Towards applied theories based on computability logic

T o w ards applied theories based on computa bilit y log ic Giorgi Japaridze Abstract Computability lo gic (C L) is a recentl y launched program for redevelo ping logic as a formal theory of computability , as opp osed to the formal theory of truth that logic has more traditionally b een. F orm ulas in it represent computational problems, “truth” means ex istence of an algorithmic solution, and pro ofs encod e such solutions. Within the line of researc h d evoted to finding axiomatizations for ever more expressive fragmen ts of CL, the presen t pap er introdu ces a new deductive system CL12 and prov es its sound ness and completeness with respect t o t h e seman tics of CL. Conserv atively extending classical predicate calculus and offering considerable additional ex pressive and deductiv e p ow er, CL12 presents a reasonable, compu tationally meaningful, constructive alternative to classical logi c as a b asis for applied theories. T o obtain a model example of such theories, this pap er rebuilds the traditional, classical-logic- based P eano arithmetic into a computabilit y- logic-based counterpart. Among the purp oses of the p resent contri bu tion is to p ro vid e a starting p oint for what, as the aut h or wishes to hope, might b ecome a n ew line of research with a potential of interesting findings — an exploration of the presumably qu ite unusual metatheory of CL-b ased arithmetic and other CL-based app lied systems. MSC : primary: 03F50; secondary: 0 3F30; 03D75; 03F50; 68Q10; 68T27; 6 8T30 Keywor ds : Co mputabilit y lo gic; Ga me semantics; Peano a rithmetic; Constructive log ic s 1 In tro du c tion Computability lo gi c (CL) is a se ma ntical platform and pr ogra m for redeveloping lo gic as a f or mal theory o f computability , as oppo sed to a f o r mal theory of truth w hich logic has mo re tr aditionally b een. It sees formulas as in tera c tive computational problems viewed a s games pla yed by a mac hine ( ⊤ ) against the en vironment ( ⊥ ), w her e logical o p erator s stand for op er ations o n games and “ tr uth” is understo o d as existence of an algorithmic winning stra tegy . Numerous paper s [1]-[14] ha ve b een published on this sub ject in recent years, and the presen t reader is a ssumed to b e familia r with the basic philosoph y , motiv ations and tec hniques of CL. Otherwise, he or she is r ecommended to take a lo ok at the first 10 sections of [13] for an in tro duction and survey . So far all technical effor ts within the CL pro ject have b een fo cused on dev elo ping CL as a pure logic, t ypica lly throug h constr ucting axio matizations for v ario us frag ments of it. The co ntribution presented in the first par t of this pap er is in the same style. It int ro duces a new reduction o pe r ation • – ≀ ≀ , similar to the earlier known ◦ – but stro nger in tha t the anteceden t is allow ed to be replicated only a finite n umber of times. With • – ≀ ≀ syntactically treated as a sepa rator of the tw o parts of a se quent, a sequent-calculus-style deductive system CL12 is then co nstructed, a nd a pro of o f its soundness and completeness with resp ect to the s emantics of CL is provided. All ato ms of the la nguage of CL12 are elementary , and its log ical vocabular y , o ther than • – ≀ ≀ , contains the ear lier k nown ¬ , ∧ , ∨ , ⊓ , ⊔ , ∀ , ∃ , ⊓ , ⊔ . It should ho wever b e r emembered that the ultimate purp ose of logic is providing an intellectual tool fo r navigating the real life. As such, fir st and foremost it should b e able to successfully ser ve as a bas is for applied (substan tial) theories. The p ossibility and exp ediency of ba s ing applied theories on CL has b e e n rep eatedly p o inted out in earlie r paper s ([1, 4 , 6, 13]), but no concr ete a ttempts ha d b een undertaken to do so until now. The seco nd part o f the present article makes long ov erdue (by the standar ds of the young but rapidly evolving CL) initial steps in this direction — the direction which app e ars to be full of pro mise, m ys ter y and the excitement of upturning virg in soil. F or the first time in the history of the pro ject, it constructs a computability-logic-based applied theory , sp ecifica lly , Peano ar ithmetic ba sed o n CL12 ins tead of classical logic. This sy stem, named CLA1 , is meant to b e a mo del example of a theory of this kind, 1 and a sta rting p oint for future explor ations that, as the autho r has rea sons to b elieve, may result in some int er e s ting findings. Generally , the nonlog ical axioms o f a computability-logic-based a pplied theor y would be any collection of (formulas e x pressing) problems whose algor ithmic solutions are known. Sometimes, together with nonlo gical axioms, we may also hav e nonlogica l rules o f inference, pr eserving the prop erty o f computability . An exa mple of suc h a r ule is the c onstructive induction rule o f CLA1 . Then, the soundness of the corresp o nding underlying axiomatizatio n of CL (such as CL12 in the present case) — which usually comes in the str ong form called u niform-c onstructive soun dness — g uarantees that every theore m T of the theo ry a lso ha s an algorithmic solution and that, furthermo re, such a solutio n can b e effectiv ely constructed from a pro of of T . Do e s not this lo o k like exactly what the constructivists hav e been ca lling for?! On the a pplied side, the ab ove fact makes CL-ba sed theories problem-so lving to ols: finding a solutio n for a given problem r educes to expr essing the problem in the lang uage of the theory , a nd finding a pro of of that problem. An alg orithmic solution for the problem then automatically comes together with suc h a pro of. 2 The dfb - reduction op erat ion, informally As noted, the reader is assumed to be familia r with the basic s of CL. Any unknown or forgotten terms and symbols used here can be lo ok ed up in [1 3], which comes with a conv enient glossar y . 1 Consider the problem ⊓ x ⊔ y ( y = x 2 ). An yone who knows the definition of x 2 (but p erhaps does not know the definition of x × y , o r is unable to compute the m ultiplication function for whatever rea son) would be able to solve the following pro blem, as it is ab out r educing the consequent to the anteceden t: ⊓ z ⊓ u ⊔ v ( v = z × u ) → ⊓ x ⊔ y ( y = x 2 ) . (1) A machine’s winning strateg y for the ab ov e go es like this. W ait till the e nvironment s p e cifies a v alue n for x , i.e. a sks “wha t is the square o f n ?”. Do no t try to immediately answer this question, but rather sp ecify the s ame v alue n for both z and u , thus asking the co unterquestion “wha t is n times n ?”. The environment will hav e to provide a co rrect a ns wer m to this co unt er question, i.e., sp ecify v a s m where m = n × n , or else it loses. Then, sp ecify y as m , and r est your case. Note that, in this s olution, the machine did not hav e to compute multiplication, doing which had b eco me environment’s r esp onsibility . The ma chine only cor rectly reduced the problem of co mputing square to the problem of computing pr o duct, which made it the winner. Even though (1) expr esses a “very ea sily solv able” problem, this for mula is still not logically v alid. The success of the reduction strategy o f the co nsequent to the antecedent that we pr ovided fo r it relies on the nonlogical fact that x 2 = x × x . That strategy would fail in a g e ne r al case where the meanings of x 2 and x × x may not necessa rily b e the same. On the o ther hand, the go al of CL as a gener al-purp os e pro ble m- solving to ol should be to allow us find purely log ic al so lutions, i.e., so lutio ns that do not r equire any sp ecial, domain- sp ecific knowledge and (thus) would b e go od no matter what the particular predica te or function symbols of the formulas mean. Any knowledge that might b e relev ant s hould b e explicitly stated a nd included either in the anteceden t of a given formula or in the s et of axioms (“implicit antecedent s” for every p otential formula) of a CL-based theory . In o ur present case, formula (1 ) e a sily tur ns into a logica lly v alid one by a dding , to its anteceden t, the definition of sq ua re in terms of multiplication: ∀ w ( w 2 = w × w ) ∧ ⊓ z ⊓ u ⊔ v ( v = z × u ) → ⊓ x ⊔ y ( y = x 2 ) . (2) The strategy that we provided for (1) is just as go o d for (2), with the differ e nc e that it is successful for (2) no matter what x 2 and z × u mean, whereas, in the case of (1), it was guar anteed to b e succes sful only under the standard arithmetic in terpr etations of the square and pro duct functions. Let us no w lo ok at the following mo dificatio n of (2): ∀ w  w 3 = ( w × w ) × w  ∧ ⊓ z ⊓ u ⊔ v ( v = z × u ) → ⊓ x ⊔ y ( y = x 3 ) . (3) 1 The glossary for the published version of [ 13] is given at the end of the b o ok (rather than article ), on pages 371-376. The reader may instead use the preprint version of [13], av ailable at http ://arxiv. or g/abs/cs.LO/05070 45 T he latter includes b oth the main text and the glossary . 2 Here is a pse udo-strateg y for (3 ) in the style of the (real) stra tegy that we gave for (1 ) and (2 ). W ait till the environment sp ecifies a v alue n for x . Then, using the reso urce ⊓ z ⊓ u ⊔ v ( v = z × u ) given in the anteceden t, mak e the environment co mpute m with m = n × n . Then, again using the same r esource , mak e the environmen t further compute m × n , and use the resulting v alue to s p e cify y in the consequent. Of course, there was s o me c hea ting her e . Namely , after the first usag e of ⊓ z ⊓ u ⊔ v ( v = z × u ), it will have evolv ed to the elementary game m = n × n and hence will b e no longer av ailable as a pro duct-c omputing resource . The ab ov e pseudo- s trategy works no t for (3 ) but for the following, weak er problem, whic h has no t one but rather t wo co pies of the pr o duct-computing resour ce in the antecedent : ∀ w  w 3 = ( w × w ) × w  ∧ ⊓ z ⊓ u ⊔ v ( v = z × u ) ∧ ⊓ z ⊓ u ⊔ v ( v = z × u ) → ⊓ x ⊔ y ( y = x 3 ) . In this pa rticular ca se we g ot lucky in finding an ea sy wa y around, as computing cub e only r equires computing pro duct a fixed num ber of times, which can b e accounted for by including, in the anteceden t, the appr opriate nu mber of copies o f the rele v an t res ource. But what if the proble m was not ab out computing cub e or squa re, but rather computing x h for a ( ⊓ -bo und) v ariable h ? Here we see a need for considering a reductio n op eration that, unlik e → , p ermits using the anteceden tal resource s any finite num b e r of times. Several ear lie r-studied weak r eduction o pe rations, including ◦ – and > – , do allow r ep eated usage of the an tecedent. But, imp osing no restrictio ns on the qua ntit y o f repetitions and thus p ermitting infinitely many r eusages, fo r mo s t purp os es they turn out to be weak er than necess ary . A t the same time, they tend to b e very hard to tame sy nt ac tica lly/axio matically . The op era tion • – ≀ ≀ that we intro duce in this pap er a nd call dfb-reduction app ea rs to b e semantically optimal in this res p ec t, a nd also syntactically b etter b ehav ed. It is applied to tw o components: ~ A , called the an teceden t , and B , ca lled the succedent . The result is written ~ A • – ≀ ≀ B . Here B is any game, w hile ~ A is an y finite sequence o f games. Mor e pr e cisely , ~ A is not r eally a seq uence but rather a binary tree with ga mes at its leaves, as will be seen later from the strict definitions. But, in the present informal expla na tion, we can sa fely ignore this technical detail and identify such a tree with the sequence of the games at its leaves. Int uitively , the meaning of A 1 , . . . , A n • – ≀ ≀ B (4) is similar to that of A 1 ∧ . . . ∧ A n → B , (5) that is , the tas k of the machine is to win in B as long as the environmen t, playing in the role of ⊤ , wins against the machine (the latter playing in the role of ⊥ ) in each A i . This amounts to solving problem B while having the problems A 1 , . . . , A n as computationa l reso urces. The difference be tw een (4) a nd (5) is that, in the former, the machine ha s the a dditional ca pability to replicate, at any time, any of the “conjuncts” o f the anteceden t (in the for m to which it ha s evolved by that time r ather than in the orig inal form), and inser t the new copy of it as a new “conjunct” next to the old co py . A r estriction here is that the machine is a llow ed to do suc h re plic a tions o nly a finite n umber of times, or else it will be co nsidered to hav e lost the g ame. As a n aside, “dfb” stands for double-finite br anching . One can define tw o other , similar but w eaker versions of r eduction: single-finite br anching (sfb) reduction • – ≀ and (simply) fin ite br anching (fb) reduction • – . What ma kes • – ≀ weak er than • – ≀ ≀ is that, in the for mer, winning in the succeden t automatically makes ⊤ the winner, even if it did infinitely many replica tions in the anteceden t. And what makes • – further weak er than • – ≀ is that, in the for mer, it do es not matter at all whether the ma chine has made finitely or infinitely many replica tions; how ever, when de ter mining the winner, o nly the copies of the a nt ecedental resource s that emerged as a result of finitely ma ny replications will b e lo ok ed a t. W e will b e exclusively concerned with dfb-reduction in this pap er, so sfb- and fb-reductions can be safely forgotten. Expressio ns of the form ~ A • – ≀ ≀ B we call se quents . If ⊤ ha s a n alg orithmic winning strateg y for the game represented by such an expr ession, we say that B is dfb-reduci b l e to ~ A , a nd c a ll such a stra tegy (just as the problem ~ A • – ≀ ≀ B its e lf ) a dfb-reduction of B to ~ A . The following sequent r epresents the pr oblem of dfb-reducing the problem of computing cube to the pr oblem of co mputing mult iplica tio n and the definition of cub e: ∀ w  w 3 = ( w × w ) × w  , ⊓ z ⊓ u ⊔ v ( v = z × u ) • – ≀ ≀ ⊓ x ⊔ y ( y = x 3 ) . (6 ) 3 Here is a winning strateg y for (6), which succeeds for any p ossible meanings of x 3 and z × u , s o not only do es (6 ) hav e an algor ithmic solution, but, in fact, it is a logica lly v alid sequent. The s tr ategy is to first wait till the en vir onment picks a v alue n for x . This brings the ga me down to ∀ w  w 3 = ( w × w ) × w  , ⊓ z ⊓ u ⊔ v ( v = z × u ) • – ≀ ≀ ⊔ y ( y = n 3 ) . Now the machine replica tes the seco nd resource of the antecedent , bringing the game down to ∀ w  w 3 = ( w × w ) × w  , ⊓ z ⊓ u ⊔ v ( v = z × u ) , ⊓ z ⊓ u ⊔ v ( v = z × u ) • – ≀ ≀ ⊔ y ( y = n 3 ) . After this, the machine sp ecifies b o th z and u a s n in the second resour ce of the anteceden t, further bringing the game down to ∀ w  w 3 = ( w × w ) × w  , ⊔ v ( v = n × n ) , ⊓ z ⊓ u ⊔ v ( v = z × u ) • – ≀ ≀ ⊔ y ( y = n 3 ) . T o av oid a loss, the environmen t will hav e to resp o nd by choos ing a (correct) v alue m for v in the same comp onent, and the game evolves to ∀ w  w 3 = ( w × w ) × w  , m = n × n, ⊓ z ⊓ u ⊔ v ( v = z × u ) • – ≀ ≀ ⊔ y ( y = n 3 ) . Now the ma chine sp ecifies z and u as m and n in the third re s ource of the anteceden t, br inging the g ame down to ∀ w  w 3 = ( w × w ) × w  , m = n × n, ⊔ v ( v = m × n ) • – ≀ ≀ ⊔ y ( y = n 3 ) . Again, the en vir o nment will hav e to resp ond by selecting a v alue k for v , a nd we get ∀ w  w 3 = ( w × w ) × w  , m = n × n, k = m × n • – ≀ ≀ ⊔ y ( y = n 3 ) . Finally , the machine s p e cifies y a s k and, ha ving br ought the play down to the following true elementary game, celebrates victory: ∀ w  w 3 = ( w × w ) × w  , m = n × n, k = m × n • – ≀ ≀ k = n 3 . Note that the succedent of the above sequent is a logical cons e quence (in the classica l se ns e) of the formu las of the ant ecedent. This c an b e se e n to imply that, a s promised, the success of our strategy in no w ay dep ends on the meanings of x 3 and z × u , so that the s tr ategy is, in fact, a “purely logica l” solution. Again, among the purp oses of computability lo gic is to serve as a to o l for finding such “purely logical” solutions, so that it can b e applied to an y doma in o f study r ather than s pe c ific domains such a s that of ar ithmetic, and to arbitrar y mea nings of nonlogical symbols ra ther than particular meanings such as that of the multiplication function for the sym b ol × . Remem b er the concept o f T ur ing r educibility . A function f ( x ) is said to b e T uring r e ducible to functions g 1 ( x ) , . . . , g n ( x ) iff there is a T uring machine that computes f ( x ) with o racles for g 1 ( x ) , . . . , g n ( x ). This is easily seen to mean nothing but the dfb-reducibilit y of ⊓ x ⊔ y  y = f ( x )  to the sequence ⊓ x ⊔ y  y = g 1 ( x )  , . . . , ⊓ x ⊔ y  y = g n ( x )  . Thu s, dfb-r educibility is a conserv ative genera lization of T uring reducibility . The former , of course, is more general, as it is not limited to problems of the form ⊓ x ⊔ y  y = h ( x )  as T uring reducibility is. 3 Dfb-reduction and a few other concepts defin ed formally Remem b er from [13] that a c onstant game is a pair A = ( Lr A , Wn A ), where Lr A is the set of legal runs of A and Wn A is a function telling us which play er ( ⊤ o r ⊥ ) has won a given run. F urther r emember that the letter ℘ is alwa ys used a s a v aria ble for pla yers, and ℘ (sometimes written as ¬ ℘ ) means ℘ ’s adv ers ary . W e also remind the reader the definition of the o p eration of prefixation: Definition 3.1 Let A b e a cons ta nt game a nd Φ a legal p o sition of A . The game h Φ i A is defined by: Lr h Φ i A = { Γ | h Φ , Γ i ∈ Lr A } ; Wn h Φ i A h Γ i = Wn A h Φ , Γ i . 4 W e say that a constant game A is fini te-depth iff there is an integer d such that no leg al run of A contains more than d (lab)mov es. The sma llest o f such in teger s d is called the depth of A . An elementary game is a game of depth 0. Our main fo cus in this pape r will be on finite-depth games. This r e striction of fo cus makes many definitions and pr o ofs simpler. Namely , in order to define a game op eration O ( A 1 , . . . , A n ) a pplied to such games, 2 it suffices to spe c ify the fo llowing: (i) Who wins O ( A 1 , . . . , A n ) if no mo ves are made, i.e., the v alue of Wn O ( A 1 ,...,A n ) hi . (ii) What are the ini tial l egal (lab)mov es , i.e., the elements of { ℘α | h ℘α i ∈ Lr O ( A 1 ,...,A n ) } , and to what game is O ( A 1 , . . . , A n ) br ought down after s uch an initial legal labmov e ℘α is made . Recall that, by saying that a given labmove ℘α br ing s a given game A down to B , w e mean that h ℘α i A = B . Then, the set of leg al runs of O ( A 1 , . . . , A n ) will be uniquely defined, and so will b e the winner in every run of the game. If, howev er, infinite legal runs may also occur , then the follo wing additional stipulation will b e required: (iii) How to determine the v alue of Wn O ( A 1 ,...,A n ) h Γ i when Γ is an infinite legal run of O ( A 1 , . . . , A n ). F or insta nc e , a few basic game op er ations of CL alr eady known from the earlier literature, ca n be (re)defined as follows: Definition 3.2 Let A , B , A 0 , A 1 , . . . b e finite-depth consta nt games, and n ∈ { 1 , 2 , . . . } . 1. ¬ A is defined b y: (i) Wn ¬ A hi = ℘ iff Wn A hi = ℘ . (ii) h ℘α i ∈ Lr ¬ A iff h ℘α i ∈ Lr A . Such an initial legal labmov e ℘α brings the game down to ¬ h ℘α i A . 2. A 0 ⊓ . . . ⊓ A n is defined by: (i) Wn A 0 ⊓ ... ⊓ A n hi = ⊤ . (ii) h ℘α i ∈ Lr A 0 ⊓ ... ⊓ A n iff ℘ = ⊥ and α = i ∈ { 0 , . . . , n } . Such an initial lega l labmov e ⊥ i brings the game down to A i . 3. A 0 ∧ . . . ∧ A n is defined by: (i) Wn A 0 ∧ ... ∧ A n hi = ⊤ iff, for each i ∈ { 0 , . . . , n } , Wn A i hi = ⊤ . (ii) h ℘α i ∈ Lr A 0 ∧ . .. ∧ A n iff α = i.β , where i ∈ { 0 , . . . , n } and h ℘β i ∈ Lr A i . Such an initia l legal labmov e ℘i.β br ing s the g ame down to A 0 ∧ . . . ∧ A i − 1 ∧ h ℘β i A i ∧ A i +1 ∧ . . . ∧ A n . 4. A 0 ⊔ . . . ⊔ A n and A 0 ∨ . . . ∨ A n are defined ex actly as A 0 ⊓ . . . ⊓ A n and A 0 ∧ . . . ∧ A n , resp ectively , only with “ ⊤ ” and “ ⊥ ” interchanged. 5. The infinite ⊓ -conjunction A 0 ⊓ A 1 ⊓ . . . is defined ex actly as A 0 ⊓ . . . ⊓ A n , only with “ i ∈ { 0 , 1 , . . . } ” instead of “ i ∈ { 0 , . . . , n } ” . Similar ly for the infinite version of ⊔ . T o define the (new, never studied befo re) ope r ation • – ≀ ≀ of dfb-reduction , we need some preliminar ies. What we ca ll a tree of games is a structure defined inductively as a n element of the smallest se t satisfying the following conditions : • E very game A is a tree of games. The one-element s equence h A i is said to b e the yield of such a tree, and the address of A in this tre e is the empt y bit string. 2 The same, in fact, holds f or the wider class of games termed in [1] p erifinite-depth . These are games with finite yet p erhaps arbitrarily long legal runs. 5 • Whenever A is a tree o f games with yield h A 1 , . . . , A m i and B is a tree of games with yield h B 1 , . . . , B n i , the pair A ◦ B is a tree of g a mes with yield h A 1 , . . . , A m , B 1 , . . . , B n i . The addre s s of ea ch A i in this tree is 0 w , where w is the address of A i in A . Similarly , the a ddr ess of each B i is 1 w , where w is the address of B i in B . Example: Wher e A, B , C, D are ga mes, ( A ◦ B ) ◦ ( C ◦ ( A ◦ D )) is a tree of g ames with y ie ld h A, B , C , A, D i . The address o f the first A o f the yield, to which we may a s w ell refer as the fir st leaf of the tr e e , is 00 , the address of the seco nd leaf B is 01, the addr ess of the third leaf C is 10, the a ddr ess o f the four th leaf A is 110, and the address of the fifth lea f D is 1 11. Note that ◦ is not a n op eration o n ga mes , but just a symbol used instead of the more c o mmon co mma to sepa rate t wo par ts of a pair. And a tree of games itself is not a g ame, but a collection of games a rrang ed int o a certain structure, just as a s equence o f g ames is not a g ame but a collection of games a rrang ed as a list. F or bit strings u and w , we will b e writing u  w to indicate that u is a (not necessarily prop er) prefix (initial segment) of w . Definition 3.3 Let A 1 , . . . , A n , B b e finite-depth co nstant games, and T b e a tr e e of games with yie ld h A 1 , . . . , A n i ( n ≥ 1). Let w 1 , . . . , w n be the addresses of A 1 , . . . , A n in T , resp ectively . The game T • – ≀ ≀ B is defined by: (i) Wn T • – ≀ ≀ B hi = ⊤ iff Wn B hi = ⊤ or, for some i ∈ { 1 , . . . , n } , Wn A i hi = ⊥ . In other w or ds, Wn T • – ≀ ≀ B hi = Wn A 1 ∧ ... ∧ A n → B hi . (ii) h ℘α i ∈ Lr T • – ≀ ≀ B iff one of the followin g conditions is satisfied: 1. ℘α = ℘S.β (here “ S ” stands for “ s uccedent”), where h ℘β i ∈ Lr B . Suc h a labmov e ℘S.β brings the game down to T • – ≀ ≀ h ℘β i B . 2. ℘α = ℘u.β , where u  w i for at least one i ∈ { 1 , . . . , n } and, for each i with u  w i , h ℘β i ∈ Lr A i . Such a la bmov e ℘u.β bring s the game down to T ′ • – ≀ ≀ B , where T ′ is the result of repla cing A i by h ℘β i A i in T for each i with u  w i . 3. ℘α = ⊤ w i :, where i ∈ { 1 , . . . , n } . W e call such a move a repli cative (lab)mov e . It br ing s the game down to T ′ • – ≀ ≀ B , whe r e T ′ is the result of replacing A i by ( A i ◦ A i ) in T . (iii) If Γ is a legal run of T • – ≀ ≀ B with infinitely many replicative mov es (which is the only case when Γ can and will be infinite), then W n T • – ≀ ≀ B h Γ i = ⊥ . Example 3. 4 L e t p , q , r , s b e any prop o sitions (elementary constant ga mes). Let G be the following ga me: ( p ⊓ q ) ⊓ ( r ⊓ s ) • – ≀ ≀ p ∧  ( q ⊓ r ) ∧ ( q ⊓ s )  . And let Γ be the r un h⊤ : , ⊥ S . 1 . 1 . 1 , ⊤ 0 . 1 , ⊤ 0 . 1 , ⊥ S. 1 . 0 . 0 , ⊤ 1 : , ⊤ 1 . 0 , ⊤ 1 0 . 0 , ⊤ 1 1 . 1 i . Γ is a legal run o f G . Below w e trace, step by step, the effects of its mov es on G . The 1st (lab)move ⊤ :, mea ning replica ting the (only) leaf of the anteceden tal tre e , with the addr e s s of that leaf being the empty bit string, brings G do wn to — in the sense that h⊤ : i G is — the following game:  ( p ⊓ q ) ⊓ ( r ⊓ s )  ◦  ( p ⊓ q ) ⊓ ( r ⊓ s )  • – ≀ ≀ p ∧  ( q ⊓ r ) ∧ ( q ⊓ s )  . The 2nd mov e ⊥ S. 1 . 1 . 1 of Γ, meaning making the mov e ⊥ 1 . 1 . 1 in the succ edent, mea ning making the mov e ⊥ 1 . 1 in the right ∧ - conjunct of the succedent , meaning mak ing the move ⊥ 1 in the right ∧ -conjunct of the right ∧ -co njunct o f the succedent, meaning selecting the right ⊓ -conjunct there, further brings the game down to — in the sense that h⊤ : , ⊥ S. 1 . 1 . 1 i G is — the following g ame:  ( p ⊓ q ) ⊓ ( r ⊓ s )  ◦  ( p ⊓ q ) ⊓ ( r ⊓ s )  • – ≀ ≀ p ∧  ( q ⊓ r ) ∧ s  . 6 The 3rd mov e ⊤ 0 . 1 of Γ, meaning making the mov e 1 in the 0-addre ssed lea f o f the antecedent a l tr ee, meaning selecting the rig ht ⊓ -co njunct ther e, further bring s the game down to — i.e., h⊤ : , ⊥ S. 1 . 1 . 1 , ⊤ 0 . 1 i G is — the follo wing game: ( r ⊓ s ) ◦  ( p ⊓ q ) ⊓ ( r ⊓ s )  • – ≀ ≀ p ∧  ( q ⊓ r ) ∧ s  . The 4th mo ve ⊤ 0 . 1 further brings the game down to s ◦  ( p ⊓ q ) ⊓ ( r ⊓ s )  • – ≀ ≀ p ∧  ( q ⊓ r ) ∧ s  . The 5th mo ve ⊥ S. 1 . 0 . 0 further brings the game down to s ◦  ( p ⊓ q ) ⊓ ( r ⊓ s )  • – ≀ ≀ p ∧ ( q ∧ s ) . The 6 th move ⊤ 1:, replica ting the 1-address e d leaf of the anteceden tal tre e , further bring s the game down to s ◦   ( p ⊓ q ) ⊓ ( r ⊓ s )  ◦  ( p ⊓ q ) ⊓ ( r ⊓ s )   • – ≀ ≀ p ∧ ( q ∧ s ) . The 7th move ⊤ 1 . 0 , whose effect is the same as the effect of the tw o consecutive moves ⊤ 10 . 0 and ⊤ 11 . 0 would b e, further brings the game down to s ◦  ( p ⊓ q ) ◦ ( p ⊓ q )  • – ≀ ≀ p ∧ ( q ∧ s ) . The 8th mo ve ⊤ 10 . 0 fur ther brings the game do wn to s ◦  p ◦ ( p ⊓ q )  • – ≀ ≀ p ∧ ( q ∧ s ) . The last, 9th mo ve ⊤ 1 1 . 1 further brings the game do wn to the even tual po s ition/game h Γ i G , whic h is s ◦ ( p ◦ q ) • – ≀ ≀ p ∧ ( q ∧ s ) . Note that this play is won by ⊤ no matter what par ticular prop ositio ns p, q , r and s ar e. If, how ever, the run had s topp ed after the 8th mov e, it co uld hav e been lost by ⊤ , namely , ⊤ w ould lose in ca se s and p are true but q is false. F or the re st of this pa pe r we fix tw o infinite sets of expr essions: the set { v 0 , v 1 , . . . } o f v ar i abl e s and the set { 0 , 1 , . . . } (decimal numerals) of constants . Without loss of generality here we a s sume that the ab ove collection of consta nts is exactly the universe o f discours e — i.e., the set ov er which the v ariables range — in a ll ca ses tha t w e consider. As a lwa ys, b y a v aluation we mean a function e that sends each v aria ble x to a constant e ( x ). In these terms, a classica l predica te p can b e understo o d as a function that sends each v aluation e to a pro p o sition, i.e. constant predica te. Simila rly , what w e call a (simply) game sends v aluations to c onstant games. Here we re pro duce a formal definition of this co ncept and asso ciated notation from [13]: Definition 3.5 A game is a function A from v aluations to constant games. W e write e [ A ] (rather than A ( e )) to deno te the constant game returned by A for v aluation e . Such a constant game e [ A ] is s aid to b e an instance of A . F o r readability , we usua lly write Lr A e and Wn A e instead of Lr e [ A ] and Wn e [ A ] . Where n is a na tural num b er, we say that a game A is n - ary iff there is ar e n v ariables such that, for any tw o v aluations e 1 and e 2 that agr ee on all those v ariables , we have e 1 [ A ] = e 2 [ A ]. Generally , a g ame that is n -ary for s ome n , is said to be finitary . W e say that a g a me A dep ends o n a v ariable x iff there are tw o v aluations e 1 , e 2 that agr ee on a ll v ariables except x such that e 1 [ A ] 6 = e 2 [ A ]. An n -ar y game thus dep ends on at mo st n v ariables. And constant games are nothing but 0-ar y games . Just as the B o olean op erations straig htf or wardly extend fro m pro p o sitions to all pr e dicates, o ur o p e r a- tions ¬ , ∧ , ∨ , → , ⊓ , ⊔ , • – ≀ ≀ extend from constant games to a ll ga mes. This is done by simply stipulating that e [ . . . ] commutes with all of those o p erations, as well as with ◦ : ¬ A is the game such that, fo r every e , e [ ¬ A ] = ¬ e [ A ]; A ⊓ B is the game such that, for every e , e [ A ⊓ B ] = e [ A ] ⊓ e [ B ]; T • – ≀ ≀ B is the game such 7 that, for every e , e [ T • – ≀ ≀ B ] = e [ T ] • – ≀ ≀ e [ B ], where e [ T ] is the result o f r eplacing in T every game A of the yield by e [ A ]; etc. The op eration of pre fixation also e xtends to nonconstant games: whenever Φ is a legal po sition of ev ery instance o f a game A , h Φ i A s hould b e unders to o d a s the unique game such that, for every e , e [ h Φ i A ] = h Φ i e [ A ]. W e remind the reader that the c hoice quantifiers can b e defined a s follows: ⊓ xA ( x ) = def A (0) ⊓ A (1) ⊓ A (2) ⊓ . . . ; ⊔ xA ( x ) = def A (0) ⊔ A (1) ⊔ A (2) ⊔ . . . . A game A is said to b e unistructural iff, fo r any t wo v aluatio ns e 1 and e 2 , we hav e Lr A e 1 = Lr A e 2 . O f course, all constant or elementary games a re unistructura l. It can also b e easily seen that all o ur game op erations preserve the unistructural prop erty of games. F or the pur p o ses of the present pap er, considering only unistructural games would b e sufficient. A unistr uctural game is sa id to b e finite-depth iff so a re all o f its instances . W e (re)define the remaining op erations ∀ and ∃ only fo r unistructural, finite-depth games : Definition 3.6 Let A ( x ) b e a unistructura l, finite-depth game. 1. ∀ xA ( x ) is defined b y stipulating that, for every v aluation e and mov e α , w e hav e: (i) Wn ∀ xA ( x ) e hi = ⊤ iff, for every c o nstant c , Wn A ( c ) e hi = ⊤ . (ii) h ℘α i ∈ Lr ∀ xA ( x ) e iff h ℘α i ∈ Lr A ( x ) e . Such an initial legal labmove ℘α br ings the game e [ ∀ xA ( x )] down to e [ ∀ x h ℘α i A ( x )]. 2. ∃ xA ( x ) is defined in exactly the same way , only with ⊤ and ⊥ interc hanged. 4 Blo ck-mo v e machine s As alwa ys, we agree that the term “ computational problem ”, o r simply “ problem ” , is a synonym o f “static game” (see Section 12 for a definition). All meaningful and reasona ble examples of ga mes — including all elementary games — ar e static, and the class of static games is closed under all of our game op erations. This fact for • – ≀ ≀ is proven in the app endix-style Section 12 of the pre sent pap er, and for all other op er ations it has been pr oven in [1] (Theorem 14.1). The pap er [1 ] defines t wo mo dels of interactiv e co mputation, called the har d-play machine ( HPM ) and the e asy-play machine ( EPM ). W e remind the r eader that b oth are sorts o f T uring ma chines with the capability of making mov es, and hav e three tap es: the or dinary read/ w r ite work t ap e , and the read-only valuation and run tap es. The v aluation tap e contains a full de s cription of some v aluation e , and its conten t remains fixed throughout the work of the machine. In this pap er we only cons ide r finitar y games, and a simplifying assumption that we are making is that the v aluatio n tap e o nly lists the v alues of finitely ma ny (relev ant) v ariables, the r est of the tap e b eing bla nk, and the v alue of e at a ll unlisted v ariables ass umed to b e 0 . As for the run ta p e , it serves as a dynamic input, a t any time s pe lling the curr ent p ositio n. E very time one o f the players makes a move, that move — with the cor resp onding la b el — is a utomatically app ended to the conten t of this tap e. In the HPM mo del, there is no r estriction on the frequency of environmen t’s mov es. In the E P M mo del, on the other hand, the machin e has full co ntrol over the s p ee d o f its adversary: the environmen t can (but is not obliga ted to) make a (one single) mov e o nly when the mach ine explicitly allows it to do so — the even t tha t we call gran ting p ermission . The only “fairness ” requirement that such a machine is e x pe cted to s atisfy is that it sho uld gr ant p ermission every once in a while; how long that “while” lasts, howev er, is totally up to the ma chine. The HPM and E PM models seem to b e tw o extremes, yet, according to Theorem 1 7.2 o f [1 ], they yie ld the same c lass of winnable static g ames. The pres ent pa p e r will only deal with the EPM model, so let us take a little clo s er lo ok a t it. Let M be an EPM. A c onfigur ation o f M is defined in the standard wa y: this is a full description of the (“current”) state of the machine, the lo cations of its three scanning heads and the conten ts of its three tap es. The initial c onfigur ation on v aluation e is the co nfig uration where M is in its sta rt state, the v aluation tap e 8 sp ells e , each scanning hea d is lo ok ing at the first c e ll of the co rresp o nding tape, and the work and run tap es are empty . A co nfiguration C ′ is said to be a suc c essor o f config uration C in M if C ′ can legally follow C in the standard — sta ndard for multitape T uring machines — sens e, based on the tr ansition function (whic h we assume to b e deterministic) of the ma chine and accounting for the p os s ibility of nondeterministic up dates — depe nding on wha t mov e ⊥ makes or whether it makes a mov e a t all — of the con tent of the run ta pe when M grants p ermissio n. T echnically , granting p er mission happ ens by en tering one of the sp ecia lly designated states called “p er mission s tates”. And M makes a move α by constructing it at the beg inning o f its work tap e and then entering o ne of the sp ecia lly designated s tates called “move states”. A computation branc h of M on e is a sequence of configur ations of M where the first configur ation is the initial configur a tion on e and every o ther co nfiguration is a success or of the previous one. Thus, the set of all computation branches on e captur es all p ossible scenar ios (on v aluation e ) co rresp onding to different behaviors by ⊥ . Such a branch is said to b e fair iff per mis sion is gra nt ed infinitely many times in it. Each computation bra nch B of M incrementally sp ells — in the obvious sense — a run Γ on the run tap e, which we call the run sp ell ed b y B . Then, for a g a me A , we write M | = e A a nd say that M wins A on v aluation e to mean that, whenever Γ is the run sp elled b y so me computation branch B of M on e and Γ is not ⊥ -illeg a l, 3 then branch B is fair and Wn A e h Γ i = ⊤ . W e write M | = A and say that M wi ns ( computes, solves ) A iff M | = e A for every v aluation e . Finally , we write | = A a nd sa y that A is computable iff there is an EPM M with M | = A . The class of sta tic games is r obust with resp ect to cho osing differen t mo dels of computation or different mo des of playing games. Spec ifically , pro cedura l decisio ns r egar ding whether players are allow ed to mov e at any time (as in the HPM mo del), or only when ⊤ so decides (as in the EPM mo del), or only when ⊥ so decides , or whether they should take alternating turns, do no t affect the class of winnable sta tic ga mes . F urther, if the players mov e in a n alternating o r der, it turns out to b e irrelev ant which player star ts the game, or how many mov es at a time either play er is a llow ed to ma ke. In the present pap er, together with the E P M mo del, we will a lso be using the new (never defined b efor e) mo del of interactive co mputation that we call BME PM (“Blo ck-Mov e E PM”). A BMEPM is the s ame as an EP M, with the o nly difference tha t her e either player can make any finite num b er of mov es at once. That is , the machine can make several mov es on a given c lo c k cyc le (s ay , by first listing tho s e mo ves on its work tap e and then e nt er ing a mov e state); and the environmen t, too , can ma ke any finite n umber of mov es (including no mov es at all) when granted p ermiss ion. All concepts a nd notations that we us e for EPMs straightforwardly extend to BME PMs a s well. The following prop osition officially establishes the equiv alence betw een EPMs and BMEPMs for static games: Prop ositi on 4.1 A s t atic game A is c omputable (i.e., ther e is an EPM that wins A ) iff ther e is a BMEPM that wins A . F u rthermor e: 1. Ther e is an effe ctive pr o c e dur e that c onverts any BMEP M M into an EPM E su ch that, for any st atic game A and any valuation e , if M | = e A , then E | = e A . 2. And vic e versa: ther e is an effe ctive pr o c e dur e that c onverts any EPM E into a BMEPM M su ch that, for any st atic game A and any valuation e , if E | = e A , then M | = e A . Pro of. Clause 1 : Consider an arbitrar y BMEPM M . The corr esp onding EPM E is a machine that, with a v aluation e sp elled on its v aluation tap e, works by simulating M with the same e sp elled on the imaginary v aluation tap e of the latter. Every time this simulation shows that M gr a nts p ermission, E also grants p ermissio n, and then feeds the environment ’s resp o nse (which can b e a single mov e or no move) back to the simulated M . And, every time the simulation shows that M makes a blo ck α 1 , . . . , α n of moves, E makes the same n mov es — only , one by one rather than the whole blo ck at once — in its real play . Obviously , the run Γ ge nerated by E in this play is a run that could hav e b e en gener ated b y M (in pre cise terms, Γ is the run sp elled by some co mputation branch of M on e ). So, if M wins a game A on v aluation e , so does E . Clause 2 : Consider an arbitra ry EP M E . The corr esp onding BME P M M is a machine that, with e sp elled on its v aluation tap e, w or k s by simulating E with the same e sp elled o n the v aluation tap e of the latter. Every time this simulation shows that E makes a mov e, M makes the same mov e. And every time the simulation shows that E g rants p ermission, so do es M . If, in re s p onse, the environment makes no move, or makes a single mov e, then the s a me resp onse is fed back to the simulated E . Supp ose now the e nvironment 3 Remem b er that a ℘ -illega l r un is an i llegal run where the last mov e of the shortest ill igal initial segment is ℘ -lab eled. 9 resp onds b y a blo ck α 1 , . . . , α n of mo ves ( n ≥ 2). In this cas e, M feeds back the r esp onse α 1 to the simulated E . It contin ues simulation (mimicking E ’s mov es in its r eal play) until E grants per mission a gain, in which case M (without g ranting p ermission in the r eal play) simply feeds the environment’s earlier r esp onse α 2 int o the simulation, and so o n until all o f the n moves ar e pro cessed in the same manner , after which M resumes the ordinary simulation. It is not har d to see that the run ∆ generated in this play by M is a ⊤ -delay of the run Γ g e ne r ated by the simulated E . So , if E wins a static g a me A on v aluation e , so do es M . ✷ 5 F orm ulas and sequen ts The fir st-order language o f (the fragment o f ) CL that we consider in this se c tio n ha s infinitely ma ny variables and c onstants — the o nes fix e d in Section 3. Next, it has infinitely many n -a r y nonlo gic al pr e dic ate symb ols and n -ary fun ction symb ols for ea ch integer n ≥ 0. Finally , it has one binary lo gic al pr e dic ate symb ol =. T erms ar e defined in the usual way , and so ar e formulas , except that the connectives now are ⊤ , ⊥ (0-a ry), ¬ (1-a r y), ∧ , ∨ , ⊓ , ⊔ (2- ary), and the quantifiers are ∀ , ∃ , ⊓ , ⊔ . Also, ¬ is only a llow ed o n atomic formulas. In an y other case it should b e understo o d as a standard abbreviation, and s o sho uld → . The definitions of free and bo und o ccurrences of v ariables are a lso standard, with ⊓ , ⊔ acting as quantifiers alo ng w ith ∀ , ∃ . T rees of form ul as are the elements of the sma llest set of ex pressions such that: • The empty e xpression is a tree of formulas, said to be empt y . • E very for mula is a ( nonempty ) tree of formulas. • If T and S a re no ne mpty tree s of formulas, then so is ( T ) ◦ ( S ). Here parentheses ca n be omitted around T and S if they are form ulas. As in the case of trees of games, the yie ld o f a tre e of form ulas is the se q uence of formulas app ear ing in it, in the left to r ig ht order. The yield of the empt y tr ee is empt y . W e will o ften wr ite a tree T o f formulas as T ( E 1 , . . . , E n ) to indicate that E 1 , . . . , E n is its yield. When n = 0, T ( E 1 , . . . , E n ) is the empt y tree. A sequen t is T ( E 1 , . . . , E n ) • – ≀ ≀ F ( n ≥ 0), where T ( E 1 , . . . , E n ), ca lled the an teceden t , is a tree of formulas, and F , called the s u ccedent , is a formula. When n = 0, such a sequent simply lo oks like • – ≀ ≀ F , and is said to be a n empt y-an tecedent sequent . Sometimes a for mula F will b e represe nt ed a s F ( x 1 , . . . , x n ), w he r e the x i are pair wise distinct v ariables. When doing so, we do no t neces sarily mea n that each such x i has a free o ccurr e nc e in F , or that every v ariable o ccurring fr ee in F is a mong x 1 , . . . , x n . In the context set by the ab ov e re pr esentation, F ( t 1 , . . . , t n ) will mean the r esult of replacing , in F , each free o ccurrence of each x i (1 ≤ i ≤ n ) by ter m t i . The same notational conv entions also apply to sequents and trees of formulas. An in terpretation 4 is a function ∗ that sends ea ch n -ary function symbol f to a function f ∗ : { 0 , 1 , 2 , . . . } n → { 0 , 1 , 2 , . . . } ; it also sends each n -ary pr edicate s y mbol p to an n -ary elementary game p ∗ ( x 1 , . . . , x n ) which do es not depend o n a ny v ariables other than x 1 , . . . , x n ; the additional condition required to be satisfied by ∗ is that = ∗ is an equiv alence re la tion on { 0 , 1 , 2 , . . . } pre s erved by f ∗ for ea ch function s ymbol f , and resp ected b y p ∗ for each nonlogica l pre dicate symbol p . 5 The ab ove uniquely extends to a mapping that sends each term t to a function t ∗ , e a ch for mula F to a game F ∗ , ea ch nonempt y tree T of for mulas to a tree T ∗ of games, and ea ch sequent X to a game X ∗ by stipulating that: 1. x ∗ = x (any v ariable x ). 2. c ∗ = c (any constant c ). 3. Where f is an n -ary function symbol and t 1 , . . . , t n are terms,  f ( t 1 , . . . , t n )  ∗ = f ∗ ( t ∗ 1 , . . . , t ∗ n ). 4. Where p is a n n -ary predicate sym b ol and t 1 , . . . , t n are terms,  p ( t 1 , . . . , t n )  ∗ = p ∗ ( t ∗ 1 , . . . , t ∗ n ). 4 The concept of an interpretat ion in CL is usuall y more general than the present one, where interpreta tions in our present sense are call ed p erfect . But here we omit the wo rd “p erfect” as we do not consider any nonp erfect inte rpr etations, an ywa y . 5 More commonly classical logic sim ply treats = as the iden tity predicate. That treatme nt of =, ho wev er, i s kno wn to be equiv alen t — in every r espect relev ant f or us — to our pr esen t one. Namely , the l atter turns int o the former by seeing any tw o = ∗ -equiv alen t constan ts as tw o different names of the same obje ct of the unive rs e, as “Evening Star” and “Morning Star” are. 10 5. ∗ commutes with ◦ , in the sense that ( E ◦ F ) ∗ = E ∗ ◦ F ∗ . 6. ∗ commutes with • – ≀ ≀ when the anteceden t is not empty: ( T • – ≀ ≀ F ) ∗ = T ∗ • – ≀ ≀ F ∗ . 7. An empt y-anteceden t sequent is simply understo o d as its succedent: ( • – ≀ ≀ F ) ∗ = F ∗ . 8. ∗ commutes with all remaining o pe rators : ⊤ ∗ = ⊤ , ( ¬ F ) ∗ = ¬ F ∗ , ( E ∧ F ) ∗ = E ∗ ∧ F ∗ , ( ⊓ xF ) ∗ = ⊓ x ( F ∗ ), etc. When O ∗ = P (whether O b e a predic a te symbol, function symbol, formula or sequent), we say that ∗ in terprets O as P . If we say “ O unde r i n terpretation ∗ ”, what we mean is O ∗ . When a given fo r mula is repr esented as F ( x 1 , . . . , x n ), we will typically write F ∗ ( x 1 , . . . , x n ) instea d of  F ( x 1 , . . . , x n )  ∗ . Let X b e a sequent. F or an HP M, EP M or BMEPM M , we say that M is a uniform sol ution for X iff M | = X ∗ for every int er pr etation ∗ . And we say that X is uni formly v alid iff there is a uniform s o lution for X . 6 Some c losure prop erties of computabilit y In this se c tion w e establish certain, mo stly rather o bvious yet impo rtant, clo sure prop erties for the c o m- putabilit y of static games. While these results can b e stated a nd prov en in mo re genera l for ms than pre s ented here, for simplicit y we r estrict them to games expressible in the formal la nguage of Section 5. By a “ rule ” in this s e ction we mean a n ( n + 1)-a ry relation R o n sequents, the insta nce s of which ar e schematically written as X 1 . . . X n X 0 , where X 1 , . . . , X n are (metav a riables for) se q uents called the premi ses , and X 0 is a seq ue nt ca lled the conclusion . Whenever R ( X 1 , . . . , X n , X 0 ) holds, w e say that X 0 follows from X 1 , . . . , X n by R . W e say that such a rule R is s ound iff it preser ves co mputability , in the sense that, whenever ea ch premise of a given instance of the rule is computable under a g iven interpretation, so is the conclusion (under the same int er pr etation). And we say that R is uniform-constructively sound iff there is an effective pr o cedure that takes any instance ( X 1 , . . . , X n , X 0 ) of the rule, a ny BME P Ms M 1 , . . . , M n and returns a BME PM M 0 such that, fo r any interpretation ∗ and v a luation e , whenever M 1 | = e X ∗ 1 , . . . , M n | = e X ∗ n , we have M 0 | = e X ∗ 0 . Below we prove several unifor m- constructive so undness results. The pro ofs will b e limited to showing how to construct M 0 from ( X 1 , . . . , X n , X 0 ) and M 1 , . . . , M n . It will b e immediately clear fr o m our descriptions of the M 0 s that they c a n b e constructed e ffectively (so that the clos ur e is “ c onstructive”), and that their work in no w ay dep e nds o n an interpretation ∗ applied to the sequents inv olved (so that the closure is “uniform”). Since an interpretation ∗ is typically irr e lev an t in such pro o fs, we will often omit it and write simply F or X where, stric tly sp eaking, F ∗ or X ∗ is meant. That is, we identif y formulas o r sequents with the g ames into which they turn once an interpretation is applied to them. Similarly , a v aluation e is often irrelev ant, and we write F or X whe r e, strictly spea king, e [ E ∗ ] or e [ X ∗ ] is mean t. W e agree that in this and subsequent sections, the expressio n E 1 , . . . , E n • – ≀ ≀ F is an abbreviation of E 1 ◦ E 2 ◦ E 3 ◦ . . . E n − 1 ◦ E n • – ≀ ≀ F which, in turn, is an abbreviatio n of E 1 ◦ ( E 2 ◦ ( E 3 ◦ . . . ( E n − 1 ◦ E n ) . . . )) • – ≀ ≀ F. When n = 0 , the sequent E 1 , . . . , E n • – ≀ ≀ F is just • – ≀ ≀ F . The ex pression “ E 1 , . . . , E n ” will often b e abbrevi- ated as ~ E . Throughout this section, n is a natural num b e r. Remember that we write T ( E 1 , . . . , E n ) for a n a rbitrary tree T of formulas with yield E 1 , . . . , E n . Again, when n = 0, T ( E 1 , . . . , E n ) is the empty tree. When T ( E 1 , . . . , E n ) = E 1 ◦ . . . ◦ E n or T is e mpty , w e say that T is standard . 11 6.1 Standardization and Destandardization Standardization and Destandardizatio n ar e the follo wing t wo rules, respe c tively: T ( E 1 , . . . , E n ) • – ≀ ≀ F E 1 , . . . , E n • – ≀ ≀ F E 1 , . . . , E n • – ≀ ≀ F T ( E 1 , . . . , E n ) • – ≀ ≀ F Standardization thus allows us to replace any tree of for mulas in the anteceden t of the premise by a standard tree with the same yield, and Destandardization do es the opp osite. Prop ositi on 6.1 Both Standar dization and Destandar dization ar e uniform-c onstructively sound. Pro of. Here we only co nsider Standardization, with the case of Destandardization be ing fully symmetric. Assume M 1 is a BMEP M that wins the pr emise under a giv en interpretation on a given v aluation. Note that the premise and the conclusion of the rule are “es sentially the s ame”, with the minor difference that the address of e a ch leaf E i in the pr emise may b e differen t from the address of the corresp onding leaf E i in the conclusion. Of cours e , one can easily figur e out a bijection f that tr anslates the a ddresses. W e co ns truct M 0 by letting it b e a machine that sim ulates M 1 , imagining that the v a luation sp elled on the v aluation tap e of the latter is the sa me as the v aluation sp elled on its own v aluation ta p e . In this simulation, M 0 acts (g rants per missions, makes moves) in the s a me wa y a s M 1 , only tr anslates any mov e w.α or w : by M 1 int o f ( w ) .α o r f ( w ): b efore mimicking it in the rea l play . Similarly , an en vir onment’s mov e w.α in the re a l play it translates int o f − 1 ( w ) .α before feeding it into the simulated play . Replicative mov es, o f co urse, change f (including its domain), but cor resp ondingly upda ting f each time is no problem. The only minor complicatio n is re lated to the fa c t that, when the environmen t makes a move u.α in the real play , or M 1 makes mov e u.α in the simulated play , u ma y not necessa rily b e the a ddress of a leaf of the current anteceden tal tree, but rather a prefix o f some of such a ddr esses. F o r instance , this is the case with the 7 th mov e of Example 3.4 . Let, in this case, w 1 , . . . , w k be all the a ddresses of the leav es o f the cur rent anteceden tal tr ee such that u is a prefix of them. Note that then, the effect of the single mov e u.α is the sa me as the effect of k consecutive moves w 1 .α, . . . , w k α . So , M 0 int er pr ets the single move u .α as the blo ck w 1 .α, . . . , w k .α before feeding it int o the simulated play or mimic king it in the real play . Remem b ering our a s sumption that M 1 wins the premise, it is not hard to s ee that M 0 wins the conclusion. ✷ 6.2 Exc hange Exchange is the following rule : ~ E , H , G, ~ K • – ≀ ≀ F ~ E , G, H , ~ K • – ≀ ≀ F Prop ositi on 6.2 Exchange is uniform-c onstructively sound. Pro of. The idea here is the same as in the pro of of the prev ious pr o p osition. M 0 plays e x actly as M 1 , only corres p o ndingly adjusting the a ddresses, and o ccasio nally interpreting a single mov e as a blo ck of mov es. ✷ 6.3 W eakeni ng W eakening is the following rule: ~ E • – ≀ ≀ F ~ E , ~ K • – ≀ ≀ F Prop ositi on 6.3 We akening is uniform-c onstructively sou n d. Pro of. Assume M 1 is a BME P M that wins the pr emise. W e co ns truct M 0 by letting it b e a machine that pla ys ex actly a s M 1 do es, b y sim ulating the latter. Only , M 0 ignores any mov es that the environment makes within the ~ K part of the anteceden t, as if that par t simply did not exist. In addition, if ~ E is empty 12 (while ~ K is no t), M 0 adds the prefix “ S. ” when mimicking moves made by M 1 in the succedent o f the premise, and deletes the same prefix when feeding ba ck in to the simulated play the environment’s mov es in the succedent of the conclusion. Obviously , M 0 wins the game. ✷ 6.4 Con traction Contraction is the following rule: ~ E , G, G • – ≀ ≀ F ~ E , G • – ≀ ≀ F Prop ositi on 6.4 Contr action is uniform-c onstructively s ound. Pro of. Again, assume M 1 is a BME PM that wins the premise. W e co nstruct M 0 by le tting it b e a machine that ma kes a re plicative move which br ing s the conclusio n down to the pr emise, and then contin ues as M 1 . ✷ 6.5 Instan tiation Instantiation is the following r ule: ~ E ( x ) • – ≀ ≀ F ( x ) ~ E ( c ) • – ≀ ≀ F ( c ) Here x is a ny v ariable, c is any cons tant, and the conclusion is the r esult of replacing, in the premise, a ll fre e o ccurrences of x by c . Prop ositi on 6.5 Instantiation is uniform-c onstructively sound. Pro of. Assume M 1 is a BMEP M that wins the premis e . W e let M 0 be the machine that, with a v aluation e spelle d on its v aluatio n tap e, plays (b y simulating M 1 ) a s M 1 would play w ith v aluatio n e ′ on its v aluation tap e, where e ′ is the v a luation that sends x to c a nd agrees with e on a ll other v ariables. ✷ 6.6 Cut Unlik e a ll previous r ules, Cut is a rule with t wo premises : ~ E • – ≀ ≀ F ~ K , F • – ≀ ≀ G ~ E , ~ K • – ≀ ≀ G Theorem 6. 6 Cut is uniform-c onstructively sound. Pro of. A detailed technical pro of of this theorem could b e lengthy , and here we essentially only g ive a pro of idea (as we, in fact, did for a ll the pre vious prop o sitions of this section as well). The ter minology we will be using b e low is even mor e r elaxed than ea r lier. Sp ecifically , it should be p ointed out that to what we may refer as “ F ” (or “ ~ E ” , etc.), str ictly sp eaking, typically will not b e the orig inal F (or ~ E , e tc.), but rather the game to whic h this comp onent has evolved by the time of the context of such a refer e nce. Assume M 1 and M 2 are t wo BME P Ms that win ~ E • – ≀ ≀ F and ~ K , F • – ≀ ≀ G , resp ectively . Let us fir st consider the simple ca se when the tw o machines never make any replicative moves in the anteceden ts of the corr esp onding ga mes. W e let M 0 be the following BME PM. Its work on a v aluation e consists of infinitely many cycles. E ach cycle cons ists in simulating, in pa rallel, the tw o machines M 1 and M 2 with the same e on their v aluation ta p e s . The simulation of either machine contin ues until it g r ants per mission. M 0 records a ll moves made by the simulated machines. O nce b oth machines hav e g r anted per mission, M 0 halts sim ulation, and all mov es made b y the tw o machines in the ~ E , ~ K and G comp onents of the premises it mimics in the same co mpo nents of the conclus ion which is play ed in the real play (after corres p o ndingly r eadjusting the addresse s of the leav es if necessary , and also p er haps interpreting cer tain 13 mov es a s whole blo cks of mov es, as we have alr eady seen in the previous pro ofs). Then M 0 grants p ermission in the r eal pla y , and records the environment’s r esp onse, whic h should be some (maybe none) mo ves within the a b ov e comp onents. Now M 0 resumes simulation, feeding these mov es back to the cor resp onding ma chines as ima g inary r esp onses to their gra nting p er mission. In a ddition, it also feeds back to M 1 all moves made by M 2 within the F comp o ne nt of the second premise , and vice versa: feeds back to M 2 all mov es made by M 1 within the F comp onent of the first pre mis e. Here the cycle ends, and the next cycle do es the same, starting (contin uing) simulation o f the tw o machin es until they bo th gra nt p ermis sion, a nd so on. With some thought and keeping in mind the cr ucial fact that the g ames we are talking ab o ut are static, one can see that M 0 wins ~ E , ~ K • – ≀ ≀ G . But, aga in, in the a b ove simplified sce na rio we did no t a ccount for the p ossibility of r eplicative mov es that M 1 and/or M 2 may make. If a s ub c o mp onent of ~ E is r eplicated (by M 1 ) o r a sub comp onent of ~ K is replicated (by M 2 ), M 0 can replicate the sa me subcomp o nent in the conclusio n, and other wise contin ue playing acco rding to the ea rlier scenario. This ca se th us do es not really create any complicatio ns . The cas e of M 2 replicating the F comp onent, th us turning the game that it plays in to ~ K , F , F • – ≀ ≀ G , is more serious though. In this case, M 0 needs to split its simulation o f M 1 int o t wo s imulations (contin uations), let us call them #1 a nd #2. It also corr esp ondingly r eplicates the whole ~ E part of the conclusion in the r eal pla y , which — mo dulo Ex change a nd Sta ndardization — will now lo ok like ~ E , ~ E , ~ K • – ≀ ≀ G . 6 F rom now on, the mov es made in the F compo nent b y M 1 in s imu la tio n #1 (resp. #2) will b e fed back int o the fir s t (r esp. second) co py of F in the game ~ K , F , F • – ≀ ≀ G play ed by the simulated M 2 , a nd vice versa: the mov es made by M 2 within the fir st (resp. s e cond) F comp onent of that ga me will b e fed back into s imulation #1 (resp. #2) of M 1 . Similarly , M 0 will be mimicking the mov es made by M 1 in simulation #1 (resp. #2) within the ~ E co mpo nent as moves in the fir st (resp. seco nd) co py of ~ E in ~ E , ~ E , ~ K • – ≀ ≀ G , and vice v ersa : feeding the environmen t’s moves made in the fir st (resp. s e cond) copy of ~ E in ~ E , ~ E , ~ K • – ≀ ≀ G into simulation #1 (r esp. #2) of M 1 . Subsequent re plications b y M 2 of an y g iven co py o f F will b e ha ndled in a s imilar w ay , cor resp ondingly further incre asing the num b e r of v arious parallel simulations of M 1 and the num b er of co pies of ~ E in the anteceden t of the conclusion. ✷ 7 Logic CL 12 The purp ose of the deductive system CL12 that we construct in this s ection is to axiomatize the set o f uniformly v alid sequents. In view o f the closure under Standardiza tion and Destandardization prov en in the pr e vious s e c tion, limiting sequents to those with standard a nt ecedents do es not yield a ny rea l loss of expressive power, 7 and this is what we do from now on: the langua g e of CL12 exclusively deals with such sequents. Corresp onding ly , from now on, when we say “ sequent ”, we exclusively mean one o f the form E 1 , . . . , E n • – ≀ ≀ F ( n ≥ 0). F or sa fety and also without lo ss of expressive pow er, we further agr ee that, from now on, the for mulas or sequents that we consider may not contain b oth b ound a nd free o c c urrences o f the same v ariable. Other than these minor revisions, the language of CL12 is the s a me as the one intro duced in Section 5. Our form ulation of CL12 relies on some terminology and notation explained below. 1. A surface o cc urrence o f a subformula is a n o ccurrence that is not in the scop e of any choice op erato rs. 2. A formula not containing c hoice op er a tors — i.e., a formula of the cla ssical la ng uage — is said to b e elementary . 6 Remem b er what was said at the b eginning of the pro of ab out our relaxed terminology: of course, neither F nor ~ E nor ~ K nor G wil l necessarily b e exactly the same as the corresponding original subgames, but rather b e whatev er games they hav e been brought down to by now. 7 Some readers may be wondering why we di d not then try to define dfb- reduction as an op eration applied to sequences rather than trees of games (in the antece dent) in the first place. There are reasons, discernable only to an expert on computab il ity logic. Among such reasons i s that any attempt to r eplace trees wi th sequences wou ld encoun ter the pr oblem of violating the (crucially imp ortant) static prop erty of games. This i s so b ecause the effect of a mov e b y ⊥ could depend on whether it was made b efore or after a replicative mov e b y ⊤ , and i t could thus cha nge when one passes f rom a r un to a ⊥ - delay of it. 14 3. A seq uent is elementary iff all of its formulas are so. 4. The e l ementar ization k F k of a formula F is the result of r eplacing in F all ⊔ - and ⊔ -subformulas b y ⊥ , and all ⊓ - and ⊓ - subformulas by ⊤ . Note that k F k is (indeed) an elemen tary formula. 5. The e l ementar ization of a sequent G 1 , . . . , G n • – ≀ ≀ F is the elementary fo rmula k G 1 k ∧ . . . ∧ k G n k → k F k . 6. A sequent is said to b e stable iff its elementarization is classically v alid. By “classica l v alidit y”, in view of G¨ odel’s completeness theorem, we mean prov abilit y in class ical fir st-order calculus with co nstants, function s y mbols a nd =, where = is tr eated a s the logica l identity predicate (so that, s ay , x = x , x = y → ( E ( x ) → E ( y )), etc. are prov able). 7. W e will be using the no ta tion F [ E ] to mean a formula F together with some (single) fixed surface o ccurrence o f a subformula E . Using this notation sets a con text, in whic h F [ H ] will mea n the res ult of re pla cing in F [ E ] the (fixed) o ccur r ence of E by H . Note that her e w e are talking ab out some o c curr enc e of E . O nly that o ccur rence g ets replaced when mo ving from F [ E ] to F [ H ], even if the for mula also had some other occur rences of E . THE R ULES OF CL12 CL12 has the s ix rules listed below, with the following additional conditions/expla nations: 1. In ⊓ -Cho ose and ⊔ -Cho ose, i ∈ { 0 , 1 } . 2. In ⊓ -Cho o se and ⊔ -Cho ose , t is either a co nstant or a v ariable with no bo und o c currences in the premise, and H ( t ) is the r e sult of r eplacing by t all free o ccurrences of x in H ( x ) (rather than v ice versa). ⊔ -C h o ose ~ G • – ≀ ≀ F [ H i ] ~ G • – ≀ ≀ F [ H 0 ⊔ H 1 ] ⊓ -C h o ose ~ G, E [ H i ] , ~ K • – ≀ ≀ F ~ G, E [ H 0 ⊓ H 1 ] , ~ K • – ≀ ≀ F ⊓ -Cho ose ~ G, E [ H ( t )] , ~ K • – ≀ ≀ F ~ G, E [ ⊓ xH ( x )] , ~ K • – ≀ ≀ F ⊔ -Cho ose ~ G • – ≀ ≀ F [ H ( t )] ~ G • – ≀ ≀ F [ ⊔ xH ( x )] Replicate ~ G, E , ~ K • – ≀ ≀ F ~ G, E , ~ K , E • – ≀ ≀ F 15 W ait X 1 , . . . , X n ( n ≥ 0), where all of the following five co nditio ns are satisfied: Y 1. ⊓ -Condition: Whenever Y has the form ~ G • – ≀ ≀ F [ H 0 ⊓ H 1 ], b oth of the sequents ~ G • – ≀ ≀ F [ H 0 ] and ~ G • – ≀ ≀ F [ H 1 ] are among X 1 , . . . , X n . 2. ⊔ -Condition: Whenever Y has the form ~ G, E [ H 0 ⊔ H 1 ] , ~ K • – ≀ ≀ F , b oth o f the sequents ~ G, E [ H 0 ] , ~ K • – ≀ ≀ F and ~ G, E [ H 1 ] , ~ K • – ≀ ≀ F ar e among X 1 , . . . , X n . 3. ⊓ -Condition: Whenever Y ha s the fo r m ~ G • – ≀ ≀ F [ ⊓ xH ( x )], for s ome v ariable y not o ccurring in Y , the sequent ~ G • – ≀ ≀ F [ H ( y )] is among X 1 , . . . , X n . Here and below, H ( y ) is the re s ult of replacing by y all free occur rences of x in H ( x ) (rather than vice v er s a). 4. ⊔ -Condition: Whenever Y has the form ~ G, E [ ⊔ xH ( x )] , ~ K • – ≀ ≀ F , fo r some v aria ble y not o ccur ring in Y , the sequent ~ G, E [ H ( y )] , ~ K • – ≀ ≀ F is a mong X 1 , . . . , X n . 5. Stabilit y Co ndition: Y is stable. A CL12-pro of of a s e quent X is a sequence X 1 , . . . , X n of sequents, with X n = X , such that, each X i follows by one o f the rules of CL12 from some (p ossibly empty in the case of W ait, and cer tainly empt y in the case of i = 1) set P of pr emises such that P ⊆ { X 1 , . . . , X i − 1 } . When a C L12 - pro of of X ex ists, we say that X is prov able in CL12 , and write CL12 ⊢ X . Example 7. 1 In this example, + is a binary function symbol. W e write t 1 + t 2 instead of +( t 1 , t 2 ). The following sequenc e of sequents is a CL12 -pr o of (of its last sequent): 1. ∀ x ( x + 0 = x ) • – ≀ ≀ w + 0 = w W ait: (no premises) 2. ∀ x ( x + 0 = x ) • – ≀ ≀ ⊔ z ( w + 0 = z ) ⊔ -Choose: 1 3. ∀ x ( x + 0 = x ) • – ≀ ≀ ⊓ y ⊔ z ( y + 0 = z ) W ait: 2 Example 7. 2 The following is a CL12 -pro of: 1. ∀ x ∀ y ∀ z  p ( x, y ) ∧ p ( y , z ) → q ( x, z )  , p ( w , v ) , p ( u, w ) • – ≀ ≀ q ( u, v ) W ait: 2. ∀ x ∀ y ∀ z  p ( x, y ) ∧ p ( y , z ) → q ( x, z )  , p ( w , v ) , p ( u, w ) • – ≀ ≀ ⊔ y q ( u, y ) ⊔ -Cho ose: 1 3. ∀ x ∀ y ∀ z  p ( x, y ) ∧ p ( y , z ) → q ( x, z )  , ⊔ y p ( w, y ) , p ( u, w ) • – ≀ ≀ ⊔ y q ( u, y ) W ait: 2 4. ∀ x ∀ y ∀ z  p ( x, y ) ∧ p ( y , z ) → q ( x, z )  , ⊓ x ⊔ y p ( x, y ) , p ( u, w ) • – ≀ ≀ ⊔ y q ( u, y ) ⊓ -Cho os e : 3 5. ∀ x ∀ y ∀ z  p ( x, y ) ∧ p ( y , z ) → q ( x, z )  , ⊓ x ⊔ y p ( x, y ) , ⊔ y p ( u, y ) • – ≀ ≀ ⊔ y q ( u, y ) W ait: 4 6. ∀ x ∀ y ∀ z  p ( x, y ) ∧ p ( y , z ) → q ( x, z )  , ⊓ x ⊔ y p ( x, y ) , ⊓ x ⊔ y p ( x, y ) • – ≀ ≀ ⊔ y q ( u, y ) ⊓ -Choo se: 5 7. ∀ x ∀ y ∀ z  p ( x, y ) ∧ p ( y , z ) → q ( x, z )  , ⊓ x ⊔ y p ( x, y ) • – ≀ ≀ ⊔ y q ( u, y ) Replicate: 6 8. ∀ x ∀ y ∀ z  p ( x, y ) ∧ p ( y , z ) → q ( x, z )  , ⊓ x ⊔ y p ( x, y ) • – ≀ ≀ ⊓ x ⊔ y q ( x, y ) W ait: 7 Example 7. 3 In this CL12 -pro of, + is as in Example 7 .1, and ′ is a unary function symbol. W e write t ′ instead of ′ ( t ). 1. ∀ x ∀ y  x + y ′ = ( x + y ) ′  , s = v ′ , u + w = v • – ≀ ≀ u + w ′ = s W ait: 2. ∀ x ∀ y  x + y ′ = ( x + y ) ′  , s = v ′ , u + w = v • – ≀ ≀ ⊔ z ( u + w ′ = z ) ⊔ -Cho ose: 1 3. ∀ x ∀ y  x + y ′ = ( x + y ) ′  , ⊔ y ( y = v ′ ) , u + w = v • – ≀ ≀ ⊔ z ( u + w ′ = z ) W ait: 2 4. ∀ x ∀ y  x + y ′ = ( x + y ) ′  , ⊓ x ⊔ y ( y = x ′ ) , u + w = v • – ≀ ≀ ⊔ z ( u + w ′ = z ) ⊓ -Cho ose: 3 5. ∀ x ∀ y  x + y ′ = ( x + y ) ′  , ⊓ x ⊔ y ( y = x ′ ) , ⊔ z ( u + w = z ) • – ≀ ≀ ⊔ z ( u + w ′ = z ) W ait: 4 16 6. ∀ x ∀ y  x + y ′ = ( x + y ) ′  , ⊓ x ⊔ y ( y = x ′ ) , ⊓ y ⊔ z ( y + w = z ) • – ≀ ≀ ⊔ z ( u + w ′ = z ) ⊓ -Cho ose: 5 7. ∀ x ∀ y  x + y ′ = ( x + y ) ′  , ⊓ x ⊔ y ( y = x ′ ) , ⊓ y ⊔ z ( y + w = z ) • – ≀ ≀ ⊓ y ⊔ z ( y + w ′ = z ) W ait: 6 8 The sound ness and completeness of CL12 Note tha t interpretations, as defined in this pa p er, are nothing but first-order models (structures ) in the classical sense with do main { 0 , 1 , 2 , . . . } . B y G¨ odel’s co mpleteness theorem, an (elementary) formula F is v alid (in the classic al sense) iff it is true in every such mo del. This, in view of the cla ssical b ehavior o f ¬ , ∧ , ∨ , ∀ , ∃ when applied to elementary for mulas, means that, for a n e le ment ar y formula, v alidity in the classical sense means the same as uniform v alidity in our sense. W e may implicitly rely on this fact in the sequel. The follo wing lemma can be verified b y s tr aightforw ar d induction o n the complexit y of F : Lemma 8.1 F or any formula F , interpr etation ∗ and valuation e , Wn F ∗ e hi = Wn k F k ∗ e hi . The above lemma e asily extends from formulas to sequents: Lemma 8.2 F or any se quent X , interpr etation ∗ and valuation e , Wn X ∗ e hi = Wn k X k ∗ e hi . Below and elsewhere, b y the l ength o f a CL12 -pro of we mean the num b er of s equents a pp e aring in the pro of. Lemma 8.3 L et X and Y b e arbitr ary s e quent s su ch that one of the fol lowing four c onditions is satisfie d: 1. X has t he form ~ E • – ≀ ≀ F [ H 0 ⊓ H 1 ] , and Y is ~ E • – ≀ ≀ F [ H i ] ( i = 0 or i = 1 ). 2. X has t he form ~ E , G [ H 0 ⊔ H 1 ] , ~ K • – ≀ ≀ F , and Y is ~ E , G [ H i ] , ~ K • – ≀ ≀ F ( i = 0 or i = 1 ). 3. X has t he form ~ E • – ≀ ≀ F [ ⊓ xH ( x )] , and Y is ~ E • – ≀ ≀ F [ H ( y )] for s ome variable y not o c curring in X . 4. X has t he form ~ E , G [ ⊔ xH ( x )] , ~ K • – ≀ ≀ F , and Y is ~ E , G [ H ( y )] , ~ K • – ≀ ≀ F for s ome variable y not o c curring in X . Then, any CL12 -pr o of P of X of length k c an b e effe ctively c onverte d into a CL12 -pr o of Q of Y of length ≤ k . F u rthermor e, if X is derive d by Wait in P , t hen the length of such a CL12 -pr o of Q of Y wil l b e strictly less than k . Pro of. Consider ar bitrary sequents X and Y , with X having a CL12 -pr o of P of length k . Using induction on k , the lemma c a n b e prov en by case s , dep ending o n which of the four co nditio ns of Lemma 8.3 is satisfied by X and Y . Case 1: X and Y satisfy conditio n 1 of the lemma. Let us fir st consider the case when X is derived in P by W ait. Note that then Y is among its pr e mises, a nd hence has a pro of of length less than k . That very pro of will be the s o ught pro of Q of Y . No w supp ose X is der ived by any of the other rules from a premise U . Note that U ha s the same form ~ E ′ • – ≀ ≀ F ′ [ H 0 ⊓ H 1 ] (some ~ E ′ , F ′ ) as X has. Let W b e ~ E ′ • – ≀ ≀ F ′ [ H i ]. W e can apply the induction hypothesis to U a nd W (in the ro le of X and Y ) and find that W has a pro o f (which can b e effectively constructed) of length less than k . But no te that Y follows from W by the same r ule as X fo llows from U . So, Y has a pro of of length at most k . Case 2: X and Y sa tisfy condition 2 of the lemma. This is similar to the previo us cas e. Case 3: X a nd Y satisfy condition 3 of the le mma . First we co nsider the c a se when X is derived in P by W ait. Note that then a cer tain sequent Y ′ , which is “essentially the same” as Y , is amo ng the premises of X , and hence has a pro of of length less than k . Here Y ′ is “essentially the same” as Y in the sense that it is either Y , o r is different from Y only in that it has a v ariable z ( z 6 = y ) where Y has y . Obviously this is not a significant difference, and the pro of of Y ′ can be turned into a pr o of of Y by just r enaming v a riables. Such a pro of of Y will b e the sought Q . Now it r emains to cons ide r the case of X b eing derived by any o f the other rules. O ur re asoning in this case will b e s imilar to the one given in Ca se 1. Case 4: X and Y sa tisfy condition 4 of the lemma. Simila r to the previo us cas e . ✷ 17 Let us say that a sequent Y is a ⊥ -dev elopm en t of a sequent X iff X a nd Y satisfy o ne of the four conditions of Lemma 8.3. And we say tha t Y is a ⊤ -developmen t o f X iff X follows from Y by o ne o f the rules of CL12 other than W ait. Now, we say that Y is a transitive ⊥ - dev elopm en t of X iff there are sequents Z 1 , . . . , Z n ( n ≥ 2) where Z 1 = X , Z n = Y and, for e a ch i with 1 ≤ i < n , Z i +1 is a ⊥ - developmen t of Z i . T ransitiv e ⊤ -dev el o pment is defined similarly . Lemma 8.4 L et X and Y b e any se quents. 1. If Y is a tr ansitive ⊥ -development of X , X has a CL12 -pr o of P of lengt h k and t he last se quent of t he pr o of ( X ) is derive d by Wait, t hen Y has a CL12 - pr o of Q of length less t han k . Such a Q c an b e effe ctively c onstruct e d fr om P . 2. If Y is a tr ansitive ⊤ -development of X and CL12 6⊢ X , then C L12 6⊢ Y . Pro of. Clause 1 is an immediate coro llary o f Lemma 8.3 . And clause 2 is straightforw ar d. ✷ Theorem 8. 5 CL12 ⊢ X iff X is uniformly valid ( any se quent X ). F urthermor e: Uniform-constructiv e soundne s s Ther e is an effe ctive pr o c e dur e that takes any CL12 -pr o of of any se quent X and c onstructs a un iform solution for X . Pro of. Conside r an arbitr ary seq uent X . Soundness: Assume X has a CL12 - pro of P X . Pick a n arbitrary interpretation ∗ . W e wan t to (show how to) construct a ∗ -indep endent EP M M X that wins X ∗ . As a n induction hypothesis , we assume that, for a ny CL12 -pro of P Y of any sequent Y , as long as P Y is shorter tha n P X , we know how to (effectively) construct a ∗ -indep endent EPM M Y that wins Y ∗ . In each case b elow, e is the (arbitrar y) v aluation s p elle d on the v aluation tap e of the to-be - constructed machine M X . As was done b efor e, we may omit either e or ∗ or bo th and, say , write simply X instead o f e [ X ∗ ]. The work of M X depe nds on b y what r ule X is der ived in P X . Case 1: X is derived from a premise Y by ⊔ -Cho ose. Then we le t M X be the machine that ma kes a mov e that brings X down to Y . F or insta nce , if X is ~ K • – ≀ ≀ E ∧ ( F ∨ ( G ⊔ H )) and Y is ~ K • – ≀ ≀ E ∧ ( F ∨ G ), then S. 1 . 1 . 0 is such a mov e. After this, M X contin ues as M Y with the sa me v aluation e sp elled on the v aluation ta p e of the latter. By the induction hypothes is, M Y wins Y , which o bviously implies that M X wins X . Case 2: X is derived from Y b y ⊓ -Cho o se. Then, as in the previo us cas e, M X is the machine that makes a move that br ing s X down to Y . F o r instance, if X is E , F ⊓ G, H • – ≀ ≀ K (i.e., E ◦ (( F ⊓ G ) ◦ H ) • – ≀ ≀ K ) and Y is E , F , H • – ≀ ≀ K (i.e., E ◦ ( F ◦ H ) • – ≀ ≀ K ), then 10 . 0 is such a mov e. After this, M X contin ues as M Y with the same v aluation e sp elled on the v aluation tap e of the latter. By the induction hypothesis, M Y wins Y , which implies that M X wins X . Case 3: X is derived from Y by ⊔ -Cho ose. Then M X is the ma chine that ma kes a mov e that brings e [ X ] do wn to e [ Y ]. F or insta nce, if X is ~ K • – ≀ ≀ E ∧ ( F ∨ ⊔ xH ( x )) and Y is ~ K • – ≀ ≀ E ∧ ( F ∨ H ( t )), then S. 1 . 1 .c is such a mov e, wher e c = t if t is a consta nt, and c = e ( t ) if t is a v ariable (in the latter case the ma chine will hav e to read c from its v aluation tap e). After this mov e, M X contin ues as M Y with the s ame v aluatio n e sp elled o n the v aluation tap e of the latter. By the induction hypo thesis, M Y wins Y , which implies that M X wins X . Case 4: X is der ived from Y by ⊓ -Cho ose. This ca se is similar to the previo us o ne(s), and we can a fford to omit details. Case 5 : X is derived from Y by Replicate. Note that Replicate is nothing but a simple co mbin a tio n of Contraction and Exchange. By the induction hypo thesis, we have a machine M Y winning the premise. Then, in view of Prop ositions 6.2 and 6.4, w e can construct the desired machine M X that wins the conclusion. Case 6 : X is derived by W ait. Then M X keeps gra nt ing p e rmission until the environment makes a mov e. If such a move is never made, then the r un that is genera ted is empty . As X is der ived by W a it, it is stable, so that k X k is cla ssically v alid. But then, in view of Lemma 8 .2 , the machine is the winner, no ma tter what v aluation e is sp elle d on its v aluation tap e. Suppos e now the environmen t makes a move. W e may assume that such a mov e is leg al, or else the machine immediately wins. With a little thought, one can see that any le g al move α by the en viro nment brings the ga me down to e ′ [ Y ] for a certain v aluation e ′ and a certa in 18 sequent Y such that Y is a tr ansitive ⊥ -developmen t of X . F or example, if X is ~ K • – ≀ ≀ ⊓ xH ( x ) and α is S. 5, then Y ca n be chosen to be ~ K • – ≀ ≀ H ( y ) for a v ariable y not o ccurring in X , and then e ′ would b e the v aluation that sends y to 5 and agre es with e o n all other v ariables, s o that e ′ [ ~ K • – ≀ ≀ H ( y )] is e [ ~ K • – ≀ ≀ H (5)], with the latter be ing the game to which e [ X ] is br ought down b y the labmove ⊥ S. 5 . As ano ther example, consider X = ⊔ xE ( x ) ◦ ⊔ xE ( x ) • – ≀ ≀ F and α = . 5 (so that the effect of α is making the mov e 5 in b oth leaves of the anteceden t). Then Y can be chosen to b e E ( y ) ◦ E ( z ) • – ≀ ≀ F for some fresh v ariables y a nd z , in whic h case e ′ would b e the v aluation that sends bo th y and z to 5, a nd a grees with e on all other v ariables. Bo th Y a nd e ′ can be effectiv ely found. F urthermo r e, by clause 1 o f Lemma 8.4, Y has a s ho rter pro of Q than X do es, a nd Q , too , can be effectively found. Then, b y the induction hypo thesis, a n EP M M Y with M Y | = Y can b e further co nstructed effectively . So, o ur M X now constructs M Y and plays the r est o f the game as M Y would play with the v aluation e ′ sp elled on its v aluation tape. This obviously mak es M X successful. Completeness: Assume CL12 6⊢ X . Here we describ e a c ounterstr ate gy for X in the fo r m of an EPM C with an or acle for CL12 -pr ov ability . “Counterstrategy” means tha t C plays as EP Ms g enerally do, with the difference that it plays in the role of ⊥ rather than ⊤ . Tha t is, the mov es it ma kes get the la b el ⊥ , and the mov es b y its adversary g et the lab el ⊤ . It will b e see n that C is alwa ys the winner for an appropr iately selected v aluation g and interpretation ∗ . In view of Lemma 20.8 of [1], this implies that X has no unifor m solution. F or a sequent Y and v aluation e , we s ay that e is Y -dis tinctiv e iff e as signs different v alues to different free v ariables of Y , a nd all those v alues are also different fro m any constants o ccurring in Y . W e fix g as an arbitra ry X -distinctiv e v aluation, which is going to b e the a b ov e-mentioned “a ppropriately s elected v aluation”. The work of C , in a sense, is s ymmetric to the w o r k of the machine M X constructed in the pro of of the s o undness part. W e define it rec ur sively . A t any time, it deals with a pair ( Y , e ), wher e Y is a CL12 - unprov able sequent and e is a Y -distinctive v aluation. The initial v alue of Y is X , and the initial v alue of e is the above X -distinctive v aluation g , which we as s ume is the v aluation sp elled on the v a luation tap e of C . How C acts on ( Y , e ) depe nds on Y . If Y is s ta ble, then there should b e a CL12 - unprov a ble sequen t Z satisfying one of the following condi- tions, for other wise Y would b e der iv able by W a it. Using its or acle for CL12 -prov ability , C finds and selects one s uch Z (say , le xicogra phically the smalle st one), and acts according to the c orres p o nding prescr iption a s given b elow. Case 1: Y has the form ~ E • – ≀ ≀ F [ G 0 ⊓ G 1 ], and Z is ~ E • – ≀ ≀ F [ G i ] ( i = 0 or i = 1 ). In this case , C makes a mov e that bring s Y down to Z , and calls itself on ( Z, e ). Case 2: Y has the form ~ E , F [ G 0 ⊓ G 1 ] , ~ K • – ≀ ≀ H , and Z is ~ E , F [ G i ] , ~ K • – ≀ ≀ H . This cas e is similar to the previous one. Case 3: Y ha s the form ~ E • – ≀ ≀ F [ ⊓ xG ( x ) ], and Z is ~ E • – ≀ ≀ F [ G ( y )], where y is a v ariable not o ccurring in Y . In this case , C makes a mo ve that brings Y down to ~ E • – ≀ ≀ F [ G ( c )] for some (say , the smallest) cons ta nt c such that c is different from any co nstant o ccurring in Y , as w ell as fr om a ny e ( z ) where z is a free v aria ble of Y . After this move, C calls itself on ( Z , e ′ ), where e ′ is the v aluation that sends y to c and a grees with e on all other v ariables. Case 4: Y has the form ~ E , F [ ⊓ xG ( x )] , ~ K • – ≀ ≀ H , and Z is ~ E , F [ G ( y )] , ~ K • – ≀ ≀ H , where y is a v ariable not o ccurring in Y . This case is similar to the previous one. Next, we consider the c ases when Y is not stable. Then C keeps gra nt ing p er mission until the environmen t makes a mov e. Case 5 : If such a move is never made, then the run that is g enerated is empty . As Y is no t stable, k Y k is not classic a lly v alid. Also, a s pr omised ear lier and as it is ea sy to verify , e is a Y -distinctiv e v aluation and hence, of cour se, it is also k Y k -distinctive. It is a co mmon knowledge from cla ssical log ic that, whenever a formula F is inv alid and e is a n F -distinctive v aluation, e [ F ] is false in some mo del. So, e [ k Y k ] is fals e in/under some model/ interpretation ∗ . This , in view o f Le mma 8.2, implies that Wn Y ∗ e hi = ⊥ and hence C is the winner in the ov erall play of X ∗ on g . Case 6 : No w ass ume the adversary makes a mov e. W e may as sume that such a mo ve is le g al, o r else C immediately wins. With a little tho ught, one can see that any lega l move α by the adversary will bring the game down to e ′ [ Z ∗ ] for a certain Z -distinctiv e v aluation e ′ and sequent Z s uch that Z is a transitive ⊤ -developmen t of Y . By cla use 2 of Lemma 8 .4, CL12 6⊢ Z . So, our machine just calls itself on ( Z , e ′ ). 19 It is clear that C wins as long a s Case 6 o ccurs only finitely many times. And, if not, C aga in wins, as then the run is infinite. ✷ 9 Natural deduction W e ar e go ing to co nstruct a CL-base d arithmetic in the form of a natura l deduction s ystem. This sec- tion explains the ba sic concepts and notational conv entions for such systems in genera l, using the (slig htly improv ed) Fitc h style. Each natura l deduction system will hav e a collection o f rules (of inference), and a deduction in suc h a system will lo ok something like the following: Steps Justifications E 1 E 2 E 3 E 4 E 5 E 6 E 7 E 8 E 9 E 10 E 11 E 12 E 13 1 . 2 . 3 . 4 . 1 . 4 . 2 . 4 . 3 . 1 . 4 . 3 . 2 . 4 . 3 . 3 . 4 . 4 . 5 . 6 . 1 . 6 . 2 . 7 . Hypo thesis Hypo thesis Rule 3 : ~ n 3 Hypo thesis Rule 5 : ~ n 5 Hypo thesis Rule 7 : ~ n 7 Rule 8 : ~ n 8 Rule 9 : ~ n 9 Rule 10 : ~ n 10 Hypo thesis Rule 12 : ~ n 12 Rule 13 : ~ n 13 Each E i here s hould b e a form ula (rather tha n sequent) o f the language of CL12 , or whatever fragment of that language is consider ed in a given system. The ab ove is a deduction of E 13 from the hypotheses E 1 and E 2 . The s teps 1 through 7 are those of the main (parent) deduction . This deduction has t wo sub deductions ( c hildren ). The first sub deduction comprises steps 4 . 1 thro ug h 4 . 4 (which can b e compressed a nd, together, called s tep 4 of the main deduction), and the second sub deduction co mprises steps 6 . 1 and 6 . 2 (or, step 6 o f the main deduction). One of the steps of the first sub de ductio n, in tur n, is itself a sub deductio n, co ns isting of thr ee steps. The vertical lines delimit (sub)deductions, and the horizo nt al line in e ach (sub)deduction separa tes the Hyp otheses from the r est of the steps. Each subdeduction thus may have its own hypotheses, in addition to thos e of its parents, grandpar ents, etc. On the o ther hand, the hypotheses of children do not count as those o f parent s, so that the deduction of E 13 that w e s ee above is a deduction from (only) the h yp o theses E 1 and E 2 . Each form ula in a deduction should co me with a justi fication . The justification for each hypothesis is the s ame: a n indication that the step is a h yp othesis (which, in fact, is r edundant as the h yp otheses can be recognized as the steps ab ove the hor izontal bar). And the justification for any other formula E i should b e an indication of the rule Rule i by which the formula is obta ined, and the step num b ers ~ n i for the premises of the corres po nding application of the rule. Here a rule is a relation of the t yp e ( { F ormulas } ∪ { Sub de ductions } ) n → { F o rmu las } ( n ma y not necessar ily b e fixed, as in the LC rule o f the following section). Such a rule has n premises, and a justification should list each of tho se premises (identified by step num ber s) together with the rule name. An imp or tant res tr iction is o n what steps can ser ve a s premis e s for a given a pplication o f a rule. First o f 20 all, each premise should be a step that o ccur s earlier than the given step. But not a ll earlie r steps c a n b e used. Sp ecifically , the steps a ppe a ring inside ea rlier sub deductions that are no longer active (are “clo sed”) cannot b e referr ed to. Only the entire c hild sub deduction (ra ther than particula r steps within it) can serve as a pre mise for a s tep in the par e nt deduction. F o r example, in the deduction o f the above figure, the steps that ~ n 12 may include a re limited to 1, 2, 3, 4, 5 a nd 6 . 1. Note that here w e hav e wr itten 4 as a reference to the entire child sub deduction 4 . 1 -4 . 4. ~ n 12 cannot include any of the par ticular s ubsteps 4 . 1 , 4 . 2, 4 . 3 or 4 . 4 of 4 , or any o f the substeps 4 . 3 . 1, 4 . 3 . 2 or 4 . 3 . 3 of substep 4 . 3 . That is b ecause all these steps are within already closed s ub deductio ns . Similarly , the steps that ~ n 13 can include a r e limited to 1 , 2, 3, 4 (as a whole sub de ductio n), 5 and 6 (as a whole subdeduction). And the steps that ~ n 9 can include are limited to 1, 2, 3, 4 . 1, 4 . 2 and 4 . 3; it cannot include 4 . 3 . 1, 4 . 3 . 2 or 4 . 3 . 3. 10 Computabilit y-logi c-b ased arithmetic There can b e v arious reasona ble systems of arithmetic based on co mputability lo gic, dep ending on what language we consider , what fra gment of CL is taken as a log ical basis, a nd wha t extra -logica l rules a nd axioms are employ ed. The system CLA1 offered in this section is o ne of them. Its langua g e, whose formulas we call CLA1-formulas and whose sequent s w e call C LA1-s equen ts , is obta ined fr o m the langua ge of CL12 by removing all nonlogica l predic a te symbo ls (thus only leaving the lo gical predicate symbol = ), all constants but 0, and all but three function symbols, which are: • su ccessor , unary . W e will write t ′ for successor ( t ). • su m , binary . W e will wr ite t 1 + t 2 for sum ( t 1 , t 2 ). • pr oduct , bina ry . W e will write t 1 × t 2 for product ( t 1 , t 2 ). W e shall also t ypically write x 6 = y for ¬ x = y . The c oncept of an interpretation defined ea rlier can now b e restricted to in terpreta tions that ar e only defined on =, ′ , + and × , as the language has no other pr edicate or function symbo ls. Of such interpretations, the s tandard in terpretation † is the one that interprets = a s the identit y r elation, ′ as the successo r ( x + 1) function, + as the sum function, and × as the pro duct function. Where F is a CLA1 -for mu la, the standard in terpretation of F is the game F † , which we typically wr ite simply as F . Similar ly for CLA1 -sequents. The axioms of CLA1 a re the ∀ -closures of the fo llowing: 1. 0 6 = x ′ 2. x ′ = y ′ → x = y 3. x + 0 = x 4. x + y ′ = ( x + y ) ′ 5. x × 0 = 0 6. x × y ′ = ( x × y ) + x 7. ⊓ x ⊔ y ( y = x ′ ) 8. F (0) ∧ ∀ x ( F ( x ) → F ( x ′ )) → ∀ xF ( x ), where F ( x ) is any elementary formula CLA1 is a natura l deduction system, a nd the ab ov e axioms are treated as r ules of inference (in the sense of the previous section) that tak e no premises . In addition, CLA1 has the following tw o rules: 21 Logical Consequence (LC) G 1 . . . G n . . . F t where CL12 ⊢ G 1 , . . . , G n • – ≀ ≀ F Constructiv e Induction (CI) F (0) . . . F ( x ) . . . F ( x ′ ) . . . ⊓ xF ( x ) t where x is a fresh v aria ble Here • indica tes the conclusion of the rule, and whatever steps we see b efor e the conclusion ar e the premises. Rule LC thus ta kes a ny n ( n ≥ 0) premises G 1 , . . . , G n and, provided that CL12 ⊢ G 1 , . . . , G n • – ≀ ≀ F , yields the conclusio n F . The CI rule takes tw o premises: a formula of the form F (0) and a sub deduction that has F ( x ) as a single premise and F ( x ′ ) as the la s t step. The “fre s hness” requirement o n x means that x sho uld not o ccur anywhere earlier in the pro of. This co ndition can b e safely weakened by only requiring that x doe s not o c cur in any ea rlier steps other than per haps those within already closed subdeductions. Deductions in this system we call CLA1-deductions . A CLA1 -pro of of a formula F is a CLA1 -deduction of F from the empt y set of hypotheses. CLA1 , how ever, can b e used not only for proving fo rmulas, but sequents as well. A CLA1-pro of of a sequent E 1 , . . . , E n • – ≀ ≀ F is a CLA1 -deduction o f F fr o m h yp otheses E 1 , . . . , E n . An e xtended CLA1 - pro of of a sequen t or for mula X is a CLA1 -pro of of X where to eac h applica tion of the LC rule is attached, as an additional justification, a CL12 -pro of of cor resp onding sequen t, i.e. of the sequent whose antecedent co nsists o f the pr emises of the rule and whose succedent is the conclusio n of the rule. When a CLA1 -pro of o f a sequent or formula X exists, we say that X is pro v able in CLA1 and write CLA1 ⊢ X . Note that CLA1 -pr ov a bility o f a formula F can b e seen as a sp ecia l c a se of the more general co ncept of prov abilit y of seque nts. Specifically , it can b e under sto o d as CLA1 -pr ov ability of the empty-an tecedent sequent • – ≀ ≀ F . CLA1 is a computabilit y-lo gic-base d co unterpart of the classica l-logic-ba sed P eano arithmeti c P A , the bes t known and most studied applied formal theory of all. F o rmulas of P A are nothing but elementary (ch oic e-op erato r-free) formulas of the languag e of CLA1 . And the axioms of P A are the same as those of CLA1 , only without the no nelementary Axiom 7 . The latter s ays simply that the success or function is computable. And the CI rule, while for mally b eing nothing but an ordina ry induction pr inc iple with just ∀ r eplaced by ⊓ , in fact, pr ovides for pr imitive recursio n. A p o int worth emphasizing her e is that, by just adding to P A the very innocuo us -lo oking Axiom 7 and rule CI, we ( CLA1 , that is) get, via computability logic, substantial informatio n ab out co mputabilit y . The informatio n obtained in this way may not b e new, but the fac t that it is obtained by purely log ical metho ds is a strong indication tha t computability logic can really do what it is in tended to do. F or typesetting consider a tions and without any danger of ambiguit y , in the following examples we con- struct CLA1 -deductions without using vertical or horizontal bar s. Example 10 .1 Pr oving ⊓ x ⊓ y ⊔ z ( y + x = z ) in CLA1 : 1. ∀ x ( x + 0 = x ) Axiom 3 2. ∀ x ∀ y  x + y ′ = ( x + y ) ′  Axiom 4 3. ⊓ x ⊔ y ( y = x ′ ) Axiom 7 4. ⊓ y ⊔ z ( y + 0 = z ) LC: 1 (Example 7.1 ) 22 5.1. ⊓ y ⊔ z ( y + w = z ) Hypo thesis 5.2. ⊓ y ⊔ z ( y + w ′ = z ) LC: 2, 3, 5.1 (Example 7.3) 6. ⊓ x ⊓ y ⊔ z ( y + x = z ) CI: 4, 5 Note that the classica l counterpart ∀ x ∀ y ∃ z ( y + x = z ) of the ab ove-pro ven for mu la is not very interesting — b eing log ic ally v alid, it carr ies no information whatso ever, and is true no t o nly for + but fo r any other function as well. As for ⊓ x ⊓ y ⊔ z ( y + x = z ), it is not trivia l at all. In view of the forthcoming soundness theorem for CLA1 , its prov a bilit y sig nifies that fo r any x and y , a z with y + x = z not mer e ly exists, but ca n also b e algor ithmically fo und; fur thermore, suc h a n algorithm, itself, ca n b e constructed from a pro o f of the formula. So , Exa mple 10.1 demonstra tes the no nt r iv ial (a lbe it not no vel) fac t that addition is a computable function; and the pro of given there actually enco des an algorithm for co mputing this function. Similarly , the following example implies the computability o f multiplication: Example 10 .2 Pr oving ⊓ x ⊓ y ⊔ z ( y × x = z ) in CLA1 : 1. ∀ x ( x × 0 = 0) Axiom 5 2. ∀ x ∀ y  x × y ′ = ( x × y ) + x  Axiom 6 3. ⊓ x ⊓ y ⊔ z ( y + x = z ) (E xample 10.1) 4. ⊓ y ⊔ z ( y × 0 = z ) LC: 1 (similar to Example 7.1) 5.1. ⊓ y ⊔ z ( y × w = z ) Hypo thesis 5.2. ⊓ y ⊔ z ( y × w ′ = z ) LC: 2, 3, 5.1 (similar to Example 7.3) 6. ⊓ x ⊓ y ⊔ z ( y × x = z ) CI: 4, 5 In view of the well k nown deduction theorem, one of the many equiv alent wa ys to define the prov ability of a for mula F in P A is to say that F is prov able in P A iff there a re a xioms E 1 , . . . , E n such that the formula E 1 ∧ . . . ∧ E n → F is pr ov able in classical predicate calculus. Supp o s e the la tter is indeed the case . Note that then the sequent E 1 , . . . , E n • – ≀ ≀ F is prov a ble in CL12 — namely , deriv able by W ait from the empty set of premises . Each E i , b eing an a xiom o f ( P A a nd hence also of ) CLA1 , is a lso prov able in CLA1 . Then, by LC, F is also prov able. T o summarize: F act 10. 3 Every formula pr ovable in P A is also pr ovable in CLA1 . Thu s, CLA1 is an extension of P A . A natur al question to which at prese nt we have no answer is whether CLA1 is a conserv ative extension of P A , that is, whether every elementary for mula F prov able in CLA1 is also prov able in P A . While, by G¨ odel’s incompleteness theorem, P A fails to prove all true a rithmetic form ulas , P A is still “practically complete”, in the sense that it prov es a ll reasona bly simple (usually ca lled “elementary”, but not in our present sense of this word) true facts a b o ut natura l num ber s. Mos t readers are well familiar with P A and hav e a go o d feel of what it can prove. This allows us to relax our pro ofs and, in presenting them, justify some “steps” (whic h, in reality , are seque nc e s of many steps r ather than single steps) b y saying that the formula in the step is prov a ble in P A . Example 10 .4 The following sequence is a CLA1 -pro of of its last formula (by the wa y , what do es that formula say?). Even ( x ) should b e understo o d as an abbre v iation o f ∃ z ( x = z + z ), a nd O dd as a n abbreviatio n of ∀ z ( x 6 = z + z ). 1. Even (0) P A 2. Even (0) ⊔ Odd (0) LC: 1 3.1. Even ( x ) ⊔ Odd ( x ) Hyp o thesis 3.2. Even ( x ) → Odd ( x ′ ) P A 3.3. Odd ( x ) → Even ( x ′ ) P A 3.4. Even ( x ′ ) ⊔ Odd ( x ′ ) LC: 3.1 , 3,2, 3.3 23 4. ⊓ x ( Even ( x ) ⊔ Odd ( x )) CI: 2 , 3 5. ∀ x ∀ y   Even ( x ) ∧ Even ( y )  ∨  Odd ( x ) ∧ Odd ( y )  → Even ( x + y )  P A 6. ∀ x ∀ y   Even ( x ) ∧ Odd ( y )  ∨  Odd ( x ) ∧ Even ( y )  → Odd ( x + y )  P A 7. ∀ y  Even ( y ) ⊔ Odd ( y ) → ⊓ x ( Even ( x + y ) ⊔ Odd ( x + y )  LC: 4, 5, 6 W e could, o f course, dire c tly find an alg orithmic solution for the problem expr e ssed by form ula 7 of the ab ov e example. But had o ur ad ho c metho ds failed to succeed — which would likely b e the case if the problem was more complex than it is — a n algor ithmic so lution for it or any other CLA1 -prov able pro blem could b e effectively found fro m a CLA1 -pro of o f it, as implied b y the following, alrea dy mentioned theorem, establishing the status of CLA1 as a systematic problem-solving to o l: Theorem 10. 5 F or any formula or se quent X , if CLA1 ⊢ X , then X (u n der t he standar d interpr etation) is c omputable. F urt hermor e, t her e is an effe ctive pr o c e dur e that takes an arbitr ary ext ende d CLA1 - pr o of of an arbitr ary formula or se quent X and c onstru cts a BMEPM (or EPM, or HPM if you like) M with M | = X . Pro of. W e prov e this theo rem b y induction on the lengths o f deductio ns. As prov ability of formulas is a specia l (or easier) case of pr ov a bility of seque nts, considering only sequents here would b e sufficient. Let X = ~ E • – ≀ ≀ F , a nd consider any CLA1 -deduction of F from h yp otheses ~ E . If F is an axio m, obviously it is computable. Sp ecifically , a ll axioms ex c ept Axiom 7 ar e true elementary formulas, and they are “c o mputed” by a machine that makes no mov es at all. And Axiom 7 is computed by a ma chine that waits for the adversary to ma ke a move n , then increments n by o ne and ma kes (the decimal string r epresenting) the resulting n umber as its only mov e in the ga me. According to our conv entions, semantically F is the same as • – ≀ ≀ F . So, we know how to compute • – ≀ ≀ F . Then, b y Pro p o sition 6.3, we als o know how to compute ~ E • – ≀ ≀ F . Next, supp ose F is obtained fro m premises G 1 , . . . , G k by L C. Obser ve that, for each G i , we hav e a shorter (shorter than that of F from ~ E ) deduction o f G i from ~ E and, o f co urse, s uch a deduction can b e effectively obta ined from the deduction of F . Thus, by the inductio n hypothesis, we know ho w to co mpute ~ E • – ≀ ≀ G i . Since F is derived by LC, we have CL12 ⊢ G 1 , . . . , G k • – ≀ ≀ F a nd hence, by Theorem 8.5, we also know how to compute G 1 , . . . , G k • – ≀ ≀ F . 8 Applying Theorem 6.6 k times (in combination w ith Pro po sitions 6.2 and 6.4), w e then also know how to compute ~ E • – ≀ ≀ F . Finally , supp ose F is obtained from premises G (0) a nd a sub deduction of G ( x ′ ) from the hypothesis G ( x ) by CI, so that F is ⊓ xG ( x ). As in the pr evious ca s e, by the inductio n hypothesis , we know how to compute ~ E • – ≀ ≀ G (0). F urther , note that G ( x ′ ) ha s a s horter (than that of F fro m ~ E ) deduction fr om the hypotheses ~ E , G ( x ). So, by the induction hypothesis, we a lso know how to co mpute ~ E , G ( x ) • – ≀ ≀ G ( x ′ ) and hence (by Prop os ition 6.5) ~ E , G (0) • – ≀ ≀ G (0 ′ ). The latter, as a g a me, is the same as ~ E , G (0) • – ≀ ≀ G (1). Then, by Theorem 6.6 (in comb inatio n with Pro p o sitions 6.2 and 6 .4), w e know how to compute ~ E • – ≀ ≀ G (1). Denote b y M 1 the machine that computes ~ E • – ≀ ≀ G (1). Co nt inuing in the sa me way , we can co nstruct machines M 2 , M 3 , . . . that c ompute ~ E • – ≀ ≀ G (2), ~ E • – ≀ ≀ G (3), . . . . Thus, for an y n , we can effectively construct a machine M n with M n | = ~ E • – ≀ ≀ G ( n ). No w, the goal pr oblem ~ E • – ≀ ≀ ⊓ xG ( x ) is computed by a machine that waits till the environmen t brings the game down to ~ E • – ≀ ≀ G ( n ) for some particula r n , after which it constr uc ts M n and, by simulating it (or “turning itself into M n ”), pla ys the rest of the game as M n would. ✷ Exercise 10 . 6 Show that CLA1 ⊢ ⊓ x ⊓ y ( x = y ⊔ x 6 = y ). W e wan t to clo se this section by observing one fa c t. 9 In view of Axiom 7, Example 10.1, E xample 10.2 and Ex e r cise 10.6, any interpretation for which the axio ms and r ules of CLA1 ar e sound (and which is 8 This i s the only place w here we rely on the condition that the pro of under consideration is an extended one. One can get b y without “extended” i s one is will i ng to l et the algorithm search f or the needed CL12 -pro ofs. Such an algori thm, ho wev er, wo uld b e dramatically more complex than the pr esen t one. 9 Po inted out by the r eferee. 24 therefore a mo del of P A ) must in terpr et ′ , + a nd × as r ecursive functions and = as a recur s ive r e la tion. So, (one of the versions of ) T ennenbaum’s well-known theo rem applies and says that the interpretation must be elementarily equiv alent to the standard mo del. Thus, co mputability lo gic go es well b eyond tra ditional first-order log ic, not only by its computation-base d semantics but also by being a ble to characterize the standard mo del of arithmetic. 11 Some v aria tions of CLA 1 Among the main purp oses of the present article was just to intr o duc e CL-bas ed applied theo ries, in the par - ticular form of the CL-bas ed ar ithmetic CLA1 . There is a tremendous amo unt of interesting a nd pr omising work in the direction of exploring such theories and their metatheories, left as a challenge for the future. In this line o f resea rch, some o ther versions o f CL-based a rithmetic might a s well b e at leas t equally interesting. One such version is the extension of CLA1 that we call CLA2 . The la ng uage of the latter is no longer limited to 0 , = , ′ , + , × , but ra ther is the full language of CL12 , having all cons ta nts and a n infinite supply o f fresh pre dicate and function symbols for each arity , with the ab ove symbols having a sp ecial status. O ther than that, the axio ms and rules of CLA2 are virtually the same a s those of CLA1 (only , Axiom 8 a nd the t wo rules no longer limited to CLA1 -formulas). By a standard i n terpretation we now mean any int er pr etation that agrees with the standard interpretation † of Section 1 0 on = , ′ , + , × . And let us say that a formula or sequent X o f the la nguage of CLA2 is CLA2-v alid iff there is a BMEPM M such that, for any standar d interpretation ∗ , M | = X ∗ . Going back to o ur pro of of the soundness of CLA1 , o ne can se e that the pr o of of Theo rem 10.5 never really relie d on the fact that the languag e of CLA1 was limited to = , ′ , + , × . So, that theor em can b e without any additional efforts strengthened to the following one: Theorem 11. 1 F or any formula or se quent X , if C LA2 ⊢ X , then X is CLA2 -valid. F urt hermor e, t her e is an effe ctive pr o c e dur e that takes an arbitr ary ext ende d CLA2 - pr o of of an arbitr ary formula or se quent X and c onstructs a BMEPM (or EPM, or HPM if you like) M s u ch t hat, for every standar d interpr etation ∗ , M | = X ∗ . Among the rea sons for wan ting to co nsider CLA2 instea d of CLA1 can b e the grea ter flexibility and conv enience that it o ffer s. Mathematicians usually do not confine themselves to strictly fixed co llections of symbols such as = , ′ , + , × , a nd feel fr ee to introduce new function o r predicate symbols in their a c tivities. CLA2 allows us to directly a ccount for that kind of prac tice . Sp ecifically , one ca n int r o duce a num ber of predicate a nd/or function symbols through definitions E 1 , . . . , E n , a nd then prov e a formula F co nt aining such symbols through proving the sequent E 1 , . . . , E n • – ≀ ≀ F in CLA2 . As a n example, b elow we show how a ll primitive-recursive functions ca n be prov en computable in CLA2 , with CLA2 thus providing a to ol for actually computing any such function. Obviously the s ame ca n b e done within CLA1 as well, as pr imitive-recursive functions can even tually b e expr e ssed only using = , ′ , + and × . But this would b e a very indirect wa y , with the resulting pr o ofs leading to terrible a lgorithms. In contrast, CLA2 pro duces natural pro ofs, yilding algorithms for computing primitive-recursive functions by directly using their primitive-recursive constr uc tio ns. W e first repr o duce so me definitions from [15]. Let f b e a function symbol o f the indicated (by the n umber of explicitly shown arguments) a rity . An absolute prim itiv e-recursi v e defini tion of f is a formula o f one of the following for ms: (I) ∀ x  f ( x ) = x ′  . (I I) ∀ x 1 . . . ∀ x n  f ( x 1 , . . . , x n ) = 0  . (I I I) ∀ x 1 . . . ∀ x n  f ( x 1 , . . . , x n ) = x i  (some i ∈ { 1 , . . . , n } ). And a relativ e primitive-recursiv e definitio n of f is a formu la of one of the following forms: (IV) ∀ x 1 . . . ∀ x n  f ( x 1 , . . . , x n ) = g  h 1 ( x 1 , . . . , x n ) , . . . , h m ( x 1 , . . . , x n )   . 25 (V) ∀ x 2 . . . ∀ x n  f (0 , x 2 , . . . , x n ) = g ( x 2 , . . . , x n )  ∧ ∀ x 1 ∀ x 2 . . . ∀ x n  f ( x ′ 1 , x 2 , . . . , x n ) = h  x 1 , f ( x 1 , x 2 , . . . , x n ) , x 2 , . . . , x n   . W e say that (IV) defines f in terms of g , h 1 , . . . , h m . Simila rly , we s ay that (V) defines f in terms of g and h . A primitive-recur si v e construction of f is a sequence E 1 , . . . , E k of formulas, where each E i is a primitive-recursive definition of so me g i , all such g i are distinct, g k = f a nd, for each i , E i is either a n absolute primitive-recursive definitio n of g i , or a relative primitive-recursive definition of g i in terms of some g j s with j < i . Prop ositi on 11.2 L et f b e an n -ary function symb ol, and ~ E = E 1 , . . . , E k a primitive-r e cursive c onstruc- tion of f . Then CLA2 ⊢ ~ E • – ≀ ≀ ⊓ x 1 . . . ⊓ x n ⊔ y  f ( x 1 , . . . , x n ) = y  . Pro of. Assume the c o nditions of the prop os ition. W e prove it by induction on k , c onsidering five po ssibilities for E k . As hypotheses of the following deductions, we only include thos e E i ( i < k ) that ar e explicitly used. The other E j , of course, ca n be v acuously added to the list of h yp othes es. Case (I) : E k has the form (I). B e low is a CLA2 -pro o f of the desired sequent: 1. ∀ x  f ( x ) = x ′  Hypo thesis 2. ⊓ x ⊔ y ( y = x ′ ) Axiom 7 3. ⊓ x ⊔ y  f ( x ) = y  LC: 1, 2 Case (II) : E k has the form (II). Be low is a CLA2 -pro of of the desired sequen t. 1. ∀ x 1 . . . ∀ x n  f ( x 1 , . . . , x n ) = 0  Hypo thesis 2. ⊓ x 1 . . . ⊓ x n ⊔ y  f ( x 1 , . . . , x n ) = y  LC: 1 Case (III) : E k has the form (II I). Be low is a CLA2 -pr o of of the desired sequent. 1. ∀ x 1 . . . ∀ x n  f ( x 1 , . . . , x n ) = x i  Hypo thesis 2. ⊓ x 1 . . . ⊓ x n ⊔ y  f ( x 1 , . . . , x n ) = y  LC: 1 Case (IV) : E k has the form (IV). Below is a (lazy) CLA2 -pro of of the desired sequent. 1. ∀ x 1 . . . ∀ x n  f ( x 1 , . . . , x n ) = g  h 1 ( x 1 , . . . , x n ) , . . . , h m ( x 1 , . . . , x n )   Hypo thesis 2. ⊓ x 1 . . . ⊓ x m ⊔ y  g ( x 1 , . . . , x m ) = y  Prov able by the induction h yp othesis 3. ⊓ x 1 . . . ⊓ x n ⊔ y  h 1 ( x 1 , . . . , x n ) = y  Prov able by the induction hypothesis · · · m+2. ⊓ x 1 . . . ⊓ x n ⊔ y  h m ( x 1 , . . . , x n ) = y  Prov able by the induction hypo thesis m+3. ⊓ x 1 . . . ⊓ x n ⊔ y  f ( x 1 , . . . , x n ) = y  LC: 1, . . . , m+2 Case (V) : E k has the form (V). Below is a (lazy) CLA2 -pro of of the desired sequent. 1. ∀ x 2 . . . ∀ x n  f (0 , x 2 , . . . , x n ) = g ( x 2 , . . . , x n )  ∧ ∀ x 1 ∀ x 2 . . . ∀ x n  f ( x ′ 1 , x 2 , . . . , x n ) = h  x 1 , f ( x 1 , x 2 , . . . , x n ) , x 2 , . . . , x n   Hypo thesis 2. ⊓ x 2 . . . ⊓ x n ⊔ y  g ( x 2 , . . . , x n ) = y  Prov able by the induction hypothesis 3. ⊓ x 1 ⊓ z ⊓ x 2 . . . ⊓ x n ⊔ y  h ( x 1 , z , x 2 , . . . , x n ) = y  Prov able by the induction hypothesis 4. ⊓ x 2 . . . ⊓ x n ⊔ y  f (0 , x 2 , . . . , x n ) = y  LC: 1, 2 5.1. ⊓ x 2 . . . ⊓ x n ⊔ y  f ( x 1 , x 2 , . . . , x n ) = y  Hypo thesis 5.2. ⊓ x 2 . . . ⊓ x n ⊔ y  f ( x ′ 1 , x 2 , . . . , x n ) = y  LC: 1, 3, 5.1 6. ⊓ x 1 ⊓ x 2 . . . ⊓ x n ⊔ y  f ( x 1 , x 2 , . . . , x n ) = y  CI: 4, 5 26 ✷ The other CL-based s ystem o f a rithmetic that we wan t to in tro duce b efore closing this se c tion is CLA3 . While CLA2 is more ex pressive than CLA1 , CLA3 is a mo dification of C LA1 in the opp osite direc tio n: the languag e of CLA3 is obtained fr o m the la nguage of CLA1 by forbidding the blind quantifiers ∀ and ∃ . The r ules of inference of CLA3 are the same LC a nd CI as in CLA1 , only now res tricted to ∀ , ∃ -free formulas. And the axioms are the follo wing: 1. ⊓ x (0 6 = x ′ ) 2. ⊓ x ⊓ y ( x ′ = y ′ → x = y ) 3. ⊓ x ( x + 0 = x ) 4. ⊓ x ⊓ y  x + y ′ = ( x + y ) ′  5. ⊓ x ( x × 0 = 0) 6. ⊓ x ⊓ y  x × y ′ = ( x × y ) + x  7. ⊓ x ⊔ y ( y = x ′ ) In view o f the obvious fact that CL12 always prov es ∀ xE ( x ) • – ≀ ≀ ⊓ xE ( x ), all of the axioms o f CLA3 , which differ from the cor resp onding axioms of CLA1 only in that they a re ⊓ - r ather than ∀ -prefix ed, ar e prov a ble in CLA1 . This, in turn, implies that CLA3 , just like CLA2 , inherits the soundness of CLA1 in the strong form of Theorem 10.5. F rom the philoso phica l point of view, the p o tential int er est in CLA3 is rela ted to the fact that it is a perfectly c onstructive system of arithmetic and, unlike s o me other systems with co nstructivistic claims such as Heyting’s intuitionistic-calculus-ba s ed arithmetic, has a clear a nd convincing constructive s emantics. What ma kes CLA3 “ pe rfectly constructive” is that it gets rid of the only p otentially “dubious” op era tors ∀ and ∃ , with all the remaining op er ators be ing fully imm une to an y do ubts or criticism from even the most radical constructivis tic p o int of view. A r elev an t obse r v ation to be made her e is that forbidding ∀ , ∃ in the formulas of CLA3 a uto matically propag ates to the underlying logic C L12 as well: it is not ha rd to see that, when the pre mises and conclusio n o f LC are ∀ , ∃ - fr ee, then s o ar e all intermediate steps p er formed within CL12 when justifying an application of LC. That mea ns that, in fact, the underlying logic of C LA3 is the ∀ , ∃ -free fragment o f CL12 rather than the full CL12 . On the mathematical s ide, an imp o r tant feature of CLA3 is that its co nc e pt of “truth” (=computability), unlike tha t o f CLA1 or just P A , can b e expressed in the lang uage of P A . Sp ecifically , one could show that the arithmetical complexity of the predicate of “truth” fo r CLA3 - formulas is Σ 3 . Replacing the classical qua ntifiers in such a predica te by their choice co unterparts ⊓ a nd ⊔ yields a formula which, in a sense, expresses the “ truth” pr edicate of CLA3 in the language o f CLA3 its e lf. This unusual phenomenon may yield un usual effects in the metatheory of CLA3 , whic h can ma ke CLA3 a n interesting sub ject for metainv estigations. 12 The closure of static games under d fb-reduction This section s ho uld b e tr eated a s an app endix, which ma ny reader s may (safely) de c ide to skip. In it we pr ov e a technical but impor tant fact, accor ding to which • – ≀ ≀ preserves the static prop er ty of games. Not passing the test for pres erving the static prop erty would disqualify a ny ga me op eratio n from being co nsidered within the c ur rent framework of CL, as only static games are “well behav ed”, and man y steps in our earlier pro ofs (spe c ifically , the pr o ofs of the closur e pro p erties of Section 6, the pro of of the soundness and completeness of CL12 , o r the pro ofs o f the s oundness of CLA1 , CLA2 , CLA3 ) would fail if the underlying games were not static. Let us remember the key r elev ant definitions first. W e say that a run ∆ is a ⊤ -delay of a r un Γ iff the following tw o conditions are satisfied: • F or either player ℘ ∈ { ⊤ , ⊥ } , er asing all ℘ -lab eled moves in Γ results in the sa me run as eras ing all ℘ -lab eled mov es in ∆. 27 • F or all k and n , if the k th ⊤ -lab eled mov e is made later than (is to the rig ht o f ) the n th ⊥ -lab eled mov e in Γ, then so is it in ∆. ⊥ -delay is defined similarly , with ⊤ a nd ⊥ interchanged. Now, we say that a constant game A is static iff, for any play er ℘ , run Γ and ℘ -delay ∆ of Γ, the following tw o conditions are satisfied: 1. If Γ is no t a ℘ - illegal run of A , then neither is ∆. 2. If Γ is a ℘ -won r un of A , then s o is ∆. And a nonconstant game is considered static iff so are all o f its instances. Remark: Another eq uiv alen t appro ach, taken in all ea rlier pap ers on CL ex cept [6], only s tipulates the second co ndition in the definition of sta tic games . That approach, how ever, assumes existence of a mov e ♠ that is illegal in every p osition of every game. As it turns out (Lemma 4.7 of [1]), the first condition of our pr esent definition o f static games is then a utomatically implied by the second co ndition, so there is no need for explicitly stating it. Without stipulating the exis tence of an a lwa ys-illega l move, how ever, b oth conditions are necessar y . Our Definition 3.3 of • – ≀ ≀ was limited to finite-depth games. The o nly reaso n for ado pting this limitation was the des ir e to keep things simple, as finite-depth ga mes were sufficient for our present treatment: every formula o f CL12 expres ses a finite- de pth game, as e le ment ar y ga mes are finite-depth, and the op erations ¬ , ∧ , ∨ , ⊓ , ⊔ , ⊓ , ⊔ , ∀ , ∃ (but not • – ≀ ≀ ) pr e s erve the finite-depth prop erty . How ever, for p ossible future needs, in this app endix-style technical section where we no long er car e ab out simplicity , it would not hurt to generalize • – ≀ ≀ to all g ames b efor e proving that it pr eserves the sta tic prop erty of g a mes. The following definition do es this job. Here are some terminolo gical and notatio nal co nven tions for Definition 12 .1 . The predecessor of a nonempty finite bit string w is the longe st prop er prefix o f w . F o r a bit string w and run Γ, Γ  w means the result of deleting from Γ all labmoves ex cept those that loo k like ℘u.α fo r some u  w , and then fur ther deleting “ u. ” in each such ℘u.α . Similar ly , Γ S. means the r esult of deleting from Γ all labmov es exc ept those that lo ok like ℘S.α , and then further deleting “ S. ” in each such ℘S.α . Definition 12.1 Let A 1 , . . . , A n and B be constant games, and T a tr ee of g a mes with yield h A 1 , . . . , A n i ( n ≥ 1). Let w 1 , . . . , w n be the addresses of A 1 , . . . , A n in T , resp ectively . The g ame T • – ≀ ≀ B is defined by: Lr : Γ is a legal run of T • – ≀ ≀ B iff the following co nditions are satisfied: 1. E very la bmov e of Γ lo oks lik e ℘S.β , ⊤ w : or ℘w .β , wher e ℘ is either pla yer, β is a mov e, and w is a finite (po ssibly empt y) bit string . 2. Whenever Γ contains ⊤ w :, it do es not hav e any other o ccurr ences o f the s a me labmov e, and either w is o ne of w 1 , . . . , w n , o r the labmove ⊤ w : is preceded by ⊤ u :, 10 where u is the predeces sor of w . W e call labmov es of the form ⊤ w : repl icativ e . 3. Whenever Γ co ntains ℘w .β , e ither w is a (not neces s arily prop er) pr efix of one of w 1 , . . . , w n , or else the labmov e ℘w.β is prec e ded b y ⊤ u :, where u is the predecessor of w . 4. Γ S. is a legal run of B . 5. F or any i ∈ { 1 , . . . , n } and any infinite bit string u , Γ  w i u is a le g al run of ¬ A i . Wn : A lega l r un Γ of T • – ≀ ≀ B is won by ⊤ iff Γ contains only finitely many replicative labmov es, and either Wn B h Γ S. i = ⊤ or, for some i ∈ { 1 , . . . , n } and some infinite bit string u , 11 Wn ¬ A i h Γ  w i u i = ⊤ . It is not hard to see that, when r estricted to finite-depth games, the ab ove definition of • – ≀ ≀ is equiv alent to Definition 3.3. Prop ositi on 12.2 L et A 1 , . . . , A n and B b e arbitr ary static games, and T b e a tre e of games with yield h A 1 , . . . , A n i ( n ≥ 1 ). Then the game T • – ≀ ≀ B is also st atic. 10 Here and b elo w, “preceded” do es not mean “immediately preceded”. Rather, it means that Γ lo oks li ke h . . . ⊤ u : . . . ⊤ w : . . . i . 11 In fact, considering only certain “sufficien tly long” finite bit stri ngs wou ld do the job here, but why bother. 28 Pro of. Supp os e A 1 , . . . , A n , B and T a r e as in the assumption of the lemma. W e may also sa fely assume tha t the n + 1 ga mes ar e co ns tant, or e ls e pick a n a rbitrary v aluation e and r e place them with e [ A 1 ] , . . . , e [ A n ] , e [ B ]. Le t ℘ b e either player, Γ an y run, and ∆ any ℘ -delay o f Γ. V erifying the first c ondition of the definition of static games . Ass ume ∆ is a ℘ -illegal r un of T • – ≀ ≀ B . W e wan t to s how that then so is Γ. Below, when w e simply say “ le gal”, “ ℘ -illegal” , etc., we mea n (b eing a) legal, ℘ -illegal, etc. r un of T • – ≀ ≀ B . Let h Ψ , ℘α i b e the shortest ℘ -illega l initial seg ment of ∆. Let h Φ , ℘α i b e the shortest initial s egment of Γ containing all the ℘ -lab eled moves of h Ψ , ℘α i , and let Θ b e the sequence of tho s e ℘ -lab eled moves of Ψ that are not in Φ. W e o bviously hav e h Ψ , ℘α i is a ℘ -delay of h Φ , ℘α, Θ i . (7) If Φ is ℘ -illegal, then so is Γ a nd we are done. Assume now that Φ is not ℘ -il le gal. (8) W e claim that then Φ is le gal. (9) Indeed, s uppo se that this was no t the case . Then, by (8), Φ sho uld b e ℘ -illegal. This would make Γ a ℘ -ille g al run with Φ as a n illeg al initia l segment which is sho rter than h Ψ , ℘α i . Then, by the induction hypothesis, any r un for which Γ is a ℘ -delay , would b e ℘ -illegal. But the fact that ∆ is a ℘ -delay o f Γ obviously implies that Γ is a ℘ -delay of ∆ (Lemma 4.6 of [1]). So , ∆ would b e ℘ -illeg al, whic h is a co ntradiction b ecause , according to our assumptions, ∆ is ℘ -illegal. W e are contin uing our pro of. There are five po ssible r easons to w hy h Ψ , ℘α i is ℘ - illegal (while Ψ b eing legal): R e ason 1 : ℘α do es not hav e the form ℘S.β , ⊤ w : or ℘w .β . B ut then, for the sa me r eason, h Φ , ℘α i is illegal. This, in view of (9), mea ns that h Φ , ℘α i is ℘ - illegal. Hence Γ, as a n ex tension of h Φ , ℘α i , is also ℘ -illegal, as desired. R e ason 2 : ℘α is ⊤ w :, but it viola tes co ndition 2 of the Lr clause of Definition 12.1 . It is not hard to see that then, for the s a me reaso n, h Φ , ℘α i is illegal. This, in turn, as in the pr e vious case, implies tha t Γ is ℘ -illegal. R e ason 3 : ℘α is ⊤ w .β , but it violates condition 3 of the Lr clause of Definition 12.1 . Ag ain, for this very reason, h Φ , ℘α i can b e seen to b e illegal. This, as in the previous cases, implies that Γ is ℘ -illegal. R e ason 4 : h Ψ , ℘α i S. is an illeg al r un of B . This means it is a ℘ - illegal run o f B , be cause we know that Ψ is a le g al r un of T • – ≀ ≀ B . Obviously (7) implies that h Ψ , ℘α i S. is a ℘ - de lay o f h Φ , ℘α, Θ i S. . Hence, a s B is static, h Φ , ℘α, Θ i S. is also a ℘ -illeg al run of B . F r om here, tak ing into account that Θ do es not hav e any ℘ -lab eled moves, we find that h Φ , ℘α i S. is a ℘ -ille gal run of B . This, together with (9), obviously implies that h Φ , ℘α i is a ℘ -illega l run o f T • – ≀ ≀ B , a nd hence so is its ex tension Γ, as desired. R e ason 5 : F or some i ∈ { 1 , . . . , n } and some infinite bit string u , h Ψ , ℘ α i  w i u is not a legal run of ¬ A i . Fix these i and u . This c a se is very similar to the pr evious one. Since Ψ is a legal run of T • – ≀ ≀ B , we should have that h Ψ , ℘α i  w i u is a ℘ -illega l run of ¬ A i . Also , (7) implies tha t h Ψ , ℘α i  w i u is a ℘ -de lay of h Φ , ℘α, Θ i  w i u . Therefore, as A i and hence ¬ A i are static, h Φ , ℘α, Θ i  w i u is a ℘ - ille gal run of it. F urther reasoning as in the pr evious case, we find that h Φ , ℘α i a nd hence Γ is a ℘ -illegal run of T • – ≀ ≀ B . V erifying the se c ond c ondition of the definition of st atic games . Here, aga in, when we simply say “ legal”, “ ℘ -illega l” , “ ℘ -won”, etc., we mean (being a) lega l, ℘ -illegal, ℘ -won etc. run of T • – ≀ ≀ B . Assume Γ is won b y ℘ . W e wan t to show tha t then so is ∆. If ∆ is ℘ -illegal, then it is won by ℘ and we are done. So, ass ume that ∆ is not ℘ -illega l. But, as noted earlier, Γ is a ℘ - delay of ∆ and, therefo r e, in view of the alrea dy prov en fact that the first condition of the definition of static ga mes is satisfied, we find that Γ is not ℘ -illegal, either . Γ also cannot b e ℘ -illeg al, for otherwise it would not b e won b y ℘ . Consequently , ∆ cannot b e ℘ -illega l either, for other w is e Γ would b e ℘ -illegal. Thus, we hav e narr ow ed down our consider ations to the ca se when bo th Γ and ∆ ar e legal. If the reaso n of Γ’s b eing won by ℘ is that Γ has infinitely many replica tive la bmov es and ℘ = ⊥ , then ∆ is w on by ℘ for exactly the same reaso n. 29 Assume no w Γ only has a finite n umber of replicative labmo ves (and hence so do es ∆). Suppo se Γ S. is a ⊤ -won r un o f B , s o that ℘ (the winner in Γ) is ⊤ . ∆’s b eing a ⊤ - delay o f Γ implies that ∆ S. is a ⊤ -delay of Γ S. , and hence, a s the latter is a ⊤ -won run of the static g ame B , so is the former . This, in turn, implies that ∆ is a ⊤ - won (i.e., ℘ -won) r un of T • – ≀ ≀ B , a s des ired. Suppo se now Γ S. is not a ⊤ -won run o f B . Then, o ur assumption that Γ is a ℘ - won run of T • – ≀ ≀ B can b e seen to imply tha t, fo r each i ∈ { 1 , . . . , n } and each infinite bit string u (if ℘ = ⊥ ), o r s ome such i a nd u (if ℘ = ⊤ ), Γ  w i u is a ℘ -won run of ¬ A i . T aking into acc ount that ∆  w i u is ob vio usly a ℘ -delay of Γ  w i u and that ¬ A i is static, the ab ov e, in turn, implies that for each i and u (if ℘ = ⊥ ), or some i and u (if ℘ = ⊤ ), ∆  w i u is a ℘ - won r un of ¬ A i . In addition, if ℘ = ⊥ , then ∆ S. is a ℘ - won r un of B , beca use so is Γ S. , B is static and ∆ S. is a ℘ -delay of Γ S. . All this implies the desired co nclusion that ∆ is a ℘ - won run of T • – ≀ ≀ B . ✷ References [1] G. Ja paridze. Int r o duction to c omputability lo gic . Annals of Pure and Applied Logic 123 (20 03), No.1-3, pp. 1-99. [2] G. Ja paridze. 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