a i | . The size of a disjunction of line ar e quations is th e total size of all linear equations in it. Since all linear equations considered in this pap er are of in tegral co eﬃcien ts, w e shall sp eak of line ar e quations when we actually mean linear equations with integ ral coeﬃcien ts. Similar to resolution, the empty disjunction is un satisﬁable and stands for the truth v alue f alse . T ranslation of clauses. As describ ed in the introdu ctio n, we can translate any CNF form ula to a collectio n of disjun ctions of linear equations in a direct manner: Every clause W i ∈ I x i ∨ W j ∈ J ¬ x j (where I and J are sets of in dices of v ariables) p ertaining to th e C NF is translated into the disjunction W i ∈ I ( x i = 1) ∨ W j ∈ J ( x j = 0). F or a clause D we d enote by e D its translation into a disjunction of linear equations. It is easy to verify that an y Bo olean assignment to the v ariables x 1 , . . . , x n satisﬁes a clause D if and only if it satisﬁes e D (where true is treated as 1 and f alse as 0). 3.2. Resolution o v er Linear Equations – R(lin). Deﬁned b elo w is our b asic p r oof system R(lin) that enables resolution to r eason with disjunctions of linear equ atio ns. As we wish to reason ab out Bo olean v ariables w e augmen t the system with the axioms ( x i = 0) ∨ ( x i = 1), for all i ∈ [ n ], called the Bo ole an axioms . Deﬁnition 3.1 ( R(lin) ) . L et K := { K 1 , . . . , K m } b e a collectio n of d isjunctions of linear equations. An R(lin)-pr o of fr om K of a disjunction of line ar e quations D is a ﬁnite sequence π = ( D 1 , ..., D ℓ ) of disju nctions of linear equations, suc h that D ℓ = D and for ev ery i ∈ [ ℓ ], either D i = K j for some j ∈ [ m ], or D i is a Bo olean axiom ( x h = 0) ∨ ( x h = 1) for some h ∈ [ n ], or D i w as dedu ced b y one of the follo win g R(lin)-inference rules, using D j , D k for some j, k < i : Resolution: Let A, B b e t wo disjunctions 3 of lin ear equations and let L 1 , L 2 b e tw o linear equations. F rom A ∨ L 1 and B ∨ L 2 deriv e A ∨ B ∨ ( L 1 + L 2 ). Similarly , f r om A ∨ L 1 and B ∨ L 2 deriv e A ∨ B ∨ ( L 1 − L 2 ). W eak ening: F rom a disjun ction of lin ear equations A d er ive A ∨ L , where L is an arbitrary linear equation o ve r X . Simpliﬁcation: F rom A ∨ (0 = k ) d eriv e A , where A is a disjun ction of lin ear equations and k 6 = 0. An R(lin) r efutation of a collectio n of disj unctions of linear equations K is a p ro of of the empt y disjunction from K . Th e size of an R(lin)-pro of π is the total s ize of all the d isjunctions of linear equations in π , den oted | π | . Similar to resolution, in case A ∨ B ∨ ( L 1 + L 2 ) is d er ived from A ∨ L 1 and B ∨ L 2 b y the resolution r ule, we say that A ∨ L 1 and B ∨ L 2 w ere r esolve d over L 1 and L 2 , r esp ectiv ely , and we 2 Alternatively , we can consider assignments of any in tegral v alues (instead of only Bo olean va lues) to the v ariables in D 1 , . . . , D m , stipulating th at the collection D 1 , . . . , D m conta ins all disjunctions of the form ( x j = 0) ∨ ( x j = 1) for all the v ariables x j ∈ X (these formula s force any satisfying assignmen t to give only 0 , 1 v alues to the v ariables). 3 P ossibly the empty d isjunction. This remark also applies t o the inference rules b elo w. 9 call A ∨ B ∨ ( L 1 + L 2 ) th e r esolvent of A ∨ L 1 and B ∨ L 2 (and similarly , when A ∨ B ∨ ( L 1 − L 2 ) is d er ived f r om A ∨ L 1 and B ∨ L 2 b y the resolution ru le; we u se the same terminology f or b oth addition and subtraction, and it should b e clear fr om the con text whic h op er ation is actually applied). W e also describ e such an application of the resolution r ule b y sa ying that L 1 was adde d (r esp., subtr acte d) to (r esp. fr om) L 2 in A ∨ L 1 and B ∨ L 2 . In ligh t of the d irect translation b et w een CNF form ulas and collections of disjun ctions of linear equations (describ ed in the previous su b section), we can consider R(lin) to b e a pr oof system for the set of unsatisﬁable CNF form ulas: Prop osition 1. The R(lin) r efutation system is a sound and c omplete Co ok-R e ckhow (se e Se c- tion 2 ) r efutation system for unsatisﬁable CN F formulas (tr anslate d into unsatisﬁable c ol le ction of disjunctions of line ar e quations). Pro of : Comp leteness of R(lin) (for th e set of unsatisﬁable CNF formulas) stems from a straigh t- forw ard sim ulation of resolution, as w e n ow s ho w. Claim 1. R(lin) p olynomially sim ulates resolution. Pro of of claim : Pro ceed by ind uction on the length of the resolution refu tatio n to s ho w that an y resolution deriv atio n of a clause A can b e translated w ith only a linear increase in size in to an R(lin) deriv ation of the corresp ondin g disj unction of linear equations e A (see the p revious subsection for the deﬁnition of e A ). The b ase c ase: An initial clause A is translated into its corresp onding disjun ction of linear equations e A . The induction step: If a resolution clause A ∨ B wa s derived by the resolution ru le f rom A ∨ x i and B ∨ ¬ x i , then in R(lin) we subtract ( x i = 0) fr om ( x i = 1) in e B ∨ ( x i = 0) and e A ∨ ( x i = 1), resp ectiv ely , to obtain e A ∨ e B ∨ (0 = 1). Then, u s ing the Simpliﬁcation r ule, we can cut-oﬀ (0 = 1) from e A ∨ e B ∨ (0 = 1), and arrive at e A ∨ e B . If a clause A ∨ B was deriv ed in r esolution from A b y the W eak ening rule, then we deriv e e A ∨ e B from e A by the W eak ening rule in R(lin). Soundn ess of R(lin) stems from the s ou n dness of the inference rules (wh ich means th at: If D w as d eriv ed from C , B by the R(lin) resolution r ule then an y assignment that satisﬁes b oth C and B also satisﬁes D ; and if D w as deriv ed from C by either the W eak enin g r ule or the Simpliﬁcation rule, then any assignment th at satisﬁes C also satisﬁes D ). The R(lin) pro of system is a Co ok-Rec kho w pro of system, as it is easy to verify in p olynomial- time whether an R (lin) pro of-line is in f erred, by an application of on e of R(lin)’s inf er en ce rules, from a p revious pro of-line (or p ro of-lines). Thus, any s equ ence of disj u nctions of linear equations, can b e chec ke d in p olynomial-time (in the size of the sequence) to d ecide whether or not it is a legitimat e R(lin) pr o of-sequence. In Section 5 we shall see that a stronger notion of completeness (that is, imp licati onal complete- ness) holds for R(lin) and its subsystems. 3.3. F ragmen t of Resolution o v er Linear E quations – R 0 (lin). Here we consider a restriction of R(lin), denoted R 0 (lin). As discussed in the in trod uction section, R 0 (lin) is roughly the fragmen t of R(lin) we kn ow h o w to p olynomially sim ulate with depth -3 m ultilinear pro ofs. By results esta blished in the sequel (Sectio ns 6.3 and 8 ) R(lin) is strictly str onger than R 0 (lin), whic h means that R(lin) p olynomially sim ulates R 0 (lin), while the con v erse do es not h old. R 0 (lin) op erates with disjun ctio ns of (arbitrarily many) linear equations with constan t co eﬃcien ts (excluding the free terms), under the follo wing restriction: Ev ery d isjunction can b e p artitio ned 10 in to a constan t num b er of sub-d isj unctions, wh ere eac h sub-disjunction either consists of linear equations that d iﬀer only in their free-terms or is a (translatio n of a) clause. As men tioned in the in tro duction, ev ery linear ine quality with Bo olean v ariables can b e rep- resen ted b y a disju n ction of linear equatio ns that diﬀer only in their free-terms. S o the R 0 (lin) pro of system resem bles, to some exten t, a pro of system op erating w ith disjunctions of constan t n um b er of linear inequalities with constan t integral co eﬃcien ts (on the other hand, it is probable that R 0 (lin) is stronger than such a p ro of system, as a disjun ction of linear equ atio ns that diﬀer only in their free terms is [expr essiv ely] stronger than a linear inequalit y [or ev en a d isjunction of linear inequalities]: the former can deﬁne the p arity f unction wh ile the latter cannot). Example of an R 0 (lin) -line: ( x 1 + . . . + x ℓ = 1) ∨ · · · ∨ ( x 1 + . . . + x ℓ = ℓ ) ∨ ( x ℓ +1 = 1) ∨ · · · ∨ ( x n = 1) , for s ome 1 ≤ ℓ ≤ n . The next section conta ins other concrete (and natural) examples of R 0 (lin)- lines. Let us deﬁne formally what it means to b e an R 0 (lin) pro of-line, that is, a pr o of-line inside an R 0 (lin) pro of, called R 0 (lin) -line : Deﬁnition 3.2 (R 0 (lin)-line) . Let D b e a d isjunction of linear equations whose v ariables ha v e constan t int eger co eﬃcien ts (the free-terms are unboun ded). Assu m e D can b e partitioned into a constan t n um b er k of su b -disjunctions D 1 , . . . , D k , where eac h D i either consists of (an un b ounded) disjunction of linear equations that diﬀer only in their f ree-terms, or is a translatio n of a clause (as deﬁned in S ubsection 3.1 ). Th en the disjun ction D is called an R 0 (lin) -line . Th us, any R 0 (lin)-line is of the follo wing general form: _ i ∈ I 1  ~ a (1) · ~ x = ℓ (1) i  ∨ · · · ∨ _ i ∈ I k  ~ a ( k ) · ~ x = ℓ ( k ) i  ∨ _ j ∈ J ( x j = b j ) , (2) where k and all a t r (for r ∈ [ n ] and t ∈ [ k ]) are integ er constant s and b j ∈ { 0 , 1 } (for all j ∈ J ) (and I 1 , . . . , I k , J are un b ounded sets of indices). Note that a disju nction of clauses can b e combined in to a single clause. Hence, without loss of generalit y we ca n assum e th at in any R 0 (lin)-line only a single (translation of a) clause o ccurs. This is d epicted in ( 2 ) (where in addition we ha v e ignored in ( 2 ) the p ossibility that the single clause obtained by com b ining seve ral clauses con tains x j ∨ ¬ x j , for some j ∈ [ n ]). Deﬁnition 3.3 (R 0 (lin)) . Th e R 0 (lin) pro of system is a r estrictio n of the R(lin) pro of system in whic h eac h p ro of-line is an R 0 (lin)-line (as in Deﬁnition 3.2 ). F or a completeness pro of of R 0 (lin) see S ectio n 5 . 4 4. Reasoning and Counting inside R( lin) and its Su bsystems In this section w e illus trate a simple wa y to reason by case-analysis inside R(lin) and its subsys- tems. Th is kind of reasoning will simplify the presenta tion of pro ofs inside R(lin) (and R 0 (lin)) in the sequel (essentia lly , a similar – th ough w eak er – kind of r easoning is applicable already in r eso- lution). W e will then demonstr ate eﬃcien t and transparent pro ofs for simple counti ng arguments that will also facilitate us in the sequ el. 4 The simulation of resolution inside R(lin) (in t he pro of of Prop ositio n 1 ) is carried on with each R (lin) pro of-line b eing in fact a translation of a clause, and hence, an R 0 (lin)-line ( notice that the Bo olean axioms of R (lin) are R 0 (lin)-lines). This already implies that R 0 (lin) is a complete refutation system for the set of unsatisﬁable CNF form ulas. In section 5 w e give a pro of of a stronger notion of completeness for R 0 (lin). 11 4.1. Basic Reasoning inside R(lin) and its Subsystems. Giv en K a collec tion of disjunc- tions of lin ear equations { K 1 , . . . , K m } and C a disjun ction of lin ear equations, denote b y K ∨ C the collec tion { K 1 ∨ C, . . . , K m ∨ C } . Recall that the formal v ariables in our pro of system are x 1 , . . . , x n . Lemma 4. L e t K b e a c ol le ction of disju nc tions of line ar e quations, and let z abbr eviate some line ar form with inte ger c o eﬃcients. L et E 1 , . . . , E ℓ b e ℓ disjunctions of line ar e quations. Assume that for al l i ∈ [ ℓ ] ther e is an R(lin) derivatio n of E i fr om z = a i and K with size at most s wher e a 1 , . . . , a ℓ ar e distinct inte gers. Then, ther e is an R(lin) pr o of of W ℓ i =1 E i fr om K and ( z = a 1 ) ∨ · · · ∨ ( z = a ℓ ) , with size p olynomial in s and ℓ . Pro of : Denote b y D th e d isjunction ( z = a 1 ) ∨ · · · ∨ ( z = a ℓ ) and by π i the R(lin) pro of of E i from K and z = a i (with size at most s ), for all i ∈ [ ℓ ]. It is easy to verify that for all i ∈ [ ℓ ] the sequence π i ∨ W j ∈ [ ℓ ] \{ i } ( z = a j ) is an R(lin) p r oof of E i ∨ W j ∈ [ ℓ ] \{ i } ( z = a j ) f r om K and D . So ov erall, giv en D and K as pr emises, there is an R(lin) d eriv ation of size p olynomial in s an d ℓ of the follo wing collect ion of disj unctions of linear equatio ns: E 1 ∨ _ j ∈ [ ℓ ] \{ 1 } ( z = a j ) , . . . , E ℓ ∨ _ j ∈ [ ℓ ] \{ ℓ } ( z = a j ) . (3) W e n o w u se th e Resolution ru le to cut-oﬀ all the equations ( z = a i ) inside all the d isjunctions in ( 3 ). F ormally , we pro v e that for eve ry 1 ≤ k ≤ ℓ th ere is a p olynomial-size (in s and ℓ ) R (lin) deriv ation from ( 3 ) of E 1 ∨ · · · ∨ E k ∨ _ j ∈ [ ℓ ] \ [ k ] ( z = a j ) , (4) and so pu tting k = ℓ , will conclude the pr oof of the lemma. W e pr oceed by in d uction on k . T he base ca se for k = 1 is immediate (from ( 3 )). F or the induction case, assu me that for some 1 ≤ k < ℓ we already h av e an R(lin) pro of of ( 4 ), with size p olynomial in s and ℓ . Consider the line E k +1 ∨ _ j ∈ [ ℓ ] \{ k +1 } ( z = a j ) . (5) W e can no w cut-oﬀ the disjun ctions W j ∈ [ ℓ ] \ [ k ] ( z = a j ) and W j ∈ [ ℓ ] \{ k +1 } ( z = a j ) from ( 4 ) and ( 5 ), resp ectiv ely , using the Resolution r ule (since the a j ’s in ( 4 ) and in ( 5 ) are disjoin t). W e w ill demonstrate this deriv ation in some detail no w, in order to exemplify a p r oof carried in side R(lin). W e shall b e less formal sometime in the sequel. Resolv e ( 4 ) with ( 5 ) o v er ( z = a k +1 ) and ( z = a 1 ), resp ectiv ely , to obtai n (0 = a 1 − a k +1 ) ∨ E 1 ∨ · · · ∨ E k ∨ E k +1 ∨ _ j ∈ [ ℓ ] \{ 1 ,k +1 } ( z = a j ) . (6) Since a 1 6 = a k +1 , w e can use the Simpliﬁcation rule to cut-oﬀ (0 = a 1 − a k +1 ) from ( 6 ), and w e arriv e at E 1 ∨ · · · ∨ E k ∨ E k +1 ∨ _ j ∈ [ ℓ ] \{ 1 ,k +1 } ( z = a j ) . (7) No w, similarly , resolv e ( 4 ) with ( 7 ) o v er ( z = a k +1 ) and ( z = a 2 ), resp ectiv ely , and use S impliﬁcation to obtain E 1 ∨ · · · ∨ E k ∨ E k +1 ∨ _ j ∈ [ ℓ ] \{ 1 , 2 ,k +1 } ( z = a j ) . 12 Con tin ue in a similar manner un til y ou arrive at E 1 ∨ · · · ∨ E k ∨ E k +1 ∨ _ j ∈ [ ℓ ] \{ 1 , 2 ,...,k ,k +1 } ( z = a j ) , whic h is precisely what we need. Under the appropriate conditions, Lemma 4 also holds for R 0 (lin) pr oofs. T his is stated in the follo w ing lemma. Lemma 5. L et K b e a c ol le ction of disjunctions of line ar e quations, and let z abbr eviate a line ar form with inte ger c o eﬃcients. L et E 1 , . . . , E ℓ b e ℓ disjunctions of line ar e quations. Assume that for al l i ∈ [ ℓ ] ther e is an R 0 (lin) derivation of E i fr om z = a i and K with size at most s , wher e the a i ’s ar e distinct i nte gers. Then, assuming W ℓ i =1 E i is an R 0 (lin) -line, ther e is an R 0 (lin) pr o of of W ℓ i =1 E i fr om K and ( z = a 1 ) ∨ · · · ∨ ( z = a ℓ ) , with size p olynomial in s and ℓ . Pro of : It can b e veriﬁed b y simple insp ection that, under the conditions sp elled out in the s tate- men t of the lemma, eac h pro of-line in the R(lin) deriv ations in the p ro of of Lemma 4 is actually an R 0 (lin)-line. 5 Abbreviations. Lemmas 4 and 5 will sometime facilita te us to pro ceed inside R(lin) and R 0 (lin) with a sligh tly less formal manner. F or example, the situation in Lemma 4 ab o ve can b e depicted b y saying that “if z = a i implies E i (with a p olynomial-size pro of ) for all i ∈ [ ℓ ], then W ℓ i =1 ( z = a i ) implies W ℓ i =1 E i (with a p olynomial-size p ro of )”. In case W ℓ i =1 ( z = a i ) ab ov e is just the Bo ole an axiom ( x i = 0) ∨ ( x i = 1), for some i ∈ [ n ], and x i = 0 imp lies E 0 and x i = 1 implies E 1 (b oth with p olynomial-size pro ofs), then to simplify the writing we s h all sometime not m en tio n th e Bo olean axiom at all. F or example, the latter situation can b e depicted by sa ying that “if x i = 0 imp lies E 0 with a p olynomial-size p ro of and x i = 1 imp lies E 1 with a p olynomial-size p ro of, then w e can derive E 0 ∨ E 1 with a p olynomial-size p ro of ”. 4.2. Basic Coun ting inside R(lin) and R 0 (lin). In this su bsection we illustrate how to eﬃ- cien tly pr o ve sev er al b asic coun ting argument s inside R(lin) and R 0 (lin). This will f acilitate us in sho wing sh ort pro ofs for hard tautologies in the sequel. In accordance with th e last paragraph in the previous su bsection, w e shall carry the pro ofs in side R(lin) and R 0 (lin) with a sligh tly less r igor. Lemma 6. L et z 1 abbr eviate ~ a · ~ x and z 2 abbr eviate ~ b · ~ x . L et D 1 b e W α ∈A ( z 1 = α ) and let D 2 b e W β ∈B ( z 2 = β ) , wher e A , B ar e two (ﬁnite) sets of inte gers. Then ther e is a p olynomial-size (in the size of D 1 , D 2 ) R(lin) pr o of fr om D 1 , D 2 of: _ α ∈A ,β ∈B ( z 1 + z 2 = α + β ) . (8) Mor e over, if ~ a and ~ b c onsist of constan t i nte gers (which me ans that D 1 , D 2 ar e R 0 (lin) -lines), then ther e is a p olynomial-size (in the size of D 1 , D 2 ) R 0 (lin) pr o of of ( 8 ) fr om D 1 , D 2 . Pro of : Denote the eleme n ts of A by α 1 , . . . , α k . In case z 1 = α i , f or some i ∈ [ k ] then we can add z 1 = α i to ev ery equation in W β ∈B ( z 2 = β ) to get W β ∈B ( z 1 + z 2 = α i + β ). Therefore, there exist k R(lin) pro ofs, eac h with p olynomial-size (in | D 1 | and | D 2 | ), of _ β ∈B ( z 1 + z 2 = α 1 + β ) , _ β ∈B ( z 1 + z 2 = α 2 + β ) , . . . , _ β ∈B ( z 1 + z 2 = α k + β ) 5 Note that when the pro ofs of E i from z = a i , for all i ∈ [ ℓ ], are all done inside R 0 (lin), t hen the linear form z ought t o ha ve c onstant co eﬃcients. 13 from z 1 = α 1 , z 1 = α 2 ,. . . , z 1 = α k , resp ectiv ely . Th us, by Lemm a 4 , w e can deriv e _ α ∈A ,β ∈B ( z 1 + z 2 = α + β ) (9) from D 1 and D 2 in a p olynomial-size (in | D 1 | and | D 2 | ) R(lin)-pro of. This concludes the ﬁrst part of the lemma. Assume that ~ a and ~ b consist of constant co eﬃcien ts on ly . Then by in sp ecting the R(lin)-pro of of ( 9 ) fr om D 1 and D 2 demonstrated ab ov e (and by using Lemma 5 ins tead of Lemma 4 ), one can v erify that this pr o of is in fact carried inside R 0 (lin). An immediate corollary of Lemm a 6 is the eﬃcien t formalization in R(lin) of the follo win g ob vious coun ting argumen t: If a linear form equals some v alue in the in terv al (of integ er num b ers) [ a 0 , a 1 ] and another linear form equals some v alue in [ b 0 , b 1 ] (for some a 0 ≤ a 1 and b 0 ≤ b 1 ), then their addition equals some v alue in [ a 0 + b 0 , a 1 + b 1 ]. More formally: Corollary 7. L et z 1 abbr eviate ~ a · ~ x and z 2 abbr eviate ~ b · ~ x . L et D 1 b e ( z 1 = a 0 ) ∨ ( z 1 = a 0 + 1) . . . ∨ ( z 1 = a 1 ) , and let D 2 b e ( z 2 = b 0 ) ∨ ( z 2 = b 0 + 1) . . . ∨ ( z 2 = b 1 ) . Then ther e is a p olynomial-size (in the size of D 1 , D 2 ) R(lin) pr o of fr om D 1 , D 2 of ( z 1 + z 2 = a 0 + b 0 ) ∨ ( z 1 + z 2 = a 0 + b 0 + 1) ∨ . . . ∨ ( z 1 + z 2 = a 1 + b 1 ) . (10) Mor e over, if ~ a and ~ b c onsist of constan t i nte gers (which me ans that D 1 , D 2 ar e R 0 (lin) -lines), then ther e is a p olynomial-size (in the size of D 1 , D 2 ) R 0 (lin) pr o ofs of ( 10 ) fr om D 1 , D 2 . Lemma 8. L et ~ a · ~ x b e a line ar form with n variables, and let A := { ~ a · ~ x | ~ x ∈ { 0 , 1 } n } b e the se t of al l p ossible values of ~ a · ~ x over Bo ole an assignments to ~ x . Then ther e is a p olynomial-size, in the size of the line ar form ~ a · ~ x , 6 R(lin) pr o of of _ α ∈A ( ~ a · ~ x = α ) . (11) Mor e over, if the c o e ﬃcients in ~ a ar e c onstants, then ther e i s a p olynom ial-size (in the size of ~ a · ~ x ) R 0 (lin) pr o of of ( 11 ). Pro of : Without loss of generalit y , assum e that all the co eﬃcien ts in ~ a are n onzero. Consider the Bo olea n axiom ( x 1 = 0) ∨ ( x 1 = 1) and the (ﬁrs t) co eﬃcien t a 1 from ~ a . Assume that a 1 ≥ 1. Add ( x 1 = 0) to itself a 1 times, and arr iv e at ( a 1 x 1 = 0) ∨ ( x 1 = 1). Then, in the r esulted lin e, add ( x 1 = 1) to itself a 1 times, un til the follo wing is reac h ed: ( a 1 x 1 = 0) ∨ ( a 1 x 1 = a 1 ) . Similarly , in case a 1 ≤ − 1 we can su btract ( | a 1 | + 1 many times) ( x 1 = 0) f rom itself in ( x 1 = 0) ∨ ( x 1 = 1), and then subtract ( | a 1 | + 1 man y times) ( x 1 = 1) from itself in the resulted lin e. In the same mann er, we can deriv e the disj u nctions: ( a 2 x 2 = 0) ∨ ( a 2 x 2 = a 2 ) , . . . , ( a n x n = 0) ∨ ( a n x n = a n ). Consider ( a 1 x 1 = 0) ∨ ( a 1 x 1 = a 1 ) and ( a 2 x 2 = 0) ∨ ( a 2 x 2 = a 2 ). F rom these tw o lines, by Lemma 6 , there is a p olynomial-size in | a 1 | + | a 2 | deriv ation of: ( a 1 x 1 + a 2 x 2 = 0) ∨ ( a 1 x 1 + a 2 x 2 = a 1 ) ∨ ( a 1 x 1 + a 2 x 2 = a 2 ) ∨ ( a 1 x 1 + a 2 x 2 = a 1 + a 2 ) . (12) In a similar fashion, no w consider ( a 3 x 3 = 0) ∨ ( a 3 x 3 = a 3 ) and apply again Lemma 6 , to obtain _ α ∈A ′ ( a 1 x 1 + a 2 x 2 + a 3 x 3 = α ) , (13) 6 Recall that the size of ~ a · ~ x is P n i =1 | a i | , that is, the size of the unary representatio n of ~ a . 14 where A ′ are all p ossible v alues to a 1 x 1 + a 2 x 2 + a 3 x 3 o v er Bo olean assignmen ts to x 1 , x 2 , x 3 . The deriv ation of ( 13 ) is of size p olynomial in | a 1 | + | a 2 | + | a 3 | . Con tin ue to consider, successiv ely , all other lines ( a 4 x 4 = 0) ∨ ( a 4 x 4 = a 4 ) , . . . , ( a n x n = 0) ∨ ( a n x n = a n ), and apply th e same reasoning. Eac h s tep uses a deriv atio n of size at most p olynomial in P n i =1 | a i | . And so ov erall we reac h the desired line ( 11 ), with a deriv ation of size p olynomial in the size of ~ a · ~ x . T his concludes the ﬁrst part of the lemma. Assume that ~ a consists of constan t co eﬃcien ts only . Then by insp ecting the R(lin)-pro of demon- strated ab o v e (and by using the second part of Lemma 6 ), one can see that this p ro of is in fact carried inside R 0 (lin). Lemma 9. Ther e is a p olynomial-size (in n ) R 0 (lin) pr o of fr om ( x 1 = 1) ∨ · · · ∨ ( x n = 1) (14) of ( x 1 + . . . + x n = 1) ∨ · · · ∨ ( x 1 + . . . + x n = n ) . (15) Pro of : W e sh o w that for ev ery i ∈ [ n ], there is a p olynomial-size (in n ) R 0 (lin) pr oof f rom ( x i = 1) of ( x 1 + . . . + x n = 1) ∨ · · · ∨ ( x 1 + . . . + x n = n ). This concludes the pro of since, by Lemma 5 , w e then can deriv e from ( 14 ) (with a p olynomial-size (in n ) R 0 (lin) pro of ) the disjunction ( 14 ) in whic h eac h ( x i = 1) (for all i ∈ [ n ]) is replace by ( x 1 + . . . + x n = 1) ∨ · · · ∨ ( x 1 + . . . + x n = n ), whic h is precisely the disjun ctio n ( 15 ) (note that ( 15 ) is an R 0 (lin)-line). Claim 2. F or ev ery i ∈ [ n ], th ere is a a p olynomial-size (in n ) R 0 (lin) pro of from ( x i = 1) of ( x 1 + . . . + x n = 1) ∨ · · · ∨ ( x 1 + . . . + x n = n ). Pro of of claim : By Lemm a 8 , for every i ∈ [ n ] there is a p olynomial-size (in n ) R 0 (lin) p ro of (using only the Bo olean axioms) of ( x 1 + . . . + x i − 1 + x i +1 + . . . + x n = 0) ∨ · · · ∨ ( x 1 + . . . + x i − 1 + x i +1 + . . . + x n = n − 1) . (16) No w add successiv ely ( x i = 1) to eve ry equation in ( 16 ) (note th at this can b e done in R 0 (lin)). W e obtain precisely ( x 1 + . . . + x n = 1) ∨ · · · ∨ ( x 1 + . . . + x n = n ). Lemma 10. Ther e i s a p olynomial-size (in n ) R 0 (lin) pr o of of ( x 1 + . . . + x n = 0) ∨ ( x 1 + . . . + x n = 1) fr om the c ol le ction of disjunctions c onsisting of ( x i = 0) ∨ ( x j = 0) , for al l 1 ≤ i < j ≤ n . Pro of : W e pro ceed b y indu ction on n . The base case for n = 1 is immediate from the Bo olea n axiom ( x 1 = 0) ∨ ( x 1 = 1). Assume w e already ha ve a p olynomial-size pro of of ( x 1 + . . . + x n = 0) ∨ ( x 1 + . . . + x n = 1) . (17) If x n +1 = 0 w e add x n +1 = 0 to b oth of the equ atio ns in ( 17 ), and reac h: ( x 1 + . . . + x n +1 = 0) ∨ ( x 1 + . . . + x n +1 = 1) . (18) Otherwise, x n +1 = 1, and so we can cut-oﬀ ( x n +1 = 0) in all the initial disjunctions ( x i = 0) ∨ ( x n +1 = 0), for all 1 ≤ i ≤ n . W e thus obtain ( x 1 = 0) , . . . , ( x n = 0). Adding together ( x 1 = 0) , . . . , ( x n = 0) and ( x n +1 = 1) w e arrive at ( x 1 + . . . + x n +1 = 1) . (19) So o v erall, either ( 18 ) holds or ( 19 ) holds; and so (using Lemma 5 ) w e arrive at the disju nction of ( 19 ) and ( 18 ), wh ic h is precisely ( 18 ). 15 5. Implica tional Comple teness of R(lin) and its Subs ystems In this section w e pro vide a pro of of the implicational completeness of R(lin) an d its sub systems. W e sh all need this p r op ert y in the sequel (see Section 6.2 ). The implicational completeness of a pro of system is a str onger prop erty than m er e completeness. Essent ially , a system is implicationally complete if whenev er something is semantic al ly implied b y a set of initial premises, then it is also derivable from the initial premises. In con trast to this, mere completeness means that an y tautology (or in case of a refutation system, an y unsatisﬁable set of initial pr emises) h as a p ro of in the system (resp ectiv ely , a refutation in the system). As a consequence, the pro of of implicational completeness in this sectio n establishes an alternativ e complete ness pro of to that obtained via sim ulating resolution (see Prop osition 1 ). Note that w e are not concerned in th is secti on with the size of th e pro ofs, but only with their existence. Recall the d eﬁnition of the semantic implicatio n relation | = from Section 3.1 . F ormally , we sa y that R(lin) is i mplic ational ly c omplete if for ev ery colle ction of disju nctions of linear equations D 0 , D 1 , . . . , D m , it holds that D 1 , . . . , D m | = D 0 implies that there is an R(lin) pro of of D 0 from D 1 , . . . , D m . Theorem 11. R(lin) is implic ationa l ly c omplete. Pro of : W e pro ceed by ind uction on n , the num b er of v ariables x 1 , . . . , x n in D 0 , D 1 , . . . , D m . The b ase c ase n = 0. W e n eed to s ho w that D 1 , . . . , D m | = D 0 implies that there is an R(lin) pro of of D 0 from D 1 , . . . , D m , where all D i ’s (for 0 ≤ i ≤ m ) ha v e no v ariables b ut only constan ts. This means that eac h D i is a disju nction of equations of the form (0 = a 0 ) for some int eger a 0 (if a linear equation ha v e no v ariables, then the left hand side of this equatio n m ust b e 0; see Section 3.1 ). There are tw o cases to consider. In the ﬁ rst case D 0 is satisﬁable . S ince D 0 has n o v ariables, this means precisely th at D 0 is the equation (0 = 0). T h us, D 0 can b e deriv ed easily from an y axiom in R(lin) (for ins tance, by su btracting eac h equation in ( x 1 = 0) ∨ ( x 1 = 1) from itself, to reac h (0 = 0) ∨ (0 = 0), whic h is equ al to (0 = 0), since w e discard dup licat e equations inside disjunctions). In the second case D 0 is unsatisﬁable . Th us, since D 1 , . . . , D m | = D 0 , there is no assignment sat- isfying all D 1 , . . . , D m . Hence, there must b e at least one unsatisﬁable disju nction D i in D 1 , . . . , D m (as a disjunction with n o v ariables is either tautological or unsatisﬁable). S uc h an uns atisﬁable D i is a disjunction of zero or more unsatisﬁable equations of th e form (0 = a 0 ), for some in teger a 0 6 = 0. W e can then u s e Simpliﬁcation to cut-oﬀ all the unsatisﬁable equations in D i to r eac h the empty disjunction. By the W eak ening ru le, we can no w deriv e D 0 from the empty d isjunction. The induction step . Assume that the theorem holds for disju nctions w ith n v ariables. Let the underlying v ariables of D 0 , D 1 , . . . , D m b e x 1 , . . . , x n +1 , and assume that D 1 , . . . , D m | = D 0 . (20) W e write the disjunction D 0 as: t _ j =1 n X i =1 a ( j ) i x i + a ( j ) n +1 x n +1 = a ( j ) 0 ! , (21) where the a ( j ) i ’s are in teg er co eﬃcien ts. W e n eed to show th at there is an R(lin) pr o of of D 0 from D 1 , . . . , D m . Let D b e a disju nction of linear equations, let x i b e a v ariable an d let b ∈ { 0 , 1 } . W e shall denote by D ↾ x i = b the disju nction D , wh ere in ev ery equation in D the v ariable x i is sub stituted by b , and the constan t terms in the left hand sides of all resu lting equations (after subs tituting b for 16 x i ) switc h sides (and c h ange signs, obviously) to the righ t h and sides of the equations (we hav e to switc h sides of constan t terms, as by deﬁnition linear equ ations in R(lin) pr oofs ha v e all constan t terms app earing only on the righ t h and sides of equations). W e no w reason (slightl y) inform ally inside R(lin) (as illustrated in Section 4.1 ). Fix some b ∈ { 0 , 1 } , and assume that x n +1 = b . T h en, from D 1 , . . . , D m w e can d eriv e (inside R(lin)): D 1 ↾ x n +1 = b , . . . , D m ↾ x n +1 = b . (22) The only v ariables o ccurrin g in ( 22 ) are x 1 , . . . , x n . F rom assu mption ( 20 ) w e clearly hav e D 1 ↾ x n +1 = b , . . . , D m ↾ x n +1 = b | = D 0 ↾ x n +1 = b . And so b y the ind uction h yp othesis there is an R(lin) deriv ation of D 0 ↾ x n +1 = b from D 1 ↾ x n +1 = b , . . . , D m ↾ x n +1 = b . So o v erall, assuming that x n +1 = b , th ere is an R(lin) deriv ation of D 0 ↾ x n +1 = b from D 1 , . . . , D m . W e no w consider the t w o p ossible cases: x n +1 = 0 and x n +1 = 1. In c ase x n +1 = 0, by the ab o v e discussion, we can deriv e D 0 ↾ x n +1 =0 from D 1 , . . . , D m . F or ev er y j ∈ [ t ], add successiv ely ( a ( j ) n +1 times) the equation x n +1 = 0 to the j th equation in D 0 ↾ x n +1 =0 (see ( 21 )). W e th us obtain pr ecisely D 0 . In c ase x n +1 = 1, again, b y the ab o v e discussion, we can derive D 0 ↾ x n +1 =1 from D 1 , . . . , D m . F or ev ery j ∈ [ t ], add successive ly ( a ( j ) n +1 times) th e equation x n +1 = 1 to th e j th equ ation in D 0 ↾ x n +1 =1 (recall that we sw itc h sides of constant terms in ev ery lin ear equation after the su bstitution of x n +1 b y 1 is p erformed in D 0 ↾ x n +1 =1 ). Again, we obtain precisely D 0 . By insp ecting the pro of of Th eorem 11 , it is p ossib le to verify that if all the disjun ctions D 0 , , . . . , D m are R 0 (lin)-lines (see Deﬁnition 3.2 ), then the pro of of D 0 in R(lin) uses only R 0 (lin)- lines as well. Therefore, w e ha v e: Corollary 12. R 0 (lin) is implic ational ly c omplete. Remark 1. Corollary 12 states that any R 0 (lin)-line that is semant ically implied by a set of initial R 0 (lin)-lines, is in fact deriv able in R 0 (lin) from the initial R 0 (lin)-lines. On th e other hand, it is p ossible that a certain p r oof of the same R 0 (lin)-line inside R(lin) will b e signiﬁcantly sh orter than the pro of in side R 0 (lin). Indeed, we shall see in Section 8 that for certa in CNF formulas R(lin) has a sup er-p olynomial sp eed-up o v er R 0 (lin). 6. Shor t Proofs for Hard T a utologies In this section w e s h o w that R 0 (lin) is already enough to admit small pro ofs for “hard” coun ting principles lik e the pigeonhole principle an d the Tseitin graph formulas for constan t degree graphs . On the other hand, as w e sh all s ee in Section 8 , R 0 (lin) inherits the same w eakness that cutting planes pro ofs hav e with resp ect to the clique-coloring tautologies. Neverthele ss, we can eﬃcien tly pro v e the clique-c oloring principle in (the stronger system) R(lin), but not by us ing R(lin) “abilit y to coun t”, rather b y using its (straigh tforw ard) abilit y to simulate Res(2) pro ofs (that is, resolution pro ofs extended to op erate with 2-DNF formulas, ins tead of clauses). 6.1. The Pigeonhole Principle T autologies in R 0 (lin). This subsection illustrates p olynomial-size R 0 (lin) pro ofs of the pigeonhole principle. Th is will allo w us to establish p olynomial-size multil inear pro ofs op erating with depth-3 multilinear formulas of the pigeonhole principle (in S ectio n 9 ). The m to n pige onhole principle states that m pigeons cannot b e mapp ed one-to-one into n < m holes. The negation of the pigeonhole principle, denoted ¬ PHP m n , is formulate d as an u nsatisﬁable CNF f orm ula as follo ws (where clauses are translated to disjunctions of linear equations): Deﬁnition 6.1. The ¬ PHP m n is the follo wing set of cla uses: (1) Pigeons axioms: ( x i, 1 = 1) ∨ · · · ∨ ( x i,n = 1), for all 1 ≤ i ≤ m ; 17 (2) Holes axioms: ( x i,k = 0) ∨ ( x j,k = 0), for all 1 ≤ i < j ≤ m and f or all 1 ≤ k ≤ n . The inte nded meaning of eac h prop ositional v ariable x i,j is that the i th p igeo n is mapp ed to the j th hole. W e n ow describ e a p olynomial-size in n refutation of ¬ PHP m n inside R 0 (lin). F or this pur p ose it is suﬃcient to p r o ve a p olynomial-size refutation of the pigeonhole pr inciple when the num b er of pigeons m equ als n + 1 (b ecause the set of clauses p ertaining to ¬ PHP n +1 n is alrea dy con tained in the set of clauses p ertaining to ¬ PHP m n , for an y m > n ). Thus, we ﬁx m = n + 1. In this subsection w e shall sa y a pro of in R 0 (lin) is of p olynomial-size , alw a ys intending p olynomial-size in n (unless otherwise stated). By Lemm a 9 , for all i ∈ [ m ] w e can deriv e fr om the Pigeon axio m (for the i th p igeo n): ( x i, 1 + . . . + x i,n = 1) ∨ · · · ∨ ( x i, 1 + . . . + x i,n = n ) (23) with a p olynomial-size R 0 (lin) pro of. By Lemm a 10 , f rom the Hole axioms w e can derive, with a p olynomial-size R 0 (lin) pro of ( x 1 ,j + . . . + x m,j = 0) ∨ ( x 1 ,j + . . . + x m,j = 1) , (24) for all j ∈ [ n ]. Let S abbr eviate the sum of all formal v ariables x i,j . In other w ords, S := X i ∈ [ m ] ,j ∈ [ n ] x i,j . Lemma 13. Ther e is a p olynomial-size R 0 (lin) pr o of fr om ( 23 ) (for al l i ∈ [ m ] ) of ( S = m ) ∨ ( S = m + 1) · · · ∨ ( S = m · n ) . Pro of : F or ev ery i ∈ [ m ] ﬁx the abbreviation z i := x i, 1 + . . . + x i,n . Thus, by ( 23 ) w e ha v e ( z i = 1) ∨ · · · ∨ ( z i = n ). Consider ( z 1 = 1) ∨ · · · ∨ ( z 1 = n ) and ( z 2 = 1) ∨ · · · ∨ ( z 2 = n ). By Corollary 7 , we can derive from these t wo lines ( z 1 + z 2 = 2) ∨ ( z 1 + z 2 = 3) ∨ · · · ∨ ( z 1 + z 2 = 2 n ) (25) with a p olynomial-size R 0 (lin) pro of. No w, co nsider ( z 3 = 1) ∨ · · · ∨ ( z 3 = n ) and ( 25 ). By Corollary 7 again, from these t wo lines we can deriv e with a p olynomial-size R 0 (lin) pro of: ( z 1 + z 2 + z 3 = 3) ∨ ( z 1 + z 2 + z 3 = 4) ∨ · · · ∨ ( z 1 + z 2 + z 3 = 3 n ) . (26) Con tin uing in the same w a y , we ev en tually arriv e at ( z 1 + . . . + z m = m ) ∨ ( z 1 + . . . + z m = m + 1) ∨ · · · ∨ ( z 1 + . . . + z m = m · n ) , whic h concludes the pro of, since S equals z 1 + . . . + z m . Lemma 14. Ther e is a p olynomial-size R 0 (lin) pr o of fr om ( 24 ) of ( S = 0) ∨ · · · ∨ ( S = n ) . Pro of : F or all j ∈ [ n ], ﬁ x the abbreviation y j := x 1 ,j + . . . + x m,j . Thus, b y ( 24 ) w e hav e ( y j = 0) ∨ ( y j = 1), for all j ∈ [ n ]. No w the p ro of is s im ilar to the pro of of Lemma 8 , except that here single v ariables are abbr eviati ons of linear form s. If y 1 = 0 then we can add y 1 to the tw o s ums in ( y 2 = 0) ∨ ( y 2 = 1), and reac h ( y 1 + y 2 = 0) ∨ ( y 1 + y 2 = 1) and if y 1 = 1 we can do the same and reac h ( y 1 + y 2 = 1) ∨ ( y 1 + y 2 = 2). So, by Lemma 5 , w e can derive with a p olynomial-size R 0 (lin) pro of ( y 1 + y 2 = 0) ∨ ( y 1 + y 2 = 1) ∨ ( y 1 + y 2 = 2) . (27) 18 No w, we consid er the three cases in ( 27 ): y 1 + y 2 = 0 or y 1 + y 2 = 1 or y 1 + y 2 = 2, and the clause ( y 3 = 0) ∨ ( y 3 = 1). W e arriv e in a similar manner at ( y 1 + y 2 + y 3 = 0) ∨ · · · ∨ ( y 1 + y 2 + y 3 = 3). W e con tin ue in the same wa y until we arriv e at ( S = 0) ∨ · · · ∨ ( S = n ). Theorem 15. Ther e is a p olynomial-size R 0 (lin) r efutation of the m to n pige onhole principle ¬ PHP m n . Pro of : By Lemmas 13 and 14 ab o v e, all w e need is to show a p olynomial-size r efutation of ( S = m ) ∨ · · · ∨ ( S = m · n ) and ( S = 0) ∨ · · · ∨ ( S = n ). Since n < m , for all 0 ≤ k ≤ n , if S = k th en using the Resolution and Simpliﬁcation rules we can cut-oﬀ all the sums in ( S = m ) ∨ · · · ∨ ( S = m · n ) and arrive at the empty clause. Thus, by Lemma 5 , there is a p olynomial-size R 0 (lin) pro of of the empty clause fr om ( S = 0) ∨ · · · ∨ ( S = n ) and ( S = m ) ∨ · · · ∨ ( S = m · n ). 6.2. Tseitin mo d p T aut ologies in R 0 (lin). This subsection establishes p olynomial-size R 0 (lin) pro ofs of Ts eitin graph tautologies (for constan t degree graphs). This will allo w u s (in Section 9 ) to extend the multili near p ro ofs of the Tseitin mo d p tautologies to an y ﬁeld of c haracteristic 0 (the pro ofs in [ R T06 ] requir ed working o v er a ﬁeld con taining a primitive p th ro ot of un it y when pro ving the T s eitin mo d p tauto logies; for more details see Section 9 ). Tseitin mo d p tautolog ies (in tro duced in [ BGIP01 ]) are generalizations of the (original, mo d 2) Tseitin graph tautologies (int ro duced in [ Tse68 ]). T o build the in tuition for the generalized v ersion, w e start by describing the (original) Tseitin mo d 2 principle. Let G = ( V , E ) b e a connected undirected graph with an o dd n umber of v ertices n . The Tseitin mo d 2 tautology states th at there is no sub-graph G ′ = ( V , E ′ ), where E ′ ⊆ E , so that for e v ery v ertex v ∈ V , the num b er of edges from E ′ inciden t to v is o dd. This statemen t is v alid, since otherwise, summ ing the degrees of all the v ertices in G ′ w ould amoun t to an od d num b er (since n is o dd), whereas this sum also coun ts ev ery edge in E ′ t wice, and s o is ev en. As men tioned ab ov e, the Tseitin mo d 2 p rinciple w as generalized b y Buss et al. [ BGIP01 ] to obtain the Tseitin mo d p principle. Let p ≥ 2 b e some ﬁxed intege r and let G = ( V , E ) b e a connected und irected r -regular graph with n v ertices and no double edges. Let G ′ = ( V , E ′ ) b e the corresp onding dir e c te d graph that results from G by r eplacing ev ery (und irected) edge in G w ith t w o opp osite directed edges. Assum e that n ≡ 1 (mo d p ). Then, the T seitin mo d p p rinciple states that there is no w a y to assign to ev ery edge in E ′ a v alue f r om { 0 , . . . , p − 1 } , so that: (i): F or ev ery p air of opp osite directed edges e, ¯ e in E ′ , with assigned v alues a, b , resp ectiv ely , a + b ≡ 0 (mod p ); and (ii): F or every v ertex v in V , the su m of the v alues assigned to th e edges in E ′ coming out of v is congruent to 1 (mo d p ). The Tseitin mo d p principle is v alid, since if we su m the v alues assigned to all edges of E ′ in pairs we obtain 0 (mod p ) (b y (i)), where summing them by v ertices we arriv e at a total v alue of 1 (mo d p ) (b y (ii) and since n ≡ 1 (mo d p )). W e shall see in what follo ws, that this simple coun ting argumen t can b e carried on in a natural (and eﬃcient) w a y already inside R 0 (lin). As an uns atisﬁable prop ositional form ula (in C NF form) the negation of the Tseitin mod p principle is f orm ulated b y assigning a v ariable x e,i for ev ery edge e ∈ E ′ and ev ery residu e i mo dulo p . The v ariable x e,i is an indicator v ariable for the fact that the edge e h as an asso ciated v alue i . The follo wing are the clauses of the Ts eitin mo d p C NF form ula (as translated to disj unctions of linear equations). Deﬁnition 6.2 ( Tseitin mo d p formulas ( ¬ Tseitin G,p )) . Let p ≥ 2 b e some ﬁxed in teger and let G = ( V , E ) b e a connected un directed r -regular graph with n ve rtices and no d ouble edges, and 19 assume that n ≡ 1 (mo d p ). Let G ′ = ( V , E ′ ) b e the corresp onding directed grap h that results from G by r ep lacing ev ery (undirected) edge in G with t w o op p osite directed ed ges. Giv en a v ertex v ∈ V , denote the edges in E ′ coming out of v b y e [ v , 1] , . . . , e [ v, r ] and deﬁne the follo w ing set of (translation of ) clauses: MOD p, 1 ( v ) := ( r _ k =1 ( x e [ v ,k ] ,i k = 0)     i 1 , . . . , i r ∈ { 0 , . . . , p − 1 } and r X k =1 i k 6≡ 1 mo d p ) . The Tseitin mo d p formula, denoted ¬ Tseitin G,p , consists of the follo win g (translation) of clauses: 1. p − 1 W i =0 ( x e,i = 1) , for all e ∈ E ′ (expresses that every edge is assigned at least one v alue from 0 , . . . , p − 1); 2. ( x e,i = 0) ∨ ( x e,j = 0) , for all i 6 = j ∈ { 0 , . . . , p − 1 } and all e ∈ E ′ (expresses that every edge is assigned at most one v alue from 0 , . . . , p − 1); 3. ( x e,i = 1) ∨ ( x ¯ e,p − i = 0) and ( x e,i = 0) ∨ ( x ¯ e,p − i = 1) , 7 for all tw o opp osite directed edges e, ¯ e ∈ E ′ and all i ∈ { 0 , . . . , p − 1 } (expresses condition (i) of the Tseitin mo d p principle ab o ve); 4. MOD p, 1 ( v ) , for all v ∈ V (expresses condition (ii) of the Tseitin mo d p principle ab o ve). Note that for eve ry edge e ∈ E ′ , the p olynomials of (1,2) in Deﬁnition 6.2 , com bined with the Bo olea n axioms of R 0 (lin), force an y collectio n of edge-v ariables x e, 0 , . . . , x e,p − 1 to cont ain exac tly one i ∈ { 0 , . . . , p − 1 } so that x e,i = 1. Also, it is easy to verify th at, giv en a vertex v ∈ V , an y assignmen t σ of 0 , 1 v alues (to the relev ant v ariables) satisﬁes b oth the disjunctions of (1,2) and the disjunctions of MOD p, 1 ( v ) if and on ly if σ corresp onds to an assignment of v alues from { 0 , . . . , p − 1 } to the edges coming out of v that su ms u p to 1 (mod p ). Un til the rest of this sub s ectio n we ﬁx an inte ger p ≥ 2 and a connecte d undirected r -regular graph G = ( V , E ) with n v ertices and n o double edges, suc h that n ≡ 1 mo d p and r is a constant . As in Deﬁnition 6.2 , we let G ′ = ( V , E ′ ) b e the corresp onding directed graph th at results from G b y r ep lacing ev ery (un directed) edge in G with t w o opp osite d irected edges. W e now pro ceed to refute ¬ Tseitin G,p inside R 0 (lin) with a p olynomial-size (in n ) refutation. Giv en a ve rtex v ∈ V , and the edges in E ′ coming out of v , denoted e [ v , 1] , . . . , e [ v , r ], deﬁne the follo w ing abbreviation: α v := r X j =1 p − 1 X i =0 i · x e [ v ,j ] ,i . (28) Lemma 16. L et v ∈ V b e any ve rtex in G ′ . Then ther e is a c onsta nt-size R 0 (lin) pr o of fr om ¬ Tseitin G,p of the fol lowing disjunction: r − 1 _ ℓ =0 ( α v = 1 + ℓ · p ) . (29) Pro of : Let T v ⊆ ¬ Tseitin G,p b e the set of all disju nctions of the f orm (1,2,4) from Deﬁn ition 6.2 that conta in only v ariables p ertaining to ve rtex v (that is, all the v ariables x e,i , where e ∈ E ′ is an edge coming out of v , and i ∈ { 0 , . . . , p − 1 } ). 7 If i = 0 then x ¯ e,p − i denotes x ¯ e, 0 . 20 Claim 3. T v seman tically implies ( 29 ), that is: 8 T v | = r − 1 _ ℓ =0 ( α v = 1 + ℓ · p ) . Pro of of claim : L et σ b e an assignment of 0 , 1 v alues to th e v ariables in T v that satisﬁes b oth th e disjunctions of (1,2) and the disju nctions of MOD p, 1 ( v ) in Deﬁnition 6.2 . As men tioned ab o v e (the commen t after Deﬁnition 6.2 ), such a σ corresp onds to an assignment of v alues from { 0 , . . . , p − 1 } to the edges coming out of v , th at sums up to 1 mo d p . This means precisely th at α v = 1 mo d p under the assignm ent σ . Thus, there exists a nonn ega tiv e int eger k , such that α v = 1 + k p und er σ . It remains to s h o w that k ≤ r − 1 (and so the only p ossible v alues that α v can get under σ are 1 , 1 + p, 1 + 2 p, . . . , 1 + ( r − 1) p ). Note that b ecause σ giv es the v alue 1 to only one v ariable from x e [ v ,j ] , 0 , . . . , x e [ v ,j ] ,p − 1 (for ev ery j ∈ [ r ]), then the m aximal v alue that α v can h a ve u n der σ is r ( p − 1). Thus, 1 + k p ≤ r p − r and s o k ≤ r − 1. F rom Claim 3 and fr om the imp licat ional completeness of R 0 (lin) (Corollary 12 ), there exists an R 0 (lin) deriv ation of ( 29 ) from T v . It remains to s h o w that this deriv ation is of constan t-size. Since the degree r of G ′ and the mo du lus p are b oth constan ts, b oth T v and ( 29 ) ha v e constan t n um b er of v ariables and constan t co eﬃcien ts (in clud ing th e free-terms). Thus, there is a constant- size R 0 (lin) deriv ation of ( 29 ) from T v . Lemma 17. Ther e is a p olyno mial-size (in n ) R 0 (lin) derivation fr om ¬ Tseitin G,p of the fol lowing disjunction: ( r − 1) · n _ ℓ =0 X v ∈ V α v = n + ℓ · p ! . Pro of : Simp ly add su ccessiv ely all th e equations p ertaining to disjun ctions ( 29 ), f or all v ertices v ∈ V . F ormally , we sh o w th at for ev ery subset of v ertice s V ⊆ V , with |V | = k , there is a p olynomial-size (in n ) R 0 (lin) deriv ation from ¬ Tseitin G,p of ( r − 1) · k _ ℓ =0 X v ∈V α v = k + ℓ · p ! , (30) and so pu tting V = V , will conclude the pro of. W e pro ceed by induction on the size of V . T he base case, |V | = 1, is immediate from Lemma 16 . Assume that we already deriv ed ( 30 ) with a p olynomial-size (in n ) R 0 (lin) pro of, for some V ⊂ V , suc h that |V | = k < n . Let u ∈ V \ V . By Lemma 16 , we ca n derive r − 1 _ ℓ =0 ( α u = 1 + ℓ · p ) (31) from ¬ Tseitin G,p with a constan t-size pr oof. No w, by Lemma 6 , eac h lin ear equation in ( 31 ) can b e added to eac h linear equation in ( 30 ), with a p olynomial-size (in n ) R 0 (lin) pro of. This results in the follo wing disjun ction: ( r − 1) · ( k +1) _ ℓ =0   X v ∈V ∪{ u } α v = k + 1 + ℓ · p   , 8 Recall that we only consider assignmen ts of 0 , 1 v alues to v aria bles when considering the semantic implication relation | =. 21 whic h is precisely what we need to co nclude the induction step. Lemma 18. L et e, ¯ e b e any p air of opp osite dir e cte d e dges in G ′ and let i ∈ { 0 , . . . , p − 1 } . L et T e ⊆ ¬ Tseitin G,p b e the set of al l disjunctions of the form (1,2,3) fr om Deﬁnition 6.2 that c ontain only variables p ertaining to e dges e, ¯ e (that is, al l the variables x e,j , x ¯ e,j , for al l j ∈ { 0 , . . . , p − 1 } ). Then, ther e is a c onstant-size R 0 (lin) pr o of fr om T e of the fol lowing disjunction: ( i · x e,i + ( p − i ) · x ¯ e,p − i = 0) ∨ ( i · x e,i + ( p − i ) · x ¯ e,p − i = p ) . (32) Pro of : First note that T e seman tically implies ( x e,i + x ¯ e,p − i = 0) ∨ ( x e,i + x ¯ e,p − i = 2) . (33) The num b er of v ariables in T e and ( 33 ) is constant. Hence, there is a constan t-size R 0 (lin)-pro of of ( 32 ) f rom T e . Also note that ( x e,i + x ¯ e,p − i = 0) ∨ ( x e,i + x ¯ e,p − i = 2) | = ( i · x e,i + ( p − i ) · x ¯ e,p − i = 0) ∨ ( i · x e,i + ( p − i ) · x ¯ e,p − i = p ) . (34) Therefore, there is also an R 0 (lin)-pro of of constant -size from T e of th e low er line in ( 34 ). W e are no w ready to complete the p olynomial-size R 0 (lin) refutation of ¬ Tse itin G,p . Using th e t w o prior lemmas, the refutation idea is simple, as w e n o w explain. Observe that X v ∈ V α v = X { e, ¯ e }⊆ E ′ i ∈{ 0 ,...,p − 1 } ( i · x e,i + ( p − i ) · x ¯ e,p − i ) , (35) where b y { e, ¯ e } ⊆ E ′ w e mean that e, ¯ e is pair of opp osite directed edges in G ′ . Deriv e by Lemma 17 the d isjunction ( r − 1) · n _ ℓ =0 X v ∈ V α v = n + ℓ · p ! . (36) This disjun ction exp r esses the fact that P v ∈ V α v = 1 mo d p (since n = 1 mo d p ). O n the other hand, u s ing Lemm a 18 , we can “sum together” all the equations ( 32 ) (for all { e, ¯ e } ⊆ E ′ and all i ∈ { 0 , . . . , p − 1 } ), to obtain a disjun ction expressing the statemen t that X { e, ¯ e }⊆ E ′ i ∈{ 0 ,...,p − 1 } ( i · x e,i + ( p − i ) · x ¯ e,p − i ) = 0 mo d p . By Equation ( 35 ), we then obtain the d esired contradictio n. This idea is formalized in the pr oof of the follo win g th eorem: Theorem 19. L et G = ( V , E ) b e an r - r e gular gr aph with n vertic es, wher e r i s a c onstant. Fix some mo dulus p . Then, ther e ar e p olynomial-size (in n ) R 0 (lin) r efutations of ¬ Tseitin G,p . Pro of : First, use Lemma 17 to derive ( r − 1) · n _ ℓ =0 X v ∈ V α v = n + ℓ · p ! . (37) Second, use Lemma 18 to derive ( i · x e,i + ( p − i ) · x ¯ e,p − i = p ) ∨ ( i · x e,i + ( p − i ) · x ¯ e,p − i = 0) , (38) for ev ery pair of opp osite d irected edges in G ′ = ( V , E ′ ) (as in Deﬁnition 6.2 ) and every resid u e i ∈ { 0 , . . . , p − 1 } . 22 W e no w reason in side R 0 (lin). Pic k a p air of opp osite d irected edges e, ¯ e and a residu e i ∈ { 0 , . . . , p − 1 } . If i · x e,i + ( p − i ) · x ¯ e,p − i = 0, then subtract this equ atio n su ccessiv ely from ev ery equation in ( 37 ). W e th us obtain a new d isjunction, similar to th at of ( 37 ), but whic h do es not con tain the x e,i and x ¯ e,p − i v ariables, and with th e same free-terms. Otherwise, i · x e,i + ( p − i ) · x ¯ e, p − i = p , then subtract this equation successive ly from ev ery equation in ( 37 ). Again, we obtain a n ew disjun ction, similar to th at of ( 37 ), bu t which do es not con tain the x e,i and x ¯ e,p − i v ariables, and suc h that p is subtracted from ev ery free-term in ev ery equation. Since, b y assu mption, n ≡ 1 m o d p , the fr ee-t erms in ev ery equatio n are (still) equal 1 mo d p . So o v erall, in b oth cases ( i · x e,i + ( p − i ) · x ¯ e,p − i = 0 an d i · x e,i + ( p − i ) · x ¯ e,p − i = p ) w e obtained a new d isjunction with all the free-te rms in equations equal 1 mo d p . W e no w con tin ue the same pro cess for ev ery p air e, ¯ e of opp osite directed edges in G ′ and ev ery residue i . Ev en tu ally , we discard all the v ariables x e,i in the equations, for ev ery e ∈ E ′ and i ∈ { 0 , . . . , p − 1 } , while all the free-terms in ev ery equation remain to b e equal 1 mo d p . Therefore, w e arrive at a disjunction of equations of the form (0 = γ ) for s ome γ = 1 mo d p . By us in g th e Simpliﬁcation ru le w e can cu t-oﬀ all such equations, and arr iv e ﬁn ally at the emp t y disjunction. 6.3. The C lique-Coloring Principle in R(lin) . In this section we observ e th at there are p olynomial-size R(lin) p ro ofs of the clique-coloring principle (for certain, w eak, parameters). T h is implies, in particular, that R(lin) d oes not p ossess the feasible m onotone interp olat ion prop erty (see more d etail s on the interp olatio n metho d in Section 7 ). A tserias, Bonet & Esteban [ ABE02 ] demonstrated p olynomial-size Res(2) r efutations of th e clique-colo ring formulas (for certain wea k parameters; Th eorem 20 ). Th us, it is su ﬃcien t to show that R(lin) p olynomially-sim ulates Res(2) pro ofs (Pr op ositio n 2 ). This can b e sho wn in a straight- forw ard manner. As noted in the ﬁrst p aragraph of S ectio n 6 , b ecause the p r oofs of the clique- coloring formula w e discuss here only follo w the pro ofs inside Res(2), then in fact these pro ofs do not tak e any adv anta ge of th e capacit y “to count” inside R(lin) (this capacit y is exempliﬁed, for instance, in S ecti on 4.2 ). W e start with the clique-coloring formulas (these f orm ulas will also b e used in Section 8 ). These form ulas expr ess the clique-coloring p rinciple that has b een w idely used in the p ro of complexit y literature (cf., [ BPR97 ], [ Pud 97 ], [ Kra97 ], [ Kra98 ], [ ABE02 ], [ K ra07 ]). This principle is based on the follo wing b asic combinatorial idea. Let G = ( V , E ) b e an un directed graph with n ve rtices and let k ′ < k b e t w o in tege rs. Then, one of the follo w ing m ust hold: (i): The graph G do es not cont ain a clique with k vertic es ; (ii): The graph G is n ot a c omplete k ′ -p artite gr aph . In other w ords, ther e is no wa y to partition G in to k ′ subgraphs G 1 , . . . , G k ′ , such that ev ery G i is an in dep enden t set, and for all i 6 = j ∈ [ k ′ ], all the v ertices in G i are connected b y edges (in E ) to all the v ertices in G j . Ob viously , if Item (ii) ab o v e is false (that is, if G is a complete k ′ -partite graph), then there exists a k ′ -coloring of the ve rtices of G ; hence the name clique-c oloring for the principle. The prop ositional form ulation of the (negatio n of th e) clique-co loring p rinciple is as follo w s. Eac h v ariable p i,j , for all i 6 = j ∈ [ n ], is an indicator v ariable for the fact that there is an edge in G b et w een vertex i and v ertex j . E ac h v ariable q ℓ,i , for all ℓ ∈ [ k ] and all i ∈ [ n ], is an indicator v ariable for the fact that the verte x i in G is the ℓ th v ertex in the k -clique. Eac h v ariable r ℓ,i , for all ℓ ∈ [ k ′ ] and all i ∈ [ n ], is an ind icato r v ariable for the fact that the verte x i in G p ertains to the indep endent set G ℓ . Deﬁnition 6.3. The negation of the clique-coloring principle consists of the follo wing unsatisﬁable collect ion of clauses (as translated to d isj unctions of linear equations), denoted ¬ clique n k ,k ′ : 23 (i) ( q ℓ, 1 = 1) ∨ · · · ∨ ( q ℓ,n = 1) , for all ℓ ∈ [ k ] (expresses th at there exists at least one vertex in G wh ic h constitutes the ℓ th v ertex of the k -clique); (ii) ( q ℓ,i = 0) ∨ ( q ℓ,j = 0) , for all i 6 = j ∈ [ n ] , ℓ ∈ [ k ] (expresses th at there exists at most one ve rtex in G w hic h constitutes the ℓ th vertex of the k -clique); (iii) ( q ℓ,i = 0) ∨ ( q ℓ ′ ,i = 0) , for all i ∈ [ n ] , ℓ 6 = ℓ ′ ∈ [ k ] (expresses that the i th v er tex of G cannot b e b oth the ℓ th and the ℓ ′ th vertex of the k -clique); (iv) ( q ℓ,i = 0) ∨ ( q ℓ ′ ,j = 0) ∨ ( p i,j = 1) , for all ℓ 6 = ℓ ′ ∈ [ k ] , i 6 = j ∈ [ n ] (expresses that if b oth the v ertices i and j in G are in the k -clique, then there is an edge in G b et w een i and j ); (v) ( r 1 ,i = 1) ∨ · · · ∨ ( r k ′ ,i = 1) , for all i ∈ [ n ] (expresses that ev ery vertex of G p ertains to at least one indep enden t set); (vi) ( r ℓ,i = 0) ∨ ( r ℓ ′ ,i = 0) , for all ℓ 6 = ℓ ∈ [ k ′ ] , i ∈ [ n ] (expresses that ev ery vertex of G p ertains to at most one in dep enden t set); (vii) ( p i,j = 0) ∨ ( r t,i = 0) ∨ ( r t,j = 0) , for all i 6 = j ∈ [ n ] , t ∈ [ k ′ ] (expresses that if th er e is an edge b et w een v ertex i and j in G , then i and j cannot b e in the same indep end ent set); Remark 2. Our formulation of the clique-coloring form ulas ab o v e is similar to the one used by [ BPR97 ], except that w e consider also the p i,j v ariables (we added the ( iv ) clauses and c hanged accordingly the ( vii ) clauses). This is done f or th e sake of clarit y of the con tradiction itself, and also to mak e it clear that the formulas are in th e appropriate form required by the in terp olation metho d (see S ectio n 7 for details on the int erp olation metho d). By resolving o v er the p i,j v ariables in ( iv ) and ( vii ), one can obtain precisely th e collection of clauses in [ BPR97 ]. A tserias, Bonet & Esteban [ ABE02 ] demonstrated p olynomial-size (in n ) Res(2) refutations of ¬ clique n k ,k ′ , when k = √ n and k ′ = (log n ) 2 / 8 log log n . Th ese are rather w eak p arameters, but they suﬃce to establish the fact that Res(2) do es n ot p ossess the feasible monotone interp olatio n prop ert y . The Res(2) p r oof system (also called 2 -DNF r esolution ), ﬁrst considered in [ Kra01 ], is resolution extended to op erate with 2-DNF form ulas, deﬁ ned as follo ws. A 2 -term is a conju nction of up to tw o literals. A 2-DNF is a disjun ction of 2-terms. The size of a 2-t erm is the n um b er of literals in it (that is, either 1 or 2). Th e size of a 2 -DN F is the total size of all the 2-terms in it. Deﬁnition 6.4 (Res(2)) . A R es( 2 ) pr o of of a 2 -DNF D fr om a c ol le ction K of 2 -DNFs is a sequence of 2-DNFs D 1 , D 2 , . . . , D s , suc h that D s = D , and ev ery D j is either from K or was deriv ed from p r evious line(s) in the sequence b y the follo wing inference rules: Cut: L et A, B b e tw o 2-DNFs. F rom A ∨ V 2 i =1 l i and B ∨ W 2 i =1 ¬ l i deriv e A ∨ B , where the l i ’s are (not necessarily d istin ct) literals (and ¬ l i is the n ega tion of the lite ral l i ). AND-in tro duction: Let A, B b e t wo 2-DNF s and l 1 , l 2 t w o literals. F rom A ∨ l 1 and B ∨ l 2 deriv e A ∨ B ∨ V 2 i =1 l i . W eak ening: F rom a 2-DNF A derive A ∨ V 2 i =1 l i , wh ere the l i ’s are (n ot necessarily dis- tinct) lite rals. A Res(2) r efutation of a collection of 2-DNFs K is a Res(2) pro of of the empty d isjunction ✷ from K (the empt y disjunction stands for f als e ). The size of a Res(2) pro of is the total size of all the 2-DNFs in it. 24 Giv en a collection K of 2-DNFs we translate it in to a collection of disjunctions of linear equations via th e follo wing translation scheme. F or a literal l , denote by b l the translation th at maps a v ariable x i in to x i , and ¬ x i in to 1 − x i . A 2-term l 1 ∧ l 2 is ﬁrst transformed int o the equ ation b l 1 + b l 2 = 2, and then movi ng the fr ee-te rms in the left hand side of b l 1 + b l 2 = 2 (in case there are su c h f ree-terms) to the right h and side; So that the ﬁnal translation of l 1 ∧ l 2 has only a single free-term in the righ t h and side. A disjun ction of 2-terms (that is, a 2-D NF) D = W i ∈ I ( l i, 1 ∧ l i, 2 ) is translated in to the disjunction of the tr anslatio ns of the 2-terms, denoted by b D . It is cle ar that ev ery assignmen t satisﬁes a 2-D NF D if and only if it satisﬁes b D . Prop osition 2. R(lin) p olynomial ly si mulates R es( 2 ). In other wor ds, if π i s a R es( 2 ) pr o of of D fr om a c ol le ction of 2 -DN Fs K 1 , . . . , K t , then ther e i s an R(lin) pr o of of b D fr om b K 1 , . . . , b K t whose size is p olynomial in the size of π . The pr o of of Prop osition 2 pr o ceeds by indu ction on th e length (that is, the n um b er of pro of- lines) in the Res(2) p ro of. This is prett y straightfo rwa rd and similar to the simulation of resolution b y R(lin), as illustrated in the pro of of Prop osition 1 . W e omit the details. Theorem 20 ([ ABE02 ]) . L et k = √ n and k ′ = (log n ) 2 / 8 log log n . Then ¬ clique n k ,k ′ has R es( 2 ) r efutations of si ze p olyn omial in n . Th us, P rop osition 2 yields the follo w ing: Corollary 21. L et k , k ′ b e as in The or em 20 . Then ¬ clique n k ,k ′ has R(lin) r e f utations of size p olynomial in n . The follo wing corollary is imp ortant (w e refer the reader to Section A in the App end ix for th e necessary relev ant deﬁnitions concerning the fe asible monotone interp olation pr op erty and to Section 7 for explanation and deﬁnitions concerning the general [non-monotone] in terp olation metho d). Corollary 22. R(lin) do es not p ossess the f e asible monotone interp olation pr op e rty. Remark 3. The pro of of ¬ clique n k ,k ′ inside Res(2) demonstrated in [ ABE02 ] (and hence, also the corresp onding pro of inside R(lin)) p ro ceeds along th e follo w in g lines. First r educe ¬ clique n k ,k ′ to the k to k ′ pigeonhole principle. F or the appropriate v alues of th e p arameters k and k ′ — and sp eciﬁcally , for the v alues in Theorem 20 — th er e is a short r e solution pro of of the k to k ′ pigeonhole p rinciple (this w as sh own b y Buss & Pitassi [ BP97 ]); (this resolution p ro of is p olynomial in the n um b er of p igeons k , but not in the num b er of holes k ′ , w hic h is exp onen tial ly smaller than k ). 9 Therefore, in ord er to conclude the refu tatio n of ¬ clique n k ,k ′ inside Res(2) (or inside R(lin)), it suﬃces to sim ulate the sh ort resolution r efutation of the k to k ′ pigeonhole principle. It is imp ortant to emp hasize this p oin t: After red u cing, inside R(lin), ¬ clique n k ,k ′ to th e pigeonhole principle, one sim ulates the r esolution refutation of the pigeonhole principle, and this has nothing to d o with the small-size R 0 (lin) refutations of the pigeonhole principle demonstrated in Section 6.1 . T his is b ecause, the reduction (inside R(lin)) of ¬ clique n k ,k ′ to the k to k ′ pigeonhole principle, r esu lts in a substitution instanc e of the p igeo nhole p rinciple formulas; in other words, the reduction r esults in a collection of disju nctions that are similar to the pigeonhole principle disjun ctions wher e e ach original pige onho le principle variable is substitute d by some big formula (and, in particular, these disjunctions are not R 0 (lin)-lines at all). (Note that R 0 (lin) d oes not admit short pro ofs of the clique-colo ring form ulas as we s ho w in Section 8 .) 9 Whenever k ≥ 2 k ′ the k to k ′ pigeonhole principle is referred to as th e we ak pige onhole principle . 25 7. Interp ola tion Resul ts for R 0 (lin) In this section we s tudy the applicabilit y of the f easible (non-monotone) int erp olation tec hnique to R 0 (lin) refutations. In p articular, w e sho w that R 0 (lin) admits a p olynomial (in terms of the R 0 (lin)-pro ofs) up p er b ound on the (non-monotone) circuit-size of in terp olan ts. In the next section w e shall giv e a p olynomial upp er b ound on the monoto ne circuit-size of in terp olan ts, b ut only in the case that the interp olan t corresp onds to the clique-coloring formulas (whereas, in this s ectio n we are in terested in the general case; that is, u p p er b ou n ding circuit-size of interp olan ts corresp onding to an y formula [of the prescrib ed t yp e; see b elo w ]). Firs t, we shortly describ e the feasible in terp olation metho d and explain ho w this m ethod can b e applied to obtain (sometime, conditional) lo w er b ounds on p ro of size. E xplicit usage of the int erp olation metho d in p ro of complexit y go es bac k to [ Kra94 ]. Let A i ( ~ p, ~ q ), i ∈ I , and B j ( ~ p, ~ r ), j ∈ J , ( I and J are sets of ind ices) b e a collectio n of formulas (for instance, a collecti on of disj unctions of linear equations) in the displa y ed v ariables only . Denote b y A ( ~ p, ~ q ) the conjunction of all A i ( ~ p, ~ q ), i ∈ I , and by B ( ~ p, ~ r ), the conjunction of all B j ( ~ p, ~ r ), j ∈ J . Assume that ~ p, ~ q , ~ r are pairwise disjoint sets of distinct v ariables, and that there is no assignment that satisﬁes b oth A ( ~ p, ~ q ) and B ( ~ p, ~ r ). Fix an assignmen t ~ α to the v ariables in ~ p . Th e ~ p v ariables are the only c ommon variables of the A i ’s and the B j ’s. Therefore, either A ( ~ α , ~ q ) is unsatisﬁable or B ( ~ α, ~ r ) is unsatisﬁable. The in terp olation tec hnique transforms a refu tatio n of A ( ~ p , ~ q ) ∧ B ( ~ p, ~ r ), in some pr oof system, in to a circu it (usu ally a Bo olean circuit) separating those assignment s ~ α (for ~ p ) for whic h A ( ~ α , ~ q ) is unsatisﬁable, from those assignmen ts ~ α for whic h B ( ~ α, ~ r ) is un satisﬁable (the tw o cases are not necessarily exclusiv e, so if b oth cases h old for an assignmen t, the circuit can output either that the ﬁrst case holds or that the second case holds). In other w ords, giv en a refutation of A ( ~ p, ~ q ) ∧ B ( ~ p, ~ r ), w e construct a circuit C ( ~ p ), called the interp olant , such that C ( ~ α ) = 1 = ⇒ A ( ~ α, ~ q ) is unsatisﬁable , and C ( ~ α ) = 0 = ⇒ B ( ~ α, ~ r ) is u nsatisﬁable . (39) (Note that if U denotes the set of th ose assignmen ts ~ α for which A ( ~ α, ~ q ) is satisﬁable , and V denotes the set of those assignments ~ α for whic h B ( ~ α, ~ r ) is satisﬁable , then U and V are disjoint [since A ( ~ p, ~ q ) ∧ B ( ~ p, ~ r ) is unsatisﬁable], and C ( ~ p ) separates U from V ; see Deﬁnition 7.2 b elo w .) Assume th at for a pro of system P the transformation from r efutations of A ( ~ p, ~ q ) , B ( ~ p, ~ r ) in to the corresp ond ing in terp olan t circuit C ( ~ p ) results in a circuit whose size is p olynomial in th e size of the r efu tatio n. Then, an exp onentia l lo w er b ound on circuits for which ( 39 ) holds, implies an exp onen tial low er b ound on P -refutations of A ( ~ p, ~ q ) , B ( ~ p , ~ r ). 7.1. In terp olation for Seman tic Refutations. W e now la y out the b asic concepts n eeded to formally describ e the feasible interpolation tec hnique. W e use the general notion of semantic r efutations (which generalizes any standard p rop ositional refu tatio n system). W e shall use a close terminology to that in [ Kra97 ]. Deﬁnition 7.1 (Semanti c refutation) . Let N b e a ﬁxed natural num b er and let E 1 , . . . , E k ⊆ { 0 , 1 } N , wher e T k i =1 E i = ∅ . A semantic r ef u tation from E 1 , . . . , E k is a sequen ce D 1 , . . . , D m ⊆ { 0 , 1 } N with D m = ∅ and such that for eve ry i ∈ [ m ], D i is either one of the E j ’s or is dedu ced from t wo previous D j , D ℓ , 1 ≤ j, ℓ < i , by the follo w ing semantic infer e nc e rule : • F rom A, B ⊆ { 0 , 1 } N deduce an y C , suc h that C ⊇ ( A ∩ B ). Observe that any stand ard prop ositional refutation (with in ference rules that deriv e fr om at most t w o pr oof-lines, a third line) can b e regarded as a seman tic refutation: ju st s ubstitute eac h refutation-line b y the set of its satisfying assignmen ts; an d by th e sound ness of the inference rules applied in the refutation, it is clea r that eac h refutation-line (considered as the set of assignmen ts that satisfy it) is deduced b y th e semantic in ference rule from previous refutation-lines. 26 Deﬁnition 7.2 (Separating circuit) . Let U , V ⊆ { 0 , 1 } n , where U ∩ V = ∅ , b e t wo disjoin t sets. A Bo olea n circuit C with n input v ariables is said to sep ar ate U fr om V if C ( ~ x ) = 1 for ev er y ~ x ∈ U , and C ( ~ x ) = 0 for every ~ x ∈ V . In this case w e also sa y that U and V are sep ar ate d by C . Con v en tion : In what follo ws we sometime iden tify a Bo olean form ula with the s et of its satisfying assignmen ts. Notation : F or t w o (or more) b inary strings u, v ∈ { 0 , 1 } ∗ , we write ( u, v ) to d enote the concate- nation of the u with v (wh er e v comes to the r igh t of u , ob viously). Let N = n + s + t b e ﬁxed from no w on. Let A 1 , . . . , A k ⊆ { 0 , 1 } n + s and let B 1 , . . . , B ℓ ⊆ { 0 , 1 } n + t . Deﬁne the follo wing t w o sets of assignmen ts of length n (formally , 0 , 1 strings of length n ) that can b e extended to satisfying assignment s of A 1 , . . . , A k and B 1 , . . . , B ℓ , r esp ectiv ely (formally , those 0 , 1 string of length n + s and n + t , that are con tained in all A 1 , . . . , A k and B 1 , . . . , B ℓ , resp ectiv ely): U A := ( u ∈ { 0 , 1 } n     ∃ q ∈ { 0 , 1 } s , ( u, q ) ∈ k \ i =1 A i ) , V B := ( v ∈ { 0 , 1 } n     ∃ r ∈ { 0 , 1 } t , ( v , r ) ∈ ℓ \ i =1 B i ) . Deﬁnition 7.3 ( p olynomial upp er b ounds on interpolants ) . Let P b e a pr op ositio nal refutation system. Assume that ~ p, ~ q , ~ r are pairwise disjoin t sets of distinct v ariables, where ~ p has n v ariables, ~ q has s v ariables and ~ r has t v ariables. Let A 1 ( ~ p, ~ q ) , . . . , A k ( ~ p, ~ q ) and B 1 ( ~ p, ~ r ) , . . . , B ℓ ( ~ p, ~ r ) b e t w o colle ctions of formulas with the d ispla yed v ariables only . Ass u me that for any such A 1 ( ~ p, ~ q ) , . . . , A k ( ~ p, ~ q ) and B 1 ( ~ p, ~ r ) , . . . , B ℓ ( ~ p, ~ r ), if th ere exists a P -r efutation of size S for A 1 ( ~ p, ~ q ) ∧ · · · ∧ A k ( ~ p, ~ q ) ∧ B 1 ( ~ p, ~ r ) ∧ . . . ∧ B ℓ ( ~ p, ~ r ) then there exists a Boolean circuit separating U A from V B of size p olynomial in S . 10 In this case w e sa y that P h as a p olynomial upp er b ound on interp olant cir cui ts . 7.1.1. The Communic ation Game T e chnique. The fe asible interp olation via c ommunic ation game te chnique is based on transformin g p ro ofs in to Bo olean circuits, where the size of the r esulting circuit dep en d s on the comm unication complexit y of eac h p ro of-line. This tec hn ique go es bac k to [ IPU94 ] and [ Razb95 ] and was su b sequen tly app lied and extended in [ BPR97 ] and [ Kra97 ] ([ IPU94 ] and [ BPR97 ] did n ot u se explicitly the n otion of in terp olation of tautologies or cont radictions). W e shall employ the in terp olation theorem of Kr a j ´ ı ˇ cek in [ Kra97 ], that demonstr ates ho w to transform a small semantic refutation with eac h p r oof-line h a vin g lo w comm unication complexit y in to a small Bo olea n circuit separating the corresp onding sets. The un derlying idea of the interpolation via comm unication game tec hnique is that a (semanti c) refutation, where eac h p ro of-line is of small (that is, logarithmic) communicatio n complexity , can b e transformed in to an eﬃcien t comm u nication proto col for the Kar chmer-Wigderson game (follo wing [ KW88 ]) for t w o pla y ers. In the Karc hmer-Wigderson game the ﬁrst pla y er kno ws some binary string u ∈ U and the second pla y er kn o w s s ome diﬀerent binary string v ∈ V , where U and V are disjoin t sets of strin gs. Th e t w o p la yers communicate by sending information bits to one another (follo win g a p rotocol previously agreed on). The goal of the game is for the tw o pla y ers to decide on an index i such that the i th b it of u is diﬀerent from the i th bit of v . An eﬃcien t Karc hmer- Wigderson p rotocol (b y whic h we mean a proto col that r equires the p la yers to exc hange at most a logarithmic n umber of b its in th e w orst-ca se) can then b e transform ed in to a s mall circuit separating 10 Here U A and V B are deﬁned as ab o ve , by iden tifying the A i ( ~ p, ~ q )’s and the B i ( ~ p, ~ r ) ’s w ith the sets of assignmen ts that satisfy them. 27 U from V (see Deﬁnition 7.2 ). This eﬃcien t transform ation from p rotocols for Karc hmer-Wigderson games (describ ed in a certain wa y) into circuits, w as demonstr ated b y Razb oro v in [ Razb95 ]. S o o v er all, giv en a seman tic refutation with pr oof-lines of lo w communicatio n complexit y , one can obtain a small circuit for separating the corresp ondin g sets. First, w e need to d eﬁne the concept of c omm unic ation c omplexity in a suitable wa y for the in terp olation theorem. Deﬁnition 7.4 (Comm u nicatio n complexit y) . Let N = n + s + t and A ⊆ { 0 , 1 } N . Let u, v ∈ { 0 , 1 } n , q u ∈ { 0 , 1 } s , r v ∈ { 0 , 1 } t . Denot e by u i , v i the i th bit of u , v , resp ectiv ely , and let ( u, q u , r v ) and ( v , q u , r v ) d enote the concat enation of strings u, q u , r v and v , q u , r v , resp ectiv ely . Consider the follo w ing three tasks: (1) Decide w hether ( u, q u , r v ) ∈ A ; (2) Decide w hether ( v , q u , r v ) ∈ A ; (3) If one of the follo wing h olds: (i) ( u, q u , r v ) ∈ A and ( v , q u , r v ) 6∈ A ; or (ii) ( u, q u , r v ) 6∈ A and ( v , q u , r v ) ∈ A , then ﬁnd an i ∈ [ n ], such that u i 6 = v i ; Consider a game b et w een t wo pla y ers, Play er I and P la yer I I, where Pla y er I knows u ∈ { 0 , 1 } n , q u ∈ { 0 , 1 } s and Pla yer I I knows v ∈ { 0 , 1 } n , r v ∈ { 0 , 1 } t . The t w o pla y ers communicate by exc han ging bits of information b etw een them (follo win g a proto col previously agreed on). Th e c ommunic ation c omplexity of A , denoted C C ( A ), is the minimal (o ver all proto cols) n um b er of bits that pla y ers I and I I need to exc hange in the w orst-case in solving eac h of T asks 1 , 2 and 3 ab o ve. 11 F or A ⊆ { 0 , 1 } n + s deﬁne ˙ A :=  ( a, b, c )   ( a, b ) ∈ A and c ∈ { 0 , 1 } t  , where a and b range o ver { 0 , 1 } n and { 0 , 1 } s , resp ectiv ely . Similarly , for B ⊆ { 0 , 1 } n + t deﬁne ˙ B :=  ( a, b, c )   ( a, c ) ∈ B and b ∈ { 0 , 1 } t  , where a and c range ov er { 0 , 1 } n and { 0 , 1 } t , resp ectiv ely . Theorem 23 ([ Kra97 ]) . L et A 1 , . . . , A k ⊆ { 0 , 1 } n + s and B 1 , . . . , B ℓ ⊆ { 0 , 1 } n + t . L et D 1 , . . . , D m b e a semantic r efutation fr om ˙ A 1 , . . . , ˙ A k and ˙ B 1 , . . . , ˙ B ℓ . Assume that C C ( D i ) ≤ ζ , for al l i ∈ [ m ] . Then, the sets U A and V B (as deﬁne d ab ove) c an b e sep ar ate d by a Bo ole an cir cuit of size ( m + n )2 O ( ζ ) . In ligh t of T heorem 23 , to demonstrate that a certain pr op ositional r efu tatio n system P p ossesses a p olynomial up p er b ound on int erp olan t circu its (see Deﬁnition 7.3 ) it su ﬃces to sh o w that any pro of-line of P ind u ces a set of assignmen ts with at most a logarithmic (in the num b er of v ariables) comm unication complexit y (Deﬁnition 7.4 ). 7.2. P olynomial Upp er Bounds on In terp olan ts for R 0 (lin). Here we apply Th eorem 23 to sho w that R 0 (lin) h as p olynomial upp er b ound s on its in terp olan t circuits. Again, in w h at follo ws w e sometime identify a disju nction of linear equations with the set of its satisfying assignments. Theorem 24. R 0 (lin) has a p olynomial upp er b ounds on interp olant cir cu its (Deﬁnition 7.3 ). According to the paragraph after Theorem 23 , all we need in ord er to establish Th eorem 24 is the follo win g lemma: 11 In oth er words, C C ( A ) is th e minimal n umber ζ , for which th ere exists a proto col, such that for every inpu t ( u, q u to Play er I and v , r v to Play er I I) and every task (from T asks 1 , 2 and 3 ), th e pla yers need to exchange at most ζ bits in order to solve the task. 28 Lemma 25. L et D b e an R 0 (lin) -line with N variables and let e D b e the set of assignments that satisfy D . 12 Then, C C ( e D ) ≤ O (log N ) . Pro of : Let N = n + s + t (and so e D ∈ { 0 , 1 } n + s + t ). F or the sak e of con v enience we shall assume that th e N v ariables in D are partitioned in to (pairwise disjoint) three groups ~ p := ( p 1 . . . , p n ), ~ q := ( q 1 , . . . , q s ) and ~ r := ( r 1 , . . . , r t ). L et u, v ∈ { 0 , 1 } n , q u ∈ { 0 , 1 } s , r v ∈ { 0 , 1 } t . Assume th at Pla y er I kno ws u, q u and Pla yer I I kno w s v , r v . By the d eﬁnition of an R 0 (lin)-line (see Deﬁnition 3.2 ) we can partition the disj u nction D into a c onstant numb er of disjun cts, w here one disjun ct is a (p ossibly empty , translation of a) clause in the ~ p, ~ q , ~ r v ariables (see Section 3.1 ), and all other disjuncts hav e the follo wing form: _ i ∈ I  ~ a · ~ p + ~ b · ~ q + ~ c · ~ r = ℓ i  , ( 40) where I is (an u n b ounded) set of ind ices, ℓ i are integ er num b ers, for all i ∈ I , and ~ a, ~ b, ~ c denote v ectors of n, s and t constant coeﬃcien ts, resp ectiv ely . Let us denote the (translation of th e) clause from D in the ~ p, ~ q , ~ r v ariables b y P ∨ Q ∨ R , where P , Q and R d en ote the (translated) sub-clauses consisting of the ~ p , ~ q and ~ r v ariables, resp ectiv ely . W e need to sho w that by exc hanging O (log N ) bits, the pla y ers can solv e eac h of T asks 1 , 2 and 3 from Deﬁn ition 7.4 , correctly . T ask 1 : The pla y ers need to decide whether ( u, q u , r v ) ∈ e D . Pla yer I I, who kn o w s r v , computes the num b ers ~ c · r v , for eve ry ~ c p ertaining to ev ery disju nct of the f orm shown in E quation ( 40 ) ab o v e. Th en, Play er I I send s th e (binary r epresen tatio n of ) these n um b ers to Pla yer I. Since there are only a constan tly man y suc h num b ers and the coeﬃcien ts in ev ery ~ c are also constants, this amoun ts to O (log t ) ≤ O (log N ) bits that Pla y er I I sends to Play er I. Pla y er I I also compu tes the truth v alue of th e su b -clause R , and sends this (single-bit) v alue to Pla y er I. No w, it is easy to see that Pla y er I h as suﬃcien t data to compute b y h erself/himself whether ( u, q u , r v ) ∈ e D (Pla y er I can then send a single b it informing Play er I I whether ( u, q u , r v ) ∈ e D ). T ask 2 : This is analogous to T ask 1 . T ask 3 : Assu me that ( u, q u , r v ) ∈ e D and ( v , q u , r v ) 6∈ e D (the case ( u, q u , r v ) 6∈ e D and ( v , q u , r v ) ∈ e D is analo gous). The ﬁr s t rounds of the proto col are completely similar to that describ ed in T ask 1 ab o v e: Pla y er I I, who kno ws r v , computes the num b ers ~ c · r v , for ev ery ~ c p ertaining to every disj unct of the form sho wn in Equation ( 40 ) ab o v e. Then, Play er II send s the (binary represen tatio n of ) these n umbers to Pla y er I. Pla yer I I also computes the tru th v alue of the sub-clause R , and send s this (single-bit) v alue to Pla yer I. Again, this amoun ts to O (log N ) bits that Pla yer I I sen d s to Pla y er I. By assumption (that ( u, q u , r v ) ∈ e D and ( v , q u , r v ) 6∈ e D ) the pla y ers n eed to deal only with the follo w ing t w o cases: Case 1: The assignmen t ( u, q u , r v ) satisﬁes the clause P ∨ Q ∨ R w h ile ( v , q u , r v ) falsiﬁes P ∨ Q ∨ R . Th us, it must b e that ~ u satisﬁes the sub -clause P while ~ v falsiﬁes P . This means that for any i ∈ [ n ] suc h that u i sets to 1 a literal in P (there ought to exist at least one su c h i ), it must b e that u i 6 = v i . Therefore, all that Pla y er I needs to d o is to sen d the (b inary r ep resen tati on of ) index i to Pla yer I I. (This amounts to O (log N ) bits that Play er I sends to Pla yer II .) 12 The notation e D has nothing to do with th e same notation used in Section 3 . 29 Case 2: Th ere is some linear equ atio n ~ a · ~ p + ~ b · ~ q + ~ c · ~ r = ℓ (41) in D , such that ~ a · u + ~ b · q u + ~ c · r v = ℓ . Note th at (b y assump tion that ( v , q u , r v ) 6∈ e D ) it m ust also hold that: ~ a · v + ~ b · q u + ~ c · r v 6 = ℓ (and so there is an i ∈ [ n ], suc h that u i 6 = v i ). Play er I can ﬁnd linear equation ( 41 ), as h e/she already receiv ed fr om Pla y er I I all the p ossible v alues of ~ c · ~ r (for all p ossible ~ c ’s in D ). Recall that the left hand side of a linear equation ~ d · ~ x = ℓ is called the line ar form of the equation. By the deﬁn ition of an R 0 (lin)-line there are only constan t many d istinct linear forms in D . S ince b oth play ers kno w these linear forms , we can assu me that eac h linear form has some index asso ciated to it by b oth pla yers. Pla yer I sends to Pla y er I I the ind ex of the linear form ~ a · ~ p + ~ b · ~ q + ~ c · ~ r from ( 41 ) in D . Since there are only c onstantly m an y suc h linear forms in D , it tak es only constant num b er of bits to send th is ind ex. No w b oth pla yers need to apply a proto col for ﬁndin g an i ∈ [ n ] s uc h that u i 6 = v i , where ~ a · ~ u + ~ b · q u + ~ c · r v = ℓ and ~ a · ~ v + ~ b · q u + ~ c · r v 6 = ℓ . T hus, it remains only to pr o ve the follo wing claim: Claim 4. Th er e is a comm unication proto col in wh ic h Pla y er I and Play er I I need at most O (log N ) bits of communicatio n in order to ﬁnd an i ∈ [ n ] suc h th at u i 6 = v i (under the ab o v e conditions). Pro of of claim : W e inv oke the well-kno wn connection b etw een Bo olean circuit-depth and com- m unication complexit y . Let f : { 0 , 1 } N → { 0 , 1 } b e a Bo olean fu nction. De note by dp( f ) the minimal d epth of a Bo olean circu it computing f . Consider a game b et w een t w o p la yers: Play er I kno ws some ~ x ∈ { 0 , 1 } N and Pla y er I I kno ws some other ~ y ∈ { 0 , 1 } N , suc h that f ( ~ x ) = 1 while f ( ~ y ) = 0. The goal of the game is to ﬁ nd an i ∈ [ N ] suc h th at x i 6 = y i . Denote b y C C ′ ( f ) the minimal num b er of bits needed for the tw o p la yers to comm unicate (in the worst case 13 ) in order to solv e th is game. 14 Then, f or any fu nction f it is known that dp( f ) = CC ′ ( f ) (see [ KW88 ]). Therefore, to conclude the pro of of the claim it is enough to establish that the fun ction f : { 0 , 1 } N → { 0 , 1 } that receiv es the input v ariables ~ p, ~ q , ~ r and computes the truth v alue of ~ a · ~ p + ~ b · ~ q + ~ c · ~ r = ℓ h as Boolean circuit of depth O (log N ). In case all the co eﬃcien ts in ~ a, ~ b, ~ c are 1, it is easy to sh o w 15 that there is a Boolean circuit of depth O (log N ) that computes the fun ction f . In the case th at the co eﬃcien ts in ~ a, ~ b, ~ c are all constan ts, it is easy to sh o w, b y a redu ction to the case wher e all co eﬃcien ts are 1, 16 that there is a Boolean circuit of depth O (log N ) that computes the function f . W e omit th e details. 8. Size Lower Bounds In this secti on w e establish an exp onen tial-siz e lo wer b oun d on R 0 (lin) r efutations of the clique- coloring f orm ulas. W e shall emp lo y the theorem of Bonet, Pitassi & Raz in [ BPR97 ] that provides exp onen tial-size lo w er b oun ds f or an y seman tic refutation of the clique-coloring form ulas, having lo w comm un icatio n complexit y in eac h refutation-line. 13 Over all inp uts ~ x, ~ y such that f ( ~ x ) = 1 and f ( ~ y ) = 0. 14 The measure C C ′ is basically the same as C C deﬁn ed earlier. 15 Using the known O (log N ) -depth Boolean circuits for the threshold fun ctions. 16 F or instance, consider the simple case where w e ha v e only a single var iable. That is, let c b e a constant and assume that we wish to constru ct a circuit that compu tes c · x = ℓ , for some integer ℓ . Then, we take a circuit t hat computes th e fun ction f : { 0 , 1 } c → { 0 , 1 } that out puts th e tru th v alue of y 1 + . . . + y c = ℓ (t h us, in f all coeﬃcients are 1’s); and to compute c · x = ℓ we only ha ve t o substitute eac h y i in the circuit with the va riable x . 30 First w e r ecall the strong lo w er b ound obtained by Alon & Boppana [ AB87 ] (impro ving o v er [ Razb85 ]; see also [ And85 ]) for th e (monotone) clique sep ar ator fun ctions, d eﬁ ned as follo ws (a function f : { 0 , 1 } n → { 0 , 1 } is called monotone if for all α ∈ { 0 , 1 } n , α ′ ≥ α implies f ( α ′ ) ≥ f ( α )): Deﬁnition 8.1 (Clique separator) . A monotone b o olean function Q n k ,k ′ is called a clique sep ar ator if it in terprets its inpu ts as th e edges of a graph on n v ertices, and outputs 1 on eve ry inp ut represent ing a k -clique, and 0 on ev ery input repr esen ting a complete k ′ -partite graph (see Section 6.3 ). Recall that a monoto ne Bo ole an cir cuit is a circuit that uses only monotone Bo olean gates (for instance, only the fan-in t w o gate s ∧ , ∨ ). Theorem 26 ([ AB87 ]) . L et k, k ′ b e inte gers such that 3 ≤ k ′ < k and k √ k ′ ≤ n/ (8 log n ) , then every monotone Bo ole an cir cuit that c omputes a clique sep ar ator function Q n k ,k ′ r e qu i r es size at le ast 1 8  n 4 k √ k ′ log n  ( √ k ′ +1 ) / 2 . F or the next theorem, we need a slightly diﬀeren t (and w eak er) v ersion of comm unicatio n com- plexit y , than that in Deﬁnition 7.4 . Deﬁnition 8.2 (Comm unication complexit y (second deﬁnition)) . Let X denote n Bo olean v ariables x 1 , . . . , x n , and let S 1 , S 2 b e a partition of X into t w o disjoint s ets of v ariables. The communicati on complexit y of a Boolean function f : { 0 , 1 } n → { 0 , 1 } is the num b er of b its needed to b e exc hanged b y tw o pla y ers, one knowing th e v alues giv en to the S 1 v ariables and the other kno wing the v alues giv en to S 2 v ariables, in the w ors t-case, o v er all p ossible p artitions S 1 and S 2 . Theorem 27 ([ BPR97 ]) . Every semantic r efutation of ¬ clique n k ,k ′ (for k ′ < k ) with m r efutation- lines and wher e e ach r efutation-line (c onsider e d as a the char acteristic fu nction of the line) has c ommunic ation c omplexity (as in D eﬁnition 8.2 ) ζ , c an b e tr ansforme d i nto a monotone cir cu it of size m · 2 3 ζ +1 that c omp utes a sep ar ating function Q n k ,k ′ . In ligh t of Theorem 26 , in ord er to b e ab le to apply T heorem 27 to R 0 (lin), and arriv e at an exp onen tial-size lo w er b ound for R 0 (lin) r efutations of the clique-co loring form ulas, it suﬃces to sho w that R 0 (lin) pro of-lines hav e logarithmic comm unication complexit y: Lemma 28. L et D b e an R 0 (lin) -line with N variables. Then, the c ommunic ation c omp lexity (as in Deﬁnition 8.2 ) of D is at most O (log N ) (wher e D is identiﬁe d her e with the char acteristic function of D ). Pro of : The pro of is similar to the p ro of of Lemma 25 for solving T ask 1 (and the analogous T ask 2 ) in Deﬁnition 7.4 . By direct calculations w e obtain the follo wing lo w er b ou n d from T heorems 26 , 27 and Lemma 28 : Corollary 29. L et k b e an inte ger such that 3 ≤ k ′ = k − 1 and assume that 1 2 · n/ (8 log n ) ≤ k √ k ≤ n/ (8 log n ) . Then, for al l ε < 1 / 3 , every R 0 (lin) r efutation of ¬ cliqu e n k ,k ′ is of size at le ast 2 Ω( n ε ) . When considering the p arameters of Theorem 20 , we obtain a sup er -p olynomial separation b e- t w een R 0 (lin) refutations and R(lin) refutations, as describ ed b elo w. F rom Theorems 26 , 27 and Lemma 28 w e h av e (by dir ect calculations): 31 Corollary 30. L et k = √ n and k ′ = (log n ) 2 / 8 log log n . Then, every R 0 (lin) r efutation of ¬ clique n k ,k ′ has size at le ast n Ω “ log n √ log log n ” . By Corollary 21 , R(lin) adm its p olynomial-size in n r efutations of ¬ clique n k ,k ′ under the p aram- eters in C orolla ry 30 . T h us we obtain the follo wing s eparation r esu lt: Corollary 31. R(lin) is sup er-p olyno mial ly str onger than R 0 (lin) . Commen t 1. Note that we do not need to assume that the co eﬃcien ts in R 0 (lin)-lines are constant s for th e low er b ound argu m en t. If the co eﬃcien ts in R 0 (lin)-lines are only p olynomially b ound ed (in the num b er of v ariables) then th e same lo w er b ound as in Corollary 30 also ap p lies. This is b ecause R 0 (lin)-lines in which co eﬃcien ts are p olynomially b oun ded inte gers, still hav e lo w (that is, logarithmic) co mm unication complexit y (as in Deﬁnition 8.2 ). 9. Applica tions to Mul tilinear Proofs In this section we arriv e at one of the main b eneﬁts of the w ork we hav e done so far; Namely , applying results on resolution o v er linear equations in order to obtain new r esu lts for m ultilinear pro of systems. Subsection 9.1 that follo ws, con tains deﬁnitions, suﬃcient for the current pap er, concerning the n otio n of m ultilinear pro ofs introd uced in [ R T06 ]. 9.1. Bac kground on Algebraic a nd Multilinear Pro ofs. 9.1.1. Arithm etic and Multiline ar F ormulas. Deﬁnition 9.1 (Arithmetic form ula) . Fix a ﬁ eld F . An arithmetic formula is a tree, with edges directed from th e lea v es to the ro ot, and with un b ounded (ﬁnite) fan-in. Ev ery leaf of the tree (namely , a no de of fan-in 0) is lab eled with either an inp ut v ariable or a ﬁeld elemen t. A ﬁ eld elemen t can also lab el an edge of the tree. Ev ery other no de of th e tree is lab eled with either + or × (in the ﬁrst case the no de is a plus gate and in the second case a pr o duct gate ). W e assume that there is only one no de of out-degree zero, called the r o ot . The size of an arithmetic f orm ula F is the total num b er of no des in its graph and is denoted by | F | . An arithmetic formula computes a p olynomial in the ring of p olynomials F [ x 1 , . . . , x n ] in th e f ollo wing wa y . A leaf just computes the input v ariable or ﬁeld elemen t that lab els it. A ﬁ eld element that lab els an edge means that the p olynomial computed at its tail (namely , the no de where th e edge is d ir ecte d fr om) is multiplied b y this ﬁeld element . A plus gate compu tes the sum of p olynomials compu ted by the tails of all incoming edges. A p r od uct gate compu tes the pro duct of th e p olynomials computed b y th e tails of all incoming edges. (Subtraction is obtained using the constan t − 1.) The outp ut of the f orm ula is the p olynomial computed by the ro ot. T h e dept h of a formula F is the maximal n um b er of edges in a p ath from a leaf to the ro ot of F . W e sa y that an arithmetic form ula has a plus (r esp., pr o duct) gate at the r o ot if the ro ot of the form ula is lab eled with a p lus (resp ., pro du ct) gate. A p olynomial is multiline ar if in eac h of its monomials the p o w er of every inp ut v ariable is at most one. Deﬁnition 9.2 (Multilinear formula) . An arithmetic form ula is a multiline ar f ormula (or equiv a- len tly , multiline ar arithmetic formula ) if the p olynomial computed by e ach gate of the formula is m ultilinear (as a formal p olynomial, that is, as an elemen t of F [ x 1 , . . . , x n ]). An additional deﬁnition we shall n eed is the follo wing linear op erator, called the multiline arization op er ator : 32 Deﬁnition 9.3 (Multilinearization op erator) . Giv en a ﬁeld F and a p olynomial q ∈ F [ x 1 , . . . , x n ], w e denote b y M [ q ] the unique multili near p olynomial equal to q mo dulo the ideal generated by all the p olynomials x 2 i − x i , f or all v ariables x i . F or example, if q = x 2 1 x 2 + ax 3 4 (for some a ∈ F ) then M [ q ] = x 1 x 2 + ax 4 . The simulatio n of R 0 (lin) by multilinea r pro ofs will rely h ea vily on the fact that multilinear symmetric p olynomials ha v e small d epth-3 m ultilinear formulas o ver ﬁelds of charac teristic 0 (see [ SW01 ] for a pro of of this fact). T o th is end w e deﬁne precisely the concept of symmetric p olyno- mials. A r e naming of the v ariables x 1 , . . . , x n is a p ermuta tion σ ∈ S n (the symmetric group on [ n ]) suc h that x i is mapp ed to x σ ( i ) for ev ery 1 ≤ i ≤ n . Deﬁnition 9.4 (Symm etric p olynomial) . Giv en a set of v ariables X = { x 1 , . . . , x n } , a symmetric p olynomial f ov er X is a p olynomial in (all the v ariables of ) X suc h that r en aming of v ariables do es n ot change the p olynomial (as a formal p olynomial). 9.1.2. Polynomial Calculus with R esolution. Here w e deﬁn e the PCR pr oof system, in tro duced b y Alekhno vic h et al. in [ ABSR W0 2 ]. Deﬁnition 9.5 (Polynomial C alculus with Resolution (PCR)) . Let F b e some ﬁxed ﬁeld and let Q := { Q 1 , . . . , Q m } b e a collectio n of multiv ariate p olynomials fr om the ring of p olynomials F [ x 1 , . . . , x n , ¯ x 1 , . . . , ¯ x n ]. Th e v ariables ¯ x 1 , . . . , ¯ x n are treated as n ew formal v ariables. Call the set of p olynomials x 2 − x , for x ∈ { x 1 , . . . , x n , ¯ x 1 , . . . , ¯ x n } , plus the p olynomials x i + ¯ x i − 1, for all 1 ≤ i ≤ n , the set of Bo ole an axioms of PCR . A PCR pr o of from Q of a p olynomial g is a ﬁnite sequence π = ( p 1 , ..., p ℓ ) of multiv ariate p olynomials from F [ x 1 , . . . , x n , ¯ x 1 , . . . , ¯ x n ] (eac h p olynomial p i is in terpreted as the p olynomial equatio n p i = 0), where p ℓ = g and for eac h i ∈ [ ℓ ], either p i = Q j for some j ∈ [ m ], or p i is a Bo olean axiom, or p i w as deduced from p j , p k , wh ere j, k < i , by one of the follo wing inference rules: Pro duct: F rom p deduce x i · p , for some v ariable x i ; F rom p deduce ¯ x i · p , for some v ariable ¯ x i ; Addition: F rom p and q dedu ce α · p + β · q , for some α, β ∈ F . A PCR r efutation of Q is a pro of of 1 (which is interpreted as 1 = 0) from Q . The nu mb er of steps in a PCR pro of is the n umber of pr oof-lines in it (that is, ℓ in the case of π ab o v e). Note that the Bo olean axioms of PCR h a ve only 0 , 1 solutions, where ¯ x i = 0 if x i = 1 and ¯ x i = 1 if x i = 0. 9.1.3. Multiline ar P r o of Systems. In [ R T06 ] the authors introd u ced a natural (seman tic) algebraic pro of system that op erates with multilinear arithm etic formulas denoted fMC (which stands for formula multiline ar c alculus ), deﬁn ed as follo w s: Deﬁnition 9.6 (F ormula Multilinear C alculus (fMC)) . Fix a ﬁeld F and let Q := { Q 1 , . . . , Q m } b e a collection of m u ltilinear p olynomials from F [ x 1 , . . . , x n , ¯ x 1 , . . . , ¯ x n ] (the v ariables ¯ x 1 , . . . , ¯ x n are treated as f ormal v ariables). Call the set of p olynomials consisting of x i + ¯ x i − 1 and x i · ¯ x i for 1 ≤ i ≤ n , the Bo ole an axioms of f MC . An fMC pr o of fr om Q of a p olynomial g is a ﬁnite sequence π = ( p 1 , ..., p ℓ ) of multilinear p olynomials fr om F [ x 1 , . . . , x n , ¯ x 1 , . . . , ¯ x n ] , such that p ℓ = g and for eac h i ∈ [ ℓ ], either p i = Q j for some j ∈ [ m ], or p i is a Bo olean axiom of fMC , or p i w as deduced b y one of the follo win g inference r ules us ing p j , p k for j, k < i : Pro duct: from p deduce q · p , for some p olynomial q ∈ F [ x 1 , . . . , x n , ¯ x 1 , . . . , ¯ x n ] such that p · q is multiline ar ; Addition: from p , q ded uce α · p + β · q , for some α, β ∈ F . 33 All the p olynomials in an fMC p ro of are represen ted as m ultilinear formulas. (A p olynomial p i in an fMC pr oof is in terpreted as the p olynomial equation p i = 0.) An fMC r efutation of Q is a pr oof of 1 (wh ic h is in terpreted as 1 = 0) from Q . The size of an fMC pro of π is deﬁned as the total sum of all the form ula sizes in π and is denoted b y | π | . Note th at the Boolean axioms ha v e only 0 , 1 solutions, where ¯ x i = 0 if x i = 1 and ¯ x i = 1 if x i = 0, for eac h 1 ≤ i ≤ n . Deﬁnition 9.7 (Depth- k F ormula Multilinear C alculus (depth- k fMC)) . F or a natural num b er k , depth- k fMC denotes a restriction of the fMC p ro of system, in whic h pr oofs consist of multilinear p olynomials from F [ x 1 , . . . , x n , ¯ x 1 , . . . , ¯ x n ] represen ted as multilinear formulas of depth at most k . 9.2. F rom R(lin) Pro ofs t o PCR Pro ofs. W e n o w demonstrate a general an d straigh tforw ard translation from R(lin) pro ofs in to PCR pro ofs o v er ﬁelds of c haracteristic 0. W e us e the term “translation” in order to distinguish it f rom a simulation ; s in ce here w e are not interested in the size of PCR pr oofs. In fact w e hav e n ot deﬁn ed the size of PCR pro ofs at all. W e shall b e intereste d only in the numb er of steps in PCR pro ofs. F r om now on, al l p olynomials and arithmetic formulas ar e c onsider e d over some ﬁx ﬁeld F of char acteristic 0. Recall that any ﬁ eld of characte ristic 0 co n tains (an isomorphic cop y of ) th e in teger n um b ers, and so w e can use in teger coeﬃcients in the ﬁeld. Deﬁnition 9.8 (P olynomial translation of R(lin) p ro of-lines) . Let D b e a disjun ction of linear equations:  a (1) 1 x 1 + . . . + a (1) n x n = a (1) 0  ∨ · · · ∨  a ( t ) 1 x 1 + . . . + a ( t ) n x n = a ( t ) 0  . (42) W e denote b y b D its translation in to the f ollo wing p olynomial: 17  a (1) 1 x 1 + . . . + a (1) n x n − a (1) 0  · · ·  a ( t ) 1 x 1 + . . . + a ( t ) n x n − a ( t ) 0  . (43) If D is the empty disjunction , w e deﬁne b D to b e the p olynomial 1. It is clear that every 0 , 1 assignmen t to the v ariables in D , satisﬁes D , if an d only if b D ev aluates to 0 u nder the assignmen t. Prop osition 3. L e t π = ( D 1 , . . . , D ℓ ) b e an R(lin) pr o of se quenc e of D ℓ , fr om some c ol le ction of initial disjunctions of line ar e quations Q 1 , . . . , Q m . Then, ther e exists a PCR pr o of of b D ℓ fr om b Q 1 , . . . , b Q m with at most a p olyn omial in | π | nu mb er of steps. Pro of : W e pro ceed by ind uction on the num b er of lines in π . The b ase c ase is the translation of the axioms of R(lin) via the translation sc heme in Deﬁnition 9.8 . An R(lin) Bo olean axiom ( x i = 0) ∨ ( x i = 1) is translated in to x i · ( x i − 1) whic h is already a Bo olea n axiom of PCR. F or the induction step , we trans late ev ery R(lin) inference r ule application in to a p olynomial-size PCR pro of sequence as follo ws . W e use the follo wing simple claim: Claim 5. Let p and q b e t w o p olynomials and let s b e the minimal size of an arithmetic f orm ula computing q . Then one can d eriv e in PCR, w ith only a p olynomial in s num b er of steps, from p the pro duct q · p . 18 Pro of of claim : By in duction on s . 17 This notation should not b e confused with the same notation in S ection 6.3 . 18 Again, note th at w e only require th at th e num ber of steps in the pro of is p olynomial. W e d o not consider here the size of the PCR pro of. 34 Assume that D i = D j ∨ L was deriv ed from D j using the W eakening in f erence ru le of R(lin), where j < i ≤ ℓ and L is some lin ear equation. Then, by Claim 5 , b D i = b D j · b L can b e deriv ed from b D j with a d eriv ation of at most p olynomial in | D j ∨ L | man y s teps . Assume that D i w as deriv ed from D j where D j is D i ∨ (0 = k ), using th e Simpliﬁcation inference rule of R(lin), where j < i ≤ ℓ and k is a non-zero in teger. Th en, b D i can b e derived from b D j = b D i · − k by multiplying with − k − 1 (via the Ad dition ru le of PCR). Th us, it remains to simulat e the r esolution rule application of R(lin). Let A, B b e t w o disju nc- tions of linear equations and assume that A ∨ B ∨ (( ~ a + ~ b ) · ~ x = a 0 + b 0 ) was derived in π f rom A ∨ ( ~ a · ~ x = a 0 ) and B ∨ ( ~ b · ~ x = b 0 ) (the case where A ∨ B ∨ (( ~ a − ~ b ) · ~ x = a 0 − b 0 ) w as d eriv ed fr om A ∨ ( ~ a · ~ x = a 0 ) and B ∨ ( ~ b · ~ x = b 0 ), is similar). W e need to derive b A · b B · (( ~ a + ~ b ) · ~ x − a 0 − b 0 ) from b A · ( ~ a · ~ x − a 0 ) and b B · ( ~ b · ~ x − b 0 ). This is done by m ultiplying b A · ( ~ a · ~ x − a 0 ) w ith b B and m ultiplying b B · ( ~ b · ~ x − b 0 ) w ith b A (using Claim 5 ), and then add ing the resulted p olynomials tog ether. Remark 4. When trans lating R(lin) pro ofs in to PCR pr oofs we actually do not make an y use of the “negativ e” v ariables ¯ x 1 , . . . , ¯ x n . Nev ertheless, th e m ultilinear pro of systems mak e use of these v ariables in order to p olynomially sim ulate PCR p r oofs (see Theorem 33 and its pro of in [ R T06 ]). W e shall need the f ollo wing corollary in the sequ el: Corollary 32. L et π = D 1 , . . . , D ℓ b e an R 0 (lin) pr o of of D ℓ , and let s b e the maximal size of an R 0 (lin) -line in π . Then ther e is a PCR pr o of π ′ of b D ℓ with p olynomial-size in | π | numb er of steps and such that every line of π ′ is a tr anslation (via Deﬁnition 9.8 ) of an R 0 (lin) -line (Deﬁnition 3.2 ), wher e the size of the R 0 (lin) -line i s p olynomial in s . Pro of : The sim ulatio n of R(lin) by PCR s h o w n abov e, can b e though t of as, ﬁ r st, considering b D 1 , . . . , b D ℓ as the “sk eleton” of a PCR pro of of b D ℓ . An d second, for eac h D i that w as dedu ced by one of R(lin)’s inference ru les from previous lines, one inserts the corresp onding PCR pro of sequence that sim ulates the appropr iate inference r ule applicatio n (as describ ed in the pro of of Prop osition 3 ). By deﬁn ition, those PCR pro of-lines that corresp ond to lines in the sk eleto n b D 1 , . . . , b D ℓ are translations of R 0 (lin)-lines (with size at most p olynomial in s ). T h us, to conclude the pro of of the corollary , one needs only to c hec k th at for an y R 0 (lin)-line D i that w as dedu ced b y one of R(lin)’s inference ru les from previous R 0 (lin)-lines (as demonstrated in the pr oof of Prop osition 3 ), the inserted corresp ondin g P C R pro of sequence uses only translations of R 0 (lin)-lines (with size p olynomial in s ). This can b e v eriﬁed by a straigh tforw ard insp ection. 9.3. F rom PCR Pro ofs to Multilinear Proofs. W e no w recall the general sim ulatio n result pro v ed in [ R T06 ] stating the follo wing: Let π b e a P C R refutation of some initial collec tion of m ultilinear p olynomials Q o v er some ﬁxed ﬁeld. Assu me that π h as p olynomially many steps (that is, the n umber of pro of lines in the PCR pro of sequence is p olynomial). If the ‘multilinea rization’ (namely , the result of app lying th e M [ · ] op erator – see Deﬁn ition 9.3 ) of eac h of the p olynomials in π has a p olynomial-siz e d epth d m ultilinear formula (with a plus gate at the ro ot), then there is a p olynomial-size d epth- d fMC refutation of Q . More formally , we ha v e: Theorem 33 ([ R T06 ]) . Fix a ﬁeld F (not ne c essarily of char acteristic 0 ) and let Q b e a set of multiline ar p olynomials fr om F [ x 1 , . . . , x n , ¯ x 1 , . . . , ¯ x n ] . L et π = ( p 1 , . . . , p m ) b e a PCR r efu tation of Q . F or e ach p i ∈ π , let Φ i b e a multiline ar formula for the p olynomial M [ p i ] . L et s b e the total size of al l formulas Φ i , that is, s = Σ m i =1 | Φ i | , and let d ≥ 2 b e the maxima l depth of al l formulas Φ i . Assume that the depth of al l the f ormulas Φ i that have a pr o duct gate at the r o ot is at most d − 1 . Then ther e is a depth- d f MC r efutation of Q of size p olynomial in s . 35 9.3.1. Depth- 3 M ultiline ar P r o ofs. Here w e show that multilinea r p ro ofs op erating with depth -3 m ultilinear formula s (that is, depth-3 fMC) o ver ﬁelds of c haracteristic 0 p olynomially sim ulate R 0 (lin) pro ofs. In ligh t of Prop osition 32 and Theorem 33 , to this end it suﬃces to sh o w that any R 0 (lin)-line D translates in to a corr esp onding p olynomial p (via the trans latio n in Deﬁnition 9.8 ) suc h th at M [ p ] has a multilinear formula of size p olynomial (in the n umber of v ariables) and d epth at most 3 (with a plus gate at the ro ot) o v er ﬁelds of charac teristic 0. W e need the follo wing p rop osition from [ R T06 ]: Prop osition 4 ([ R T06 ]) . L et F b e a ﬁeld of char acteristic 0 . F or a c onstant c , let X 1 , . . . , X c b e c ﬁnite sets of variables (not ne c essarily disjoint), wher e Σ c i =1 | X i | = n . L et f 1 , . . . , f c b e c symmetric p olynomials over X 1 , . . . , X c (over the ﬁe ld F ), r esp e ctively. Then, ther e is a depth- 3 multiline ar formula for M [ f 1 · · · f c ] of size p olynomial (in n ), with a plus gate at the r o ot. The follo wing is the k ey lemma of the sim ulatio n: Lemma 34. L et D b e an R 0 (lin) -line with n variables and let p = b D (se e De ﬁnition 9.8 ). Then, M [ p ] has a depth- 3 multiline ar formula over ﬁelds of char acteristic 0 , with a plus gate at the r o ot and size at most p olynomial in the size of D . Pro of : Assu me that the und erlying v ariables of D are ~ x = x 1 . . . , x n . By the deﬁnition of an R 0 (lin)-line (see Deﬁnition 3.2 ) w e can partition th e disju nction D into a c onstant numb er of disjuncts, where one disju nct is a (p ossibly empt y , translation of a) clause C , 19 and all other disjuncts ha v e th e follo wing form: m _ i =1 ( ~ a · ~ x = ℓ i ) , (44) where the ℓ i ’s are in tege rs, m is not necessarily b ounded and ~ a denotes a v ector of n c onstant in teger coeﬃcients. Let us denote by q the p olynomial represen ting the clause C . 20 Consider a disjun ct as sh o wn in ( 44 ). Since the co eﬃcien ts ~ a are co nstan ts, ~ a · ~ x can b e written as a sum of constant num b er of linear forms, eac h with the same constant co eﬃcien t. In other w ords, ~ a · ~ x can b e written as z 1 + . . . + z d , for s ome constan t d , where for all i ∈ [ d ]: z i := b · X j ∈ J x j , (45) for some J ⊆ [ n ] and some constan t int eger b . W e shall assume without loss of generalit y that d is the same constan t for every disjunct of the form ( 44 ) inside D (otherwise, take d to b e the maximal suc h d ). Th us, ( 44 ) is translated (via the translation scheme in Deﬁnition 9.8 ) in to: m Y i =1 ( z 1 + ... + z d − ℓ i ) . (46) By f u lly expand ing the pro du ct in ( 46 ), w e arr iv e at: X r 1 + ... + r d +1 = m α r d +1 · d Y k =1 z r k k ! , (47) 19 If there is more than one clause in D , w e simply combine all the clauses into a single clause. 20 C is a translation of a clause (that is, disjunction of literals) into a disjunction of linear eq uations, as deﬁned in S ection 3.1 . The p olynomial q is then th e p olynomial translation of this disjunction of linear equations, as in Deﬁnition 9.8 . 36 where the r i ’s are n on-negativ e integ ers, and where the α r ’s, for eve ry 0 ≤ r ≤ m are just in teger co eﬃcien ts, formally d eﬁned as f ollo ws (this deﬁn ition is not essen tial; we present it only for the sak e of concreteness): α r := X U ⊆ [ m ] | U | = r Y j ∈ U ( − ℓ j ) . (48) Claim 6. The p olynomial b D (the p olynomial tran s latio n of D ) is a linear com bination (o v er F ) of p olynomially (in | D | ) man y terms, su c h that eac h term can b e wr itten as q · Y k ∈ K z r k k , where K is a collect ion of a constan t num b er of in d ices, r k ’s are non-negativ e in tegers, and th e z k ’s and q are as ab ov e (that is, the z k ’s are linear form s , where eac h z k has a single co eﬃcien t f or all v ariables in it, as in ( 45 ), and q is a p olynomial translation of a clause). Pro of of claim : Denote the total num b er of disju ncts of the form ( 44 ) in D by h . By deﬁ n ition (of R 0 (lin)-line), h is a constan t. Consider the p olynomial ( 47 ) ab o v e. In b D , w e actually n eed to m ultiply h many p olynomials of the f orm sh o wn in ( 47 ) and the p olynomial q . F or ev ery j ∈ [ h ] w e write the (single) linear form in the j th d isjunct as a sum of constan tly man y linear form s z j, 1 + . . . + z j,d , wher e eac h linear form z j,k has th e s ame co eﬃcien t for ev ery v ariable in it. Thus, b D can b e written as: q · h Y j =1        X r 1 + ... + r d +1 = m j α ( j ) r d +1 · d Y k =1 z r k j,k ! | {z } ( ⋆ )        , (49) (where the m j ’s are n ot b ounded, and the coeﬃcien ts α ( j ) r d +1 are as d eﬁ ned in ( 48 ) except that h ere w e add the index ( j ) to denote that they dep end on the j th disjun ct in D ). Denote the maximal m j , for all j ∈ [ h ], by m 0 . The size of D , denoted | D | , is at least m 0 . Note that s ince d is a constan t, the num b er of summands in eac h (middle) sum in ( 49 ) is p olynomial in m 0 , whic h is at most p olynomial in | D | . Thus, by expanding the outermost pro du ct in ( 49 ), w e arrive at a su m of p olynomially in | D | many summ an d s. Eac h summand in this sum is a pr o du ct of h terms of the form ( ⋆ ) multiplied by q . It remains to apply the multilinea rization op erator (Deﬁnition 9.3 ) on b D , and v erify that the re- sulting p olynomial has a depth -3 multilinea r form ula w ith a plus gate at the ro ot and of p olynomial- size (in | D | ). S ince M [ · ] is a linear op erator, it suﬃces to sh o w that w h en applying M [ · ] on eac h summand in b D , as d escrib ed in Claim 6 , one obtains a (m ultilinear) p olynomial th at has a depth-3 m ultilinear formula with a plus gate at the ro ot, and of p olynomial-size in the n um b er of v ariables n (note that clearly n ≤ | D | ). Th is is established in the follo wing claim: Claim 7. The p olynomial M  q · Q k ∈ K z r k k  has a d epth-3 multilinear formula of p olynomial-size in n (the o verall num b er of v ariables) and with a plus gate at the ro ot (o v er ﬁ elds of c haracte ristic 0), under the same notation as in Claim 6 . Pro of of claim : Recall that a p o w er of a sy m metric p olynomial is a s y m metric p olynomial in itself. Since eac h z k (for all k ∈ K ) is a symm etric p olynomial, then its p o w er z r k k is also symm etric. The p olynomial q is a translation of a clause, hence it is a pro du ct of t w o symm etric p olynomials: the 37 symmetric p olynomial that is the translation of the d isjunction of literals with p ositiv e signs, and the symmetric p olynomial that is the translation of the disjun ctio n of literals with negativ e s igns. Therefore, q · Q k ∈ K z r k k is a pro duct of constan t num b er of symmetric p olynomials. By Prop osition 4 , M  q · Q k ∈ K z r k k  (where her e the M [ · ] oper ator op erates on the ~ x v ariables in the z k ’s and q ) is a p olynomial f or which there is a p olynomial-size (in n ) depth-3 m ultilinear formula with a plus gate at the ro ot (o ver ﬁelds of charac teristic 0). W e no w come to the main corolla ry of this sectio n. Corollary 35. M ultiline ar pr o ofs op er ating with depth- 3 multiline ar formulas (that is, depth- 3 fMC pr o ofs) p olyno mial ly-simulate R 0 (lin) pr o ofs. Pro of : Im mediate fr om Corollary 32 , Theorem 33 and Prop osition 34 . F or the sak e of cla rit y we rep eat th e c hain of transformations needed to pro v e th e simulatio n. Giv en an R 0 (lin) pro of π , we ﬁrst use Corollary 32 to transform π in to a PCR pro of π ′ , with n umber of steps that is at most p olynomial in | π | , and where eac h line in π ′ is a p olynomial translation of some R 0 (lin)-line w ith size at most p olynomial in the maximal lin e in π (which is clearly at most p olynomial in | π | ). Thus, b y Prop osition 34 eac h p olynomial in π ′ has a corr esp onding multi linear p olynomial w ith a p olynomial-size in | π | dep th-3 multilinea r f ormula (and a plus gate at the ro ot). Therefore, by T heorem 33 , w e can transform π ′ in to a depth-3 fMC pro of with only a p olynomial (in | π | ) in crease in size. 9.4. Small Depth- 3 Multilinear Pro ofs. Since R 0 (lin) admits p olynomial-size (in n ) refutations of the m to n pigeonhole principle (for an y m > n ) (as deﬁn ed in 6.1 ), Corollary 35 and Theorem 15 yield: Theorem 36. F or any m > n ther e ar e p olynomial-size (in n ) depth- 3 fMC r efutations of the m to n pige onhole principle PHP m n (over ﬁelds of char acteristic 0 ). This improv es o v er the result in [ R T06 ] that demonstrated a p olynomial-size (in n ) depth-3 fMC refutations of a we ak er principle, namely the m to n func tional pigeonhole principle. F urth ermore, corollary 35 and Theorem 19 yield: Theorem 37. L et G b e an r -r e gular gr aph with n vertic es, wher e r is a c onstant, and ﬁx some mo dulus p . Then ther e ar e p olynomial-size (in n ) depth- 3 fMC r efutations of Tseitin mo d p formulas ¬ Tseitin G,p (over ﬁelds of char acteristic 0 ). The p olynomial-size refu tatio ns of Tseitin graph tautolog ies here are diﬀerent than those demon- strated in [ R T06 ]. T heorem 37 establishes p olynomial-size refutations ov er any ﬁeld of charact eristic 0 of Tseitin mo d p f orm ulas, whereas [ R T06 ] required the ﬁeld to contai n a pr imitiv e p th ro ot of unit y . On the other hand, the refutations in [ R T06 ] of T seitin mo d p formulas do not mak e an y u se of th e seman tic n ature of the fMC pro of sys tem, in the sense that they do not utilize the fact that the base ﬁeld is of c haracteristic 0 (which in turn enables one to eﬃcien tly represent any symmetric [m ultilinear] p olynomial b y a depth-3 m ultilinear f orm ula). 10. Rela tions with Exte nsions o f Cutting Planes In this sectio n w e tie some lo ose ends b y sho wing that, in full generalit y , R(lin) p olynomially sim ulates R(CP) with p olynomially b ound ed coeﬃcients, denoted R(CP*). First w e d eﬁne the R(CP*) pro of system – introdu ced in [ Kra98 ] – whic h is a common extension of resolution and CP* (the latter is cutting planes with p olynomially b ounded co eﬃcien ts). The system R(CP*), th us, is essen tially r esolution op erating with disjun ctions of linear inequalities (with p olynomially b ounded integral coeﬃcien ts) augmente d with the cutting p lanes inference rules. 38 A lin ear inequalit y is written as ~ a · ~ x ≥ a 0 , (50) where ~ a is a v ector of integral co eﬃcien ts a 1 , . . . , a n , ~ x is a v ector of v ariables x 1 , . . . , x n , and a 0 is an int eger. The si ze of the linear inequalit y ( 50 ) is the su m of all a 0 , . . . , a n written in unary notation (this is similar to the size of linear equations in R(lin)). A disjunction of line ar ine qualities is ju st a d isj unction of inequalities of the form in ( 50 ). The semanti cs of a disjunction of inequalities is the natural one, that is, a disjun ctio n is true under an assignmen t of integ ral v alues to ~ x if and only if at least one of the inequ alitie s is true und er the assignment. The size of a disju nc tion of line ar ine qualities is the total size of all linear inequalities in it. W e can also add in the ob vious w a y linear inequalities, that is, if L 1 is the linear inequality ~ a · ~ x ≥ a 0 and L 2 is the linear inequality ~ b · ~ x ≥ b 0 , then L 1 + L 2 is the linear inequalit y ( ~ a + ~ b ) · ~ x ≥ a 0 + b 0 . The pr o of s y s tem R(CP*) op erates with disjunctions of linear inequalities with int egral co eﬃ- cien ts (written in unary repr esen tat ion), and is deﬁ n ed as follo ws (our form ulation is similar to that in [ Ko j 07 ]): 21 Deﬁnition 10.1 (R(CP*)) . Let K := { K 1 , . . . , K m } b e a collection of disj unctions of linear in- equalities (wh ose co eﬃcien ts are written in unary representati on). An R(CP*)-pr o of fr om K of a disjunction of line ar ine qualities D is a ﬁn ite sequence π = ( D 1 , ..., D ℓ ) of disju nctions of linear inequalities, suc h that D ℓ = D and for eac h i ∈ [ ℓ ]: either D i = K j for some j ∈ [ m ]; or D i is one of the follo wing R(CP*)-axioms : (1) x i ≥ 0, for any v ariable x i ; (2) − x i ≥ − 1, for any v ariable x i ; (3) ( ~ a · ~ x ≥ a 0 ) ∨ ( − ~ a · ~ x ≥ 1 − a 0 ), where all co eﬃcien ts (including a 0 ) are inte gers; or D i w as deduced fr om previous lines b y one of the follo w ing R(CP*)-infer enc e rules : (1) Let A, B b e tw o disjun ctio ns of linear inequalities and let L 1 , L 2 b e tw o linear inequalities. 22 F rom A ∨ L 1 and B ∨ L 2 deriv e A ∨ B ∨ ( L 1 + L 2 ). (2) Let L b e some lin ear equation. F rom a disjunction of lin ear equations A deriv e A ∨ L . (3) Let A b e a disjunction of linear equations F rom A ∨ (0 ≥ 1) derive A . (4) Let c b e a non-negativ e integ er. F rom ( ~ a · ~ x ≥ a 0 ) ∨ A deriv e ( c ~ a · ~ x ≥ ca 0 ) ∨ A . (5) Let A b e a disjunction of linear inequalities, and let c ≥ 1 b e an integ er. F rom ( c ~ a · ~ x ≥ a 0 ) ∨ A deriv e ( a · ~ x ≥ ⌈ a 0 /c ⌉ ) ∨ A . An R(CP*) r efutation of a collection of disj unctions of linear in equalitie s K is a pro of of th e empt y disjunction from K . Th e size of a pro of π in R(CP*) is the total size of all the disjunctions of linear inequalities in π , denoted | π | . In order for R(lin) to simulat e R(CP*) p ro ofs, we need to ﬁx the follo wing translation sc heme. Ev ery linear inequality L of the form ~ a · ~ x ≥ a 0 is translated into the f ollo wing disjunction, den oted b L : ( ~ a · ~ x = a 0 ) ∨ ( ~ a · ~ x = a 0 + 1) ∨ · · · ∨ ( ~ a · ~ x = a 0 + k ) , (51) where k is suc h that a 0 + k equals the s u m of all p ositiv e co eﬃcien ts in ~ a , that is, a 0 + k = max ~ x ∈{ 0 , 1 } n ( ~ a · ~ x ) (in case the sum of all p ositiv e co eﬃcien ts in ~ a is less than a 0 , then we put k = 0). An inequ ality with no v ariables of the form 0 ≥ a 0 is translate d in to 0 = a 0 in case it is false (that 21 When we allow co eﬃcien ts to b e written in binary r epr esentat ion , instead of un ary representation, the resulting proof system is denoted R(CP). 22 In all R(CP*)-inference rules, A, B are p ossibly the empty disjunctions. 39 is, in case 0 < a 0 ), and into 0 = 0 in case it is true (that is, in case 0 ≥ a 0 ). Note that since the co eﬃcien ts of linear inequalities (and linear equ atio ns) are written in unary r epresen tatio n, an y linear inequalit y of size s translates into a disjunction of linear equatio ns of size O ( s 2 ). Clearly , ev ery 0 , 1 assignmen t to th e v ariables ~ x satisﬁes L if and only if it satisﬁes its translatio n b L . A disjunction of linear inequ alitie s D is translated int o the disju nction of the translations of all the linear inequalities in it, d en oted b D . A collectio n K := { K 1 , . . . , K m } of disjunctions of lin ear inequalities, is tran s late d in to the col lection n b K 1 , . . . , b K m o . Theorem 38. R(lin) p olynomial ly-simulates R(CP*). In other wor ds, if π is an R(CP*) pr o of of a line ar i ne quality D fr om a c ol le ction of disjunctions of line ar ine qu alities K 1 , . . . , K t , then ther e is an R(lin) pr o of of b D fr om b K 1 , . . . , b K t whose size is p olynomial in | π | . Pro of : By induction on the num b er of pro of-lines in π . Base c ase: Here w e only need to show that the axioms of R(CP*) translates in to axioms of R(lin), or can b e derived w ith p olynomial-size (in the size of the original R(CP*) axiom) R(lin) deriv ations (from R(lin)’s axio ms). R(CP*) axiom num b er (1): x i ≥ 0 translates into the R(lin) axiom ( x i = 0) ∨ ( x i = 1). R(CP*) axiom num b er (2): − x i ≥ − 1, translates in to ( − x i = − 1) ∨ ( − x i = 0). F rom the Bo olea n axiom ( x i = 1) ∨ ( x i = 0) of R(lin), one can deriv e with a constant-siz e R(lin) p r oof the line ( − x i = − 1) ∨ ( − x i = 0) (for instance, by subtracting twice eac h equation in ( x i = 1) ∨ ( x i = 0) from itself ). R(CP*) axiom num b er (3): ( ~ a · ~ x ≥ a 0 ) ∨ ( − ~ a · ~ x ≥ 1 − a 0 ). The inequalit y ( ~ a · ~ x ≥ a 0 ) translates in to h _ b = a 0 ( ~ a · ~ x = b ) , where h is th e maximal v alue of ~ a · ~ x o v er 0 , 1 assignmen ts to ~ x (that is, h is ju st the sum of all p ositiv e co eﬃcien ts in ~ a ). The in equalit y ( − ~ a · ~ x ≥ 1 − a 0 ) translates into f _ b =1 − a 0 ( − ~ a · ~ x = b ) , where f is th e maximal v alue of − ~ a · ~ x o ver 0 , 1 assignmen ts to ~ x (that is, f is just the sum of all negativ e co eﬃcien ts in ~ a ). Note that one can alwa ys ﬂ ip th e sign of any equation ~ a · ~ x = b in R(lin). Th is is done, f or instance, b y subtracting t wice ~ a · ~ x = b from itself. So o verall R(CP*) axiom n um b er (3) translates in to h _ b = a 0 ( ~ a · ~ x = b ) ∨ f _ b =1 − a 0 ( − ~ a · ~ x = b ) , that can b e con v er ted inside R(lin) in to a 0 − 1 _ b = − f ( ~ a · ~ x = b ) ∨ h _ b = a 0 ( ~ a · ~ x = b ) . (52) Let A ′ := {− f , − f + 1 , . . . , a 0 − 1 , a 0 , a 0 + 1 , . . . , h } and let A b e the set of all p ossible v alues that ~ a · ~ x can get o ve r all p ossible Bo olean assignment s to ~ x . Notic e th at A ⊆ A ′ . By Lemma 8 , for any ~ a · ~ x , there is a p olynomial-size (in the size of the linear form ~ a · ~ x ) deriv ation of W α ∈A ( ~ a · ~ x = α ). By using the R(lin) W eak ening ru le we can then deriv e W α ∈A ′ ( ~ a · ~ x = α ) whic h is equal to ( 52 ). Induction step: Here we simply need to show ho w to p olynomially simulate inside R(lin) every inference rule applicatio n of R(CP*). 40 Rule (1): Let A, B b e tw o d isjunctions of linear inequalities and let L 1 , L 2 b e t w o linear inequalities. Assume w e already hav e a R(lin) pro ofs of b A ∨ b L 1 and b B ∨ b L 2 . W e need to deriv e b A ∨ b B ∨ \ L 1 + L 2 . Corollary 7 sho ws that there is a p olynomial-size (in the size of b L 1 and b L 2 ; whic h is p olynomial in the size of L 1 and L 2 ) deriv ation of \ L 1 + L 2 from b L 1 and b L 2 , f rom wh ic h the desired der iv ation immediately follo w s . Rule (2): The sim ulation of this rule in R(lin) is done using the R(lin) W eake ning rule. Rule ( 3): Th e simulatio n of this rule in R(lin) is done us ing the R(lin) Simpliﬁcation rule (remem- b er that 0 ≥ 1 translates in to 0 = 1 und er our translation sc h eme). Rule (4): Let c b e a non-negativ e inte ger. W e need to derive \ ( c ~ a · ~ x ≥ ca 0 ) ∨ b A from \ ( ~ a · ~ x ≥ a 0 ) ∨ b A in R(lin). This amount s only to “adding together” c times the disju nction \ ( ~ a · ~ x ≥ a 0 ) in \ ( ~ a · ~ x ≥ a 0 ) ∨ b A . This can b e ac hiev ed b y c man y applications of Corollary 7 . W e omit the details. Rule (5): W e need to derive \ ( ~ a · ~ x ≥ ⌈ a 0 /c ⌉ ) ∨ b A , from \ ( c ~ a · ~ x ≥ a 0 ) ∨ b A . Consider th e disjunction of linear equations \ ( c ~ a · ~ x ≥ a 0 ), whic h can b e written as: ( c ~ a · ~ x = a 0 ) ∨ ( c ~ a · ~ x = a 0 + 1) ∨ . . . ∨ ( c ~ a · ~ x = a 0 + r ) , (53) where a 0 + r is the maximal v alue c ~ a · ~ x can get ov er 0 , 1 assignmen ts to ~ x . By Lemma 8 there is a p olynomial-size (in the size of ~ a · ~ x ) R(lin) pro of of _ α ∈A ( ~ a · ~ x = α ) , (54) where A is the set of all p ossible v alues of ~ a · ~ x o v er 0 , 1 assignment s to ~ x . W e now use ( 53 ) to cut-oﬀ from ( 54 ) all equations ( ~ a · ~ x = β ) for all β < ⌈ a 0 /c ⌉ (this will giv e us the desired d isj unction of linear equations). Consider the equation ( ~ a · ~ x = β ) in ( 54 ) for some ﬁxed β < ⌈ a 0 /c ⌉ . Use the resolution rule of R(lin) to add this equation to itself c times inside ( 54 ). W e thus obtain ( c ~ a · ~ x = cβ ) ∨ _ α ∈A\{ β } ( ~ a · ~ x = α ) . (55) Since β is an in teg er and β < ⌈ a 0 /c ⌉ , w e hav e cβ < a 0 . Thus, the equation ( c ~ a · ~ x = cβ ) do es not app ear in ( 53 ). W e can th en su ccessive ly resolv e ( c ~ a · ~ x = cβ ) in ( 55 ) with eac h equatio n ( c ~ a · ~ x = a 0 ) , . . . , ( c ~ a · ~ x = a 0 + r ) in ( 53 ). Hence, we arriv e at W α ∈A\{ β } ( ~ a · ~ x = α ). Ov erall, we can cut-oﬀ all equations ( ~ a · ~ x = β ), for β < ⌈ a 0 /c ⌉ , from ( 54 ). W e then get the disjun ctio n _ α ∈A ′ ( ~ a · ~ x = α ) , where A ′ is the s et of all element s of A greater or equal to ⌈ a 0 /c ⌉ (in other w ords, all v alues greater or equal to ⌈ a 0 /c ⌉ that ~ a · ~ x can get o v er 0 , 1 assignmen ts to ~ x ). Using the W eak enin g rule of R(lin) (if necessary) w e can arrive ﬁ nally at the desired disjunction \ ( ~ a · ~ x ≥ ⌈ a 0 /c ⌉ ), whic h concludes the R(lin) sim ulation of R(CP*)’s inference Rule (5). Appendix A. Feas ible Monoto ne Int erpola tion Here we formally deﬁ n e the feasible monotone interp olatio n prop ert y . The deﬁnition is tak en mainly from [ Kr a97 ]. Recall that for t w o binary strings of length n (or equiv alen tly , Bo olean assignmen ts f or n prop o- sitional v ariables) α, α ′ , w e denote by α ′ ≥ α that α ′ is bitwise greater than α , that is, that for all i ∈ [ n ], α ′ i ≥ α i (where α ′ i and α i are the i th bits of α ′ and α , resp ectiv ely). Let A ( ~ p, ~ q ) , B ( ~ p, ~ r ) b e t w o collectio ns of formulas in the display ed v ariables only , where ~ p, ~ q , ~ r are pairwise disjoin t 41 sequences of d istinct v ariables (similar to the notation at the b eginning of Section 7 ). Assume that there is no assignment that satisﬁes b oth A ( ~ p, ~ q ) and B ( ~ p, ~ r ). W e sa y that A ( ~ p, ~ q ) , B ( ~ p , ~ r ) are monotone if one of the follo w ing conditions hold: (1) If ~ α is an assignment to ~ p and ~ β is an assignment to ~ q such that A ( ~ α, ~ β ) = 1, then f or an y assignmen t ~ α ′ ≥ ~ α it holds that A ( ~ α ′ , ~ β ) = 1. (2) If ~ α is an assignmen t to ~ p and ~ β is an assignmen t to ~ r such that B ( ~ α, ~ β ) = 1, th en for any assignmen t ~ α ′ ≤ ~ α it holds that B ( ~ α ′ , ~ β ) = 1. Fix a certain p ro of sys tem P . Recall the d eﬁnition of the in terp olan t function (corresp onding to a giv en unsatisﬁable A ( ~ p, ~ q ) ∧ B ( ~ p, ~ r ); that is, fun ctions for whic h ( 39 ) in Section 7 hold). Assume that for eve ry monotone A ( ~ p, ~ q ) , B ( ~ p , ~ r ) there is a transformation from every P -refutation of A ( ~ p, ~ q ) ∧ B ( ~ p, ~ r ) in to the corresp on d ing int erp olan t monotone Bo olean circuit C ( ~ p ) (that is, C ( ~ p ) uses only monotone gates 23 ) and whose size is p olynomial in the size of th e refutation (note that for ev ery monotone A ( ~ p, ~ q ) , B ( ~ p, ~ r ) the corresp onding interp olan t circuit must compu te a monotone function; 24 the in terp olan t circuit itself, ho w ev er, might n ot b e monotone, namely , it may u se n on- monotone gates). In su c h a case, w e say that P has the fe asible monotone interp ola tion pr op e rty . This means that, if a pr o of sys tem P has the feasible monotone in terp olatio n prop ert y , then an exp onen tial lo w er b ound on monotone circuits that compute the in terp olan t fun ction corresp onding to A ( ~ p, ~ q ) ∧ B ( ~ p, ~ r ) implies an exp on ential- size lo w er b ound on P -refutations of A ( ~ p, ~ q ) ∧ B ( ~ p, ~ r ). Deﬁnition A.1 ( F easible monotone in terp olation prop ert y ) . Let P b e a prop ositional refu- tation system. Let A 1 ( ~ p, ~ q ) , . . . , A k ( ~ p, ~ q ) an d B 1 ( ~ p, ~ r ) , . . . , B ℓ ( ~ p, ~ r ) b e tw o collections of formulas with the d isp la yed v ariables only (where ~ p h as n v ariables, ~ q has s v ariables and ~ r has t v ariables), suc h th at either (the set of satisfying assignments of ) A 1 ( ~ p, ~ q ) , . . . , A k ( ~ p, ~ q ) meet condition 1 ab o v e or (the set of satisfying assignmen ts of ) B 1 ( ~ p, ~ r ) , . . . , B ℓ ( ~ p, ~ r ) meet condition 2 ab ov e. Assume that for any suc h A 1 ( ~ p, ~ q ) , . . . , A k ( ~ p, ~ q ) and B 1 ( ~ p, ~ r ) , . . . , B ℓ ( ~ p, ~ r ), if there exists a P -r efu tatio n for A 1 ( ~ p, ~ q ) ∧ · · · ∧ A k ( ~ p, ~ q ) ∧ B 1 ( ~ p, ~ r ) ∧ . . . ∧ B ℓ ( ~ p, ~ r ) of size S then there exists a monotone Bo olea n circuit separating U A from V B (as deﬁn ed in S ectio n 7.1 ) of size p olynomial in S . In this case w e sa y that P p ossesses the fe asible monotone interp olation pr op erty . 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Resolution over Linear Equations and Multilinear Proofs

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