Cirquent calculus deepened

Cirquen t calculus deep ened Giorgi Japaridze Abstract Cir quent c alculus is a new proof-th eoretic and semantic framew ork, whose main distinguishing feature is being b ased on circuit-st yle structures (called cir quents ), as opp osed to the more traditional approaches that deal with tree-like ob jects such a s form ulas, sequ en ts or hyp ersequen ts. Among its adv antages are greater eﬃciency , ﬂexibilit y and expressiv eness. This paper presents a detailed elaboration of a d eep- inference cirquent logic, whic h is naturally an d inherently resource conscious. It shows that classical logic, b oth sy ntactically and semantica lly , can b e seen to b e just a special, conserv ativ e fragmen t of this more general and, in a sense, more basic logic — the logic of resources in th e form of cirquen t calculus. The reader will ﬁnd v arious argumen ts in fav or of switc hing to the new framew ork, suc h as arguments sho wing the insuﬃciency of the exp ressi ve p o w er of linear logic or other form ula-based approac hes to devel oping resource logics, exponential improvemen ts ov er the traditional approac hes in b oth rep resen tational and pro of complexities oﬀered by cirquen t calculus (includ ing the existence of p olynomial size cut-, substitut ion- and extension-free cirquent calculus pro ofs for the n otorio usly hard pigeonhole principle), and more. Among the main purposes of this pap er is t o pro vide an in tro ductory- style starting point for what, as t he author wishes to hop e, might hav e a c hance to become a new line of researc h in proof theory — a proof theory based on circuits instead of formulas. MSC : primar y: 03B47; seco ndary: 03B70; 03F03 ; 03F20; 68T15. Keywor ds : Pro of theory; Cir quen t calculus; Resource semantics; Deep infer ence; Computability logic; P i- geonhole principle. 1 In tro duction Among the main ob jectives o f the introductor y section of a well-written pap er should b e to help the reader determine whether he or s he is willing to in vest time into reading the rest of it. The follo wing rhetorical question 1 may con tribute to making such a determination in the pres en t case: What is the most natur al r epr esentation of Bo ole an functions, formulas or cir cuits? Those who ar e not quite clear ab out the meaning of the w ord “natural”, ma y try to replace it with “direct”, “reaso nable” or “eﬃcient ”, and think ab out wher e the co mputer industry would b e at present if, fo r some strange rea son, computer engineers had insisted on tree- rather than graph-s tyle cicuitries. Or ask why one do es not hear theoretica l computer scientists sp eak a bout formula complexity nea rly as often as ab out circ uit complexity . Should then pro of-theoreticians contin ue sticking to for m ulas, especia lly now that log ic is increas ingly CS-oriented, and eﬃcienc y is of muc h gr e ater concer n than it w as in the da ys o f F rege, Hilber t and Ge ntzen? The a utho r believes that there ar e no go o d reasons for such conserv atism other than habit and tra dition, if not laziness. And this paper is for those who might feel po tentially ready to accept the same view, or b e curious eno ugh to b e willing to ta k e a lo ok at what happe ns when one gives the idea a tr y . It is devoted to (re)introducing a nd adv a ncing the foundations of cir quent c alculus , the circuit-ba sed pro of theory . Unlik e the mor e traditiona l syntactic a ppr oaches that manipulate tree- or forest-like o b jects such as formulas or sequents and where pro ofs are o ften also tre e s, cirquent calculus deals with circuit-style co nstructs called ci r quents , in which children ma y b e sha red b et ween diﬀer en t parent no des. F urthermore, b eing 1 Aske d by Alessio Guglielmi at htt p://news.gmane.org/gmane.science.mathe matics.frogs on Septem b er 17, 2007 during a discussion of th e preliminary ve rsion of the present paper. 1 int rinsica lly a deep inference (see later) appr oach, it makes p ossible combining, within a single cirquent, what would otherwis e b e diﬀerent parallel no des (form ulas, sequen ts) of a pro of tree, meaning that ev entually not only sub cirquen ts, but also subtransformatio ns are amenable to b eing shared. Sha ring th us allows us to achiev e higher eﬃciency , whether it be the compactness of repre sen tations o f Boo lean functions o r other ob jects of study , or the num b ers o f steps in der iv ations a nd pro ofs. Indeed, in natural situations, speciﬁca lly ones a rising in the world of computing, pr ohibitiv ely lo ng formulas typically owe their sizes to reo ccurr ing subformulas, and explosively large pro of trees often emerge a s a result of the necess it y to p erform ide ntical or simila r steps ov er and over again. The p ossibilit y of compr essing for m ulas or pro ofs is not the only — in fact, no t even the primar y — app eal of cirquen t calculus. Genera lit y , ﬂe x ibilit y and e xpressiveness are other, more fundamen tal, adv ant age s to po in t o ut. Cir q uen t calculus is mo r e genera l than the calculus of structures (Guglielmi et al. [3, 4 , 9, 10]); the latter is more general than hyper sequen t calculus (Avron [1], P ottinger [17]); a nd the latter, in turn, is more general than sequent calculus (Gentzen). Ea ch fra mew ork in this hierar c hy permits to successfully axio matize certain logics that th e pr edecessor fr amew orks fail to tame. Cirquent ca lculus itself w as o riginally intro duced as a deductiv e system for the resource-conscio us computabilit y logic [12, 13, 1 5] a fter it had b ecome evident that neither sequent calculus nor the more ﬂexible and promising c alculus of structur es w ere s uﬃcient to axiomatize it. While in classical logic circ uits do not oﬀer any additional expr essiv e p o wer, they — more precisely , cir quents that are mor e general tha n circuits — turn out to be properly more expre s siv e than form ulas w he n it co mes to ﬁner semantical approaches such as r esource logics, with c omputability lo gic ([12 , 13, 15, 16]) and abstr act reso ur c e semantics ([14]) b eing t wo examples. Eﬃciency considerations totally aside, it was exactly this expressive power tha t in [14] made a diﬀerence betw een axio matizabilit y and unaxiomatizabilit y for computability lo gic o r abs tr act re s ource semantics: ev en if o ne is only trying to set up a deductive system that prov es all (and only) v alid formulas , intermediate steps in pro ofs of such formulas still inherently requir e using ob jects (cirquents) that cannot b e wr itten as formulas. Switc hing fro m for mulas to cirque nts indeed b ecomes imp erative — not only s yn tactically but also seman- tically — if o ne wan ts to s ystematically develop resource logics. Fine-level reso urce-semantical approaches int rinsica lly require the abilit y to acco un t for the possibility of r esour c e sharing , the ability that linear logic or other formula- or sequent-based approa c hes do not and cannot possess. The following naive example may provide some insights. W e are talking ab out a vending machine that has slots for 25-cent (25 c ) coins, with ea c h s lot taking a s ingle coin. Coins can be authentic or counterfeited. Let us instead use the mor e generic terms t rue and false here, as there are v a r ious particular situatio ns naturally and inevita bly emerging in the world of resource s co rresp o nding to thos e t wo opposite v alues. Below are a few examples of r eal-world r e sources and the p ossible meaning s of the tw o semant ical v a lue s for them: • A ﬁnancial debt, which may (true) or may not (false) b e eventually paid; • an electr ical outlet or a battery , which ma y (true) or no t (false) actually hav e suﬃcient pow er in it; • a sta ndard task p erformed by a company’s emplo yee or an AI agent, which, even tually , may (true) or not (false) be success fully completed; • a sp eciﬁed amount of computer memory requir ed by a pro cess, whic h may (true) or not (false) b e av ailable at a given time; • a pro mise, which ma y b e kept (true) or broken (false). See Section 8 of [14] for detailed elab orations of these intuitions, as w ell as strict deﬁnitions of the co ncepts of the asso ciated formal semantics, whic h is the ea r lier-ment ioned abstract r esource semantics. Contin uing the de s cription of our v ending machine, inserting a false co in into a slot ﬁlls the slo t up (so that no o ther coins can b e inserted in to it un til the op eration is co mplete), but otherwise do es not fo ol the machine int o thinking that it has r eceiv ed 25 cents. A candy co sts 50 cents, a nd the machine will disp ense a candy if at least tw o of its slots receive true c oins. Pr essing the “dispe nse” button while having inser ted anything less th an 50 cen ts, suc h as a single coin, or one true a nd tw o false coins, results in a non-recoverable loss. 2 Victor has three 25 c -coins, and he knows that tw o of them are true while one is p erhaps false (but he has no w ay to tell which one is false). Could he get a candy? W ell, expec ted or no t, the answer dep ends on how man y slots the machine has. Consider t wo cases: machine M 2 with tw o slo ts, and machine M 3 with three slots. Victor would hav e no problem with M 3 : he can insert his three coins in to the three slots, and the mac hine, ha ving received ≥ 50 c , will dispens e a candy . With M 2, how ever, Victor is in trouble. He can try inserting a rbitrary t wo of his three coins into the t wo slots of the machine, but there is no guara n tee that one of tho s e t w o co ins is not false, in which case Victor will end up with no ca ndy and only 25 cents remaining in his p o c ket. Both M 2 and M 3 can b e understo o d as resources — resources turning coins in to a candy . A nd note that these tw o reso urces are not the same: M 3 is obviously strong er (“better”), as it allows Victor to get a candy whereas M 2 do es no t, while, a t the same time, an yone rich enoug h to b e able to make M 2 dispens e a candy w ould be able to do the sa me with R 3 as well. Y et, formulas fail to capture this imp ortant diﬀerence. With → , ∧ , ∨ here a nd later standing for mult iplicative-style connectives (called p ar al lel c onne ctives in computability logic), M 2 and M 3 can be written a s R 2 → Candy and R 3 → Candy , resp ectiv ely: they consume a certa in r e source R 2 or R 3 and pr oduce Candy . W hat ma kes M 3 stro nger than M 2 is that the subres ource R 3 that it consumes is weaker (easier to supply) than the subresource R 2 consumed by M 2 . Speciﬁca lly , with one false and tw o tr ue coins, Victor is able to sa tisfy R 3 but not R 2. The reso ur ce R 2 can b e represented as the following cirquent: ✧ ✧ ✧ ❜ ❜ ❜ 25 c 25 c ∧ ❥ which, due to b eing tree-like, can also b e a dequately written as the formula 25 c ∧ 25 c. As for the resource R 3, either one o f the follo wing t wo cir quen ts is an adequate representation of it, with one of them pro bably sho wing the relev ant part of the actual physical circuitry used in M 3: ❍ ❍ ❍ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ 25 c 25 c 25 c ∧ ❥ ∧ ❥ ∧ ❥ ∨ ❥ Figure 1: Two equiv alen t cir quen ts for the res ource R 3 ❍ ❍ ❍ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ 25 c 25 c 25 c ∨ ❥ ∨ ❥ ∨ ❥ ∧ ❥ Unlik e R 2, how ever, R 3 ca nnot b e repre s en ted through a formula. 25 c ∧ 25 c do es no t ﬁt the bill, for it represents R 2 whic h, as we already agreed, is not the same as R 3. In another a ttempt to ﬁnd a form ula, w e might try to rewrite one of the ab ov e t wo cirquen ts — let it b e the one on the right — into an “equiv alent ” formula in the standa rd w ay , by duplicating and s eparating shared no des. This results in (25 c ∨ 25 c ) ∧ (25 c ∨ 25 c ) ∧ (25 c ∨ 25 c ) (1) which, how ever, is not any more adeq uate than 25 c ∧ 25 c . It expresses not R 3 but the reso urce cons umed by a machine with six coin slots group ed in to three pairs, where (at least) one slot in each of the three pairs needs to receive a true coin. Such a machine th us dispe nses a candy for ≥ 75 rather than ≥ 50 cents, whic h makes Victor’s reso urces insuﬃcien t. The trouble here is related to the inability of for m ulas to explicitly acco un t for resource sharing o r the absence thereof. The c irquen t on the rig h t o f Figure 1 stands for a co njunction of thr ee r esources, each conjunct, in turn, being a disjunction of tw o subres ources of type 2 5 c . How ever, altogether there a re three 3 rather than six 25 c -type s ubr esources, each one b eing shared b et ween tw o diﬀerent co njuncts of the main resource . F or mula (1) is inadequate b ecause, for example, it fails to indicate that the ﬁrst and the third o ccurrences of “25 c ” s tand for the same r esource while the seco nd and the ﬁfth (as well as the four th a nd the sixth) o ccurrences stand for ano ther resource, alb eit a r esource of the same 25 c -t yp e. F rom t he resource- philosophical p oint of view, cla s sical logic and linea r logic are tw o imp erfect extremes. In the former, all o ccurrences of a same subform ula mean “the sa me” (represent the same res ource), i.e., everything is shar e d that can b e shared; and in the latter, each o ccurrence stands for a separate resource, i.e., nothing is shar e d at all. Neither a pproach do es th us p ermit to ac c oun t fo r mixed case s where certain o ccurrences a re meant to represent the same r esource while so me other o ccurrence s stand for diﬀerent resource s of the s ame type. And it is an absolute shame tha t linear logic or similar — naive from the per spective of cirquent calc ulus and abstract resourc e sema ntics — reso urce-oriented approa c hes fail to express simple, natural and unavoidable things such as the “t wo out of three” com bination expressed b y the cirquents of Figure 1 . It w as mentioned earlier that cirquents are more gene r al than cir c uits — otherwise there would be no need to inven t a sp ecial name for them after a ll. In str uc tur es commonly referred to as Bo olean circuits, the lab el of eac h input is unique, while cir q uen ts ma y hav e an y n um b er of “inputs” (called p orts in abstract resource semantics) of any g iv en typ e (lab el). So, strictly sp eaking, what w e see in Figure 1, having three 25 c -p orts, are cirquents but not circuits. Of course, in an attempt to ma k e them mea ningful as cir cuits in the traditional sense, one could think of renaming the three 25 c -p orts in to, say , P , Q and R . But then a crucial piece of info r mation w ould b e lost, speciﬁca lly the information ab out all inputs being of the same t yp e 25 c , as opp osed to, say , the three diﬀere nt types 25 c , 10 c , 5 c . This would make it impo ssible to match Victor’s re sources with those inputs. Among the main ideolo gical merits of the present contribution is the uniﬁcation and r econciliation of classical logic and the logic of res ources on the bas is of one s ingle semantics a nd one single syn tax. The rather unsettling situation of conﬂict a nd disag reemen t b et ween cla s sical and r e source logics , familiar from linear logic 2 or ev en the predecessor [14] of the pr esen t pa per, now gives way to a perfect pe ace an harmony . Spec iﬁc a lly , we make the p oin t that the la nguage o f class ical logic can and should be s een as a prop er fragment of the lang uage of r esource log ics, obtained by considering o nly cir cuits , i.e., cirquents where m ultiple iden tical-lab el p orts ar e not allow ed. With this view, ther e is no need to hav e separate s e ma n tics and syntax for classica l lo g ic: they turn out to b e the same as t hose of our cirquent c alculus, only restricted to the cirquents that are circuits. That is, our resource log ic (cirquent calculus) is simply more expressive — and th us more gener al — than classical lo g ic, but otherwis e there are no semantic or syntactic diﬀerences or disagreements betw een the tw o: the former is a conserv ative extension of the latter. The following example may help get a feel of this. Let us consider the formula ¬ P ∨ ( ¬ Q ∧ P ) ∨ ( P ∧ ¬ R ) ∨ ( Q ∧ R ) . (2) This for mula is v alid in clas sical logic while linear lo gic, o r the approa c h of [1 4], w ould consider it inv alid. But is it or is no t it v alid according to our presen t a pproach, whic h w as promised to eliminate a n y disa greement s betw een the c lassical and resource- c onscious views? It s hould b e remember ed that w e have dismissed formulas as imp e rfect means o f express ion. So, the intended meaning o f (2) m ust b e ﬁrst expre s sed throug h a cirquen t befo re answ ering or ev en asking a question about its v alidit y . And, as we have not (not yet, a t least) agreed on a n y standar d w ay of tra nslating formulas int o cirquents, only who ever wrote (2) can tell wha t he o r she wan ted to express. F rege would pr obably explain his meaning thro ugh the left cirquent of the following Figure 2, while Gira rd through the right cirquen t: 2 What is mean t by line ar lo gic here and throughout this paper is the multiplicativ e fragment of linear logic, whose connectiv es are wri tte n using the classical symbols. 4 ¬ P ¬ Q P ¬ R Q R ✱ ✱ ∧ ❥ ❧ ❧ ∧ ❥ ❧ ❧ ∧ ❥ ❳ ❳ ❳ ❳ ✘ ✘ ✘ ✘ ✘ ✘ ❧ ❧ ✱ ✱ ∨ ❥ Figure 2: Two po ssible meanings o f (2) ¬ P ¬ Q P P ¬ R Q R ❆ ❆ ✁ ✁ ∧ ❥ ❆ ❆ ✁ ✁ ∧ ❥ ❆ ❆ ✁ ✁ ∧ ❥ ❳ ❳ ❳ ❳ ❳ ✘ ✘ ✘ ✘ ✘ ✘ ❧ ❧ ✱ ✱ ∨ ❥ Then, rega rding the question on v alidity , as will b e seen later , we would answer “Y es” to F rege and “No” to Gira rd. And o ur neg ativ e answer in the se c ond ca se do es no t at all conﬂict with the seemingly po sitiv e answer of classical logic. The right cirquent of Figure 2, for the reas o n of having tw o sepa rate P -p orts and th us no t being a c ircuit, is simply not a meaningful or lega l — let alone v a lid — expression for cla s sical lo gic. The idea of cirquent calculus was b orn very re cen tly in [14]. That s o far the only pa p er on the sub ject int ro duced cir q uen t calculus in a spec ia l, “sha llo w” form, where all cirquents were require d to b e of depth t wo, with a conjunctive gate at the ro ot and disjunctiv e ga tes a s second-level no des. While the shallow ver- sion of cirquent calculus w as suﬃcien t to achiev e the main goal of that pap er — axiomatizing the otherwise unaxiomatizable ba sic fra gmen t of computability logic — the pap er also outlined the p ossibility and exp e - diency of studying more gener al, deep v ersions of cirquent calculi. The present article con tains a rea lization of that outline. It elab o rates a deep cirquent calculus system CL8 fo r computability logic, which happ ens to coincide with the lo gic induced b y abstract re s ource s e man tics, a nd which is a conser v ative extens io n of c la ssical logic , so that CL8 is also an alternative system for classical lo gic. CL8 per mits cir quen ts of arbitrar y depths and forms, which natura lly invites inference rules that modify cir q uen ts at an y lev el rather than o nly around the ro ot as is the cas e in sequent calculus. This is c alled de ep infer enc e , a nd is one of the central ideas in the earlier men tioned calculus of str uctures. The present pap e r also b orrows many other useful ide a s and techniques from the calculus o f structures, which is the near est precurso r of cirquent ca lculus in its present, general form. The rest of the paper is organized as follows. In Section 2 we (re)intro duce the notion of c irquen ts whic h generalizes the cir quen ts fro m [14] by removing any restr ictions on the depths and for ms of cirquents. Section 3 int ro duces and explains the rules of inference of the deep cirq uen t calculus system CL8 . Section 4 deﬁnes the notions of deriv ation, pr oof, a nd a dmissibilit y for CL8 and similar systems. Section 5 gener alizes b oth the s e ma n tics of classical logic and the abstract resource semantics of [14] to a common, unifying resource semantics (still called abstr act r esour c e semantics ) for all cirquents, a nd proves the corresp onding soundness and co mpleteness result for CL8 . Section 6 discusses s ome po ssible v ariations of cirquent calculus sy stems, including a version of CL8 where each rule o f infer ence comes with its dual (symmetric) one, and systems that deal with cirquents with non- standard types of g a tes. Section 7 discus ses the relation of CL8 to classical sequents calculus (showing the p-simulation of the latter b y the for mer) a nd the tw o shallow c irquen t ca lculus systems co ns tructed earlier in [14]. Fina lly , Section 8 presen ts p olynomial size CL 8 -pro o fs of the notoriously hard-to-pr o ve family of tautolo gies known a s the pige onhole principle . These ar e so far the o nly known eﬃcient proofs of that family that e mplo y neither cut nor extension or substitution — the rules undesirable for their b eing “highly no n-analytic”. 2 Cirquen ts, form ulas and h yp erform ulas W e ﬁx some set of syn tactic ob jects called atoms , for which w e w ill be using P , Q, R, S, T as metav ariables . An atom P and its negation ¬ P ar e called literals . The tw o literals P and ¬ P are sa id to b e opp osi te . Let us agree that in this pap er a graph means a dir ected acyclic graph whose every no de is lab eled with either a literal or ∧ or ∨ . The ∧ - and ∨ -lab eled no des of (such) a graph we ca ll gates , and the no des lab eled with literals we c a ll p orts . Sp eciﬁcally , a no de la beled with a literal L is said to b e a an L -p ort ; an ∧ -lab eled node is said to be a conjunctiv e g ate ; and an ∨ -labeled no de is said to b e a disjunctiv e gate . When there is an edge from a no de a to a no de b , we say that b is a c hild of a a nd a is a paren t of b . The relations “ de scendan t ” and “ ancestor ” a re the transitive closures of the relations “ch ild” and “parent”, resp ectively . The meanings of some other standar d r elations such as “grandchild”, “grandpa r en t”, etc. sho uld also b e clear . 5 A cirquen t is a graph (in the ab ov e sense) satisfying the following t w o c o nditions: • Ports ha ve no children. • There is a no de, ca lled the ro ot , whic h is an ance stor of all other no des in the gra ph. A cirquent is said to b e a circuit iﬀ all o f its p orts hav e diﬀerent lab els (here the t wo opp osite lab els P and ¬ P coun t as diﬀerent). Graphically , we represent a por t thro ug h the corre s ponding literal, a conjunctive ga te through ◦ , and a disjunctiv e gate through • . W e agree tha t the direction of an edge is a lw ays up ward, which allows us to draw lines rather than ar ro ws fo r edg e s. Be low is an example of a cirquent with 4 p orts and 8 ga tes. Note that not only p orts but also gates ca n be childless. A childless disjunctive gate s eman tically corr esponds to ⊥ , and a childless conjunctive gate cor responds to ⊤ . P Q ¬ P ¬ P ❜ ❜ ❜ ✧ ✧ ✧ ❜ ❜ ❜ ✧ ✧ ✧ ❜ ❜ ❜ ✧ ✧ ✧ ✧ ✧ ✧ ❜ ❜ ❜ ✧ ✧ ✧ ❜ ❜ ❜ ✧ ✧ ✧ ❜ ❜ ❜ t t ❞ t ❞ t ❞ t In the intro ducto ry sec tio n, we emphas ized the need for r ejecting formulas in favor of cirquent s: fr om the p ersp ectiv e of cir quen t calculus, formulas are inco mplete, ineﬃcient and no t the most natur al means o f expression. In the pr ocess of for malizing a pie c e of the rea l w orld, a natural way to repr esen t a Boo lean function or w ha tev er similar ob jects w e study is to do so directly thr o ugh a cirquent. It w ould be somewhat o dd to ﬁrst try to write it through a formula (if p ossible a t all) and then tra nslate that formula int o a cirquent. So, in pr inc iple , we hav e no “legal oblig ation” to deﬁne the precise meaning s of formulas in terms of cirquents, as there is no need for formulas at all. But even if not “lega l”, w e still do hav e a “moral” duty to agr ee on so me standard way of tra nslating formulas in to c ir quen ts, to pay tribute to the ﬁrmly established logical traditio n of dealing with formulas. By a form ula in this pap er w e mean that of the language of classical prop ositional logic, built from literals a nd v ar iable-arity op erator s ∨ , ∧ in the standar d w ay . The disjunction of F 1 , . . . , F n can b e written as either F 1 ∨ . . . ∨ F n or ∨{ F 1 , . . . , F n } . Similarly for conjunction. ⊥ is considered an abbrevia tion of the empt y disjunction ∨{} , a nd ⊤ an abbrev iation of the empt y conjunction ∧{} . F urther, we treat E → F as an abbr e v iation of ¬ E ∨ F , and ¬ H , when H is not an a tom, as an abbreviation deﬁned by: ¬¬ F = F ; ¬ ( F 1 ∨ . . . ∨ F n ) = ¬ F 1 ∧ . . . ∧ ¬ F n ; ¬ ( F 1 ∧ . . . ∧ F n ) = ¬ F 1 ∨ . . . ∨ ¬ F n . W e agree to understand (“tra ns late”) each formula used in this pap er a s — and iden tify with — the cirquent whic h is nothing but the parse tree for that fo rm ula. Mor e precisely , we hav e: • A literal L is understo o d as the cirquent whose only no de (=ro ot) is an L -p ort. • Let F 1 , . . . , F n be a ny form ulas, and let gr aph G be the disjoint union of those form ulas understo od as cirquents. Then: – F 1 ∨ . . . ∨ F n is understo o d as the cirquent obtained by adding a new disjunctive gate (ro ot) to G , and connecting it with an edge to each of the n par en tless no des of G . – Similarly fo r F 1 ∧ . . . ∧ F n , with the diﬀerence that the ro ot gate here will be a c onjunctiv e one. Note that since w e r equire the ab ov e G to b e a disjoin t union, every formula is a tree - lik e cirquent, with each non-roo t no de having exactly one parent. The ab o ve way o f t ra ns lating form ulas int o c ir quen ts is th us in the spirit of formula-based r esource log ic s (such as linear lo gic) rather than clas sical log ic. F or example, formula (2) of Section 1 translates in to the right rather than the left cirquent of Figur e 2. So, we still need to separately clar ify how to translate form ulas when they app ear in the cont ext of class ical log ic (as they 6 mostly do in the literature) ra ther tha n in the context of reso urce logics (as they alwa ys do in the pr esen t pap er). F or that purp ose, we ﬁrst gener alize form ulas to what w e ca ll “h yp erformulas”. A h yp erform ula is the same as a formula, with the only diﬀerence that some subformulas in it may be overline d (double overlines are not allow ed). Hyperfor m ulas, just lik e form ulas, are understo od as cirquen ts. T o translate a hyperfo r m ula F into a corresp onding cirquent, one sho uld ﬁrst ig no re all ov erlines in F and translate it in to a tree-like cirquent acco rding to the ea rlier prescriptions , and then mer ge all s ubcirquents that corr espond to (origina te from) identical overline d subformulas of F . Rather than trying to turn this semiformal explanatio n in to a strict deﬁnition (which is certainly p ossible), b elow w e just provide a few examples that should make the meaning of the a b ov e-said pe rfectly clear . The hyperfor mula ( Q ∨ R ) ∧ ( R ∧ Q ) means (is translated in to) the cirquent Q R Q t ❅ ❞ ❅ ❅ ❞ Here the t wo o ccurrences of R in the hyper fo rm ula are consider ed “the same” as b oth ar e o verlined; on the other hand, the tw o o ccurrences of Q did not mer ge becaus e they were not ov erlined. At the same time, each of the hyperfo r m ulas ( Q ∨ R ) ∧ ( R ∧ Q ), ( Q ∨ R ) ∧ ( R ∧ Q ), ( Q ∨ R ) ∧ ( R ∧ Q ), ( Q ∨ R ) ∧ ( R ∧ Q ) stands for the sa me tree-like cirquent Q R R Q t ❅ ❞ ❅ ❍ ❍ ✟ ✟ ❞ as there is nothing to mer ge within or a cross the overlined subexpress ions. On the le ft of the following ﬁgure w e see an ov erline-free hype r form ula (i.e., a formula) a nd the cor re- sp onding tree-like c ir quen t; and, o n the righ t, we see the same (hyper)formula fully ov erlined, resulting in a m uch mor e co mpressed cir q uen t. Note tha t in this ca s e not o nly all identical-label p orts have merged, but also the (tw o) iden tical-conten t g ates, as b oth of the corresp onding subformulas ¬ P ∨ P were found under an ov erline. ¬ P P ¬ P P ❅ ❅   ❅ ❅   t t P ❳ ❳ ❳ ❳ ❳ ❳ ✘ ✘ ✘ ✘ ✘ ✘ ❞ ¬ P P ✑ ✑ t ❩ ❩ ❩ ❞ ( ¬ P ∨ P ) ∧ ( ¬ P ∨ P ) ∧ P ( ¬ P ∨ P ) ∧ ( ¬ P ∨ P ) ∧ P An ywa y , now we ar e rea dy to explain how fo rm ulas should be tr anslated into cirquents when they are used in the co ntext of cla ssical logic. W e a gree to see each formula F of classical logic as the h yp erformula — and hence the corresp onding cirquent — o btained from F b y o verlining all (and only) 3 literals. Such a hyperformula will b e denoted by underlining (rather than overlining) F : F . 3 Nothing would go wr ong if we dropped this “ only l iterals” condition, as long as the “ al l literals” condition i s retained. F or example, ov erlining the entire F (which would automatically include all li terals) would work for our purp oses just as we ll. But o verlining only li terals i s easier, so wh y b othe r. 7 So, for e x ample, if formula (2) of Section 1 is found in a textb oo k on classica l (as opp osed to linea r) log ic, it should be understo o d as a “lazy way” to write the hyperformula ¬ P ∨ ( ¬ Q ∧ P ) ∨ ( P ∧ ¬ R ) ∨ ( Q ∧ R ) , i.e., the hyperformula ¬ P ∨ ( ¬ Q ∧ P ) ∨ ( P ∧ ¬ R ) ∨ ( Q ∧ R ) , and, co rresp ondingly , should b e tra nslated as the left rather than the right cirquent of Figure 2. Generally , a notatio nal synchronization of any traditional piece of wr iting on or in clas sical logic with o ur approach would tak e as little as just underlining — explicitly or implicitly — every form ula a ppearing in it. Before c lo sing this section, we wan t to make the obser v ation that, while hyperfo r m ulas ar e more expressive than formulas, they are still far fr o m b eing expr e s siv e enough to b e able to r e presen t all cirq ue nts. F or example, the cirquents of Figur e 1 cannot b e wr itten as hyper form ulas. 3 The rules of CL8 Con v en tion 3. 1 By a rule of inference in this section we mean a set R of (2 + m + n )-tuples ( A, B , a 1 , . . . , a m , Π 1 , . . . , Π n ) (ﬁxed m, n for a given rule), called i nstances or applications of R , where: 1. A and B are cirque nts, s aid to b e the premise and the conclusion (of the given a pplication of R ), resp ectiv ely . 2. a 1 , . . . , a m , said to b e cen tral parameters , ar e pa irwise distinct no des, each one b eing a no de of either the premise or the co nclusion or b oth. 3. E a c h Π i , sa id to b e a periph e ral parameter , is a set o f no des not containing a n y central parameter s, such that every b ∈ Π i is a parent or a c hild of some cent ral para meter in either the premise or the conclusion or bo th. 4. All children and parents of e a c h central parameter , whether it b e in the premise or in the conclusio n, are among the elements of { a 1 , . . . , a m } ∪ Π 1 ∪ . . . ∪ Π n . When ( A, B , a 1 , . . . , a m , Π 1 , . . . , Π n ) ∈ R , we sa y that B follows from A by rule R with parameter s a 1 , . . . , a m , Π 1 , . . . , Π n , or — if lazy to sp ecify the par ameters — simply that B follows from A by R . Thu s, each application of a r ule has one premise and o ne conclusion. The conclusio n is usually obtained from t he premise (or vice versa) through modifying only a ce r tain part, while leaving the rest of the cirquent unch ange d. Speciﬁca lly , dep ending on the particular stipulations of a given rule, some (and only) central- parameter no des may a ppear or disappea r when moving from the premise to the conclusio n or vice versa, or may change their lab els (say , turn from a co njunctive gate into a disjunctive one). Similarly , some (and only) ar cs p oint ing to or from cen tral para meters may app ear or disapp ear. No other no des or edges a re aﬀected. The only exc e ptio n is when deleting arc s from a ce n tral par ameter to some o f its children leav es those c hildren parent less. As we do not allow non-ro ot “o rphan” no des in cirquents, such no des (together with the arcs inciden t with them, of course) should then a lso b e deleted, alo ng with their p o ssibly further orphaned children, g randc hildren, etc. Such a chain of dele tions may delete no des tha t are not among the central pa rameters and per haps not even within the pe r ipheral parameters. O ther than this, we rep eat, any no de o f the cirquent tha t do es no t happ en to b e a ce ntral parameter, and any arc of the cirquent that do es not happ en to b e inciden t with a central par ameter, remains unaﬀected when moving from premise to conclusion o r vice versa. In v ie w of conditions 3 a nd 4 of Con ven tion 3.1 , the role of p eripheral parameter s is to list all parent s and child ren of cen tral para meters that do not themselves happen to be cen tral parameter s. The additional purp ose that they sometimes ser v e is dividing those parents or children into gr oups for reference pur poses, as will b e seen later . 8 In general, the in tuitive role pla yed by parameters is telling us “where the rule is exactly applied” in the cirquent. Without this piece of infor mation, determining whether one cirquent indeed follows from another by a given rule can be harder than it has to b e. When drawing cir quen ts, we typically do not b other to assign (unique) names to their no des, as this is also commonly done in the liter ature when dealing with gra phs in g eneral: mor e often than no t, one do es not diﬀere ntiate b et ween isomorphic graphs — gr aphs that only diﬀer in the names of their no des — a s such graphs b eha ve in the same ways in all relev an t asp ects, and assigning names to their no des can usually be done in an arbitrary fashion if and when necessary . Howev er, when dealing with rules of inference, or a n y graph-tra nsformation pro cedures, ha ving names for no des becomes necessary in order to b e able to proper ly deﬁne and apply (mac hine-implement) those rules or pro cedures. Indeed, if we contin ue see ing cirquents not as particular gr aphs but rather as isomorphis m cla s ses of graphs (as was implicitly done in the pr eceding section, whether the r eader noticed it o r not), then even deciding whether o ne cirquent is “the same as” another cir quen t would take quite some work, let alone deciding whether one cirquent follo ws fro m another one by a g iv en rule. So, oﬃcia lly we require that, when applying rules , all no des of the inv olved cirquents had names, and that each tra nsition from a premise to a co nclusion b e justiﬁed by not merely indicating the name of the corres p onding rule, but a lso indicating the precise v a lues of each of the parameters of the rule. With this requirement, the question on whether an y given step (transition from a premise to a co nclusion) is leg al in a cirquent-calculus pro of or der iv ation essentially reduces to nothing but chec king whether the indicated parameters of the pr e mis e and the conclusion sa tis fy a ll conditions of the indica ted rule, which, in the case of the r ule s of C L8 or any o ther rules discussed in this pa per, can be seen to b e a rather eas y (cer tainly po lynomial time do able) task. W e will b e s c hematically r epresen ting rules of inference in the form X Y where X sta nds for the relev ant p ortion of the premise and Y stands for the r elev ant por tion of the con- clusion. Her e “ relev a n t p ortion” is the fragment of the cirq ue nt that c on tains all cen tral parameters , a ll per ipheral parameters, all edges incident with the central parameters , and no other edges or no des. In such a representation, the letters a, c, b, Γ , ∆ , Π , Σ , Ω , Θ will b e used a s v ariables for the parameter s o f the rule. As noted, the conclusio n is obtained from the premise (or vic e versa) through replacing the X part b y Y (or vice versa), leaving the r e s t of the cirquent unchanged. While X and Y represent no t the pr emise a nd the conclusion but only the to-be-mo diﬁed parts of those, b y abuse of terminology , w e may still sometimes refer to them as the “premise” a nd the “conclus ion”. Below is a full list of the rules of inference of CL8 represented schematically for the conv enience of quick future re ferences. Certain neces sary explanations of their meaning s and examples of a pplications follow. RESTRUCTU RING RULES: Deep ening Γ ∆    ❅ ❅ ❅ ❞ q Θ b a a q ❞ Γ ∆  ❅ ❅ ❅ q ❞ Θ Flattening Globalization Γ ❞ q   Θ Ω ❅ ❅ q ❞ ❅ ❅ q ❞   Γ Θ Ω c b a Lo calization Lengthening a  ❅ ❅ Θ  q ❞ a Θ Shortening b Ω Ω Γ Γ ❅ ❅ 9 MAIN RULES: Coupling ❞ Θ ¬ P P  ❅ t Θ a a b c W eak ening Γ t Θ ∆ Γ   ❅ ❅ t Θ a a Pulldown Γ Π ✑ ✑ ✑ ✑ t ◗ ◗ ◗ ◗ ∆ ❞ ◗ ◗ ◗ ◗ Σ t Θ Γ t Σ ◗ ◗ ◗ ◗ ∆ Π ❞ ◗ ◗ ◗ ◗ ✑ ✑ ✑ ✑ t Θ a a b b c c The double names and double ho rizon tal lines in the restructuring rules indicate that these rules work in bo th top-down a nd b o ttom-up dir ections. The na me on the top is for the dir ection where the top part is the premise, and the na me at the bottom is for the dir ection where the b ottom part is the premise . F urther mo re, ⊙ is a v aria ble over {• , ◦ } . This means that each restructur ing r ule co mes in tw o versions: o ne for • and one for ◦ . So, alto gether there ar e 12 restructuring r ules. The following conv ention provides additional ex planations and co nditions, some e ssen tially just reitera ting (for safety) certa in earlier- ma de stipulatio ns: Con v en tion 3. 2 1. Lowercase La tin letters s tand for the central parameters of the rule. 2. Upp ercase Greek letters stand for the per ipheral parameters of the rule. W e do not require the per ipheral pa r ameters to b e non-empt y , or — when there are sev eral peripheral parameter s in the r ule — disjoint or even no n- iden tical. W e use a double r ather than a single arc to indicate the prese nce of an arc b etw een a given cen tral par ameter and each no de of a given per ipher al parameter. 3. P s tands for any atom. 4. W e assume that central parameters ha ve no children and parents other than those ex plicitly indica ted (through sing le or do uble arcs) in the s c hematic repr esen tation of the rule. Hence, for example, as w e see b and c only in the conclusio n o f the rule of coupling, these tw o no des are simply absent in the premise. 5. On the o ther ha nd, the nodes of p eripheral para meters may hav e additional par e nts a nd/ or children, not shown in the schematic representation of the rule. According to the earlier explana tio ns, it is understo od that the connections b et ween suc h no des with their invisible parents a nd/or c hildren, just as all other invisible (“ c o n textual”) connections and no des, will b e prese rv ed when moving from premise to co nclusion or vice versa. So, for exa mple, while we do not see ∆ in the premise of weakening, this do es not necessarily mean that the nodes of ∆ disapp ear when moving from conclusion to premise: those nodes may r emain prese nt in the premise if (and only if ) they had some additional, invisible parents. Below come explanations of all rules. Such explanations can be provided either by saying ho w to obtain a conclusion from the premis e , o r saying how to obtain a pr emise from the conclusion. W e choo se one o r 10 the o ther way dep ending o n which one app e a rs to b e more in tuitive a nd conv enient . F or the same reason, for each of the three pair s of restructur ing rules, we expla in only one, with the o ther r ule of the pair b eing symmetric, obtaine d by in terchanging premise with conclusion. 3.1 Deep ening As can b e seen from the schematic repr esen tation, this rule has tw o central par ameters a, b and three per ipheral parameters Γ , ∆ , Θ. Its meaning is that if a cir quen t has a gate b with exactly one parent a suc h that b and a ar e o f the same type (both conjunctive, or b oth disjunctive), then a premise ca n be obtained by deleting b a nd connecting its children ∆ dir ectly to a . Γ ∆    ❅ ❅ ❅ ❞ q a Θ q ❞ b Γ ∆  ❅ ❅ ❅ q a ❞ Θ Below are several examples of a pplications of this rule. Example 1 P ¬ P  ❅ t 1 3 4 t 2 P ¬ P  ❅ 1 t 3 4 Example 2 P ¬ P  ❅ t 1 3 4 3 4 t 2 ❛ ❛ ❛ P ¬ P  ❆ ❆ ❆ 1 t Example 3 P ¬ P  ❅ t 1 t 2 ❛ ❛ ❛ P ¬ P ✪ ✪ ✪ ✟ ✟ ✟ 1 t 3 4 3 4 Example 4 P ¬ P  ❅ t 1 t 2  ❅ P ¬ P 1 t 3 4 3 4 Example 5 P ¬ P  ❅ t 1 t 1 4 ✟ ✟ ❍ ❍ P ¬ P t 3 4 3 2 As w e re mem ber , a justiﬁcation of an application of a r ule sho uld include a sp e ciﬁcation of the v alues of its par ameters. Here are such speciﬁcatio ns: • In Exa mple 1: a = 1, b = 2, Γ = { 3 } , ∆ = { 4 } , Θ = {} . • In Exa mple 2: a = 1, b = 2, Γ = { 3 } , ∆ = { 3 , 4 } , Θ = {} . • In Exa mple 3: a = 1, b = 2, Γ = { 3 , 4 } , ∆ = { 3 , 4 } , Θ = { } . • In Exa mple 4: a = 1, b = 2, Γ = {} , ∆ = { 3 , 4 } , Θ = {} . • In Exa mple 5: a = 1, b = 2, Γ = { 3 , 4 } , ∆ = ∅ , Θ = {} . The following is an example of deepening applied to a bigger cirquent. Her e w e have a = 1 , b = 2 , Γ = { 5 } , ∆ = { 6 , 7 } a nd Θ = { 3 , 4 } : 11 Example 6 ❞ ❍ ❍ P Q R S ✱ ✱ ✱ ✁ ✁ ✁ ❅ 1 t 5 ❞ ❍ ❍ ❞ ❞ ✟ ✟ t t ✟ ✟ ❍ ❍ 3 t ✟ ✟ ❍ ❍ 4 ❞ ✟ ✟ ❍ ❍ t ❞ 2 ❍ ❍ ✟ ✟ P Q R S  ❅ 1 t 5 ❞ ❍ ❍ ❞ ❞ ✟ ✟ t t ✟ ✟ ❍ ❍ 3 t ✟ ✟ ❍ ❍ 4 ❞ ✟ ✟ ❍ ❍ 6 7 6 7 Most rea ders, no doubt, would (still) feel more comfortable with f or mulas than with cirquent s. Therefor e it w ould no t hurt to a lso see a co uple of e x amples where bo th the premise and the conclusion are (tree-like and hence can be written as) for m ulas. W e are not providing the v alues o f the ﬁv e pa rameters for these instances of the r ule, which a re easy to guess anyw ay . F urthermore, note that Ex ample 8 is simply the s ame as exa mple 5. Example 7 P ∧  Q ∨ R ∨ S  P ∧  Q ∨ ( R ∨ S )  Example 8 P ∨ ¬ P , i.e., P ∨ ⊥ ∨ ¬ P ∨{ P, ¬ P } ∨{ P, ∨{} , ¬ P } 3.2 Lo calization According to this rule, if a cirquent has tw o co njunctiv e or tw o disjunctive ga tes a, b with exactly the sa me children Γ (but not necessa r ily the same parents), then a premise can b e obtained by merging a and b and calling the resulting no de c . Here “merg ing” means that c has the same t yp e and sa me children as a a nd b hav e, and the set of the par en ts of c is the union of those of a and b . Γ ❞ q c   Θ Ω ❅ ❅ q ❞ b ❅ ❅ q ❞ a   Γ Θ Ω Here ar e four exa mples of applications of this r ule. 12 Example 1 ❞ t 3 ❞ ❞ ❞ ✟ ✟ ❅ Q Q P ¬ P  ❅ t ✟ ✟  ❅ ❍ ❍ ❍ ❍ 4 5 6 7 6 7 ❞ t t ❞ ❞ ❞ ✟ ✟ 1 Q Q P ¬ P  ❅ 2 t ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ 4 5 Example 2 ❞ t 3 ❞ ❞ ❞ ✟ ✟ ❅ Q Q P ¬ P  ❅ t ✟ ✟  ❅ ❍ ❍ ❍ ❍ 4 5 ❞ t t ❞ ❞ ❞ ✟ ✟ 1 Q Q P ¬ P  ❅ 2 t ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ✟ ✟ 4 5 6 7 6 7 Example 3 ❞ t 3 ❞ Q Q P ¬ P ✟ ✟  ❅ ❍ ❍ ❞ ✏ ✏ ✏ P P P 4 Q Q P ¬ P ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❞ t t ❞ 1 2 ❞ ✏ ✏ ✏  ❅ P P P 4 6 7 6 7 Example 4 t 3 R S ❞ ✏ ✏ ✏ P P P 4 R S t t 1 2 ❞ ✏ ✏ ✏  ❅ P P P 4 • In Exa mple 1: a = 1, b = 2, c = 3, Γ = { 6 , 7 } , Θ = { 4 } , Ω = { 5 } . • In Exa mple 2: a = 1, b = 2, c = 3, Γ = { 6 , 7 } , Θ = { 4 } , Ω = { 4 , 5 } . • In Exa mple 3: a = 1, b = 2, c = 3, Γ = { 6 , 7 } , Θ = { 4 } , Ω = { 4 } . • In Exa mple 4: a = 1, b = 2, c = 3, Γ = {} , Θ = { 4 } , Ω = { 4 } . The following is the same as Exa mple 1 , o nly with the premise and the conclusio n wr itten as hyper form ulas (whic h, b y g oo d luck, is p ossible her e, ev en thoug h the sa me could not be done using just formulas):  ( Q ∧ P ) ∧ ( P ∨ ¬ P )  ∨  ( P ∨ ¬ P ) ∧ ( ¬ P ∧ Q )   ( Q ∧ P ) ∧ ( P ∨ ¬ P )  ∨  ( P ∨ ¬ P ) ∧ ( ¬ P ∧ Q )  3.3 Lengthening According to this rule, if a cirquent has a gate b with exac tly one child a , then a premise can be obtained by deleting b a nd connecting a directly to the parents Θ of b . a  ❅ ❅ Θ  q ❞ a Θ b Ω Ω Γ Γ ❅ ❅ Here ar e so me illustrations: 13 Example 1 P ❞ 2 1 1 P Example 2 ❞ t 2 ❞ 1 1 Example 3 ❞ ✧ ✧ ✧ ❜ ❜ ❜ 1 ✁ ✁ ✁ ❆ ❆ ❆ ❞ ❞ ❅ t t ❅ ✏ ✏ ✏ P P P t  P Q 3 4 3 4 5 6 7 8 5 6 7 8 t 2 ❞ ✧ ✧ ✧ ❜ ❜ ❜ 1 ❅ ❞ ❞ ❅ t t ❅ ✏ ✏ ✏ P P P t  P Q • In Exa mples 1 and 2: a = 1, b = 2, Γ = {} , Θ = {} and Ω = {} . • In Exa mple 3: a = 1, b = 2, Γ = { 7 , 8 } , Θ = { 3 , 4 } a nd Ω = { 5 , 6 } . In terms of formulas, lengthening simply re places a subformula F by ∨{ F } or ∧{ F } . Shortening, of course, seems to b e do ing a more useful job than lengthening when it comes to formulas: it remov es a “dummy” disjunction or conjunction that is a pplied to a single co njunct or disjunct. 3.4 Coupling According to this rule, if a cirquent has a childless conjunctive ga te a , then a conclusion ca n b e obtained through making a a dis junctive gate and adding to it t wo children b and c which are por ts with oppos ite lab els. ❞ Θ ¬ P P  ❅ t Θ a a b c An impor tan t co ndition here is that the a b ov e b and c should b e new no des not present in the premise. That is, one cannot utilize so me alr eady existing no de to make a child of a . E xample 3 b elow viola tes this condition, and hence is an exa mple of a wrong “applicatio n” of coupling. 14 Example 1 ❞ 1  ❅ t 1 2 3 P ¬ P Example 2 ¬ Q Q  ❅ t ❍ ❍ ❍ ✟ ✟ ✟ ❞ ❞ P ¬ P  ❅ t ❍ ❍ ❍ ✟ ✟ ✟ ❞ ❍ ❍ ❍ ✟ ✟ ✟ ❞ 1 4 5 2 3 ¬ Q Q  ❅ t ❍ ❍ ❍ ✟ ✟ ✟ ❞ ¬ P P  ❅ t P ¬ P  ❅ t ❍ ❍ ❍ ✟ ✟ ✟ ❞ ❍ ❍ ❍ ✟ ✟ ✟ ❞ 1 4 5 Example 3 ¬ Q Q  ❅ t ❍ ❍ ❍ ✟ ✟ ✟ ❞ ❞ P ¬ P  ❅ t ❍ ❍ ❍ ✟ ✟ ✟ ❞ ❍ ❍ ❍ ✟ ✟ ✟ ❞ 1 4 5 ¬ Q Q  ❅ t ❍ ❍ ❍ ✟ ✟ ✟ ❞ P ✘ ✘ ✘ ✘ ❅ t P ¬ P  ❅ t ❍ ❍ ❍ ✟ ✟ ✟ ❞ ❍ ❍ ❍ ✟ ✟ ✟ ❞ 1 4 5 2 3 3 WR ONG !!! • In Exa mple 1: a = 1, b = 2, c = 3 and Θ = { } . • In Exa mple 2: a = 1, b = 2, c = 3 and Θ = { 4 , 5 } . • Example 3 (with the same par ameters as Exa mple 2) is wrong b ecause no de 3 was alr eady in the premise. Below w e see Example 2 rewritten using hyperformulas (another “lucky case” w her e this is p ossible):  ( Q ∨ ¬ Q ) ∧ ⊤  ∧  ⊤ ∧ ( ¬ P ∨ P )   ( Q ∨ ¬ Q ) ∧ ( P ∨ ¬ P )  ∧  ( P ∨ ¬ P ) ∧ ( ¬ P ∨ P )  3.5 W ea k ening According to this rule, a premise can b e obtained from the conclusion b y deleting ar cs from a disjunctive gate a to so me c hildren ∆ of it. Γ Θ t Θ ∆ Γ   ❅ ❅ t a a Deleting arc s from a may make some children of a pa ren tless. As noted ea rlier, our pre s en t approa c h considers non-ro ot parentless no des (“or phan no des”) meaningles s and do es not oﬃcially a llo w them in cirquents. So , deleting the arcs from a to the no des of ∆ s hould b e follow ed by (p erhaps r epeatedly) deleting all orpha ns as well, as done in Exa mples 1 and 3 below. 15 Example 1 ¬ P P  ❅ t 1 2 3 Q ¬ P P ✘ ✘ ✘ ✘ ✘  ❅ t 1 2 3 4 Example 2 ¬ Q Q  ❅ t ¬ P P  ❅ t 1 ❍ ❍ ❍ ✟ ✟ ✟ 5 ❞ 2 3 ¬ Q Q  ❅ t ¬ P P ✘ ✘ ✘ ✘ ✘  ❅ t 1 ❍ ❍ ❍ ✟ ✟ ✟ 5 ❞ 2 3 4 4 Example 3 ¬ Q Q  ❅ t ¬ P P  ❅ t 1 5 ❍ ❍ ❍    6 ❞ 2 3 ¬ Q Q R  ❅ ❞ 4 5  ❅ t ¬ P P ✦ ✦ ✦ ✦ ✘ ✘ ✘ ✘ ✘ ✘  ❅ t 1 ❍ ❍ ❍    6 ❞ 2 3 • In Example 1: a = 1, Γ = { 2 , 3 } , ∆ = { 4 } and Θ = {} . The a rc from 1 to 4 was deleted (when moving from conclusio n to premise), and so w as no de 4 be c ause it had no o ther parents. • In Ex ample 2 : a = 1, Γ = { 2 , 3 } , ∆ = { 4 } and Θ = { 5 } . The arc from 1 to 4 was deleted but 4 was preserved, as it had another parent in the cirquent. • In Example 3 : a = 1, Γ = { 2 , 3 } , ∆ = { 4 , 5 } and Θ = { 6 } . The arcs from 1 to 4 a nd 5 w ere deleted. This made 4 an or phan, a nd 4 was also deleted. But deleting 4 made the R - labeled no de an orpha n, which resulted in further deleting that no de as well. When applied to a (cirquen t represented by a ) formula, w eakening can delete any num b er of disjuncts from a disjunctiv e subformula of the co nc lus ion, as illustrated b elow: P ∨  Q ∧ ( R 1 ∨ R 2 ∨ R 3 ∨ R 4 )  P ∨  Q ∧ ( R 1 ∨ R 3 )  3.6 Pulldo wn This r ule applies when the conclusion (as well as the pr emise) has a disjunctive gate a with a sing le conjunctive parent b , whic h, in turn, has a single disjunctive parent c . Then a premise can b e obtained by passing some (any) c hildren Π from c to a . a a b b c c Γ Π ✑ ✑ ✑ ✑ t ◗ ◗ ◗ ◗ ∆ ❞ ◗ ◗ ◗ ◗ Σ t Θ Γ t Σ ◗ ◗ ◗ ◗ ∆ Π ❞ ◗ ◗ ◗ ◗ ✑ ✑ ✑ ✑ t Θ When perfor ming the above children-passing transfor mation, some o r all no des of Π may still remain children of c . This is so beca use, acco r ding to Co n ven tion 3.2 , Π a nd ∆ do no t necessa rily hav e to b e 16 disjoint. Simila r ly , Π and Γ do not have to b e disjoin t, mea ning that so me nodes o f Π may simply stop being children of c without acquiring a as a new parent, as a alr e ady was a parent of them. Example 1 S R P T t Q ◗ ◗ ◗ ◗ ◗ ◗ ❞ ◗ ◗ ◗ ✑ ✑ ✑ t 4 5 6 7 8 1 2 3 S R P T Q t ◗ ◗ ◗ ◗ ◗ ◗ ❞ ◗ ◗ ◗ ✑ ✑ ✑ t 4 5 6 7 8 1 2 3 Example 2 S R P T t Q ◗ ◗ ◗ ◗ ◗ ◗ ❞ ◗ ◗ ◗ ✑ ✑ ✑ ✑ ✑ ✑ t 4 5 6 7 8 1 2 3 S R P T Q t ◗ ◗ ◗ ◗ ◗ ◗ ❞ ◗ ◗ ◗ ✑ ✑ ✑ t 4 5 6 7 8 1 2 3 Example 3 S R P T t Q ◗ ◗ ◗ ◗ ◗ ◗ ❞ ◗ ◗ ◗ ✑ ✑ ✑ t 4 5 6 7 8 1 2 3 S R P T Q t ◗ ◗ ◗ ◗ ◗ ◗ ✑ ✑ ✑ ❞ ◗ ◗ ◗ ✑ ✑ ✑ t 4 5 6 7 8 1 2 3 • In Exa mple 1: a = 1, b = 2, c = 3, Γ = { 6 , 7 } , ∆ = { 4 } , Π = { 8 } , Σ = { 5 } , Θ = { } . • In Exa mple 2: a = 1, b = 2, c = 3, Γ = { 6 , 7 } , ∆ = { 4 , 8 } , Π = { 8 } , Σ = { 5 } , Θ = {} . • In Exa mple 3: a = 1, b = 2, c = 3, Γ = { 6 , 7 , 8 } , ∆ = { 4 } , Π = { 8 } , Σ = { 5 } , Θ = {} . Example 1 (and only this example) can a ls o be written using formulas: P ∨  Q ∧ ( R ∨ S ∨ T )  P ∨  Q ∧ ( R ∨ S )  ∨ T 4 Deriv abilit y , pro v abilit y and admissibilit y A CL8-deriv ation of a cir quen t A from a cirquent B is a sequence C 1 , . . . , C n of cirq uen ts such that C 1 = B , C n = A and each C i +1 follows from C i by o ne of the r ules o f CL 8 . A deriv ation is usually required to come with an — even if only implicit — justiﬁcation , which is an indica tion of b y whic h rule any giv en cirquent C i +1 follows from C i and wher e that rule is applied, i.e., wha t the v alues of the para meters of the rule are. A CL8-proof of a cirquen t A is a CL8 -deriv ation of A from ◦ . Thus, th e single-no de cirquen t ◦ , i.e ., ⊤ , is the (only) axiom of CL8 . When a CL8 -pro of o f a cirquent A exis ts, we say that A is prov able in CL8 a nd write CL8 ⊢ A . Similar terminology applies to an y other cirquent calculus system as w ell. When CL8 is the only s ystem we deal with in a given context (suc h as the present section), we usually omit “ CL8 -” and simply say “der iv ation”, “prov able” etc. Below is a CL8 - proo f of the left cir quen t o f Figure 2 in full deta il, serving the purp ose of giving the reader a better syntactic feel of cirquent calculus. All p orts of eac h cirquent of the proo f hav e unique labels, which allows us to unambiguously refer to those p orts (in justiﬁcations ) by their lab els, w itho ut as signing names to them a s w e did in the examples of the pr evious section. axiom ❞ 4 17 deepening: a = 4, b = 2, Γ = {} , ∆ = {} , Θ = {} ❞ ❜ ❜ ❞ 4 2 deepening: a = 4, b = 3, Γ = { 2 } , ∆ = {} , Θ = {} ❞ ❜ ❜ ✧ ✧ ❞ ❞ 4 3 2 coupling: a = 2, b = Q , c = ¬ Q , Θ = { 4 } ¬ Q ❞ ❜ ❜ ✧ ✧ Q ❛ ❛ ❛ ❞ t 4 3 2 coupling: a = 3, b = R , c = ¬ R , Θ = { 4 } ¬ Q ¬ R ❞ ❜ ❜ ✧ ✧ Q R ✦ ✦ ✦ ❛ ❛ ❛ t t 4 3 2 lengthening: a = 4, b = 1, Γ = { 2 , 3 } , Θ = {} , Ω = {} ¬ Q ¬ R ❞ ❜ ❜ ✧ ✧ Q R ✦ ✦ ✦ ❛ ❛ ❛ t t t 1 4 3 2 pulldown : a = 2, b = 4, c = 1, Γ = { Q } , ∆ = {} , Π = {¬ Q } , Σ = { 3 } , Θ = {} ¬ Q ¬ R ❞ ❜ ❜ ✧ ✧ ❳ ❳ ❳ ❳ ❳ ❳ Q R ✦ ✦ ✦ t t t 1 4 3 2 pulldown : a = 3, b = 4, c = 1, Γ = { R } , ∆ = {¬ Q } , Π = {¬ R } , Σ = { 2 } , Θ = {} ¬ Q ¬ R ❞ ❜ ❜ ✧ ✧ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳ Q R t t t 1 4 3 2 shortening: a = Q , b = 2, Γ = {} , Θ = { 4 } , Ω = {} ¬ Q ¬ R ❞ ❜ ❜ ✧ ✧ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳ Q R t t 1 4 3 18 shortening: a = R , b = 3, Γ = {} , Θ = { 4 } , Ω = {} ¬ Q ¬ R ❞ ❜ ❜ ✧ ✧ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳ Q R t 1 4 lengthening: a = ¬ Q , b = 2, Γ = {} , Θ = { 1 } , Ω = {} ¬ Q ¬ R ❞ ❞ ❜ ❜ ✧ ✧ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳ Q R t 1 4 2 lengthening: a = ¬ R , b = 3, Γ = {} , Θ = { 1 } , Ω = {} ¬ Q ¬ R ❞ ❞ ❜ ❜ ✧ ✧ ❞ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳ Q R t 1 4 2 3 deepening: a = 2, b = 5, Γ = {¬ Q } , ∆ = {} , Θ = { 1 } ¬ Q ¬ R ❞ ✧ ✧ ❞ ❜ ❜ ✧ ✧ ❞ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳ ❞ Q R 5 t 1 4 2 3 deepening: a = 3, b = 6, Γ = {¬ R } , ∆ = {} , Θ = { 1 } ¬ Q ¬ R ❞ ✧ ✧ ❞ ❜ ❜ ✧ ✧ ❞ ❜ ❜ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳ ❞ ❞ Q R 5 6 t 1 4 2 3 just r edrawing the cir quen t ¬ Q ¬ R Q R ❞ ✧ ✧ ❞ ❜ ❜ ❞ ❜ ❜ ❜ ❜ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳ ❞ ❞ 5 6 t 1 3 2 4 globalization: a = 5, b = 6, c = 7, Γ = {} , Θ = { 2 } , Ω = { 3 } ¬ Q ¬ R Q R ❞ ✧ ✧ ✧ ✧ ❞ ❜ ❜ ❜ ❜ ❞ ❜ ❜ ❜ ❜ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳ t ❞ 1 3 2 4 7 19 coupling: a = 7, b = P , c = ¬ P , Θ = { 2 , 3 } ¬ P ¬ Q P ¬ R Q R ❞ ✧ ✧ ✧ ✧ ❞ ❜ ❜ ❜ ❜ ❞ ❜ ❜ ❜ ❜ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳  ❅ t t 1 3 2 4 7 lo cali zat ion: a = 5, b = 6, c = 7, Γ = {¬ P , P } , Θ = { 2 } , Ω = { 3 } ¬ P ¬ Q P ¬ R Q R ❞ ✧ ✧ ✧ ✧ ❞ ❜ ❜ ❜ ❜ ❞ ❜ ❜ ❜ ❜ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳   ❅ ❅ t t t 1 3 2 4 5 6 pulldown : a = 6, b = 3, c = 1, Γ = { P } , ∆ = { 2 , 4 } , Π = {¬ P } , Σ = {¬ R } , Θ = {} ¬ P ¬ Q P ¬ R Q R ❞ ✧ ✧ ✧ ✧ ❞ ❜ ❜ ❜ ❜ ❞ ❜ ❜ ❜ ❜ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ✥ ✥ ✥ ✥ ✥ ✥   t t t 1 3 2 4 5 6 pulldown : a = 5, b = 2, c = 1, Γ = { P } , ∆ = {¬ P , 3 , 4 } , Π = {¬ P } , Σ = {¬ Q } , Θ = {} ¬ P ¬ Q P ¬ R Q R ❞ ✧ ✧ ✧ ✧ ❞ ❜ ❜ ❜ ❜ ❞ ❜ ❜ ❜ ❜ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ t t t 1 3 2 4 5 6 shortening: a = P , b = 6, Γ = {} , Θ = { 3 } , Ω = {} ¬ P ¬ Q P ¬ R Q R ❞ ✧ ✧ ✧ ✧ ❞ ❜ ❜ ❜ ❜ ❞ ❜ ❜ ❜ ❜ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ t t 1 3 2 4 5 shortening: a = P , b = 5, Γ = {} , Θ = { 2 } , Ω = {} ¬ P ¬ Q P ¬ R Q R ❞ ✧ ✧ ✧ ✧ ❞ ❜ ❜ ❜ ❜ ❞ ❜ ❜ ❜ ❜ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ t 1 3 2 4 This is the last time in this pap er that we provide a pro of in all details. Subsequen t pro ofs will b e “lazie r ”, with sev eral steps often co m bined toge ther , and with justiﬁcations t ypically r educed to indicating the names of the rules us ed, without indicating the (usua lly easy to guess) v alues of the cor respo nding parameters. 20 The reader may wan t to try to see why and where the a bov e pro of fa ils if the tar get is the right rather than the left cirque nt o f Figure 2. That cir quen t s imply has no pro of. The diﬀerence between one shared P -p ort and tw o s eparate P -p orts is thus cruc ia l here. The left cir quen t of Figure 2, unlike the right cirquent, is a cir c uit — each po rt in it has a uniq ue lab el. How ev er, this in not at all the rea son why the former is prov a ble and the latter is not. The left cirquent of the following Figure 3 is not a cir cuit but its pro of can be mechanically o btained from the ab o ve pr oof by replacing a ll atoms by P . On the other hand, the right cirquent o f Fig ure 3, just like its pr edecessor from Figure 2, can b e shown to hav e no pro of: ¬ P ¬ P P ¬ P P P ✱ ✱ ❞ ❧ ❧ ❞ ❧ ❧ ❞ ❳ ❳ ❳ ❳ ✘ ✘ ✘ ✘ ✘ ✘ ❧ ❧ ✱ ✱ t Figure 3: CL8 prov es the left but not the r igh t cirquent ¬ P ¬ P P P ¬ P P P ❆ ❆ ✁ ✁ ❞ ❆ ❆ ✁ ✁ ❞ ❆ ❆ ✁ ✁ ❞ ❳ ❳ ❳ ❳ ❳ ✘ ✘ ✘ ✘ ✘ ✘ ❧ ❧ ✱ ✱ t An (atomic-level) instance of a cirquent is the res ult of renaming (all, some or no) atoms in it. Here, of course, diﬀerent o ccurr ences (in the lab els of diﬀerent p orts) of the sa me ato m ar e required to b e renamed int o the same atom, but it is also po ssible that diﬀerent ato ms are renamed into the same atom. E xample: the tw o cir quen ts of Figur e 3 are instances of the tw o cirquents of Fig ure 2. W e noted ab ov e tha t the left cirquent of Figure 3 is prov able bec a use so is its more gener a l predecessor from Figure 2 . In other w ords, the former is pr o v able be c ause it is an instance of the la tter which, in turn, has alr eady been seen to hav e a pr oof. The following lemma g eneralizes this observ ation: Lemma 4.1 If a cir quent is pr ovable, then so ar e al l of its instanc es. Pro of. Consider an arbitra ry cirquent C , and let C ′ be a n instance o f it, resulting from renaming each atom P of C in to an atom P ′ . Assume P is a pro of of C . No te that no c ir quen t in P cont ains an y atom that do es not o ccur in C . So, let P ′ be the result of r enaming each atom P in to P ′ in each cirquent o f P . It is not har d to see that P ′ is a pro of of C ′ . ✷ By a transition w e mean an y binary relation T o n cirquen ts. When A T B , w e say t hat B follows from A by T , and ca ll A and B the prem ise and the conclus ion of the given application of the transition, resp ectiv ely . T ransitions a re the same as rules of inference, only in a more relaxed se ns e than th e strict sense of Section 3. Of course, ev ery rule R of inference induces — and can often b e ident iﬁed with — a transition T , such that B follows from A by T iﬀ B follows from A by R with so me (whatever) para meters. W e may not a lw ays b e very str ic t in terminolo gically diﬀerentiating betw een tr ansitions and r ules . A transition is s aid to b e strongly admis sible in a given system if, whenever B follows from A by that transition, there is also a deriv ation of B from A . And a transition is weak ly admissible iﬀ, whenever B follows from A by that transition and A is prov able in the sy s tem, B is also pr o v able. One of the useful stro ngly admissible transitions is des tandardization . T o obtain a premise fr om the conclusion A of destandardization, we a pply to A — in the bo ttom-up sense — a series of globalizatio ns un til every non- roo t gate has exactly one parent; then we a pply a ser ie s of deepening s until no conjunctiv e gate ha s conjunctive children a nd no dis junctiv e gate has disjunctive children; ﬁnally , we apply a ser ies of lengthenings until there a re no gates that hav e exactly one c hild. It is ea sy to see tha t this pro cedure applied to A yields a unique (mo dulo isomorphism) cirquent B , to which we will be referring as the standardization of A . Then we say that such a B follows from A by destandardiza tion. The same transition but with premise and co nclusion interc hanged we a ls o call standardization . Of course, standardizatio n, just like destandardizatio n, is among the strongly admissible transitio ns in CL8 . Another strongly admissible transition for which we ha ve a sp ecial name is restructuring , which works in b oth top-down and b ottom-up dir ections. W e say that a cirquent B follows from a cirquent A by restructuring, or that “ A can be r estructured into B ”, if there is a deriv ation of B from A that uses o nly restructuring r ules . Destandardiza tion and standardization ar e th us sp ecial cases of restructuring . One more strongly admiss ible transition that we are going to rely on is trade . It is given b y 21 c 1 c n t ... t ✑ ✑ ◗ ◗ Π ◗ ◗ ◗ ◗ ✑ ✑ ✑ ✑ ❞ Θ c 1 ... c n ✑ ✑ ◗ ◗ Π ❞ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ◗ ◗ ◗ ◗ ✑ ✑ ✑ ✑ ◗ ◗ ◗ ◗ t Θ Ω 1 Ω n Γ 1 Γ n Ω 1 Ω n Γ 1 Γ n a a b b 1 b n where n ≥ 0, and the co n ven tions o f Section 3 contin ue to be in for c e, except that, as we see, here the nu mber o f parameter s is not ﬁxe d (so that tr ade is not just a single r ule in the strict sense of Section 3 but rather a co llection of rules, one for each n ∈ { 0 , 1 , 2 , . . . } ). Be lo w is an example of an application of tra de where b oth the premise a nd the co nclusion can b e written a s h yp erformulas: ( P ∨ R ) ∧ ( Q ∨ R ) ( P ∧ Q ) ∨ R Referring to the no des of the ab o ve cir quen ts b y the corres ponding subformulas, in this application of trade n = 2, Π = { R } , the other p eriphera l parameters are empty , c 1 = P , c 2 = Q , b = ( P ∧ Q ) ∨ R , b 1 = P ∨ R , b 2 = Q ∨ R , and a is P ∧ Q in the conclusion and ( P ∨ R ) ∧ ( Q ∨ R ) in the premise. The a gate of the c o nclusion of tr ade w ill b e said to be the principal gate of a giv en application of this rule. Note that when n = 0, i.e., when the pr incipal gate is childless, trade is simply ❞ Θ Π ❞ ✑ ✑ ✑ ✑ t Θ a a b whose stro ng admissibilit y is seen fro m the following transfor mations: ❞ Θ a lengthening ❞ t Θ b a wa kening Π ❞ ✑ ✑ ✑ ✑ t Θ b a 22 And the following transformations show the stro ng admissibilit y of trade for the case n ≥ 1: c 1 c n t ... t ✑ ✑ ◗ ◗ Π ◗ ◗ ◗ ◗ ✑ ✑ ✑ ✑ ❞ Θ ✑ ✑ ✑ ✑ ◗ ◗ ◗ ◗ Ω 1 Ω n Γ 1 Γ n a b 1 b n lengthening c 1 c n t ... t ✑ ✑ ◗ ◗ Π ◗ ◗ ◗ ◗ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ◗ ◗ ◗ ◗ Ω 1 Ω n Γ 1 Γ n a b 1 b n b ❞ Θ t pulldown ( n times) c 1 c n t ... t ✑ ✑ ◗ ◗ Π ✟ ✟ ✟ ✟ ✑ ✑ ✑ ✑ ◗ ◗ ◗ ◗ Ω 1 Ω n Γ 1 Γ n a b 1 b n b ❞ Θ t shortening ( n t imes) c 1 ... c n ✑ ✑ ◗ ◗ Π ❞ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ◗ ◗ ◗ ◗ t Θ Ω 1 Ω n Γ 1 Γ n a b 5 Seman tics In this section we deﬁne a semantics for cirquents, termed abstract resource seman tics . This is a generaliza tion, to all cir quen ts, of the same-name s eman tics introduced in [14] for the ea rlier mentioned sp ecial, “shallow”, class of cirquents. The ma in purp ose of a go o d semantics sho uld b e s erving a s a bridge betw een the real world a nd the otherwise meaningles s for mal expressions of logic. And, corresp ondingly , the v alue of a semantics should be judged b y how successfully it a c hieves this purpo se, which, in turn, dep ends on how natura lly and adequately it formalizes cer tain basic int uitions connecting logic with the outside world. Suc h intuitions behind abstract resource semantics have b een amply expla ine d a nd illustrated in Section 8 of [14]. The reader is strongly recommended to get familiar with that piece of literature in order to appreciate the claim of abs tract reso ur ce semantics that it is a “ r eal” se man tics of re s ources, fo r malizing the reso urce philoso phy traditionally (and, as a rgued in [14], somewhat wrongly) asso ciated with linear lo gic and its v ariations. In this pap er we just provide formal deﬁnitions, o nly o ccasionally making very brief intuit ive comments, and otherwise fully relying on [14] for extended explanations of the in tuitions, motiv a tions and philosophy underlying the semantics. 23 Abstract resour ce se man tics can b e seen as a cons erv ative generaliz a tion o f the semantics of c lassical log ic from circuits to all cirq uen ts. The starting point of the semantics is the concept of a truth assignment for a given cir q uen t C . This is a function that assigns o ne o f tw o v alues — true or false — to ea c h p ort of C . Any such function f is a legitimate truth assignment, including the cases when f assig ns diﬀeren t truth v a lues to por ts that have identical lab els. Intuitiv ely this is p erfectly meaningful in the world o f resources bec ause, say , one 25 c -p ort (slot of the vending mac hine) may receive a true co in while the other 25 c - port may receive a false coin or no coin at all. Each tr uth assignment for a cirquen t extends from its por ts to all gates and the cirquent itself in the following, expe c ted, wa y: • A disjunctive gate is true iﬀ it ha s at lea st one true c hild. • A conjunctive gate is true iﬀ so are all of its c hildren. • The cirq ue nt is true iﬀ so is its ro ot. An allo cation fo r a g iv en cirquent C is an unorder e d pair { a, b } of p orts o f C with opp osite lab els (lab els P and ¬ P for some — the same — atom P ). And an arrangement for C is an y set of pairwise disjoint allo cations for C . W e call the condition requir ing all allo cations to b e disjoint the monogami cit y condition . Let C be a cirquent, f a truth assignment for C , and α an arrang emen t for C . W e say that f is consis ten t with α iﬀ, for every allo cation { a, b } ∈ α , f ( a ) 6 = f ( b ). Tha t is, if p orts a and b are allo cated to each other (meaning that { a, b } ∈ α ), a truth assignment consistent with α should ass ign o pposite truth v a lues to a and b . 4 And w e sa y that α is v alidating 5 (for C ) iﬀ C is t rue under ev ery truth assignment consistent with α . T o see a n example, consider the following cirquent: 1 2 3 4 5 6 7 8 ¬ P ¬ P ❅ t ¬ P ¬ P ❅ t ❞ ❳ ❳ ❳ ✘ ✘ ✘ P P ❅ t P P ❅ t ❞ ❳ ❳ ❳ ✘ ✘ ✘ ❤ ❤ ❤ ❤ ❤ ❤ ✭ ✭ ✭ ✭ ✭ ✭ t Figure 4: A v a lid cirquent And consider the fo llowing tw o arr angemen ts for this cirquent: α = {{ 1 , 5 } , { 2 , 6 } , { 3 , 7 } , { 4 , 8 } } ; β = {{ 1 , 5 } , { 2 , 7 } , { 3 , 6 } , { 4 , 8 }} . Here α is not a v a lidating a rrangement. Sp eciﬁcally , the following truth assignment f , while obviously consistent with α , makes the cirquent false: f (1) = f (2) = f (7 ) = f (8 ) = fa lse ; f (3) = f (4) = f (5 ) = f (6) = true . This ass ig nmen t, on the other hand, is not consistent with β . Moreov er, with so me tho ug h t, o ne can see that no truth assignment that makes the cirq uen t o f Figur e 4 false can be co nsisten t with β . This means that β , unlike α , is a v a lida ting arrang e men t f or that cir quen t. As explained a nd illustrated in [14], our formal co ncept of an allo cation co rresp onds to the in tuition o f allo cating one re s ource to another: a coin (25 c ) to a coin-receiving slot ( ¬ 25 c ), a memory (100 M B ) to a memory-re q uesting pro cess ( ¬ 100 M B ), a p ow er source (100 w ) to a pow er-co nsuming utensil ( ¬ 100 w ), an USB-interface externa l device ( U S B ) to an USB p ort of a co mputer ( ¬ U S B ), etc. A j ustiﬁcation behind the monogamicity condition for ar rangements is that if a res o urce a is used by (allo cated to) b , then it cannot be also used b y (allo cated to) another c 6 = b . And the in tuition behind a v alidating arr angement is that of a successful res ource-management s trategy/solution. 4 In [14], a we aker condition was adopted, according to which at least one (but possibl y both) of the nodes a, b should b e assigned true . It is easy to see th at either condition yields th e same concept of v alidity , so that this diﬀerence is unimpor tan t. 5 The corresp onding term u sed in [14] was “ t rivializing ”. 24 Deﬁnition 5.1 W e say that a cirquent is a v alid 6 (in abstract r esource semantics) iﬀ there is a v alidating arrang emen t for it. F or example, naming the ports (in the left to right order ) of the cirquents o f Figure 2 by the c o nsecutiv e nu mbers 1 , 2 , ... , the set {{ 1 , 3 } , { 2 , 5 } , { 4 , 6 }} is a v alidating arrangement for the left c ir quen t, which makes that cirquen t v alid. On the other hand, with a little thought, o ne can see that no p ossible arra ng emen t for the right cirquent of the same ﬁgure is v alidating, so that that cirquent is no t v a lid. Note that the mono g amicit y condition plays a crucial ro le in precluding the right cirquent o f Figur e 2 from b eing v alid: b ecause of mo nogamicity , o ne o f the t wo P -p orts of that cirquent will hav e to be left una llo cated. The ab o ve arr angement is also v alidating for the left cirquent of Figur e 3 . And, aga in with so me (this time a little more) thought, one can see that the right cirque nt of the sa me ﬁgure ha s no v alidating arrang emen t, th us be ing non-v alid. The following cirquent is not v alid, either, even though it lo oks so “similar” to the v alid cir quen t of Figure 4: ¬ P ¬ P ❅ t ¬ P ¬ P ❅ t ¬ P ¬ P ❅ t ❞ ❵ ❵ ❵ ❵ ❵ ✥ ✥ ✥ ✥ ✥ P P ❅ t P P ❅ t P P ❅ t ❞ ❵ ❵ ❵ ❵ ❵ ✥ ✥ ✥ ✥ ✥ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ t Figure 5: A non-v alid cirquent Again, as illustrated in [1 4 ], v alid cirquents are resour c e-management problems (such a s the problem of getting a candy from a vending mach ine with a given co lle ction of av aila ble coins) that hav e succes sful solutions. And among the po ten tial pra ctical v alues of sound and complete deductive sy stems such a s our CL8 or the s ystem CL5 construc ted in [14] is that they pre s en t to ols for systematically ﬁnding such solutions. Let C be a circuit, and µ be the set of all p ossible allo cations for C . This set satisﬁes the mo nogamicity condition and hence is an arrang emen t for C , b ecause the latter, b eing a c ircuit, has at mos t one P -p ort and at most o ne ¬ P -por t f or a n y given a tom P . Of cour se, a n y other arra ngemen t for C will b e a subset of µ , whic h, in turn, easily implies that C is v alid if and only if the arrangement µ is v alida ting for it. In other words, C is valid iﬀ it is true under every truth assignment c onsistent with µ . (3) But notice that truth as signmen ts consistent with µ ar e nothing but truth a ssignment s in the kind old classical sense, meaning functions that a ssign opp osite truth v a lues to P and ¬ P , for any atom P . In view of (3), w e th us ﬁnd that: F act 5. 2 V ali dity in our sense and validity (t autolo gicity) in t he classic al sense me an the same for cir cuits, and henc e for formulas of classic al lo gic understo o d as cir cu its ac c or ding to the s t ipulatio ns of Se ction 2. So, as promised, o ur a bstract resource semantics is a conserv ative extension o f classical semantics from circuits to all cir quen ts. Lemma 5.3 A cir quent is valid iﬀ it is an instanc e of a valid cir cuit. Pro of. Co ns ider an arbitr a ry cirquen t A . ( ⇐ :) Assume A is an instance of a v alid circuit B . Let α b e a v a lidating arra ngemen t fo r B . It is not hard to see that then the same α is a lso a v alidating ar rangement for A , so tha t A is v alid. ( ⇒ :) Suppo se A is v alid. Let α be a v alidating arrangement for it. Let then B b e the result of re naming the o ccurrences o f atoms within the lab els of the po rts of A in such a way th at no atom (with or without a 6 The corresp onding term u sed in [14] was “ t rivial ”. 25 negation) o ccurs in the labels of tw o diﬀerent po rts a and b unles s { a, b } ∈ α , in which case both occur rences of the (same) atom in the lab els o f a, b within A ar e renamed into the same a tom. Th us, B is a circuit. With a little thought, one can also see that the same ar rangement α remains v alidating for B , s o that B is, in fact, a v alid circuit. Now, it remains to notice that A is an instance o f B . ✷ Theorem 5.4 A cir quent is pr ovable in C L8 iﬀ it is valid in abstr act r esour c e s emantics. Pro of. Let C b e an arbitra ry cirquen t. Soundness : Assume CL8 ⊢ C . Let P be a CL8 -pro of of C . Let us r ename the atoms (o ccurring in the lab els) o f the cirquen ts of P in such a wa y that every time coupling is used, the atom P it intro duces is new, in the sense that the pr emise do es not ha ve any p orts labele d with P or ¬ P . Let us further rename the atoms of P so that ev ery time w eakening int ro duces so me new por ts (ones tha t did not exist in the premise), the labels o f such po rts ar e new and diﬀerent f rom eac h o ther. Let us call the resulting sequence of cirquents P ′ . It is no t hard to see that t hen P ′ is a pro of of a cirquen t C ′ such that C is an instance of C ′ . The axiom ◦ is , of course, a circuit, and every rule of inference obviously preserves the circuit prop erty (“ cir cuit n ess ”) of cirquents except coupling a nd weakening. But with the conditions that we impose d o n thos e tw o rules when o bta ining P ′ from P , all of the cirquen ts in P ′ are circuits. It is also eas y to se e that all inference rules preserve truth and hence v alidit y of cir c uits. Th us, all cirquents in P ′ are v alid circuits, including C ′ . And, as C is an instance of C ′ , Lemma 5.3 implies that C is v alid. Completeness : Assume C is v alid. Then, by Lemma 5 .3, there is a v alid circuit C ′ such that C is an instance o f C ′ . Fix this C ′ . W e ar e g o ing to show that C ′ is prov able, which, by Lemma 4.1 , immediately implies that C is also prov able. W e construct, b ottom-up, a pro of of C ′ as follows. First, applying (b ottom-up) destandar dization, we pro ceed from C ′ to its s tandardization J . Let us ﬁx a “suﬃciently lar ge” in teger s , such that s ≥ 2 a nd s exceeds the total n umber of no des in J . Given a cirquent H , w e deﬁne an activ e gate of H to b e a disjunctiv e g ate a of H tha t has no disjunctive ancestors. W e deﬁne the rank o f suc h an a to be s m , wher e m is the n um b er of co njunctive ga tes that are des cendan ts of a . And w e deﬁne the rank of H to be the s um of the ranks of its active gates. Our construction of a pro of of C ′ contin ues up w ard from J as follows. W e repe at the following t w o steps while there are non- roo t conjunctive ga tes in the curren t (topmost in the so far constructed proof ) cirquent: Step 1. Pick an arbitra ry conjunctiv e child c of an arbitra ry active no de o f the c ur ren t cirquent, a nd apply (b ottom-up) trade so that c is the principal g a te of the a pplica tion. Step 2. Apply (b ottom-up) destandar dization to the resulting cirq uen t. With some thought, one can see that every time the a bov e tw o steps are p erformed, the rank of the current (topmost) cirquent decreases. Hence, the proc e dure will end s ooner or later, and the r esulting cirquent D will hav e no non-r o ot conjunctive gates. It is easy to see that destandardiza tio n a nd tra de preserve b oth v a lidit y and circuitness (no t only in the to p-do wn but a lso) in the b ottom-up directio n. So, D is a v alid circuit. The pathologica l case when D has no conjunctive gates is simple and we do no t consider it here. Otherwise, D is a circuit with a conjunctiv e ro ot, wher e ea c h c hild of the ro ot is a disjunctiv e gate a nd eac h grandchild of the r oot is a p ort, as shown in the following example: D : ¬ Q Q ✟ ✟ P P P P ❵ ❵ ❵ ❵ ❵ ❵ t t ❍ ❍ ✟ ✟ R ¬ P P ✘ ✘ ✘ ✘ ✘  ❅ t ❍ ❍ ❍   ✏ ✏ ✏ ✏ ❞ The v alidity o f D ob viously implies that among the c hildren of each disjunctive g ate is a pair of p orts with opp osite lab els. W e select one such pair for each disjunctive ga te, and remove all o ther child ren using weak enings. 7 Now, the resulting cirquent E has a conjunctive g ate at its ro ot, whose every child is a disjunctive gate with exactly tw o children, with those tw o children b eing p orts with o pposite labels, as shown below: 7 At this point we see that w eake ning in CL8 can b e restricted to port weak ening , i.e., the version of wea kening that perm its deleting only arcs to ports (rather tha n an y no des). This is relev ant to the c laim made in Subsection 6.2. 26 E : ¬ Q Q ✟ ✟ t t ❍ ❍ ¬ P P  ❅ t ❍ ❍ ❍   ✏ ✏ ✏ ✏ ❞ F urthermore, E , of course, inherits circuitness from D . And E ’s being a circuit obviously implies tha t whenever tw o disjunctive gates s ha re a c hild, they shar e b oth of their children. Applying (bo ttom-up) lo calizations to E , w e pro ceed from E to F , where F is just like E , only without any s haring o f children betw een diﬀerent disjunctiv e g a tes: F : ¬ Q Q  ❅ t ¬ P P  ❅ t ❍ ❍ ❍ ✟ ✟ ✟ ❞ Now, applying (b ottom-up) couplings to F , we replace in it each disjunctiv e gate by a child less conjunctive gate, obta ining a cirq uen t G wher e all no des ar e conjunctiv e g ates: G : ❞ ❞ ❍ ❍ ❍ ✟ ✟ ✟ ❞ Applying (b ottom-up) to G a series o f deepenings yields the axio m cirquent ◦ . ✷ An alternativ e pro of of the completeness of CL8 could rely on the forthcoming Theo rem 7.1. The latter, in vie w of the known completeness of the system G considere d there, implies that, for ev ery tautological formula F of classica l logic, CL8 ⊢ F . The cirquent J co nstructed in our pr oof o f Theorem 5.4 can be seen to be F for some tautology F and hence, in view of Theorem 7.1, CL8 -pr o v able. How ever, s uc h a pro of, alb eit shorter , would not be as dir ect as the one pr esen ted ab ov e. It should b e remembered that, as noted earlier, the initial impulse to cirquent calculus w as g iv en by the needs of computabilit y lo gic. Therefore , this pap er would not b e complete without oﬃcia lly es tablishing a connection b et ween the latter and CL8 . The or ig inal semantics of co mputability logic deals with for m ulas rather than cirq uen ts. And, as shown in [14], the class o f formulas (in the sense of o ur Section 2) v alid in computability logic coincides with the class of form ulas v alid in abstract resour ce semantics. This, in view of Theo rem 5.4, means that: Theorem 5.5 A formula (in our pr esent sense) is valid in c omputabi lity lo gic iﬀ it — se en as a tr e e-like cir quent ac c or ding to t he st ipula tions of Se ction 2 — is pr ovable in CL8 . [14] further show ed how to ex tend the s eman tics of computability logic from f ormulas to cirquen ts. While “cirquents” there only mea n t sp ecial sorts of cirquents in o ur pre s en t, mor e general, sense, the genera lization of the semantics of computability logic o utlined in [14] almost automatically extends to all cirquents in o ur present sense as well: details can b e very easily ﬁlled by a n yone familiar with computability log ic. And we claim without a pr o of that, with this gener alized semantics o f computability logic in mind, Theorem 5.5 ca n be streng thened b y r eplacing “formula” with “cirquent”. Those familiar with computability lo gic will also re mem ber that the language o f the latter has tw o sorts of a toms: P , Q, R, S, . . . , calle d gener al , and p, q , r , s, . . . , called elementary . The t wo sorts of ato ms hav e t wo diﬀerent se man tic in terpretations, whic h result in a resource- conscious logical behavior o f general atoms and class ic a l b eha vior of elementary atoms. In this pape r , which is notationally fully sy nc hronized with computability lo gic, we hav e b een using uppercase ra ther than lo wercase letters fo r atoms. Hence, “ form ula” in Theorem 5.5, as a form ula of computabilit y logic, is to b e understo od as one where all atoms are general. But, a ccording to the following claim that we fur ther make without a pro o f, CL8 in fact captures a m uch more expr essiv e fragment of computability logic than implied b y Theo r em 5.5: 27 Claim 5.6 L et F b e a formula of the ¬ , ∧ , ∨ -fr agment of t he language of c omputability lo gic, which may c ontain either sorts of atoms. F or simpli city, her e we assume that F is written in a form wher e ¬ is only applie d to atoms. L et then ˘ F b e the cir quent r epr esente d — ac c or ding t o the stipulations of Se ction 2 — by the hyp erfo rmula obtaine d fr om F thr ough overlining al l elementary (but not gener al) atoms and their ne gations, with p, q , . . . , along with P, Q, . . . , now t r e ate d as or dinary atoms of the language of CL8 . Then F is valid in c omp utability lo gic iﬀ CL8 ⊢ ˘ F . 6 Other deep cirquen t calculus systems 6.1 A symmetric version of CL8 The dual o f a given infer ence rule is obtained b y in terchanging premise with co nclusion and conjunctiv e gates with disjunctive gates. Each res tructuring r ule c o mes together with its dual, a s tho se rules w ork in bo th directions a nd for either sort o f gates. System CL8S that we deﬁne here is a fully symmetric version of CL8 , obta ine d b y adding to the latter the duals of the main rules : DUALS O F THE MAIN RULES: a a b b c c a a a a b c Co coupling t Θ ¬ P P  ❅ ❞ Θ Co w eak ening Γ Θ ❞ Θ ∆ Γ   ❅ ❅ ❞ Copulldown Γ Π ✑ ✑ ✑ ✑ ❞ ◗ ◗ ◗ ◗ ∆ t ◗ ◗ ◗ ◗ Σ ❞ Θ Γ ❞ Σ ◗ ◗ ◗ ◗ ∆ Π t ◗ ◗ ◗ ◗ ✑ ✑ ✑ ✑ ❞ Θ It is easy to see that each of the ab o ve three rules preserves v alidit y . Ther efore, in view of the a lready prov en completeness, these rules are weakly admissible in CL8 . The negation ¬ C o f a g iv en cirq ue nt C is obtained by changing the label of each p ort to its opp o site ( P to ¬ P and vice versa), and changing the t yp e (conjunctive/disjunctiv e) of each gate to the other t yp e. The r ule of c ocoupling can also be ca lled cut , sp eciﬁcally , p ort cut . It would not b e ha rd to show that cut remains w eakly admissible in CL 8 when extended from por ts P , ¬ P to an y subcirq uents A, ¬ A . In fact, non-p ort cut is strongly admiss ible in CL8S , for it ea sily (=p olynomially) reduce s to the p ort (“atomic”) version as is the cas e in the calculus of structures (see [3, 4]). An interesting question to which at present we ha ve no answ er is whether cut can b e eliminated witho ut an exp onen tial increa se of pro of sizes. This question is known to hav e a negative answ er for o rdinary sequent calculus. The top- do wn symmetry in the st yle o f the o ne enjoy ed by CL8S w as ﬁr st a c hieved a nd exploited within the framework o f the ca lculus of structures (see, again, [3, 4]). Such a sy mmetr y g enerates a num b er of nice eﬀects, so me similar to those enjo yed b y na tural deduction systems. Below w e obser v e only one such eﬀect. A refuta tion of a given cirq ue nt C is a deriv atio n of • from C . When suc h a deriv ation exists, C is said to b e refutable . The following fact — which, note, do es not hold for CL8 — is obvious in view of the full symmetry of the r ules of CL8S : F act 6. 1 In CL8S , a cir quent is pr ovable iﬀ its ne gatio n is r efutable. 28 Unlik e CL8 , ho wev er, CL8S is non-analytic , in any reaso nable sense of this w ord. Often in the literature analyticity is just under s too d as enjoying the subformula pr op erty , according to which everything in the premise of any given applica tion of any of the rules of the sy stem is a subformula o f (some formula of ) the conclusion. The subformula prop erty is meaningful for sequent calculi b ecause there the pre mis e s and the conclusion are not formulas but ra ther collectio ns (sequences, multisets o r sets) of formulas. But in cirquent calculus, wher e the premise is a sing le cir quen t a nd so is the conclusion, the subformula (“sub cirquent”) prop ert y hardly makes any sense. Indeed, if it is under stoo d literally — as the requirement that ev erything in the pr emise be a sub cirquent of “something in the conclusion”, then simply the whole pr emise itself would hav e to b e a subcirquent of the conclusio n. This w ould fully retard a n y cirquen t calculus system, essen tially limiting its rules to the one that (in the bo ttom- up view) just deletes the ro ot and jumps to one of its children. And it is not only cirquent calculus wher e the s ubfor m ula prop erty is no longer meaningful. The same holds for deep inference sy stems in gener al, s uc h as the ca lc ulus of str uctures. F or this rea son, [3] uses the term “a nalytic” in a more rela x ed sens e, simply meaning the absence of cut, substitution, extension or r ules in the st yle of our co weak ening. T he common undesirable feature of those rejected rules is that, when moving from a conclusion to a premise, they int ro duce some new compo ne nts, as oppose d to the rules deemed in [3] analytic (and all rules of CL8 would also qualify as analytic by similar sta ndards), which merely regr oup some a lr eady exis ting comp onen ts without cr eating new components. What “comp onents” or “ regrouping” sho uld exactly mean here, ho wev er, certainly doe s requir e some additional a nd probably nontrivial e x planations. T o summarize, there appears to be no well-agreed-up on concept of anaiticity in the literature. T o av oid accus ations of taking excessive terminologica l lib erties, here we introduce the new term “ interfac e analyticity ”, whose meaning well migh t be the b est that one can achieve in an attempt to deﬁne a cirquent- calculus counterpart of the more traditional meaning of the word “analyticity”. F ollowing [14], b y th e in terface of a giv en cirquen t C w e mean the set of all of its po rts. In tuitively , this is the visible par t of the resour c e C , such as the collection o f a ll input/output p orts on the back a nd front panels of one’s p ersonal co mputer. This collection indeed present s the ac tive “ in terface” of the resour ce, with the rest of it — the gates and internal wiring, that is — b eing ﬁxed, hidden a nd una v ailable in the pro cess o f reso urce mana g emen t, which, as we remember, means setting up allo cations b et ween p orts (and by no means b etw een gates). Imagine a circuit optimization problem. Its t ypical goal would b e generating a b etter cir cuit that, how ever, co mputes the sa me Bo olean function — and henc e has the same c o llection of inputs (sa me interface) — as the original one. Ther e a re certa in quite similar intuitiv e reasons for w anting rules of inference to preserve — more pre cisely , not to expand — the interface of the conclus ion when moving to a pr e mise. Having noted this, w e say that a rule of infer ence is interfac e-analytic , or i-analytic for shor t, iﬀ , in any application of the r ule, the in terface of the premise is a subset of tha t of the conclusion (with the lab els of all por ts preserved). And a system is i-a nalytic iﬀ a ll of its r ules are so. Note that CL8 is i-analytic. On the other hand, the rules of cocoupling (cut) and cow eakening o f CL8S a re not i-analytic. Nor w ould b e the rules of substitution ([7]) or extens io n ([6]) if they were present in whatever form in our system. The same can b e said ab out the r ule o f contraction, traditiona lly co nsidered analytic. As a matter of fact, one could question the compliance of contraction with our normal, no matter ho w v ague, in tuition of analy ticity . That is b ecause, when moving from conclusion to premise, cont rac tion do es intro duce some new material, ev en if only in the form of new c o pies of old (sub)form ulas. Y et, this non-analytic b eha vior of contraction is not noticeable in sequen t calculus, b ecause, when used “r easonably”, contraction, while certa inly introducing new material fro m the per s pective of the whole pr o of tr e e , do es no t really do so from the p ersp ective of an y particular br anch of that tree. Her e by “ using con traction reaso nably” we mean applying it (in the b ottom- up view of pr oofs) only b efore using ∧ -intro duction, to just make s ure that each br a nc h of the pro of tree gets its own c o pies of side formulas. But in cir quen t calculus or deep inference sys tems in g eneral, where all branches a re com bined within one cirquent o r f ormula, contraction loses its apparent a nalytic innocence. In any case, unlik e the formula-based deep inference approa c hes suc h as the calculus o f structures, fortunately there is no need for contraction in cirquent calculus. If this rule (in whatever precise form) was adopted by CL8 , it would certainly stop b eing i-analytic. 29 6.2 V ersions with the lo calit y prop ert y Certain easy mo diﬁcations of CL8 or CL8S yie ld v ersio ns that are lo c al , meaning that each inference r ule only aﬀects a bounded p ortion of the cirquent. More precisely , a lo cal rule mo diﬁes (deletes, creates , or changes the label in the case of nodes) only a b o unded num b er of nodes and arc s when mo ving fr om premise to co nclusion or vice v ersa . Lo cality is a desir a ble prop ert y in computer implementations. The only r eason why in this pa per we hav e not chosen lo cal ax iomatizations has b een str iv ing to minimize bur eaucracy . T o see what we mean by “easy mo diﬁcations”, let us just consider weakening a nd pulldo wn a s t wo examples. W eakening is not lo cal b ecause the num ber of the arcs of the conclusion that it can delete is not b ounded. But nothing can b e easier than to “ﬁx ” this pro blem. Speciﬁca lly , w e could adopt a new — loc a l — v ersion of w eakening tha t deletes exactly one arc. That is, the ∆ parameter of w eakening now w ould be r equired to be a singleton. Then, a n applica tion of the old weakening rule that de le tes n a r cs can b e simulated with n applications of the new w eakening rule. F urthermor e, as p ointed o ut in a foo tnote when proving Theorem 5.4, weak ening can b e further restricted by re q uiring the deleted ar c to b e p oin ting at a p ort rather than any no de. This would eliminate the p ossibility that deleting an arc may result in an unbounded c hain of further de le tions of orphaned no des. Similarly , pulldown is not local as it is allowed to mov e around an un b ounded num ber of arcs . W e co uld start requiring that only a single arc be mo ved, that is, req uir ing the Π parameter t o b e a singleton. Just as in the case of w eakening, an application o f the old rule of pulldo wn can then alw ays be simulated by several applications o f the new, lo cal version of it. 6.3 W eak ening the w eak ening rule The r esource philosophy asso ciated with CL8 and CL8S is that o ne cannot use mo r e resource s tha n av ailable. A more radical p osition is that one also has to use all av a ilable reso urces (nothing should b e “wasted”). Under this ex tr eme philosophy familiar fro m linear log ic , the weak ening rule and its dual cow eakening b ecome wr o ng. Removing these rules co uld as well b e necessar y when constructing sys tems for relev ance log ic . How ev er, mech anica lly deleting weakening (and its dual, if pr esen t) from a given s ystem may r esult in throwing o ut the ba b y with the bath w ater. So, rather than disca rding the rule altogether a s done in linear lo g ic, o ne would apparently w ant to simply r eplace weak ening by certain weaker v ersions o f it — versions that, on one hand, are cons isten t with the ab o ve radical resource philo s oph y and, on the other hand, allo w us to retain all inno c e n t principles . Reasonable candidates for suc h a replacement for w eakening and cow eakening are the following rules : Merging Γ ∆ Θ Ω   ❅ ❅ t Θ ∆ Γ   ❅ ❅ t t Ω Comerging Γ ∆ ❞ Θ Ω Γ ∆   ❅ ❅ ❞ ❞   ❅ ❅ Θ Ω b c a b c a Let us lo ok at Blass’s [2] principl e  ( ¬ P ∨ ¬ Q ) ∧ ( ¬ R ∨ ¬ S )  ∨  ( P ∨ R ) ∧ ( Q ∨ S )  . Resources are p erfectly ba lanced in this formula, and ther e are hardly an y g oo d reas ons for rejecting it even from the most radical resour ce-philosophical point of v iew. It is therefore embarrassing that Bla ss’s principle is not prov able in linear logic and not e ven in a ﬃne lo gic: as shown in [14], every pr o of o f it in 30 ordinary sequent calculus w ould require b oth cont ra c tion and w eakening. This for m ula cannot be prov en in CL8 without weak ening, either. Its prov abilit y ca n b e how ev er retained with the fully r esource-fair rule o f merging instead of weakening, as shown below: ❞ deepening (6 times) ❞ ❳ ❳ ❳ ❳ ❳ ✘ ✘ ✘ ✘ ✘ ❞ ❞ ❞ ❞ ❞ ❅ ❞  ❅ coupling (4 times) ¬ P ¬ Q ❍ ❍ ✟ ✟ ¬ R ¬ S ❍ ❍ ✟ ✟ ❞ ❳ ❳ ❳ ❳ ❳ ✘ ✘ ✘ ✘ ✘ P Q t t R S t t ❞ ❅ ❞  ❅ lengthening (3 times) ¬ P ¬ Q ❍ ❍ ✟ ✟ t ¬ R ¬ S ❍ ❍ ✟ ✟ t ❞ ❳ ❳ ❳ ❳ ❳ ✘ ✘ ✘ ✘ ✘ P Q t t R S t t ❞ ❅ ❞  ❅ t pulldown (4 times) ¬ P ¬ Q ❍ ❍ ✟ ✟ t ¬ R ¬ S ❍ ❍ ✟ ✟ t ❞ ❳ ❳ ❳ ❳ ❳ ✘ ✘ ✘ ✘ ✘ P Q t t R S t t ❞ ❅ ❞  ❅ t pulldown (twice) ¬ P ¬ Q ❅ t ¬ R ¬ S ❅ t ❞ ❳ ❳ ❳ ✘ ✘ ✘ P Q t t R S t t ❞ ❅ ❞  ❅ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✟ ✟ ✟ t merging (twice) ¬ P ¬ Q ❅ t ¬ R ¬ S ❅ t ❞ ❳ ❳ ❳ ✘ ✘ ✘ P R ❅ t Q S ❅ t ❞ ❤ ❤ ❤ ❤ ❤ ❤ ❞ ✭ ✭ ✭ ✭ ✭ ✭ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✟ ✟ ✟ t globalization ¬ P ¬ Q ❅ t ¬ R ¬ S ❅ t ❞ ❳ ❳ ❳ ✘ ✘ ✘ P R ❅ t Q S ❅ t ❞ ❳ ❳ ❳ ✘ ✘ ✘ ❤ ❤ ❤ ❤ ❤ ❤ ✭ ✭ ✭ ✭ ✭ ✭ t 31 6.4 Cirquen ts with man y ro ots Some future treatments may ca ll for cons idering cirquents tha t allow multiple ro ots (pare n tless no des). F or example, the methods of cirquent calculus could b e p otent ially used in verifying circuit equiv alence, optimizing cir cuits, or other re lated pro blems arising in digital design. And it s ho uld be remembered that circuits in actual computer hardware typically have not only multiple inputs (p orts), but als o m ultiple outputs (ro ots). Of course, there can also b e ma n y other r easons, including theoretical ones, for studying these mo r e general sor ts of cirquents. 6.5 Cirquen ts with additional sorts of gates and arcs As we alrea dy know, the in tro duction of cirquent calculus w as orig inally motiv ated by the needs of com- putabilit y logic. Cirquent calculus in the form present ed in this paper captures only the modes t ( ¬ , ∧ , ∨ )- fragment of computability logic though. Ex tending cirquent calculus so as to accommo date incr emen tally more e x pressive frag ments of c omputabilit y log ic would r equire consider ing cir quen ts with gates for choice connectives, and gates and/ o r arcs for recurr e nce connectives. Accounting for the more recently ([16]) in- tro duced non-c o mm utative sequential op erators o f computabilit y logic w ould a ls o r equire linearly ordering the outgoing edges of the corresp onding ga tes . Ther e is a tremendous amoun t of interesting and challenging work to do in this direction. 7 CL8 v ersus sequen t calculus and shallo w cirquen t calculus sys- tems This section is devoted to ce r tain asp ects of the relation b etw een CL8 and Gentzen-st yle se q uen t calculus systems, as well as the sha llo w cirque nt calculus sy stems CL5 and CCC pres en ted in [1 4]. Spec iﬁc a lly , w e ﬁrst want to compare CL8 with the classical cut-free sequent calculus system G deﬁned below. One diﬀer e nc e that we already know is the greater expressiveness of CL8 . B ut e ven if we are only concerned with ob jects that the languages of b oth s ystems can express — Bo olean functions pr esen ted in the for m of clas s ical formulas or (the corresp onding) circuits, that is — CL8 still has distinctiv e adv antages, related to eﬃciency . In Section 8 we will see the existence o f polynomia l size CL8 -pro ofs for the pige onhol e principle , the class of tautologie s known to have o nly exponential s ize pr oofs in G or similar systems. T o appreciate this po in t, it would b e necessar y to also sho w that, on the other hand, no class of tautologies admits in G considerably shorter proo fs than in CL8 . In other words, w e need to see that CL8 can p- simulate G , meaning that there is a p olynomial function p such that, for any for m ula F of classica l logic, whenever F has a G -pro of of s iz e n , it — more precisely , the cirquent F — also has a CL8 -pro of of size ≤ p ( n ). Then and only then w e can oﬃcially decla re that CL8 oﬀers an exp onent ial sp eedup (in pro of eﬃciency) over G . System G deals with sequents understo o d as nonempty ﬁnite sets of formulas. This version is known to b e equiv alen t — in the strong sense of mutual p-simulation — to the probably mo r e common v ersio ns of cut-free sequent ca lculi for clas s ical logic where sequents are sequences or m ultisets (r ather than sets) o f formulas. An adv antage of G ov er such systems is the a bsence of structura l rules. Below Γ stands for an y set of formulas, P for a n y atom, and E , F for a n y for m ulas. F ollowing the standard pr actice, an expres sion suc h as “Γ , E , F ” should b e unders too d as Γ ∪ { E , F } . The axioms of G are any sequents of the fo r m Γ , ¬ P, P , in addition to which the sy stem (only) has the following tw o rules of inference: ∨ -in tro duction Γ , E , F Γ , E ∨ F ∧ -in tro duction Γ , E Γ , F Γ , E ∧ F 32 The deﬁnition of pr ov abilit y of a sequen t Γ in G is standard: this means existence of a tre e of sequents — called a pro of tree for Γ — with Γ a t its ro ot, in which every lea f of the tree is an axiom and every non-leaf node follows from its child or c hildren b y one of the rules o f G . A formula F is consider ed prov able in G iﬀ F , viewed as a one-element sequen t, is pr o v able. Since we will b e dealing with complexit y issues, w e need to agree on what the size of a formula, cir quen t, sequent, der iv ation or pro of means . W e assume some reas onable enco ding (computer repres en tation) of these ob jects to be ﬁxed, and agree that the size of any suc h ob ject is the amount of bits taken b y its co de when written in computer memory . It is under stoo d that all “rea sonable” enco dings are po lynomially equiv alent (the diﬀerences in their eﬃciencies ar e a t most po ly nomial, that is) and, since in this pap er w e o nly care ab out po lynomialit y versus exp onen tiality , it is not imp ortant which particula r “reasonable” enco ding we hav e in mind. Theorem 7.1 CL8 p-simulates G . Pro of. Consider an arbitra ry G -pro of tr e e T for an arbitra ry formula F . Below we des cribe a pro cedure for conv erting T into a C L8 - proo f T ∗ of F . It will b e clear from our description that the size of T ∗ is po lynomial in the size o f T . By abuse of terminology , in the present pro of we w ill b e often identifying a node of T with the corre- sp onding sequent , even though it should b e remembered that the same sequent may b e “ sitting” at more than o ne no de. W e cons tr uct the CL8 -pro of T ∗ of F bo ttom-up. The last three cir quen ts of T ∗ are ∧{∨{ F }} , ∨{ F } and F . F follows from its predece ssor ∨{ F } by shortening , and so doe s ∨{ F } from its predecessor ∧{ ∨{ F } } . Thu s, the topmost cir quen t of the bo ttom fragment of T ∗ that we hav e constructed so far is ∧{∨{ F }} . Let us call this cirquent A 1 . W e asso ciate the ro ot of T with the ∨{ F } sub cirquen t of ∧{∨{ F }} . A 1 is only the ﬁrst cir quen t of a certain ser ies A 1 , A 2 , A 3 , . . . of cirq ue nts that w e ar e going to co ns truct one after one a nd include in our evolving (in the upw ard direction) T ∗ . Any such A i will lo ok like ∧{∨{ E 1 1 , . . . , E 1 k 1 } , . . . , ∨{ E n 1 , . . . , E n k n }} , i.e., ( E 1 1 ∨ . . . ∨ E 1 k 1 ) ∧ . . . ∧ ( E n 1 ∨ . . . ∨ E n k n ) , where with eac h conjunct ( E j 1 ∨ . . . ∨ E j k j ) , as in A 1 , is asso ciated a no de of T such t hat the sequent at that no de is E j 1 , . . . , E j k j . W e describ e the wa y of g enerating the A i s and including them in T ∗ inductively . A 1 has a lr eady been generated. Supp ose now we ha ve alr eady cons tr ucted the b ottom p ortion of T ∗ such that A i is the top cirquent. F ur ther suppose that ther e is a conjunct o f A i such that the asso ciated node of T is not an axiom of G (i.e., not a leaf of T ). W e may assume here that the last co njunct E n 1 ∨ . . . ∨ E n k n of A i is suc h. How w e proceed from A i up ward in our co nstruction of T ∗ depe nds on whether the asso ciated sequent E n 1 , . . . , E n k n is o btained b y ∨ -intro duction or ∧ -introductio n in T . Suppo se E n 1 , . . . , E n k n is o btained b y ∨ -intro duction, meaning tha t it loo ks lik e E n 1 , . . . , E n k n − 1 , G ∨ H (4) and the premise is E n 1 , . . . , E n k n − 1 , G, H. (5) W e then choose A i +1 to b e the cirquent ( E 1 1 ∨ . . . ∨ E 1 k 1 ) ∧ . . . ∧ ( E n − 1 1 ∨ . . . ∨ E n − 1 k n − 1 ) ∧ ( E n 1 ∨ . . . ∨ E n k n − 1 ∨ G ∨ H ) . 33 The no des o f T asso ciated with the conjuncts of A i +1 remain the same as in A i , with the exce ptio n of the last ( n th) conjunct, with which w e now asso ciate the premise (5) of (4). Note that A i , which is ( E 1 1 ∨ . . . ∨ E 1 k 1 ) ∧ . . . ∧ ( E n − 1 1 ∨ . . . ∨ E n − 1 k n − 1 ) ∧ ( E n 1 ∨ . . . ∨ E n k n − 1 ∨ ( G ∨ H )) , follows from A i +1 by deepening . So, we include A i +1 in fro n t (on to p) of A i in our b ottom-up construction of T ∗ , and justify the transitio n from A i +1 to A i by deepening . Suppo se now E n 1 , . . . , E n k n is obtained by ∧ -in tro duction, meaning that it lo oks like E n 1 , . . . , E n k n − 1 , G ∧ H (6) and the t wo premises of it in T ar e E n 1 , . . . , E n k n − 1 , G (7) and E n 1 , . . . , E n k n − 1 , H. (8) In this case we choos e A i +1 to b e the cirquent ( E 1 1 ∨ . . . ∨ E 1 k 1 ) ∧ . . . ∧ ( E n − 1 1 ∨ . . . ∨ E n − 1 k n − 1 ) ∧ ( E n 1 ∨ . . . ∨ E n k n − 1 ∨ G ) ∧ ( E n 1 ∨ . . . ∨ E n k n − 1 ∨ H ) . The no des of T a ssoc ia ted with the ﬁrst n − 1 conjuncts of A i +1 remain the same as in A i . And with the last t wo conjuncts of A i +1 we asso ciate the premis e s (7) and (8) o f (6), re s pectively . It is not ha rd to see that A i , which is ( E 1 1 ∨ . . . ∨ E 1 k 1 ) ∧ . . . ∧ ( E n − 1 1 ∨ . . . ∨ E n − 1 k n − 1 ) ∧ ( E n 1 ∨ . . . ∨ E n k n − 1 ∨ ( G ∧ H )) , follows fro m A i +1 by trade in combination with some stra igh tforward restructuring . So, w e a dd the corr e- sp onding (b ounded n um b er o f ) cirquents together with the appropriate justiﬁcations in fron t (on top) of A i , with the new top cirquent of our bo ttom-up construction of T ∗ now being A i +1 . W e co ntin ue extending T ∗ up ward by adding new A i s in the ab ov e wa y until we hit the point where the topmost A m is suc h that all no des of T ass ociated with its conjuncts ar e leav es. It is not hard to see that this m w ould b e nothing but the total n um b er of no des of T . Thus, the now to pmo st cirquent of the evolving T ∗ is A m = ( E 1 1 ∨ . . . ∨ E 1 k 1 ) ∧ . . . ∧ ( E n 1 ∨ . . . ∨ E n k n ) , where each E j 1 , . . . .E j k j is an axiom o f G and hence contains at least one pair P, ¬ P of o pp osite liter a ls. W e choose one suc h pa ir o f literals in ea c h conjunct of A m , and delete the arc s to all other nodes from the cor respo nding disjunctiv e ga te using (b ottom up) a series of weak enings. This results in a cir quen t B = ( P 1 ∨ ¬ P 1 ) ∧ . . . ∧ ( P n ∨ ¬ P n ) , where eac h P j is a n atom. Not all P i and P j with i 6 = j may b e diﬀ erent a toms here though. If this is indeed the cas e, w e further apply (bottom-up) a ser ies of lo calizations to B and get a cir q uen t C = ( Q 1 ∨ ¬ Q 1 ) ∧ . . . ∧ ( Q e ∨ ¬ Q e ) ( e < n ), where each Q j is an a tom (one of the o ld atoms P 1 , . . . , P e ) diﬀerent from a n y Q i with i 6 = j . Next we apply (b ottom-up) coupling to C e times, which res ults in a cirquent where all non-ro ot no des are childless co njunctiv e g ates. Suc h gates can b e eliminated by a pplying (b ottom-up) a s eries of deep enings, and we end up with the axiom cir quen t ◦ . ✷ In a similar wa y o ne could show that CL8 p-simulates the cut-free versions of the m ultiplicative linear and aﬃne logics. Ho wev er, as already mentioned, those are not conserv ative fra gmen ts of CL8 . F or example, 34 the CL8 -prov a ble Bla s s’s pr inciple (see Section 6.3), or the cirquent of Figure 4, are bo th expressible in the language of linear log ic, but neither linear lo g ic nor the str onger aﬃne logic prov e them. F urthermore, our pro of of Theorem 7.1 can b e rather easily mo diﬁed into pro ofs of the facts that CL8 also p-simulates the shallow cirquen t calculus systems CL5 and CCC of [14]. A t the same time, the known pro ofs o f the nonexis tence of p olynomial size pr oofs of the pigeonhole principle in G -style systems can b e mo diﬁed so a s to show the nonexis tence of such pro ofs in CCC . And a so mewha t similar argument, based on a certain r esource-conscio us version o f the pigeonhole principle (no literal has more tha n one occurre nce), can be used to also sho w a n exp onent ial sp eedup o ver CL5 oﬀered by CL8 . Th us, CL8 is certa inly an improv ement over CCC and CL5 from the p e rspective of eﬃciency . But there is a m uch mo re signiﬁcan t diﬀerence betw een our present approach and the approa c h taken in [14]. While [14] is the oﬃcial birth place of the idea s of cirquent ca lculus a nd abstract reso urce semantics, the par ticular systems ela bora ted in detail in [14] stopp ed only half wa y on the road of fully and consisten tly materializing those ideas. This w as rela ted to the limited s yn tax adopted there, which was a somewhat unnatural mixture of circuit-style and tr ee-st yle str uctures. Sp eciﬁcally , a s men tioned earlier, the depths of cirquents w ere limited to tw o, with the ro ot of each suc h cirquent required to b e a conjunctive gate and its children required to b e disjunct ive g ates. This w as a signiﬁcant limitation of expres siv eness and, to partially comp ensate for it, the “input” no des (g randc hildren of the ro ot) were allow ed to be any for m ulas rather than only litera ls as in our present treatment . And so, pos sible sharing of children b et ween diﬀeren t pa ren ts was taking place only at one single (ro ot’s c hildren) level of cir quen ts. Even thoug h [14] prov ed (Theorem 20) that sha llo w cirquents, unlike formulas, were suﬃcient to repr esen t all abstr act r esour c es (which, roughly , are the sa me to abstract resour ce sema n tics as Bo olean functions to the sema n tics of clas s ical logic), such representations were generally very ineﬃcient, essentially requir ing every a bstract resource to be expressed in co njunctiv e no rmal form. 8 F rom cla s sical logic we k no w that conjunctive no rmal forms, while complete as means of expressing all B oolea n functions, can gener ally b e exp onen tially longer than other, more relaxed representations. Similar reaso ns apply to a bstract reso urce semantics as well, meaning that the ob jects of our study (abs tract re s ources) are exp onen tially harder to express — let alone prov e — in CL5 or CCC than in CL8 . But the most decisiv e improv ement of the present approach ov er the approach of [14] is turning classica l logic into just a s p ecial fra gmen t o f the mo r e general logic of r e source, thus eliminating conﬂicts b etw een the classical and res ource-conscious view s , with b oth the semantics a nd the syntax of CL8 b eing single unifying and reconciling frameworks for the tw o diverging philosophical traditions in logic. This was impossible to achiev e under the shallow cir q uen t calculus a pproach of [14], fo r the reaso n of the limitations of the expressive power of shallow cirquen ts. And this is ex actly why [14] had to c o nstruct t wo diﬀeren t logics : one — CCC — for classical seman tics and the other — CL5 — for abs tract res ource semantics, and corres pondingly prov e tw o sepa r ate completeness theorems. The t w o systems had the same language but diﬀeren t seman tics, and disagreed o n many principles expressible in that co mmon languag e. Speciﬁcally , CCC w as proper ly stronger than CL5 , obtained f ro m the latter by adding (a cirquen t ca lculus version of ) con tractio n to it, the rule that we criticized a while ag o as be ing not “tr uly analytic”. The main purp ose of the pr esen t pap er is to provide a starting p oin t a nd an initial impulse for wha t ( as the author wishes to hope) ma y become a new line of resea rc h in proo f theor y and r esource logics — namely , a pro of theory and a resource semantics based on cir cuit-st yle (rather than fo rm ula-style) construc ts . [1 4], with its limited and not fully consisten t (in that it s till contin ued to rely on formulas) mater ia lization of this idea, had signiﬁcantly lo wer c hances to b e succes sful in serving this purp ose. 8 The pigeonhole principle The (pr opositio nal) pigeonhole principle is a family of classical tautologies that is k no wn to hav e no p oly- nomial size pro ofs in resolution systems or ana lytic s e q uen t calculus systems (Haken [11]). And existence of polynomia l size pro ofs for this family in the c ut- and substitution-free calculus of structures is a n op en problem, conjectur e d to ha ve a negativ e solution (s e e [3]). While p olynomial size pro ofs for it in F rege- and Gent zen-style systems have b een found (Coo k and Rec hko w [6], Buss [5]), those pro ofs re ly on cut and, in 8 It should be noted that v alid cirquen ts in conjunctiv e normal form , where c hildren may be shared b etw een diﬀerent disjunctive no des, are not as trivial to pr o v e as v al i d conjunctive-normal-form form ulas in classi cal l ogic. 35 the case of [6], also on an extension rule. Mo re recent ly (Finger [7]), cut-free p olynomial size pro ofs for the pigeo nhole principle w ere a lso constructed, which, how ever, rely on a substitution rule. 9 All known po lynomial siz e pro ofs of the pige o nhole principle thus use extension, cut, or substitution — the “ hig hly non-analytic” rules. This section presents polyno mial size CL8 -pro ofs for the pigeonhole pr inciple. They stand out as the ﬁr s t known “r easonably analy tic” — a t least in the precise sense o f i-a nalyticit y — tractable pro ofs of this class o f tautologies. Our construction partly explo its certain technical ideas from [6]. Throughout this section, n is an ar bitrary but ﬁxed p ositiv e in teger. When w e say “po lynomial” or “exp onen tial”, it sho uld b e understo od as p olynomial or exponential in n . As before, out of laziness, we will only b e concer ned with p olynomiality versus exp o nen tiality , le aving a more accurate as ymptotic analysis as an exercise for an interested reader. Suc h an analys is, of cours e , would req uir e a more precis e sp eciﬁcation of the meaning o f the concept of pro of size tha n the one we gav e in Section 7. The (h yp er)formulas and cir quen ts that we consider are built from ( n + 1) × n atoms denoted P i,j , one p er each i ∈ { 0 , . . . , n } (the set o f pigeons) a nd j ∈ { 1 , . . . , n } (the s e t of pigeonho le s). The meaning asso ciated with P i,j is “ pigeon i is sitting in hole j ”. The n -pige o nhole principle is expres s ed b y the hyperformula P H P n = ∨ {¬ P i, 1 ∧ . . . ∧ ¬ P i,n | 0 ≤ i ≤ n } ∨ ∨ { P i,j ∧ P e,j | 0 ≤ i < e ≤ n, 1 ≤ j ≤ n } (there is no need to ov erline the negative occurr ences of atoms becaus e there is only one such oc currence for each atom). Its left disjunct asser ts that ther e is a pigeon i that is not sitting in any hole. And the r ig h t disjunct asserts that th ere is a hole j in which so me tw o distinct pigeons i a nd e are sitting. This is the s ame as to say that if every pigeon is sitting in some hole, then there is a hole with (a t least) tw o pigeo ns. F or each i, j with 0 ≤ i ≤ n a nd 1 ≤ j ≤ n , we deﬁne the formulas X n i,j = P i,j ; Y n i,j = ¬ P i,j . Next, for each k , i , j with 1 < k ≤ n , 0 ≤ i ≤ k − 1 and 1 ≤ j ≤ k − 1 , we deﬁne the formulas X k − 1 i,j = ( X k i,j ∨ X k i,k ) ∧ ( X k i,j ∨ X k k,j ); Y k − 1 i,j = Y k i,j ∧ ( Y k i,k ∨ Y k k,j ) ∧ ( X k i,k ∨ Y k i,k ) ∧ ( X k k,k ∨ Y k k,k ) . Finally , for each k with 1 ≤ k ≤ n , we deﬁne the fo rm ulas B k = ∧ { X k i,j ∨ Y k i,j | 0 ≤ i ≤ k , 1 ≤ j ≤ k } ; C k = ∨ { Y k i, 1 ∧ . . . ∧ Y k i,k | 0 ≤ i ≤ k } ∨ ∨ { X k i,j ∧ X k e,j | 0 ≤ i < e ≤ k , 1 ≤ j ≤ k } . The sizes of the formulas (cirq ue nts) B k and C k are obviously exp onential. How ever, due to sharing, the sizes of their “full compress io ns” B k and C k can b e seen to b e o nly po lynomial. In wha t follows, we pr o ve a num b er o f statements claiming existence o f cer tain po ly nomial size deriv ations. In o ur pro ofs of those statements we usually restrict ourselves to de s cribing the deriv ations, without an y further explicit analys is o f their sizes. Such descriptions alone will be suﬃcient for an experience d re a der to immediately s ee that the deriv ations are indeed of p olynomial sizes. F rom now on, a “der iv ation” o r “pro of ” means a deriv a tio n or proo f in CL8 . When justifying steps in deriv ations, w e often omit ex plicit references to res tructuring, and indicate only one of the three main rule s , even though that rule needs to b e combined with so me res tructuring steps to yield the co nclusion. Mostly such a “rule” is going to be pulldown and, to indicate that pulldo wn is com bined with some straightf orward restructuring, we will write “ pul ldo wn* ” instead o f just “ pulldo wn”. Similarly for “ weak ening* ” and “ coupling* ” . Lemma 8.1 C n = P H P n . 9 Finger’s approac h also requires to switc h back to the m ore traditional F rege- and Hi lbert-style understanding of pro of s (only applied to sequen ts rather than for m ulas), where pro ofs are seen not as t rees but as sequences (which, of course, can als o be viewed as DA Gs) of form ulas, with the p ossibility f or each formu la to serv e as a pr emise for any num b er of later formulas in the sequ ence. The relate d pa p er [8] further illustrates the eﬃciency adv antag es of proof sequences ov er proof t rees. 36 Pro of. Immediate, as Y n i,j = ¬ P i,j and X n i,j = P i,j . ✷ Lemma 8.2 B n has a p olynomial size pr o of. Pro of. B n , which is the same as B n , has O ( n 2 ) conjuncts, ea c h c onjunct being X n i,j ∨ Y n i,j for some 0 ≤ i ≤ n, 1 ≤ j ≤ n . The latter is nothing but P i,j ∨ ¬ P i,j , which can be introduced by coupling* . ✷ Lemma 8.3 Ther e is a p olynomial size deriv ation of C 1 fr om B 1 . Pro of. B 1 is ( X 1 0 , 1 ∨ Y 1 0 , 1 ) ∧ ( X 1 1 , 1 ∨ Y 1 1 , 1 ), and C 1 is — more prec is ely , can be restructured into — Y 1 0 , 1 ∨ Y 1 1 , 1 ∨ ( X 1 0 , 1 ∧ X 1 1 , 1 ). The latter follows from the former by pulldo wn* applied twice. ✷ Lemma 8.4 F or e ach k with 1 < k ≤ n , t her e is a p oly nomial size derivation of B k − 1 fr om B k . Pro of. B k is ∧ { X k i,j ∨ Y k i,j | 0 ≤ i ≤ k , 1 ≤ j ≤ k } , which ca n also b e written as ∧ { X k i,j ∨ Y k i,j , X k k,j ∨ Y k k,j , X k i,k ∨ Y k i,k , X k k,k ∨ Y k k,k | 0 ≤ i < k , 1 ≤ j < k } . (9) W e introduce the abbr eviation D k i = ( X k i,k ∨ Y k i,k ) ∧ ( X k k,k ∨ Y k k,k ) and restr ucture (9) into the following cirque nt: ∧ { ( X k i,j ∨ Y k i,j ) ∧  ( X k i,k ∨ Y k i,k ) ∧ ( X k k,j ∨ Y k k,j )  ∧ D k i | 0 ≤ i < k , 1 ≤ j < k } . (10) Next, for each of the O ( k 2 ) conjuncts of (10 ), in turn, we pe r form the following transfor mation, leaving the res t of the cirquent unc hanged 10 while doing so.  X k i,j ∨ Y k i,j  ∧  ( X k i,k ∨ Y k i,k ) ∧ ( X k k,j ∨ Y k k,j )  ∧ D k i pulldow n* (twice)  X k i,j ∨ Y k i,j  ∧  ( X k i,k ∧ X k k,j ) ∨ Y k i,k ∨ Y k k,j  ∧ D k i w eakening* (t wice)   ( X k i,j ∨ X k i,k ) ∧ ( X k i,j ∨ X k k,j )  ∨ Y k i,j  ∧  ( X k i,k ∧ X k k,j ) ∨ Y k i,k ∨ Y k k,j  ∧ D k i w eakening* (t wice)   ( X k i,j ∨ X k i,k ) ∧ ( X k i,j ∨ X k k,j )  ∨ Y k i,j  ∧   ( X k i,j ∨ X k i,k ) ∧ ( X k i,j ∨ X k k,j )  ∨ Y k i,k ∨ Y k k,j  ∧ D k i globalizatio n (3 times)   ( X k i,j ∨ X k i,k ) ∧ ( X k i,j ∨ X k k,j )  ∨ Y k i,j  ∧   ( X k i,j ∨ X k i,k ) ∧ ( X k i,j ∨ X k k,j )  ∨ Y k i,k ∨ Y k k,j  ∧ D k i the same as ( X k − 1 i,j ∨ Y k i,j ) ∧ ( X k − 1 i,j ∨ Y k i,k ∨ Y k k,j ) ∧ D k i pulldow n* (twice) X k − 1 i,j ∨  Y k i,j ∧ ( Y k i,k ∨ Y k k,j ) ∧ D k i  the same as X k − 1 i,j ∨  Y k i,j ∧ ( Y k i,k ∨ Y k k,j ) ∧  ( X k i,k ∨ Y k i,k ) ∧ ( X k k,k ∨ Y k k,k )   10 I.e., copying and pasting th ose unaﬀec ted parts from on e cirquen t to t he next one in the deriv ation. 37 restructuring X k − 1 i,j ∨  Y k i,j ∧ ( Y k i,k ∨ Y k k,j ) ∧ ( X k i,k ∨ Y k i,k ) ∧ ( X k k,k ∨ Y k k,k )  the same as X k − 1 i,j ∨ Y k − 1 i,j After re p eating the ab ove transformation for a ll ( i, j ), we end up with the target cirquent B k − 1 = ∧ { X k − 1 i,j ∨ Y k − 1 i,j | 0 ≤ i < k , 1 ≤ j < k } . ✷ Lemma 8.5 F or e ach k with 1 < k ≤ n , t her e is a p oly nomial size derivation of C k fr om C k − 1 . Pro of. C k − 1 can b e written as ∨ {∧ { Y k − 1 i,j | 1 ≤ j < k } | 0 ≤ i < k } ∨ ∨ { X k − 1 i,j ∧ X k − 1 e,j | 0 ≤ i < e < k , 1 ≤ j < k } . (11) This is how w e derive C k from (11). At the b eginning, for each of the O ( k 3 ) subcir quen ts X k − 1 i,j ∧ X k − 1 e,j of (11), in turn, we do the following transformatio n, lea ving the r est of the cirquent unc hanged: X k − 1 i,j ∧ X k − 1 e,j the same as  ( X k i,j ∨ X k i,k ) ∧ ( X k i,j ∨ X k k,j )  ∧  ( X k e,j ∨ X k e,k ) ∧ ( X k k,j ∨ X k e,j )  trade  X k i,j ∨ ( X k i,k ∧ X k k,j )  ∧  ( X k e,j ∨ X k e,k ) ∧ ( X k k,j ∨ X k e,j )  restructuring   X k i,j ∨ ( X k i,k ∧ X k k,j )  ∧  X k e,j ∨ X k e,k   ∧  X k k,j ∨ X k e,j  pulldow n*  X k i,j ∨  ( X k i,k ∧ X k k,j ) ∧ ( X k e,j ∨ X k e,k )   ∧  X k k,j ∨ X k e,j  pulldow n*  X k i,j ∧ ( X k k,j ∨ X k e,j )  ∨  ( X k i,k ∧ X k k,j ) ∧ ( X k e,j ∨ X k e,k )  restructuring  X k i,j ∧  X k k,j ∨ X k e,j   ∨  X k i,k ∧  ( X k e,j ∨ X k e,k ) ∧ X k k,j   pulldow n*  X k i,j ∧  X k k,j ∨ X k e,j   ∨  X k i,k ∧  X k e,k ∨ ( X k e,j ∧ X k k,j )   pulldow n*  X k e,j ∧ X k k,j  ∨  X k i,j ∧ ( X k k,j ∨ X k e,j )  ∨  X k i,k ∧ X k e,k  pulldow n*  X k e,j ∧ X k k,j  ∨  X k i,j ∧  X k e,j ∨ ( X k i,j ∧ X k k,j )   ∨  X k i,k ∧ X k e,k  pulldow n* ( X k i,j ∧ X k k,j ) ∨ ( X k e,j ∧ X k k,j ) ∨ ( X k i,j ∧ X k e,j ) ∨ ( X k i,k ∧ X k e,k ) 38 After repe a ting the above trans fo rmation for all subcirquents X k − 1 i,j ∧ X k − 1 e,j of (11), the latter turns into the following cirquent: ∨{ ∧{ Y k − 1 i,j | 1 ≤ j < k } | 0 ≤ i < k } ∨ ∨{ ( X k i,j ∧ X k k,j ) ∨ ( X k e,j ∧ X k k,j ) ∨ ( X k i,j ∧ X k e,j ) ∨ ( X k i,k ∧ X k e,k ) | 0 ≤ i < e < k , 1 ≤ j < k } . (12) Next, for each of the k sub cirquents ∧{ Y k − 1 i,j | 1 ≤ j < k } of (12), one after one, we p erform the following transformatio n, lea ving the r est of the cirquent unc hanged: ∧{ Y k − 1 i,j | 1 ≤ j < k } the same as ∧{ Y k i,j ∧ ( Y k i,k ∨ Y k k,j ) ∧ ( X k i,k ∨ Y k i,k ) ∧ ( X k k,k ∨ Y k k,k ) | 1 ≤ j < k } restructuring  ∧ { Y k i,j | 1 ≤ j < k }  ∧  X k i,k ∨ Y k i,k  ∧  ∧ { Y k i,k ∨ Y k k,j | 1 ≤ j < k } ∧ ( X k k,k ∨ Y k k,k )  trade  ∧ { Y k i,j | 1 ≤ j < k }  ∧  X k i,k ∨ Y k i,k  ∧   Y k i,k ∨ ∧ { Y k k,j | 1 ≤ j < k }  ∧  X k k,k ∨ Y k k,k   pulldow n*  ∧ { Y k i,j | 1 ≤ j < k }  ∧  X k i,k ∨ Y k i,k  ∧   ∧ { Y k k,j | 1 ≤ j < k } ∧ ( X k k,k ∨ Y k k,k )  ∨ Y k i,k  pulldow n*  ∧ { Y k i,j | 1 ≤ j < k }  ∧  X k i,k ∨ Y k i,k  ∧   ∧ { Y k k,j | 1 ≤ j < k } ∧ Y k k,k  ∨ X k k,k ∨ Y k i,k  restructuring  ∧ { Y k i,j | 1 ≤ j < k }  ∧  X k i,k ∨ Y k i,k  ∧  ∧ { Y k k,j | 1 ≤ j ≤ k } ∨ X k k,k ∨ Y k i,k  restructuring  ∧ { Y k i,j | 1 ≤ j < k }  ∧   X k i,k ∨ Y k i,k  ∧  ∧ { Y k k,j | 1 ≤ j ≤ k } ∨ X k k,k ∨ Y k i,k   pulldow n*  ∧ { Y k i,j | 1 ≤ j < k }}  ∧  Y k i,k ∨  ∧ { Y k k,j | 1 ≤ j ≤ k }  ∨  ( X k i,k ∨ Y k i,k ) ∧ X k k,k   pulldow n*  ∧ { Y k i,j | 1 ≤ j < k }}  ∧  Y k i,k ∨  ∧ { Y k k,j | 1 ≤ j ≤ k }  ∨  X k i,k ∧ X k k,k   pulldow n*  ∧ { Y k i,j | 1 ≤ j < k } ∧ Y k i,k  ∨  ∧ { Y k k,j | 1 ≤ j ≤ k }  ∨  X k i,k ∧ X k k,k  After repe a ting the abov e pro cedure for all subcirquents ∧ { Y k − 1 i,j | 1 ≤ j < k } of (12), the latter turns into the following cirquent: ∨{  ∧ { Y k i,j | 1 ≤ j < k } ∧ Y k i,k  ∨  ∧ { Y k k,j | 1 ≤ j ≤ k }  ∨  X k i,k ∧ X k k,k  | 0 ≤ i < k } ∨ ∨{ ( X k i,j ∧ X k k,j ) ∨ ( X k e,j ∧ X k k,j ) ∨ ( X k i,j ∧ X k e,j ) ∨ ( X k i,k ∧ X k e,k ) | 0 ≤ i < e < k , 1 ≤ j < k } . (13) Finally , we restructure (1 3) in to ∨ { Y k i, 1 ∧ . . . ∧ Y k i,k | 0 ≤ i ≤ k } ∨ ∨ { X k i,j ∧ X k e,j | 0 ≤ i < e ≤ k , 1 ≤ j ≤ k } , which is nothing but the desired C k . ✷ Theorem 8.6 Ther e is a p olynomial size pr o of of P H P n . Pro of. Lemmas 8 .2 , 8.4, 8.3 and 8.5 imply that there is a p olynomial size pro of of C n , which, by Lemma 8.1, is the same a s P H P n . ✷ 39 References [1] A. Avr on. A c onstructive analysis of RM . Journal of Sym b olic Logic 5 2 (1 987), No.4, pp. 939 -951. [2] A. Blass . A game semantics for line ar lo gic . Annals of Pure and Applied Logic 5 6 (1992), pp. 183-2 20. [3] P . Brusco li and A. Guglielmi. On the pr o of c omplexity of de ep infer enc e . A CM T ransactions on Computational Logic (to app e ar). [4] K. Br ¨ unnler. Deep Inference and Symmetry in Cl assical Pro ofs . Logos V erla g , Berlin, 2004. [5] S. Buss. Polynomial size pr o ofs of t he pr op ositional pige onhole principle . 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