Many concepts and two logics of algorithmic reduction

Man y concepts and t w o logics of algor ithmic reduction Giorgi Japaridze Institute of Artificial In telligence, Xiamen Univ ersit y and Departmen t o f Computing Sciences, Villano v a Univ ersit y Abstract Within the program of finding axiomatizations for v arious parts of c omputability l o gic , it w as prov en earlier that the logic of in teractive T uring reduction is exactly the implicativ e fragmen t of Heyting’s intuitionistic calculus. That sort of reduction p ermits unlimited reusage of the computational res ource represen ted by the anteceden t. A n at least equally basic and natural sort of algorithmic reduction, h o wev er, is the one t h at do es n ot allo w such reusage . The present article shows th at t urning the logic of t h e first sort of reduction into th e logic of the second sort of reduction takes nothing m ore than just deleting the contractio n rule from its Gentzen-st yle axiomatization. The first (T uring) sort of interac tive reduction is also sho wn to come in three n atural versions. While those three versions are very different from eac h other, their logical b ehaviors (in isolation) turn out to b e indistinguishable, with that common behavior b eing precisely captured by implicative intuitionistic logic. Among the other contri b u tions of the present article is an informal introduction of a series of n ew — fin ite and b ounded — v ersions of recurrence operations and th e asso ciated reduction op erations. MSC : primary: 03B4 7; secondar y: 03F50; 0 3B70; 68 Q10; 68T2 7; 68T30 ; 91A05 Keywor ds : Computabilit y logic; Intuitionistic logic; Affine logic; Linea r logic; Interactive c ompu- tation; Game semantics. 1 In tro du ction This ar ticle is a new addition to the evolving lis t of pap e rs [7, 8, 9, 1 0, 12, 13, 14, 15, 16, 18, 19, 20] devoted to finding axiomatizatio ns for v arious fra gments of c omputabili ty lo gic . The la tter is a progra m fo r redeveloping logic as a formal theor y of computability , as opp os ed to a formal theory of tr uth which it has mo r e traditionally b e e n. Under the approach of computability logic, formulas ex press interactive computational prob- lems defined as ga mes betw een the tw o play ers ⊤ ( machine ) and ⊥ ( envir onment ), with log ic al op erator s standing fo r basic op erations on games . “T ruth” of a pro ble m/ game means existence of an algorithmic solution, i.e. ⊤ ’s effective winning stra tegy . And v alidit y of a logic a l formula is understo o d as (such) tr uth under every particular interpretation o f a toms. With this semantics, computability log ic provides a s ystematic answer to the fundamen tal ques tion “ what c an b e c om- pute d? ”, just as clas sical logic is a systematic to ol for telling what is true . F urthermore, as it turns out, in p os itive cases “ what can b e computed” always allows itself to b e repla ced by “ how can be computed”, which makes co mputability lo gic of p otential interest in not only theoretical com- puter science, but many mor e applied areas as well, including in tera ctive knowledge base systems, resource oriented systems for planning and action, o r decla r ative pr ogra mming la nguages . On the lo gical side, computability log ic can ser ve as a constr uctive and co mputationally meaningful alternative to classical log ic a s a basis for applied theories. The first concrete steps in the direction of materializing this p o tential have bee n ma de very rec ently in [19], where a co mputabilit y- logic- based sy stem of ar ithmetic was co nstructed — a formal theory whose every formula expr esses a computational problem and every pro of enco des an algorithmic s olution for such a pr oblem, th us fully reducing problem-s olving to theorem-pr oving. 1 Having sa id the ab ov e, motiv ationally or technically (re)introducing co mputability logic is no t within the sco pe of the present paper . This jo b has b een done in [6, 11, 17], and the present pa p er , whose goa l is merely putting one more brick in to the foundation of the edifice under constructio n, primarily tar gets r eaders alrea dy familia r with the bas ics of computability logic. Y et, as it happ ens, the pro of of the main technical result of the pape r, given in Sections 2-4, can be understo o d in full detail without knowing muc h (if a nything at all) ab out computability lo gic. Those with no pr ior acquaintance with the sub ject may benefit from browsing the rest of the pap er just a s well. Even though doing so w ould b e certainly insufficien t for getting full insights into the pro ject, c hances are that such a reader may at least start feeling cur ious enough to b e willing to lo ok a t so me additional literature. The most recommended rea ding for familiarity with the basic philoso phy , motiv ations, concepts and techniques of computability lo gic is the tutorial-style [17]. Here we very quickly review, in a simplified form, ce r tain basic concepts on the ga mes used in c omputability logic, to r efresh the memory of those previo usly ex po sed to the sub ject, and to provide some clues to those who hav e never seen it. A m o ve means a finite string ov er so me fixed alphab et, s uch as the standar d keyb o ard alphab et. A lab el ed mov e is a move prefixed with ⊤ or ⊥ . The meaning of such a prefix (“ lab el ”) is to indicate which of the tw o play ers has made the mov e. A run is a (finite or infinite) sequence o f lab eled mov es, and a p ositi o n is a finite run. Runs (a nd p ositions a s sp ecia l cas es of r uns) ar e th us records o f interaction histories, sp elling out what moves, in what order and by which players hav e b een made during a given play of a g ame. A game 1 is a pair ( Lr , Wn ) consis ting o f what a re called its structure ( Lr ) a nd conten t ( Wn ). One o f the many equiv alent wa ys to de fine the structure co mpo nent of a ga me is to say that it is a binary relation b etw een po sitions and lab eled mov es. Then the intu itive meaning of Lr (Φ , ℘α ) is that α is a le g al mo ve by play er ℘ in p ositio n Φ. A run where a ll mov es a re lega l (in the p ositio ns preceding those moves) is said to b e a l egal run . T he empty run is thus alwa ys trivially legal. As e x p e c ted, “ ille gal ”, whether it b e a move or a run, means “not le g al”. As for the conten t Wn of a game, it can b e defined as a set of leg al runs , whose elements are said to b e (and in tuitively thoug ht of as ) the runs won by play er ⊤ (and hence lost by ⊥ ), with all o ther leg al runs consider ed l ost b y ⊤ (and hence won by ⊥ ). As for illegal runs, they ar e alwa ys considered to b e lost b y the player who made the firs t illegal mov e. Note the relaxed nature o f such g ames. There are no c o nditions on the order in which mov es should or could b e made (such a s , say , s trict alter nation o f players’ turns), and g e nerally either play er may hav e leg al mov es in a given p o sition/situation. This makes the games of computability logic a rather direct (without any “bureaucr atic p ollutants”) a nd flex ible to ol for mo deling in ter- action, including asynchronous interactions. The re la xed nature of our g ames makes it imposs ible to unders ta nd ga me- playing strategies as functions from p ositions to mov es, a s this is typical for most other game mo dels . Ins tead, ( ⊤ ’s effective) stra tegies ar e understo o d a s in tera c tive machines. Such a ma chine is nothing but a T ur ing machine with the additional capability of mak ing moves. The a dversary can a lso move at any time, with suc h mo ves b eing the o nly nondeterministic even ts from the machine’s p ersp ective. The play is fully visible to the machine through an additional, read-only ru n tap e whic h, at any time, spe lls the “cur rent p ositio n” of the play . W e say tha t such a machine wi ns a g iven game iff, no matter how the adversary acts (what mov es it makes and when it makes them), the run incrementally sp elled on the run tap e is won by ⊤ . A universal-utility ga me semantics should b e ab o ut interaction, w he r eas functions ar e inher- ent ly non-interactive. The ab ov e-mentioned traditiona l, str ate gies-as-functions , approach miss es this imp o r tant p oint and cr eates a hybrid of interactive (games) and non-interactive (functions) ent ities. T o see the resulting loss, it would b e sufficient to reflect on the b ehavior of one’s per sonal computer. The jo b of your computer is to play o ne long — p otentially infinite — ga me ag a inst you. Now, hav e you noticed y o ur “a dversary” getting slow er ev er y time you use it? Pro bably not. That is b ecause the co mputer is s mart enough to follow a non-functiona l strategy in this game. If 1 T o what we r efer as a “game” in this pap er, is in fact called a “constan t game” in computabilit y l ogic, and the term “game” is reserved for a slight ly more general concept. Considering only constant games is sufficient for our presen t purposes thoug h and, to keep things simple, we are using the term “game” for them. 2 its strateg y w as a function from p ositio ns (in ter a ction histories) to mov es, the res po nse time would inevitably keep w or sening due to the need to rea d the entire — c o ntin uously lengthening a nd, in fact, pr actically infinite — interaction his tory every time b efor e r e sp onding. Defining s trategies as functions o f o nly the latest mov es (rather than entire interaction histo ries) in Abra msky a nd Jagadees an’s [1] tradition is a lso no t a wa y out, as typically mo re than just the last move matters. Back to your p ers onal computer, its actions certainly dep end on more than your la st keystroke. Thu s, the difference b etw een the traditio nal functional strategies and the p ost-functional strateg ies of computability logic is not just a matter of taste or con venience. It will beco me es p e c ially imp or- tant when it comes to (y et to b e developed) in tera ctive complexity theor y : ha rdly any meaningful int er active complexity theor y ca n be done with the s trategies- a s-functions appro ach. And com- plexity issues will inev ita bly come forward when computability logic or similar approa ches achiev e a cer ta in degree of maturity: nowada ys, 95% of the theory of computation is ab out co mplexity rather tha n just co mputability . Time has not yet matur e d for serio usly addressing co mplexity issues within the framework of co mputability logic tho ugh, and the latter , including the pr esent pap er, contin ues to be fo cused on just computability , which still ab ounds with op en questions waiting for answers. In the ab ove outline, w e describ ed interactiv e T uring machines in a relaxed fashion, leaving to the rea de r filling technical details ab out, say , how, exactly , mov es are made by the machine, how many mov es either play er can make at once, what happ ens if b oth players attempt to mov e “si- m ultaneo usly”, etc. As it turns out, all rea sonable design choices yield the sa me cla ss of winnable games as lo ng as we consider a certain natur al sub class o f g ames called s tatic . Such g a mes are obtained by imp os ing a cer tain simple forma l condition on g ames (s e e, e.g., Section 5 of [17]), which we do not repro duce here a s nothing in this pap er relies on it. W e will only po int out that, intu- itively , static games are in tera c tive tasks whe r e the relative sp eeds of the play er s ar e irrelev ant , as it never h urts a player to p ostp one making mo ves. In o ther words, static games ar e the games that are contests of intellect ra ther than co ntests of sp eed. And o ne of the theses tha t computability logic philoso phically relies on is tha t static games pr esent an a dequate formal c ounterpart of our int uitive co ncept of “pure”, s p e e d-indep endent int er active co mputational problems . Cor resp ond- ingly , computability logic restricts its attention (mo r e sp ecifica lly , p ossible interpretations of the atoms of its for mal languag e) to static games. Needless to say , the clas s of static games is close d under all game op erations studied in computability logic. Among the most interesting of such op erations are s everal v er s ions of r e duction . The simplest form of reduction (of B to A ) is A → B . This is a parallel play of the tw o games A and B with the roles of ⊤ and ⊥ interc hanged in the a nt ecedent. Winning a g iven run of this g ame for ⊤ means that whenever the adversary wins A , ⊤ has to win B . More formally , ev ery lega l mov e of A → B has to b e prefixed with one of the tw o string s “0 . ” or “1 . ” to indicate in which o f the tw o comp onents the mov e is made. The effect of a move 0 .α is making mov e α in the anteceden t, and the effect of 1 .α is making mov e α in the consequent. In order for such a mov e 0 .α o r 1 .α to b e legal, α should b e a lega l move in (the co rresp o nding po sition of ) the co rresp o nding comp onent A or B . Then a legal r un Γ of A → B is co nsidered won b y ⊤ iff Γ 1 . is a ⊤ -won r un of B or ¬ Γ 0 . is a ⊥ -won run of A . Here Γ 1 . means the r esult of deleting fro m Γ all moves except tho se o f the for m 1 .α , and then further deleting the prefix “1 . ” in such mov es . Similarly for Γ 0 . . And ¬ Γ 0 . means the result of turning upside down (so tha t ⊤ bec omes ⊥ and vice versa) all lab els in Γ 0 . . As can b e felt from the ab ov e passa ge, formal definitions may not b e as nice to work with as informal o r intuit ive explanatio ns. F o r this reason, our subsequent ex planations of game op erations in this section will be limited to informa l ones, keeping in mind that they cer tainly can b e turned int o strict tec hnica l definitions. Since the r o les of the play er s ar e switched in the anteceden t of A → B , the A compo nent, a s a computational pro blem from ⊥ ’s p ersp ective, b ecomes a computational reso ur ce for ⊤ . Namely , ⊤ can o bserve how the adversary is solving /playing (a single s ession) of A , a nd utilize that informa tion in its own solving/playing B . The following example illus trates the ab ove-said. Let H b e the halting pr oblem , which ca n b e understo o d as a game of depth 2 (i.e., no legal r un has mor e than tw o mov es). 3 In the initial (empty) p osition of this ga me, only ⊥ has legal mo ves, and such a mov e should be the phrase “Does T ur ing machine m halt on input i ?”, where m is a leg itimate descr iption of a T uring machine and i a p ossible input for it. After such a mov e is made, the second and last leg al mov e is by ⊤ , which should b e either “Y es” or “ No ”. ⊤ wins iff it cor rectly answers the question a sked by ⊥ . The fa ilur e b y ⊥ to make an initial mov e is considered ⊤ ’s win, as there was no question to answer. And, if such a mo ve is made, then the failure of ⊤ to r esp ond is considered ⊥ ’s w in. The ac c eptanc e pr oblem A is s imilar, o nly it is ab out whether a g iven machine accepts (ra ther than halts on) a given input. Neither H nor A is decidable, which o bviously means that these problems, as g ames, hav e no alg orithmic winning strategies . Ho wever, A is a lgorithmically reducible to H . Spec ific a lly , ⊤ do es hav e a n effective winning strateg y in the game H → A , which go es like this. W ait till, in the co nsequent, ⊥ asks a question reg arding whether a certain ma chine m accepts a certain input i . The n, in the a nteceden t, a sk a counterquestion reg arding whether m ha lts on i (the same m and i ). If a n a nswer to this counterquestion is “ No”, answer “No” to the o riginal question in the consequent and rest your case , as not halting implies no t a ccepting. Otherwis e , if the answer in the anteceden t is “Y es”, simulate machine m o n input i unt il it ha lts, a nd say “Y es” or “No” in the consequent de p ending on whether the simulation accepted or r ejected. (Of course, the p os sibility that the simulation go es on for ever is not ruled out here; but this would mean that m do e s not r eally halt on i , and having lied in the a nteceden t would make ⊥ lo se the game r egardles s of what happe ns in the co nsequent). In fact, → is not only the simplest but also the strongest form of reduction. In this res p e c t, at the other extreme is the w ea kest reductio n ◦ – . The game A ◦ – B c an be characteriz e d in the same intuitiv e terms as A → B , with the differe nce that, in A ◦ – B , unlike A → B , ⊤ is allowed to reus e A (as a c o mputational reso urce) a ny n umber of times, with “r euse” here understo o d in the str ongest algorithmic sense poss ible. Namely , at any time, ⊤ can temp or arily aba ndon a given po sition of A (while reser ving the right to c ome back to it la ter), backtrack to any of the earlier po sitions o f it and try a differ ent contin uation from there , th us forcing ⊥ to play m ultiple parallel sessions of A against such a capricio us adv er sary in this most unfair g ame: the failure of ⊥ to win A in al l se ssions of it a utomatically res ults in ⊤ ’s victo ry . A while ag o we saw how to reduce the accepta nce problem to the halting pro blem in the stro ng sense of → . W e would not hav e b een just as successful if instead of the acceptance problem A we had taken the Kolmo gor ov c omplexity pr o blem K , where the initial mov e o f the form “ What is the Kolmog orov complexity of num b er n ?” is by the environmen t, o bligating the ma chine to resp ond with a move “ m ” such that m is (indeed) the K olmogor ov complexity of n , i.e., m is the size of the s mallest T uring machine that retur ns n on input 0. One can show that, unlike H → A , the g ame H → K do es not have an alg orithmic winning strateg y . But the weak er game H ◦ – K certainly do es, due to the fact that, in it, the reduction is allow ed to use the anteceden t rep eatedly . Such a strateg y go es like this. W ait to hear a question ab out the Kolmo gorov complexity of a nu mber n in the consequent. Then, s ta rting fro m m = 0, do the following. Duplicate the original anteceden t and sav e one copy of it for future usag e (further duplications). In the other copy , ask the co unt er question r egarding whether the mach ine (enco ded by) m halts on input 0. If you hear “No”, increment m to m + 1 and rep eat the s tep. Otherwise, if you hear “ Y es”, simulate m on input 0; if the simulation shows that m returns n on input 0 , answer | m | (where | m | is the size of m ) to the or iginal question in the conseq ue nt, and w as h your hands. In any other cas e, increment m to m + 1 and rep eat the step. There is a whole spectr um of natural reduction oper ations of in termediate strength b etw een → and ◦ – . O nly some of those ha ve b een officially in tro duce d within the fr amework of co mputabilit y logic so far , with mor e to b e probably defined la ter dep ending on pa r ticular needs, motiv ations and tastes. It has b een re pe a tedly p ointed out earlier that the forma lism of computability logic is op en-ended, w elco ming any meaningful a ugmentations. Among the most natural and simple reductio n o pe r ations of int er mediate strength is > – . Jus t like A ◦ – B and unlike A → B , the g a me A > – B allows ⊤ to reuse A infinitely many times. But the form of reusa ge is les s flexible here: ⊤ is r e quired to r estart A from the very b eginning every time it wan ts to reuse it, meaning that it es sentially cannot utilize the adv antages of backtracking 4 per mitted in A ◦ – B . Sp ecifically , unles s ⊥ plays in exac tly the same ways in different parallel sessions of A , ⊤ has no p oss ibility to exp er iment with differe nt r eactions to the same actions by the a dversar y . Thu s, the difference betw een A > – B a nd A ◦ – B is in the allowed typ e of reusage of A , with the quantity of r eusages be ing otherwise unlimited. Y et, as it happ ens, this difference in the types o f reusage automa tica lly yields a difference in the qua nt ities a s w ell. Spec ifically , in A > – B at most countably many pa rallel runs of A ca n b e gener ated, while in A ◦ – B , when A has infinitely lo ng legal r uns, that quantit y can be a contin uum. A very s imple mo difica tion in the formal definition of ◦ – , g iven later in Section 5, turns it into a definition of the Blass-style [2] reduction ◦ – ℵ 0 , by its s tr ength strictly b etw een > – and ◦ – . The type of r eusage of A in A ◦ – ℵ 0 B is the same as in A ◦ – B , but the qua ntit y of reusag es is limited to the co untably infinite cardinal ℵ 0 . The o p e r ation ◦ – ℵ 0 is a pparently the weakest nontrivial str engthening of ◦ – . Both ◦ – and > – can b e further strengthened to ◦ – F and > – F by a llowing ⊤ to reuse the a nt ece de nt only a finite (yet unbounded) num b er o f times. In turn, the op er a tions ◦ – F and > – F can b e further str engthened to b ounded versions o f ◦ – , > – . The simples t form of a b o unded version of ⊃∈ { > – , ◦ – } would b e ⊃ n , wher e n is a natural num be r. It means the same as ⊃ , only the n umber of allowed (r e)usages of the a nt ece de nt is limited to n , s o that A ⊃ 0 B is nothing but simply B , and A ⊃ 1 B is no thing but A → B . B ut b o unds do not necessar ily hav e to b e natur al num b ers. Reasonable transfinite ordinals could b e interesting to study as well, such as ordinals less than ǫ 0 . F or example, wher e ω is the smallest infinite or dinal, A ⊃ ω B would mean a g ame where ⊤ has to declare a num b er n be fore s tarting using A , a fter which the ga me contin ues a s if it was A ⊃ n B . This genera liz e s to A ⊃ kω B fo r any k ≥ 0, w he r e k (rather than just one) declarations n 1 , . . . , n k are made. The first declaration n 1 op ens n 1 copies o f A for usa ge; the second declaratio n n 2 , which can b e made a ny time la ter when the previous ly “activ ated” copies o f A a re p erhaps a lready at adv anced stages, creates the po ssibility to use n 2 additional copies; the third declar ation a ctiv ates n 3 additional copies, etc., with the ov er a ll n umber of (re)usa ges of A thus even tually no t exceeding the finite n 1 + . . . + n k . Next, A ⊃ ω 2 B would b e a ga me where ⊤ has to declare a n umber n b efo r e starting using A , after which the play contin ues as it would pro c e ed in A ⊃ nω B . This can b e further generaliz ed to A ⊃ ω k B for any k ≥ 0. Then A ⊃ ω ω B co uld b e characterized as a game where ⊤ ’s initial choice of n turns it into a game that pr o ceeds as A ⊃ ω n B . And s o on a nd so on. F urthermor e, ◦ – has an even greater v ariety o f b ounded versions of po tential interest, esp ecially in the (y et to b e develop ed) area of interactive computational complexity theory . One may wan t to differentiate betw een just b ounds on the overall num ber o f reusa ges o f the anteceden t and bounds on, say , the “depths” of reusages . Roughly , the depth o f r e usages here means the maximum nu mber of ancestor p ositions of any given run of the anteceden t at which re s tarts (“forking s”, “replications ”) happ ened. In more precise terms — for those familiar with the relev an t fo rmal definitions — such b ounds would mean b ounds on the heights o f the co rresp o nding underlying bitstring trees (see [17]). F or > – , o n the other hand, the ab ove concept of depth is not meaningful as it automatically trivializes to 1 (or to 0, depending on whether or not only prop er reusage s count). Finite or bo unded versions of reduction op era tions, e x cept the “mo st finite” and “most b ounded” → , hav e never b een studied, and a t this p oint we do not know what log ics they induce. In what follows our fo cus is only on → , > – , ◦ – ℵ 0 , ◦ – . Of these fo ur op erations , ◦ – stands out as , in a sense, most natura l and imp ortant. What makes ◦ – sp ecial is that it has go o d claims to precisely ca pture everything that a nyone would ever call (interactive) alg orithmic reduction. That is in the same sense as T uring computability of functions ca ptur es our in tuitive concept of effectiveness. What a lso makes ◦ – natural is that, as sugges ted by the ab ov e characterizations, definitions of other reductions can b e e asily obtained from the definition of ◦ – by imp o s ing cor resp onding r estrictions on the form a nd qua ntit y of reusage o f the a nteceden t, with → b eing the most extreme nontrivial ca se, where any prop er reusage is simply forbidden a ltogether. Alternatively , we can consider → rather than ◦ – as the basic sor t of reduction, and define all weak er versions of reductio n in terms o f → and what a re c alled r e curr enc e op er a tions ( ∧ | , ◦ | ℵ 0 , ◦ | , . . . ), 5 in their gener al spir it r esembling the storag e op er ator ! o f linear logic. This is exactly the approa ch that computability logic has prefered to take so far. 2 F or instance, [17] treats A > – B and A ◦ – B as abbreviatio ns of ∧ | A → B a nd ◦ | A → B , resp ectively . This so rt of a decomp ositio n of weak implication-style op erator s lo oks well familia r from linear lo gic [5], or the even earlier work [2] by B lass. So, the a bove discussion o f v arious new sorts of reduction can b e in fact co nsidered an informal introduction of the corres po nding series o f ne w r ecurrence o p erations. F or this reaso n, and also for the (rela ted) reas on of b eing the only fully resour ce-conscio us reduction, the ope ration → is at least as impo rtant, basic a nd natura l as ◦ – . The op era tions > – , ◦ – ℵ 0 and ◦ – equally enjoy the status of b eing co nserv ative generaliza tions of T uring reduction for the in terac tive context. Sp ecifically , when A and B ar e tra ditional sorts of problems such as decision pro blems or problems o f computing a function, effective winnabilit y of any o f the three games A > – B , A ◦ – ℵ 0 B , A ◦ – B turns o ut to coincide with T uring reducibility of B to A , with the subtle differences b e t ween > – , ◦ – ℵ 0 , ◦ – bec oming relev ant only when these op erator s are applied to problems with higher deg rees of interactivit y . The sa me does not extend to A → B though: (ev en) when A and B a re traditiona l sor ts of problems, effective winna bilit y of A → B mea ns something pr o p erly strong er. It means ex istence of a T ur ing machine that solves B with a n oracle fo r A where the ora cle can b e queried only once (while, as we k now, ordinary T uring reducibility do es not imp ose an y limits on how many times the orac le can b e queried). The ear lier mentioned finite v er sions > – F and ◦ – F of weak reductions can a lso b e seen to b e c o nserv ative genera lizations of T uring reduction. As for the b ounded versions of weak reductions, they genera lize certain prop er stre ngthenings of T uring reduction, obtained b y imp osing (finite or transfinite) b ounds on the num b er of p ossible queries of the or acle. Going ba ck to our Kolmogo rov complexity ex ample, the game H ⊃ K has a n algor ithmic winning stra tegy for each ⊃∈ { > – , ◦ – , ◦ – ℵ 0 , > – F , ◦ – F } . F urthermor e, with s ome thought and keeping in mind the known fact that the K olmogo rov complexity of n never exceeds n itse lf (for the exception of a finite nu mber o f “very small” n ’s), one can see that H ⊃ K remains algor ithmically so lv able with either ⊃∈ { > – ω , ◦ – ω } as well. As it turns out, the logica l b ehaviors of > – , ◦ – ℵ 0 and ◦ – are indistinguisha ble when these op erator s are tak en in isola tion, and that common b ehavior is precisely ca ptured by the implicative fragment In t ⊃ of Heyting’s intuitionistic calculus. F or > – and ◦ – , a pro of of this fact w a s given in [1 2]. And the pre sent pa p er ex tends that result to ◦ – ℵ 0 as well. As for → , it turns out that its logical b ehavior is captured by CL7 , which is (the Gentzen-st yle axiomatizatio n of ) In t ⊃ with just the contraction rule deleted. In other words, CL7 is nothing but the implicative fra gment of affine logic. A pro of of this re s ult is the main technical co ntribution o f the pre s ent ar ticle. And this is not a result tha t could b e taken for granted. As shown in [17], a ffine log ic in its full la nguage, while sound, is fa r fro m being complete with resp ect to the sema nt ics of computability logic. In fact, even just the ( → , ¬ )-fra g ment of computability logic is not the same as the corres p o nding fragment of a ffine logic, no r do es it a pp e ar to b e a xiomatizable in traditiona l pro o f theory . Another w ay to summarize the main technical r esult of the present pap er is to say that the s e t of implica tive binar y tautologies and their substitutional instances is precisely des crib ed by CL7 . Here binary tautolo gies mean tautologies of cla s sical logic wher e no atom o ccur s more than twice, and implic ative binary tautolo gies a re binary tautolo gies that contain no co nnectives others than → . Bina r y tautolo gies and their instances hav e a r isen in the past as a class of formulas so und and complete with r esp ect to several natural semantics, most notably B lass’s game semantics for linear logic [3], Blass’s resource- c o nscious semantics for classical logic [4], the semantics of computability logic [6], and abstract resour ce semantics [10, 16 ]. This class of for mulas has stubb ornly resisted any axiomatizatio n attempts within the framework of traditional deductive approaches and, as argued by Blass in [3], apparently this phenomenon is not q uite a n a ccident. A r easonable a xiomatization for the s et of binar y tautolo gies and their ins ta nces was even tually found in [10], but it to ok switching to a substantially new deductive fra mework called cir quent c alculus (roughly , it is s equent calculus where formulas may b e shared b etw een different s equents), indirectly corr ob orating Bla s s’s thesis that binary tautologies a re for eign to traditional pro o f theor y . Against this background, the 2 In fact, computability logic further decomposes → , defining A → B as ¬ A ∨ B . 6 fact that the implicativ e fra gment of that wild class can still be tamed with traditional means such as substr uctural se quent ca lculus in which CL7 is constructed, is worth re c e iving our atten tion. 2 Logic CL7 The la nguages that we c o nsider in this pa pe r hav e infinitely many nonlo gical prop o sitional atoms for which we use the metav ariables P , Q , and have no logica l atoms. Where ⊃ ∈ {→ , > – , ◦ – ℵ 0 , ◦ – } , by a ⊃ -formula we mea n a for m ula built from atoms and (the binary) ⊃ in the standard wa y . W e will b e using E , F, G, H as metav ar iables for formulas, and Γ , ∆ as metav ariables for (p ossibly empt y) m ultisets of formulas. As usua l, we write Γ , ∆ or Γ , F instead of Γ ∪ ∆ or Γ ∪ { F } . A (t wo-sided) ⊃ -s equen t is a pa ir Γ ⇒ F , where Γ is a finite multiset of ⊃ -formulas and F is a ⊃ -formula. Here Γ is s aid to b e the an tecedent of the s equent, and F is said to b e the succeden t . W e axiomatize CL7 using tw o- sided → -sequents. A ( → -) formula H is c o nsidered prov able in this sy stem (wr itten CL7 ⊢ H ) iff the empty-an tecedent s equent ⇒ H is so. The axiom s of CL7 ar e all → -sequents of the form Γ , F ⇒ F. And the sys tem only has the following tw o rules of inference : Γ , E ⇒ F Right → Γ ⇒ E → F Γ , F ⇒ G ∆ ⇒ E Left → Γ , ∆ , E → F ⇒ G W e say that a for mula of classica l propo sitional logic (with → -formulas here also seen as such) is binary iff no atom o ccurs in it more than twice. The concepts of b eing binary , tauto lo gical, true or false extend from for mulas to seq uent s by understanding each s equent E 1 , . . . , E n ⇒ F as the for mula E 1 ∧ . . . ∧ E n → F . A (substitutional) instance o f a given formula F , as usual, means the r esult o f replacing atoms in F b y any for mulas, with a ll o ccur rences of the sa me ato m being replaced by the same for mu la , of course. Theorem 2. 1 F or any → -formula H , the fol lowing c onditions ar e e quivalent: (i) CL7 ⊢ H . (ii) H is an instanc e of a binary tautolo gy. (iii) H is valid in c omputability lo gic, whether it b e in t he or dinary s en se of validity or in t he str onger sense of what is c al le d “uniform validity” (se e [17]). Pro of. The equiv alence b etw een (ii) and (iii) in a stro ng er form which is not r estricted to just → -formulas, has been proven in [10 ]. 3 So, to prov e the present theorem, it w o uld b e sufficien t to show that (i) implies (iii) (call this soundness ) and that (ii) implies (i) (call this c omple ten ess ). This w ill b e done in the following t wo sections. ✷ 3 The soun dness of CL7 W e can rewrite CL7 into a clearly eq uiv alent system that uses one-si ded sequents , her e restr icted to finite multisets o f for m ula s of classical prop os itional lo gic without → , where negation is a pplied only to ato ms. This is done by rewriting e ach → - s equent E 1 , . . . , E n ⇒ F as ¬ E 1 , . . . , ¬ E n , F , and then iter a tively rewriting e a ch (sub)formula E → F as ¬ E ∨ F , each subfor m ula ¬ ( E ∨ F ) as 3 A game-seman tical soundness and completeness of the class of substitutional i nstances of binary tautologies was first prov en with r esp ect to Blass’s game semant ics in [3]. 7 ¬ E ∧ ¬ F , each subformula ¬ ( E ∧ F ) a s ¬ E ∨ ¬ F and each s ubformula ¬¬ E a s E . The axioms o f the r esulting system a re all s e quents of the form Γ , ¬ F , F , 4 and the r ules of inference now read as follows: Γ , ¬ E , F Right → Γ , ¬ E ∨ F Γ , ¬ F, G ∆ , E Left → Γ , ∆ , E ∧ ¬ F, G Among several equiv alent axiomatiza tio ns of the (mu ltiplicative frag ment of the) well known affine lo gic is the one that uses one-side d sequents in our present se ns e. It has the same axiom scheme Γ , ¬ F, F . And the a b ov e Right → and Left → rules are specia l cases of the ∨ -in tro duction and ∧ -int r o duction rules of that s y stem, resp ectively , where ∧ , ∨ a re seen as m ultiplicatives. 5 Thu s, understanding E → F a s an abbreviation of ¬ E ∨ F , affine log ic proves ev er y → -for mula prov able in CL7 . But, as prov en in [17], affine lo gic is so und with resp ect to the s emantics co mputabilit y logic, and the latter sees no difference between E → F and ¬ E ∨ F . So, cla use (i) of Theo rem 2.1 implies c la use (iii), as desired. 4 The c ompleteness of CL7 W e define the head of a → -formula as follows: • Every atom is its own head. • The head of E → F is that of F . In other w or ds, the head of a for mula is the atom with the r ig htmost o ccurre nc e in the for mula — the (unique) o cc ur rence that is not in the anteceden t of any subformula. Consider any bina ry → -sequent Γ ⇒ F . W e define the relev an t formulas o f this sequent to be the elements of the smallest set S such that: • Every formula of Γ whose hea d o ccur s in F is in S . • Every formula of Γ whose hea d o ccur s in some element of S is also in S . The for mulas o f Γ that a re not relev ant will b e said to b e irrelev an t . Lemma 4. 1 A ssume Γ ⇒ F is a binary tautolo gic al → - se quent, and ∆ is the r esult of deleting fr om Γ al l irr elevant formulas of Γ ⇒ F . Then the se quent ∆ ⇒ F is also tautolo gic al (and, of c ourse, r emains binary). Pro of. Let Γ , ∆ , F be as ab ov e. In what follows, by a “relev ant fo r mula” we alwa ys mean a relev ant fo r mula of Γ ⇒ F . Similar ly for “irrelev ant”, “ a nteceden t”, “succedent”. Suppo se that ∆ ⇒ F is not tauto logical. Consider a truth ass ig nment tha t makes it false, i.e., makes ∆ true a nd F false. E xtend it to a ll formulas of Γ by stipulating that, if an a tom do es not o ccur in ∆ ⇒ F , it is true. Obviously the hea d o f every ir relev ant form ula is true under this extended as signment and hence every irrelev ant formula is true. All relev ant formulas of the anteceden t a lso remain true . And the succedent remains false. So, Γ ⇒ F is false a nd hence non-tautolog ic al. ✷ 4 Of course, it does not matter whether here and later we write Γ or ¬ Γ, with ¬ Γ m eaning the multiset of the negations of the elements of Γ. 5 In fact, writing E instead of ¬ E , Right → is simply the same as the ∨ - introduction rule of affine logic. 8 Lemma 4. 2 A ssume Γ ⇒ E and Γ ⇒ F ar e binary se quents, wher e E and F do not shar e any atoms. Then t he sets of r elevant formulas of the two se quents ar e disjoint. Pro of. Assume the conditions of the lemma. Co ns ider a n a rbitrar y rele v ant formula G of Γ ⇒ E . Let P b e the hea d of G . If the re a son for G ’s relev ance is that P o ccur s in E , then (as E and F share no a toms) P do es no t o ccur in F , no r does it o c c ur in a ny formula of Γ other than G bec ause o f the binarity of the se quent. This , by the definition of relev ance, means that G is not a relev ant fo r mula of Γ ⇒ F . Suppo se now the reas on for G ’s b eing a rele v ant formula of Γ ⇒ E is that P o ccurs in some relev ant formula H of Γ ⇒ E . The relev ance of H has thus b ee n established earlie r than that o f G and hence, by the inductio n h yp o thesis, H is not a relev ant formula of Γ ⇒ F . But, in v iew o f binarity , the only t wo places w he r e P o ccurs (whether it b e within Γ ⇒ E or Γ ⇒ F ) a re in G and H . Hence G cannot b e a relev ant for m ula of Γ ⇒ F . ✷ Lemma 4. 3 C L7 pr oves every binary taut olo gic al → -se quent . Pro of. Consider an arbitra ry binary tautological se quent. W e may assume that its succedent is a n atom P , for otherwis e, if the succe dent is E → F , mo ve E to the anteceden t of the s e q uent, and rep eat the same until the succedent has b eco me atomic; in view of the pre s ence o f Right → in CL7 , prov abilit y of the r esulting seq uent implies prov abilit y of the o riginal o ne . If P is one o f the formulas of the anteceden t, then the sequent we deal with is a n axio m and th us CL7 prov es it. Otherwise, the anteceden t sho uld contain a formula E → F whos e head is P , o r else the sequent could b e falsified by the truth assignment which makes P false and makes all other atoms true. T hus, the sequent we ar e talking a b out lo oks like Γ , E → F ⇒ P , where P o ccurs in F and hence o ccurs in neither E nor Γ, as the se quent is binary . O bviously the tautolo gicity o f this sequent implies the tautologicity of Γ , F ⇒ P . Since E do es not co nt a in P , the tautologicity of Γ , E → F ⇒ P also implies the tautologicity o f Γ ⇒ E . Indeed, assume that some truth assignment falsifies Γ ⇒ E . Extend that as signment to a ll ato ms o f Γ , E → F ⇒ P in such a wa y that it makes P false. Obviously such an extended assig nment fa ls ifies Γ , E → F ⇒ P , contradicting o ur a s sumption that this sequent is ta utological. Th us, Γ , F ⇒ P and Γ ⇒ E are binary tautologica l sequents, and their succede nts do not share any atoms. Let Γ 1 and Γ 2 be the submu ltisets of Γ cons isting of all r elev an t formulas of Γ , F ⇒ P and Γ ⇒ E , resp ectively . By Lemma 4.2, Γ 1 and Γ 2 are dis joint. Also , b y Lemma 4.1, Γ 1 , F ⇒ P and Γ 2 ⇒ E a r e tautological. Hence, by the induction hypo thes is (where induction is o n the n umber of connectives o cc urring the sequent), these t wo seq uent s are prov able. Then, by Left → , the sequent Γ 1 , Γ 2 , E → F ⇒ P is also prov able. This can be easily seen to imply the prov abilit y of the origina l sequent Γ , E → F ⇒ P , as CL7 is obviously closed under the weakening r ule “fro m ∆ ⇒ G conclude ∆ , H ⇒ G ”. 6 ✷ In view of the e vident fact that CL7 is closed under substitution of atoms by whatever for m ula s , Lemma 4.3 immediately implies the desired co nc lus ion that, w he ne ver H is a → -for mula which is an instance of some bina r y tautology , H is prov able in CL7 . 5 The three v ersions of w eak reduction As noted in Section 1, the three weak reduction o p erations > – , ◦ – ℵ 0 and ◦ – can b e defined in terms of → and the corresp o nding three r e curr enc e op er ations ∧ | , ◦ | ℵ 0 , ◦ | by A > – B = def ∧ | A → B ; A ◦ – ℵ 0 B = def ◦ | ℵ 0 A → B ; A ◦ – B = def ◦ | A → B . 6 In the present v ersi on of CL7 , we akening i s “hidden” in axioms. Alternatively , we could ha ve chosen the axioms of C L7 to b e just F ⇒ F , with weak ening explicitly s tipulated as one of the inference rules. It is kno wn that either c hoice yields the same set of prov able formulas, whether it be classi cal, affine or i ntuitionistic logic. 9 (Recurrences have the hig hes t precedence, so ∧ | A → B should b e r e a d as ( ∧ | A ) → B , and similar ly for ◦ | ℵ 0 , ◦ | .) W e r efer to ∧ | as parallel recurrence , and refer to ◦ | ℵ 0 and ◦ | as branc hing recur- rences . Namely , ◦ | ℵ 0 can b e called counta bl e bra nching rec ur rence, and ◦ | called uncountable branching recurre nce . ◦ | and ∧ | hav e b een defined in some ea rlier literature o n computability logic (see, e.g., [17]). On the other hand, ◦ | ℵ 0 , a s a full-fledge d citizen of computability logic, is first officially int ro duced in the present paper (see also “Historical remar ks” at the e nd o f this sectio n). Let us start with taking a closer (than do ne in Section 1) int uitive lo ok at how ◦ | and ∧ | compare. Imagine a computer that has a pro gram successfully playing Chess . The r esource that such a computer pr ovides is obviously strong er than just Chess : a reaso nable o p e r ating system would allow to simultaneously run as ma ny pa rallel s essions o f Chess as the user needs, while Chess , as such, only assumes a single play . This is what is ca ptured by the para llel recurre nc e ∧ | Chess . A more adv anced o pe r ating system, howev er, would in addition also make it p os s ible to branch/replicate each par ticular stag e o f each particular session, i.e. create any num ber of “copies” of a ny alrea dy reached p ositio n o f the multiple parallel plays o f Chess , thus giving the user the p oss ibilit y to try different co ntin uations fro m the sa me p osition. What cor resp onds to this intuition is the branching r ecurrence ◦ | Chess . As no ted earlier, the user of the r e source ◦ | A do es not hav e to r e s tart A from the very b eginning every time it wan ts to reuse it; ra ther, it is (essentially) a llow ed to backtrack to any of the previous — not nece s sarily starting — p ositio ns and try a new co ntin uation from there, thus depriving the adversary o f the p ossibility to reconsider the mov es it has a lready made in that p osition. This is in fact the type of reusage every purely softw ar e resource allows or would allow in the pre s ence of an adv anced op erating system and unlimited memor y: one can start running pro ces s A ; then fork it at any stage thus creating tw o threads that hav e a commo n past but po ssibly diverging futures (with the p ossibility to treat one of the threads as a “backup co py” and preserve it for backtracking purp oses); then further fork any of the bra nches a t a ny time; and so on. The less flexible type of reusag e o f A assumed by ∧ | A , on the o ther hand, is clos er to wha t infinitely ma ny autonomous ph ys ical resources would natura lly o ffer, such a s an unlimited num ber of indep endently ac ting rob ots each per forming task A , or an unlimited num ber of computers with limited memor ie s, each one only capable of and resp onsible for running a s ingle thre ad of pro cess A . Here the e ffect of replicating/for king a n adv anced s tage of A cannot be achieved unless, by go o d luc k, there are tw o ident ica l copies o f the stage, meaning that the co rresp onding tw o ro b o ts or computers have so fa r acted in precis ely the s ame ways. The difference b etw een the countable ( ◦ | ℵ 0 ) and uncountable ( ◦ | ) versions of branching re c ur- rence app ears to b e m uch more s ubtle than the difference betw een the par allel ( ∧ | ) and branching ( ◦ | or ◦ | ℵ 0 ) sorts o f r ecurrence. In fact, the a bove intuitiv e-level discussion of ∧ | vs. ◦ | is just as v alid for ∧ | vs. ◦ | ℵ 0 . Y et, ◦ | and ◦ | ℵ 0 turn out to induce dr amatically different logics, even if those logics coincide when ◦ – ℵ 0 or ◦ – (or > – ) is the only connective in the logical vo cabulary . The following is a n example of a princ iple which could b e shown to b e v alid with ◦ | but inv alid with ◦ | ℵ 0 as well as with ∧ | : ◦ | ◦ | P → ◦ | ◦ | P ( ◦ | abbreviates ¬ ◦ | ¬ , where ¬ is the “r o le switch” op er a tion). And, as we started discussing differences b etw een the principles v alidated by the different so rts o f recurr ences, her e comes an example of a principle which can b e s hown to b e v alid with ∧ | but inv alid with either ◦ | or ◦ | ℵ 0 : P ∧ ∧ | ( P → Q ∧ P ) → ∧ | Q ( ∧ , called p ar al lel c onjunction , is a computabilit y-lo gic counterpart o f the tensor of linear log ic . A ∧ B mea ns a pa rallel play of A a nd B , where ⊤ has to win in b oth plays to be the w inner in the o verall game). These are just isola ted e x amples, and finding a systematic deductive characteriza tion of all v alid principles tha t involv e recur r ence op era tions remains a grea t challenge in computability logic. Before we mov e to more exa mples illus trating differences be tw een ∧ | , ◦ | ℵ 0 and ◦ | , it would b e a go o d idea to first define the thr ee ope r ations under question. 10 F orma lly , ∧ | A is defined as the infinite conjunction A ∧ A ∧ A ∧ . . . , wher e A 0 ∧ A 1 ∧ A 2 ∧ . . . is a straig htforward gener alization of the just-mentioned para llel co njunction ∧ op eratio n from the binary ca se to the infinite cas e. Defining the branching recurrence s takes more work. In semifor mal terms , a play of ◦ | A sta r ts as an ordinar y play o f g ame A . At any time, how ever, play er ⊥ is allow ed to make a “replicative mov e”, which cr eates t wo copies o f the current p ositio n Φ o f A . F ro m tha t p oint on, the g ame turns into t wo games play ed in parallel, ea ch contin uing from p os itio n Φ. W e us e the bits 0 a nd 1 to de no te tho se t wo threads, that — using our earlier words — hav e a common past (p osition Φ) but p ossibly diverging futures. Again, at a ny time, ⊥ c an further branch either thread, creating t wo copies of the current p osition in that thread. If thr ead 0 was br anched, the re s ulting t wo threads will b e denoted by 00 and 01; and if the branched thread was 1, then the r esulting threa ds will b e denoted by 10 and 11. And so o n: at any time, ⊥ may split any of the exis ting threa ds w int o tw o threads w 0 a nd w 1. Each thread in the even tual run of the g ame will b e thus denoted by a (p ossibly infinite) bit string. The ga me is considered won by ⊤ if it wins A in ea ch of the threads; o therwise the winner is ⊥ . T o each infinite bit str ing w may th us co rresp ond a separa te run of A in thread (represented by) w a nd, as ther e are uncountably many infinite bit strings , uncountably many paralle l runs of A may be genera ted when playing ◦ | A up. Let us call a bit string w essentially finite if it contains o nly a finite num b er of 1 s; o therwise we say that w is es sen tiall y infinite . W e extend these ter ms from bit strings to the corre s po nding threads in the play of ◦ | A . The definition of ◦ | A th us requir es from ⊤ to win A in a ll — whether they b e essentially finite or e s sentially infinite — threads. All it takes to turn that definition into a definition of ◦ | ℵ 0 is to relax that require ment and, when determining the winner, only lo ok a t essentially finite threads. Since ther e ar e only countably many essentially finite bit s tr ings, o nly co unt a bly many runs of A ar e gener a ted — more precise ly , only co unt a bly many runs of A a re of relev ance — in ◦ | ℵ 0 A . This completes our semiformal definition/explanation of ◦ | ℵ 0 . In fully formal terms, bo th ◦ | A and ◦ | ℵ 0 A hav e the same structures ( Lr co mp o ne nts). Ther e are t wo types of lega l mov es in (legal) p o sitions o f either game: (1) replicative and (2) non- replicative. T o define these, let us a gree that by an active no de 7 of a p osition Φ we mean a bit string w such that w is either e mpty , 8 or else is u 0 or u 1 for some bit string u such that Φ contains the move u :. A replicative mov e can only b e made by (is only lega l for ) ⊥ , and such a mov e in a given po s ition Φ should b e w :, where w is an active no de of Φ and Φ do es not alr eady contain the same move w : . 9 As for non-replica tive moves, they can b e made b y either pla yer. Such a mo ve by a play er ℘ in a given p osition Φ should b e w.α , wher e w is a n active no de of Φ and α is a mo ve such that, for any infinite bit string v , α is a legal move b y ℘ in p osition Φ  w v of A . 10 Here and later , for a r un Θ and bit string x , Θ  x means the result of deleting from Θ a ll mov es except tho s e that lo ok like u.β for some initial seg ment u of x , and then further dele ting the prefix “ u. ” from such moves. 11 As for the conten ts (the W n comp onents) o f these ga mes, a lega l run Γ o f ◦ | A is conside r ed w on by ⊤ iff, fo r every infinite bit string v , Γ  v is a ⊤ -won r un of A . And a leg al r un Γ of ◦ | ℵ 0 A is considered won by ⊤ iff, for every infinite but essent ially finite bit string v , Γ  v is a ⊤ -won run o f A . This completes our definition of ◦ | and ◦ | ℵ 0 . As we just saw, the definition of ◦ | ℵ 0 is obtained from the definition o f ◦ | by merely inser ting the words “but ess entially finite”. But, again, trying to analy z e the rather technical definition given in the a bove para graph may not b e a go o d idea for a reader of this pap er. Relying, instea d, on 7 In tuitively , an activ e node is (the name of ) an already existing thread of a play ov er A . 8 In tuitively , the empt y string is the name/address of the initial thread; all other threads wi ll b e descendant s of that thread. 9 The intuitiv e meaning of mov e w : i s splitting thread w into w 0 and w 1, thus “activ at i ng” these t wo new nodes/threads. 10 The intuitiv e m eaning of such a mov e w .α is making mov e α i n thread w and all of its (current or future) descendan ts. 11 In tuitively , Θ  x is the run of A that has been play ed in thread x , i f suc h a thread exists (has b een generated); otherwise, Θ  x is the run of A that has b een play ed in (the unique) existing thread whic h (whose name, that is) i s some initial segment of x . 11 the infor mal explana tions that we provided should b e sufficient. T o see the distance b etw een ◦ | and ◦ | ℵ 0 , following V ereshc hag in [2 0], let us consider any set S of natural num b ers (identified with their decima l repr esentations), s uch that S is no t recur sively enum er able. Let A be the game w her e only ⊤ has legal mov es, each lega l move b eing a(ny) natural nu mber . A g iven run of this game is cons idered won b y ⊤ iff the set of the moves it makes in it equals S . In other words, ⊤ wins iff it enumerates S . Now let us lo ok at the games ◦ | A and ◦ | ℵ 0 A , where ◦ | = ¬ ◦ | ¬ a nd ◦ | ℵ 0 = ¬ ◦ | ℵ 0 ¬ . That is, ◦ | A is the s ame a s ◦ | A , only here it is ⊤ rather than ⊥ who can create new threa ds (make r eplicative mov es), and who se adversary needs to win A in each of the thr eads to b e the winner in the overall game. Similar ly for ◦ | ℵ 0 . Obviously ⊤ has an effective winning strategy for ◦ | A , consisting in enumerating all of the 2 ℵ 0 sets of natura l num b ers, one p er each of the 2 ℵ 0 threads that it may crea te in ◦ | A . On the other ha nd, ⊤ do es no t hav e an effective winning s trategy for ◦ | ℵ 0 A . Otherwis e, o ne would b e a ble to recursively enumerate S by selecting the (essentially finite) bit string w represe nt ing a winning threa d, and then listing the mov es made in that thread. Our discussio n would not b e complete without als o seeing a sp ecific example illus trating the distance b etw een the parallel and bra nching versions of recurrence . The g ame ∨ | A , whic h is a dual of ∧ | A in the same se nse as ◦ | A is a dual of ◦ | A , is defined as A ∨ A ∨ A ∨ . . . . This ca n b e thought of as a para llel play of game A o n infinitely many b oar ds: #0, #1, #2, . . . . ⊥ wins it iff it wins A on each of the b oar ds. Where f ( x ) is a total function from natural num b e r s to natural num be r s, ⊓ x ⊔ y  y = f ( x )  denotes a g ame ev er y lega l run of whic h consists of (a t most) tw o mov es. 12 The first mo ve is by ⊥ , and the move is a n arbitra ry num b er m . The second mo ve is b y ⊤ , who should name a num b er n . ⊤ wins iff n equals f ( m ). ⊤ has an effective winning str ategy that works for b oth ◦ | ⊓ x ⊔ y  y = f ( x )  and ◦ | ℵ 0 ⊓ x ⊔ y  y = f ( x )  . It consists in waiting till the adversary makes a mov e m , after whic h ⊤ creates infinitely (but countably) many threads , and tries all po ssible resp onses — all p ossible v alues for n , that is — in those threads, one resp onse per thread. Similarly , where B ( x ) is a predicate, ⊓ x ( ¬ B ( x ) ⊔ B ( x )) denotes a game where the fir st move (again), consisting o f cho osing a num be r m , is b y ⊥ . The second move is by ⊤ , who should choo se betw een 0 and 1. ⊤ wins iff B ( m ) is false a nd 0 was chosen, or B ( m ) is true and 1 was chosen. ⊤ ’s effective winning strategy for bo th ◦ | ⊓ x ( ¬ B ( x ) ⊔ B ( x )) a nd ◦ | ℵ 0 ⊓ x ( ¬ B ( x ) ⊔ B ( x )) is that it waits till the adversary makes a move m , after which ⊤ creates tw o threads , ma king mov e 0 in one thread and move 1 in the o ther threa d. The same trick, ho wever, fails with ∨ | ⊓ x ( ¬ B ( x ) ⊔ B ( x )). F or example, it fails when ⊥ chooses m = 0 on b oar d #0, m = 1 on b o ard #1, m = 2 on b oar d #2, etc. Let us call this strategy of ⊥ the diversifying str ate gy . Now, for any effective strateg y M of ⊤ , using diagonaliza tion, we can construct a particular pr edicate B ( x ) such that ⊤ loses ∨ | ⊓ x ( ¬ B ( x ) ⊔ B ( x )) aga inst the diversifying strategy . Namely , we ca n define B ( i ) (any i ) to be true if M makes the mo ve 0 on b oard # i when playing agains t the diversifying str ategy , a nd false otherwise. This guar a ntees that M ’s all r esp onses to the adversary ’s moves ar e “wr ong”. A simila r idea could b e employ ed in showing that, for an appro pr iately selected f , the problem ∨ | ⊓ x ⊔ y  y = f ( x )  has no algorithmic solution. Historical rem arks. Blass [2] was apparently the fir st to consider a n op era tor in the style of the exp onential op er ator ! of linea r logic. He called it the r ep etition op er ator R . The ga me- semantical co ntext in which R was intro duced was limited compared with the co ntext that com- putabilit y logic o p erates in. The main contextual difference is that Bla ss’s ga mes are strict , mea ning games where in each p o s ition only one pla yer may hav e (legal) mo ves. Computability logic, on the other hand, deals with the already ment io ne d mo re g eneral type of static games . As opp osed to strict games, static games a r e fr e e , in the s ense that g e ne r ally b o th play ers may have legal mov es in a given pos ition. F ur ther more, the recurrence op e r ations (as w ell as the non-recurr e nce para llel op erations ∧ , ∨ , ∧ , ∨ ) of computability logic generate pr op erly free games even when applied to strict g a mes, while B lass’s op eratio ns, of cours e , preserve the strict prop erty of games. Ho wev er , 12 In computabilit y l ogic, ⊓ is called choic e universal quantifier , and ⊔ called choic e exist ential quantifier . The smaller versions ⊓ and ⊔ of the same sym b ols stand for what are called c hoic e c onjunction and choic e disjunction , resp ectively . The choice operators of computab i l ity logic are reminiscent of the additiv e operators of linear logic. 12 if we disrega rd this difference and try to bring Blas s ’s games and static ga mes to so me reason- able co mmon denominato r , Bla ss’s r ep etition op eration R would apparently transla te (whatever “translate” should precisely mean her e) into ◦ | ℵ 0 . The reason why it would not translate into ∧ | is that R is a bra nching op era tio n in the pr op er sense, a llowing effects such as backtracking. And the reas on why R would b e less than an adequate counterpart o f ◦ | is that ◦ | A a llows ⊥ to try a contin uum of different runs of A , while that quantit y is automatica lly limited to ℵ 0 in RA (and artificially limited to ℵ 0 in ◦ | ℵ 0 A , as w e saw from the definition). 6 Implicativ e in tuiti onistic logic Where ⊃ is o ne of the op erato rs > – , ◦ – ℵ 0 or ◦ – , a Gentzen-st yle axioma tization o f the c or- resp onding implic ative (fra gment of ) intuitionistic lo gic , denoted by Int ⊃ , is CL7 — only with ⊃ -sequents instead o f → -s e quents, of course — plus the following single additional rule Γ , E , E ⇒ F Con traction Γ , E ⇒ F Alternatively , In t ⊃ could b e chosen to be formulated exa ctly as CL7 , with the o nly difference that the anteceden ts of s equents in In t ⊃ are seen as sets ra ther than multisets of formulas, which eliminates the need for explicitly stating contraction as a n inference rule . Theorem 6. 1 L et ⊃ b e any one of the op er ators > – , ◦ – ℵ 0 or ◦ – . F or any ⊃ - formula H , the fol lowing c onditions ar e e quivalent: (i) Int ⊃ ⊢ H . (ii) H is valid in c omputability lo gic, whether it b e in t he or dinary sense of validity or in the sense of uniform validity. Pro of. F or Int > - and In t ◦ - , this theorem was officially es tablished in [12]. As for In t ◦ - ℵ 0 , as it happe ns, the pro of of the soundness and completeness of In t ◦ - given in [12], in fact, is also a pro of of the soundnes s and completeness o f Int ◦ - ℵ 0 : a s imple re-r eading of that pro of re veals that v irtually no step in it r elies o n the fa ct that we deal with the uncountable rather tha n the countable v er sion of reduction. ✷ Historical remarks and further discussions. The ab ov e-mentioned r esult of [1 2] for In t ◦ - was further streng thened in [15], where soundness and completeness (with r e sp ect to the semantics of c o mputability logic) was prov en for the full prop ositional fra g ment of intu itionis tic lo gic. With only one or tw o months’ delay , V ereshc hagin [20] came up with a n alternative and shor ter pr o of of the same re s ult. I t should b e noted, ho wev er , tha t in his work V ere shchagin mo dified the “canonical” definitions o f computability logic q uite a bit, whic h essentially resulted in interpreting int uitionistic implicatio n as ◦ – ℵ 0 rather than ◦ – . Moreover, in an a ttempt to simplify things, V ereshchagin further limited ga mes to strict ones, essentially defining the ◦ | ℵ 0 comp onent of ◦ – ℵ 0 as s omething clo ser to Bla ss’s rep etition o pe r ator R than to ◦ | ℵ 0 in our present precise sense. In view of (and despite) the ab ov e- said, V ereshchagin’s pro of, with certain technical a djustment s, can be consider ed an alternative pro of o f the In t ◦ - ℵ 0 part of Theore m 6.1. The soundness pro of for In t ◦ - found in [12], in fact, can b e dr amatically simplified when we are concerned with ◦ – ℵ 0 rather than ◦ – . Sp e cifically , a lemma on which the soundness pro of given in [1 2] (a s well as similar pr o ofs given in [2] and [20]) relies is a b out the v alidity of the principle ◦ | A → ◦ | ◦ | A . And a strict pro of of that lemma , presented in [17], takes several pages . On the o ther 13 hand, a pro o f o f the v alidity o f the same principle for ◦ | ℵ 0 instead of ◦ | would not take more than just a para graph, as it did (for Blas s’s or V ereshchagin’s versions of ◦ | ℵ 0 ) in [2] o r [20]. The ab ov e-said a lso a pplies to the pr o of of the soundness and completenes s of the full pro p o si- tional intuitionistic lo g ic given in [15]. Tha t pro of officially is for the case when the int uitionistic implication is read a s ◦ – . How ever, the sa me pro o f is just as go o d for ◦ – ℵ 0 as well. Similarly , V ereshchagin’s [20] completeness pro of can b e ada pted to either interpretation ◦ – , ◦ – ℵ 0 of in tu- itionistic implication. The same cannot be s aid abo ut V ereshc hag in’s soundness pro of though: as noted ab ove, proving soundnes s when int uitionis tic implication is r ead as ◦ – rather than ◦ – ℵ 0 takes considerably g r eater effor ts. Finally , for r easons similar to the ab ov e, the soundness pr o of for the full first-o rder intu itionis tic calculus given in [14], is eq ually go o d for either re ading ◦ – , ◦ – ℵ 0 of intuitionistic implica tion. 7 Conclusion Computability log ic is a sema ntically conceived approa ch with the a mbit ion to b e “a formal theory of computability in the same sense as cla ssical lo g ic is a formal theo ry o f tr uth” ([7]). As such, it do es not yet have a sufficiently develope d syntax, and among the main curr ent o b jectives of computability logic as a resear ch pr ogra m is to find axiomatizatio ns for v arious natural frag ment s of the se t of formulas v alidated by its semantics. The langua g e of computabilit y logic, with logical op erator s standing for op er ations on computational pro blems, is very rich and, in fact, open- ended. Ident ifying the mo s t natura l and p otentially useful new op er ators to b e included in it is another direction on which efforts within the pro ject contin ue to b e fo cused. The present pap er contributes to b oth of the ab ove directions. Within the second direction, it officially introduces the Blass-s tyle ([2, 3]) c ountable r e curr enc e o p erator ◦ | ℵ 0 and the asso ciated reduction o p er ator ◦ – ℵ 0 . It als o o utlines the idea of finite and b ounded versions of recurr ence op erator s together with the asso ciated reduction op era tors. T he three other main reductio n op er- ators → , > – and ◦ – studied in the pap er hav e b een introduced earlier . The v ariety of r eduction op erator s captures v ario us flav ors of our intuit io n of algo rithmically reducing one co mputational problem to another, with → b eing the stronges t form of reduction and ◦ – being the weakest form. Within the first direction, this pap er establishes tw o results. According to one result, a grea ter part of which was established ea rlier, the lo gic induced by ea ch of > – , ◦ – ℵ 0 , ◦ – is exactly the implicative fragment of Heyting’s in tuitionistic calculus. And, according to the o ther theor em, the logic induced b y → is the same calculus but with the con tra ction rule removed. The philosophical summary of these results is that, des pite the sig nificant semantical v arieties within the ba sic group of reduction o p erations, they (when taken in isolation) only gener ate tw o kinds o f lo gical behavior, dep ending on whether resour ce/anteceden t r eusage is allowed ( > – , ◦ – ℵ 0 , ◦ – ) or no t ( → ). Syntactically , those tw o b ehaviors ar e precisely accounted for b y the mer e presence or absence of c o ntraction in Gentzen-st yle axiomatizations . References [1] S. Abramsky a nd R. Jagadees an. Games and ful l c ompleteness for mu ltiplic ative line ar lo gic . Journal of Symb olic Logic 59 (2) (1994), pp. 543- 574. [2] A. Blass. D e gr e es of indeterminacy of games . F undamenta Mathematicae 77 (197 2), pp. 151-1 66. [3] A. Blass. A game semant ics for line ar lo gic . Annals of Pure and Applied Logic 56 (1992 ), pp. 1 8 3-22 0. [4] A. Blass. Reso u r c e c onsciousness in classic al lo gic . 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