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Interactive Learning Based Realizability and 1-Backtracking Games Federico Aschieri DipartimentodiInformatica Universita`diTorino Italy SchoolofElectronicEngineeringandComputerScience QueenMary,UniversityofLondon UK Weprovethatinteractivelearningbasedclassicalrealizability(introducedbyAschieriandBerardifor firstorderarithmetic[1])issoundwithrespecttoCoquandgamesemantics.Inparticular,anyrealizer ofanimplication-and-negation-freearithmeticalformulaembodiesawinningrecursivestrategyfor the1-BacktrackingversionofTarskigames. Wealsogiveexamplesofrealizerandwinningstrategy extractionfor some classical proofs. We also sketch some ongoingwork abouthow to extend our notionofrealizabilityinordertoobtaincompletenesswithrespecttoCoquandsemantics,whenitis restrictedto1-Backtrackinggames. 1 Introduction In this paper we show that learning based realizability (see Aschieri and Berardi [1]) relates to 1- Backtracking Tarski games as intuitionistic realizability (see Kleene [8]) relates to Tarski games. It iswellknowthatTarskigames(see,definition12below)arejustasimplewayofrephrasingtheconcept of classical truth in terms of a game between two players - the first one trying to show the truth of a formula,theseconditsfalsehood-andthatanintuitionisticrealizergivesawinningrecursivestrategyto thefirstplayer. Theresultisquiteexpected: sincearealizergivesawayofcomputingalltheinformation about thetruth ofaformula, the player trying toprove the truth ofthat formula has arecursive winning strategy. However, not at all any classically provable arithmetical formula allows a winning recursive strategy for that player; otherwise, the decidability of the Halting problem would follow. In [5], Co- quandintroduced agamesemanticsforPeanoArithmeticsuchthat,foranyprovableformulaA,thefirst player has a recursive winning strategy, coming from the proof of A. The key idea of that remarkable result is to modify Tarski games, allowing players to correct their mistakes and backtrack to a previ- ous position. Hereweshow that learning based realizers have direct interpretation aswinning recursive strategies in 1-Backtracking Tarski games (which are a particular case of Coquand games see [4] and definition 11 below). The result, again, is expected: interactive learning based realizers, by design, are similar to strategies in games with backtracking: they improve their computational ability by learning from interaction and counterexamples in a convergent way; eventually, they gather enough information aboutthetruthofaformulatowinthegame. AninterestingsteptowardsourresultwastheHayashirealizability[7]. Indeed,arealizerinthesense of Hayashi represents a recursive winning strategy in 1-Backtracking games. However, from the computational point of view, realizers do not relate to 1-Backtracking games in a significant way: Hayashi winning strategies work by exhaustive search and, actually, do not learn from the game and from the interaction with the other player. As a result of this issue, constructive upper bounds on the length of SteffenvanBakel,StefanoBerardi,UlrichBerger(Eds.): (cid:13)c F.Aschieri ClassicalLogicandComputation2010(Cl&C’10) Thisworkislicensedunderthe EPTCS47,2011,pp.6–20,doi:10.4204/EPTCS.47.3 CreativeCommonsAttributionLicense. F.Aschieri 7 gamescannot beobtained, whereas using ourrealizability itispossible. Forexample, inthecase ofthe 1-Backtracking Tarski game for the formula ∃x∀yf(x)≤ f(y), the Hayashi realizer checks all the natural numbers until an n such that ∀yf(n)≤ f(y) is found; on the contrary, our realizer yields a strategy which bounds the number of backtrackings by f(0), as shown in this paper. In this case, the Hayashi strategy is the same one suggested by the classical truth of the formula, but instead one is interested in theconstructive strategysuggested byitsclassical proof. Since learning based realizers are extracted from proofs in HA+EM (Heyting Arithmetic with ex- 1 cluded middle over existential sentences, see [1]), one also has an interpretation of classical proofs as learning strategies. Moreover, studying learning based realizers in terms of 1-Backtracking games also sheds light on their behaviour and offers an interesting case study inprogram extraction and interpreta- tioninclassical arithmetic. The plan of the paper is the following. In section §2, we recall the calculus of realizers and the main notion of interactive learning based realizability. In section §3, we prove our main theorem: a realizer of an arithmetical formula embodies a winning strategy in its associated 1-Backtracking Tarski game. In section §4, weextract realizers from two classical proofs and study their behavior as learning strategies. Insection §5,wedefineanextension ofourrealizability andformulate aconjecture about its completeness withrespectto1-Backtracking Tarskigames. T 2 The Calculus and Learning-Based Realizability Class The whole content of this section is based on Aschieri and Berardi [1], where the reader may also find fullmotivationsandproofs. Werecallherethedefinitionsandtheresultsweneedintherestofthepaper. The winning strategies for 1-Backtracking Tarski games willbe represented by terms of T (see [1]). Class T is a system of typed lambda calculus which extends Go¨del’s system T by adding symbols for non Class computable functions and a new type S (denoting a set of states of knowledge) together with two basic operationsoverit. ThetermsofT arecomputedwithrespecttoastateofknowledge,whichrepresents Class afiniteapproximation ofthenoncomputable functions usedinthesystem. For a complete definition of T we refer to Girard [6]. T is simply typed l -calculus, with atomic types N(representing theset Nofnatural numbers) andBool(representing thesetB={True,False}of booleans), producttypesT×U andarrowstypesT →U,constants0:N,S:N→N,True,False:Bool, pairs h.,.i, projections p ,p , conditional if and primitive recursion R in all types, and the usual 0 1 T T reduction rules(b ),(p ),(if),(R)forl ,h.,.i,if ,R . Fromnowon,ift,uaretermsofTwitht =uwe T T denote provable equality inT. Ifk∈N, the numeral denoting k isthe closed normal term Sk(0) oftype N. Allclosed normal terms of type N are numerals. Anyclosed normal term of type Bool in T is True orFalse. We introduce a notation for ternary projections: if T =A×(B×C),with p ,p ,p werespectively 0 1 2 denote the terms p , l x:T.p (p (x)), l x:T.p (p (x)). If u=hu ,hu ,u ii:T, then pu=u in T for 0 0 1 1 1 0 1 2 i i i=0,1,2. Weabbreviate hu ,hu ,u ii:T withhu ,u ,u i:T. 0 1 2 0 1 2 Definition1(StatesofKnowledgeandConsistentUnion) 1. A k-ary predicate of T is any closed normaltermP:Nk →BoolofT. 2. An atom is any triple hP,~n,mi, where P is a (k+1)-ary predicate of T, and~n,m are (k+1) numerals,andP~nm=TrueinT. 3. TwoatomshP,~n,mi,hP′,~n′,m′iareconsistent ifP=P′ and~n=~n′ inTimplym=m′. 4. Astateofknowledge, shortlyastate,isanyfinitesetSofpairwiseconsistent atoms. 8 InteractiveLearningBasedRealizabilityand1-BacktrackingGames 5. TwostatesS ,S areconsistent ifS ∪S isastate. 1 2 1 2 6. Sisthesetofallstatesofknowledge. 7. TheconsistentunionS US ofS ,S ∈SisS ∪S ∈SminusallatomsofS whichareinconsistent 1 2 1 2 1 2 2 withsomeatomofS . 1 ForeachstateofknowledgeSweassumehavingauniqueconstantSdenotingit;ifthereisnoambiguity, wejustassumethatstateconstantsarestringsoftheform{hP,n~ ,m i,...,hP,n~ ,m i},denotingastateof 1 1 k k knowledge. WedefinewithTS=T+S+{S|S∈S}theextension ofTwithoneatomictypeSdenoting S,andaconstant S:SforeachS∈S,andnonewreduction rule. Computation onstateswillbedefined byasetofalgebraic reduction ruleswecall“functional”. Definition2(Functionalsetofrules) LetCbeanysetofconstants, eachoneofsometypeA →...→ 1 A →A, for some A ,...,A ,A∈{Bool,N,S}. We say that R is a functional set of reduction rules for n 1 n C if R consists, for all c∈C and all closed normal terms a1 :A1,...,an :An of TS, of exactly one rule ca1...an7→a,wherea:AisaclosednormaltermofTS. We define two extensions of TS: an extension TClass with symbols denoting non-computable maps XP : Nk →Bool,F :Nk →N (for each k-ary predicate P of T) and no computable reduction rules, another P extensionT ,withthecomputableapproximations c ,f ofX ,F ,andacomputablesetofreduction Learn P P P P rules. X andF areintendedtorepresentrespectivelytheoraclemapping~ntothetruthvalueof∃xP~nx, P P and a Skolem function mapping~n to an element m such that ∃xP~nx holds iff P~nm = True. We use the elements of T to represent non-computable realizers, and the elements of T to represent a Class Learn computable “approximation” ofarealizer. Wedenote termsoftypeSbyr ,r ′,.... Definition3 Assume P:Nk+1 →Bool is a k+1-ary predicate of T. We introduce the following constants: 1. c :S→Nk →Boolandj :S→Nk →N. P P 2. X :Nk →BoolandF :Nk →N. P P 3. ⋒:S→S→S(wedenote⋒r r withr ⋒r ). 1 2 1 2 4. Add :Nk+1→Sandadd :S→Nk+1→S. P P 1. X S isthesetofallconstants c P,j P,⋒,addP. 2. X isthesetofallconstants X ,F ,⋒,Add . P P P 3. T =TS+X . Class 4. Atermt ∈T hasstate0/ ifithasnostateconstant different from0/. Class Let~t =t ...t . Weinterpret c s~t andj s~t respectively asa“guess” forthe values oftheoracle and the 1 k P P Skolem map X and F for ∃y.P~ty, guess computed w.r.t. the knowledge state denoted by the constant P P s. There is no set of computable reduction rules for the constants F ,X ∈ X , and therefore no set P P of computable reduction rules for T . If r ,r denotes the states S ,S ∈S, we interpret r ⋒r as Class 1 2 1 2 1 2 denotingtheconsistentunionS US ofS ,S . Add denotesthemapconstantlyequaltotheemptystate 1 2 1 2 P 0/. add S~nmdenotestheemptystate0/ ifwecannotaddtheatomhP,~n,mitoS,eitherbecausehP,~n,m′i∈S P for some numeral m′, or because P~nm=False. add S~nm denotes the state {hP,~n,mi} otherwise. We P defineasystem T withreduction rulesoverX S byafunctional reduction setRS. Learn Definition4(TheSystemT ) Lets,s ,s bestateconstantsdenotingthestatesS,S ,S . LethP,~n,mi Learn 1 2 1 2 beanatom. RS isthefollowingfunctional setofreductionrulesforX S: F.Aschieri 9 1. IfhP,~n,mi∈S,then c s~n7→Trueandj s~n7→m,else c s~n7→Falseandj s~n7→0. P P P P 2. s ⋒s 7→S US 1 2 1 2 3. add s~nm 7→ 0/ if either hP,~n,m′i ∈ S for some numeral m′ or P~nm = False, and add s~nm 7→ P P {hP,~n,mi}otherwise. WedefineT =TS+X S+RS. Learn Remark. T isnothingbutTSwithsome“syntacticsugar”. T isstronglynormalizing,hasChurch- Learn Learn Rosserproperty forclosedtermofatomictypesand: Proposition 1(NormalFormPropertyforT ) AssumeAiseitheranatomictypeoraproducttype. Learn Then any closed normal termt ∈T of type A is: a numeral n:N, or a boolean True,False:Bool, Learn orastateconstant s:S,orapairhu,vi:B×C. Definition5 Assumet ∈T andsisastateconstant. Wecall“approximation oft atstates”theterm Class t[s] of T obtained from t by replacing each constant X with c s, each constant F with j s, each Learn P P P P constant Add withadd s. P P Ifs,s′arestateconstantsdenotingS,S′∈S,wewrites≤s′forS⊆S′. Wesaythatasequence{si}i∈N ofstateconstants isaweaklyincreasing chainofstates(isw.i. forshort), ifs ≤s foralli∈N. i i+1 Definition6(Convergence) Assumethat{si}i∈N isaw.i. sequence ofstateconstants, andu,v∈TClass. 1. uconverges in{si}i∈N if∃i∈N.∀j≥i.u[sj]=u[si]inTLearn. 2. uconverges ifuconverges ineveryw.i. sequence ofstateconstants. Ourrealizability semanticsreliesontwoproperties ofthenoncomputabletermsofatomictypeinT . Class First,ifwerepeatedlyincreasetheknowledgestates,eventuallythevalueoft[s]stopschanging. Second, ift hastypeS,andcontainsnostateconstantsbut0/,thenwemayeffectivelyfindawayofincreasingthe knowledge statessuchthateventually wehavet[s]=0/. Theorem1(StabilityTheorem) Assumet ∈T isaclosedtermofatomictypeA(A∈{Bool,N,S}). Class Thent isconvergent. Theorem2(FixedPointProperty) Let t : S be a closed term of T of state 0/, and s = S. Define Class t (S)=S′ ift[S]=S′,and f(S)=S∪t (S). 1. For any n ∈ N, define f0(S) = S and fn+1(S) = f(fn(S)). There are h ∈ N, S′ ∈ S such that S′= fh(S)⊇S, f(S′)=S′ andt (S′)=0/. 2. Wemayeffectively findastateconstant s′≥ssuchthatt[s′]=0/. Definition7(ThelanguageL ofPeanoArithmetic) 1. The terms of L are all t ∈ T, such that N N t :NandFV(t)⊆{x ,...,x }forsomex ,...,x . 1 n 1 n 2. The atomic formulas of L are all Qt ...t ∈T, for some Q:Nn →Bool closed term of T, and 1 n sometermst ,...,t ofL. 1 n 3. TheformulasofL arebuiltfromatomicformulasofL bytheconnectives ∨,∧,→∀,∃asusual. Definition8(Typesforrealizers) For each arithmetical formula A wedefine atype |A|of T byinduc- tion on A: |P(t ,...,t )|= S, |A∧B|= |A|×|B|, |A∨B|= Bool×(|A|×|B|), |A → B|= |A|→ |B|, 1 n |∀xA|=N→|A|,|∃xA|=N×|A| 10 InteractiveLearningBasedRealizabilityand1-BacktrackingGames We define now our notion of realizability, which is relativized to a knowledge state s, and differs from Kreisel modified realizability for a single detail: if we realize an atomic formula, the atomic formula doesnotneedtobetrue,unlesstherealizer isequaltotheemptysetins. Definition9(Realizability) Assumesisastateconstant,t ∈T isaclosedtermofstate0/,A∈L is Class aclosedformula, andt :|A|. Let~t =t ,...,t :N. 1 n 1. t (cid:14) P(~t)ifandonlyift[s]=0/ inT impliesP(~t)=True s Learn 2. t (cid:14) A∧Bifandonlyifp t (cid:14) Aandp t (cid:14) B s 0 s 1 s 3. t(cid:14) A∨Bifandonlyifeither p t[s]=TrueinT and p t(cid:14) A,or p t[s]=FalseinT and s 0 Learn 1 s 0 Learn p t (cid:14) B 2 s 4. t (cid:14) A→Bifandonlyifforallu,ifu(cid:14) A,thentu(cid:14) B s s s 5. t (cid:14) ∀xAifandonlyifforallnumeralsn,tn(cid:14) A[n/x] s s 6. t (cid:14) ∃xAifandonlyforsomenumeraln,p t[s]=ninT andp t (cid:14) A[n/x] s 0 Learn 1 s Wedefinet (cid:14)Aifandonlyift (cid:14) Aforallstateconstants s. s Theorem3 IfAisaclosed formula provable inHA+EM (see [1]), then there existst ∈T suchthat 1 Class t (cid:14)A. 3 Games, Learning and Realizability Inthissection,wedefinethenotionofgame,its1-BacktrackingversionandTarskigames. Wealsoprove ourmaintheorem,connecting learningbasedrealizability and1-Backtracking Tarskigames. Definition10(Games) 1. AgameGbetweentwoplayersisaquadruple(V,E ,E ,W),whereV isa 1 2 set,E ,E aresubsetsofV×V suchthatDom(E )∩Dom(E )=0/,whereDom(E)isthedomain 1 2 1 2 i ofE,andW isasetofsequences, possiblyinfinite, ofelementsofV. TheelementsofV arecalled i positions of the game; E , E are the transition relations respectively for player one and player 1 2 two: (v ,v )∈E meansthatplayericanlegallymovefromtheposition v totheposition v . 1 2 i 1 2 2. Wedefine aplaytobeawalk,possibly infinite, inthegraph(V,E ∪E ),i.e. asequence, possibly 1 2 void, v ::v ::...::v ::... of elements ofV such that (v,v )∈E ∪E for every i. A play of 1 2 n i i+1 1 2 theform v ::v ::...::v ::...issaid tostart from v . Aplay issaid tobecomplete ifitiseither 1 2 n 1 infinite or is equal to v ::...::v and v ∈/ Dom(E ∪E ). W is required to be a set of complete 1 n n 1 2 plays. If pisacompleteplayand p∈W,wesaythatplayeronewinsin p. If pisacompleteplay and p∈/W,wesaythatplayertwowinsin p. 3. LetP be the set of finite plays. Consider a function f :P →V; aplay v ::...::v ::... issaid G G 1 n tobe f-correctif f(v ,...,v)=v foreveryisuchthat(v,v )∈E 1 i i+1 i i+1 1 4. Awinningstrategyfrompositionvforplayeroneisafunctionw :P →V suchthateverycomplete G w -correctplayv::v ::...::v ::...belongstoW. 1 n Notation. Iffor i∈N,i=1,...,n wehave that p =(p) ::...::(p) isafinite sequence of elements i i 0 i ni oflengthn,with p ::...:: p wedenotethesequence i 1 n (p ) ::...::(p ) ::...::(p ) ::...::(p ) 1 0 1 n1 k 0 k nk F.Aschieri 11 where(p) denotes the j-thelementofthesequence p. i j i Suppose that a :: a :: ... ::a is a play of a game G, representing, for some reason, a bad situation 1 2 n forplayerone(forexample,inthegameofchess,a mightbeaconfigurationofthechessboardinwhich n player one has just lost his queen). Then, learnt the lesson, player one might wish to erase some of his movesandcomebacktothetimetheplaywasjust,say,a ,a andchoose,say,b inplaceofa ;inother 1 2 1 3 words, player one might wish to backtrack. Then, the game might go on as a ::a ::b ::...::b and, 1 2 1 m once again, player one might want to backtrack to, say, a ::a ::b ::...::b, with i<m, and so on... 1 2 1 i Asthere is no learning without remembering, player one must keep in mind the errors made during the play. Thisistheideaof1-Backtracking games(formoremotivations, wereferthereaderto[4]and[3]) andhereisourdefinition. Definition11(1-BacktrackingGames) LetG=(V,E ,E ,W)beagame. 1 2 1. Wedefine1Back(G)asthegame(P ,E′,E′,W′),where: G 1 2 2. P isthesetoffiniteplaysofG G 3. E′ :={(p::a, p::a::b)| p,p::a∈P ,(a,b)∈E }and 2 G 2 E′ :={(p::a, p::a::b)| p,p::a∈P ,(a,b)∈E }∪ 1 G 1 {(p::a::q::d, p::a)| p,q∈P ,p::a::q::d∈P ,a∈Dom(E ) G G 1 d ∈/Dom(E ),p::a::q::d∈/W}; 2 4. W′ isthesetoffinitecompleteplays p ::...:: p of(P ,E′,E′)suchthat p ∈W. 1 n G 1 2 n Note. Thepair(p::a::q::d, p::a)inthedefinitionaboveofE′ codifiesabacktrackingmovebyplayer 2 one(andwepointoutthatq::d mightbetheemptysequence). Remark. Differentlyfrom[4],inwhichbothplayersareallowedtobacktrack,weonlyconsiderthecase inwhichonlyplayeroneissupposed dothat(asin[7]). Itisnotthatourresultswouldnothold: clearly, the proofs inthis paper would work just asfinefor thedefinition of1-Backtracking Tarski gamesgiven in[4]. However,asnotedin[4],anyplayer-onerecursivewinningstrategyinourversionofthegamecan be effectively transformed into a winning strategy for player one in the other version the game. Hence, addingbacktracking forthesecondplayerdoesnotincreasethecomputational challenge forplayerone. Moreover,thenotionofwinnerofthegamegivenin[4]isstrictlynonconstructive andgamesplayedby player one with the correct winning strategy may even not terminate. Whereas, with our definition, we canformulateourmaintheoremasaprogramterminationresult: whateverthestrategychosenbyplayer two, the gameterminates withthewinofplayer one. Thisisalso the spirit ofrealizability and hence of thispaper: theconstructive information mustbecomputedinafiniteamountoftime,notinthelimit. InthewellknownTarskigames,therearetwoplayersandaformulaontheboard. Thesecondplayer - usually called Abelard - tries to show that the formula is false, while the first player - usually called Eloise-triestoshowthatitistrue. Letusseethedefinition. Definition12(TarskiGames) LetAbe aclosed implication and negation free arithmetical formula of L. WedefinetheTarskigameforAasthegameT =(V,E ,E ,W),where: A 1 2 12 InteractiveLearningBasedRealizabilityand1-BacktrackingGames 1. V isthesetofallsubformula occurrences ofA;thatis,V isthesmallestsetofformulassuchthat, ifeitherA∨BorA∧Bbelongs toV,thenA,B∈V;ifeither ∀xA(x)or∃xA(x)belongs toV,then A(n)∈V forallnumeralsn. 2. E is the set of pairs (A ,A )∈V ×V such that A =∃xA(x) and A =A(n), or A =A∨B and 1 1 2 1 2 1 eitherA =AorA =B; 2 2 3. E is the set of pairs (A ,A )∈V ×V such that A =∀xA(x) and A =A(n), or A =A∧B and 2 1 2 1 2 1 A =AorA =B; 2 2 4. W isthesetoffinitecompleteplaysA ::...::A suchthatA =True. 1 n n Note. Westress that Tarski games are defined only for implication and negation free formulas. Indeed, 1Back(T ),whenAcontains implications, wouldbemuchmoreinvolved andlessintuitive (foradefini- A tionofTarskigamesforeveryarithmetical formulaseeforexampleBerardi[2]). Whatwewanttoshowisthatift(cid:14)A,tgivestoplayeronearecursivewinningstrategyin1Back(T ). A The idea of the proof is the following. Suppose we play as player one. Our strategy is relativized to a knowledge stateandwestartthegamebyfixingtheactualstateofknowledgeas0/. Thenweplayinthe same way as we would do in the Tarski game. For example, if there is ∀xA(x) on the board and A(n) is chosen by player two, we recursively play the strategy given by tn; if there is ∃xA(x) on the board, wecalculate p t[0/]=nand play A(n) and recursively the strategy given by p t. Ifthere isA∨B on the 0 1 board, we calculate p t[0/], and according as to whether it equals True or False, we play the strategy 0 recursivelygivenby p t or p t. Ifthereisanatomicformulaontheboard,ifitistrue,wewin;otherwise 1 2 weextendthecurrentstatewiththestate0/⋒t[0/],webacktrack andplaywithrespect tothenewstateof knowledgeandtryingtokeepascloseaspossibletothepreviousgame. Eventually,wewillreachastate largeenough toenableourrealizer togivealwayscorrectanswersandwewillwin. Letusconsider first anexampleandthentheformaldefinition ofthewinningstrategyforEloise. Example (EM ). Given a predicate P of T, and its boolean negation predicate ¬P (which is repre- 1 sentable inT),therealizerE of P EM :=∀x.∃yP(x,y)∨∀y¬P(x,y) 1 isdefinedas la NhX a , hF a , 0/i, l mN Add a mi P P P According to the rules of the game 1Back(TEM ), Abelard is the first to move and, for some numeral n, 1 chooses theformula ∃yP(n,y)∨∀y¬P(n,y) NowistheturnofEloiseandsheplaysthestrategygivenbytheterm hX n, hF a , 0/i, l mN Add nmi P P P Hence,shecomputesX n[0/]=c 0/n=False(bydefinition 4),sosheplaystheformula P P ∀y¬P(n,y) F.Aschieri 13 andAbelardchoosesmandplays ¬P(n,m) If¬P(n,m)=True,Eloisewins. Otherwise,sheplaysthestrategygivenby (l mN Add a m)m[0/]=add 0/nm={hP,n,mi} P P So,thenewknowledge stateisnow{hP,n,mi}andshebacktracks totheformula ∃yP(n,y)∨∀y¬P(n,y) Now,bydefinition4,X n[{hP,n,mi}]=Trueandsheplaystheformula P ∃yP(n,y) calculates theterm p hF n, 0/i[{hP,n,mi}]=j {hP,n,mi}n=m 0 P P playsP(n,m)andwins. Notation. Inthefollowing, weshalldenote withupper caseletters A,B,C closed arithmetical formulas, with lower case letters p,q,r plays of T and with upper case letters P,Q,Rplays of 1Back(T )(and all A A those letters maybe indexed by numbers). Toavoid confusion withthe plays ofT ,plays of1Back(T ) A A willbedenotedas p ,...,p ratherthan p ::...:: p . Moreover,ifP=q ,...,q ,thenP,p ,...,p will 1 n 1 n 1 m 1 n denotethesequence q ,...,q ,p ,...p . 1 m 1 n Definition13 Fixusuchthatu(cid:14)A. Let pbeafiniteplayofT starting withA. Wedefinebyinduction A onthelengthof patermr (p)∈T (readas‘therealizeradaptto p’)inthefollowingway: Class 1. If p=A,thenr (p)=u. 2. If p=(q::∃xB(x)::B(n))andr (q::∃xB(x))=t,thenr (p)=p t. 1 3. If p=(q::∀xB(x)::B(n))andr (q::∀xB(x))=t,thenr (p)=tn. 4. If p=(q::B ∧B ::B)andr (q::B ∧B )=t,thenr (p)=p t. 0 1 i 0 1 i 5. If p=(q::B ∨B ::B)andr (q::B ∨B )=t,thenr (p)= pt. 1 2 i 1 2 i GivenaplayP=Q,q::Bof1Back(T ),wesetr (P)=r (q::B). A Definition14 Fix u such that u(cid:14)A. Let r be as in definition 13 and P be a finite play of 1Back(T ) A starting with A. Wedefine by induction on the length of P astate S (P)(read as ‘the state associated to P’)inthefollowing way: 1. IfP=A,thenS (P)=∅. 2. IfP=(Q,p::B,p::B::C)andS (Q,p::B)=s,thenS (P)=s. 3. If P=(Q,p::B::q,p::B) and S (Q,p::B::q)=s and r (Q,p::B::q)=t, then if t :S, then S (P)=s⋒t[s],elseS (P)=s. Definition15(Winningstrategy for1Back(T )) Fixusuchthatu(cid:14)A. Letr andS berespectively as A in definition 13 and 14. We define a function w from the set of finite plays of 1Back(T ) to set of finite A playsofT ;w isintended tobearecursivewinningstrategy fromAforplayeronein1Back(T ). A A 14 InteractiveLearningBasedRealizabilityand1-BacktrackingGames 1. Ifr (P,q::∃xB(x))=t,S (P,q::∃xB(x))=sand(p t)[s]=n,then 0 w (P,q::∃xB(x))=q::∃xB(x)::B(n) 2. Ifr (P,q::B∨C)=t andS (P,q::B∨C)=s,thenif(p t)[s]=Truethen 0 w (P,q::B∨C)=q::B∨C::B else w (P,q::B∨C)=q::B∨C::C 3. IfA isatomic,A =False,r (P,A ::···::A )=t andS (P,A ::···::A )=s,then n n 1 n 1 n w (P,A ::···::A )=A ::···::A 1 n 1 i whereiisequaltothesmallest j<nsuchthatr (A ::···::A )=wandeither 1 j A =∃xC(x)∧A =C(n)∧(p w)[s⋒t[s]]6=n j j+1 0 or A =B ∨B ∧A =B ∧(p w)[s⋒t[s]]=False j 1 2 j+1 1 0 or A =B ∨B ∧A =B ∧(p w)[s⋒t[s]]=True j 1 2 j+1 2 0 Ifsuch jdoesnotexist,weseti=n. 4. Intheothercases, w (P,q)=q. Lemma1 Suppose u(cid:14)Aandr ,S ,w asindefinition 15. LetQbeafinitew -correctplayof1Back(T ) A starting withA,r (Q)=t,S (Q)=s. IfQ=Q′,q′::B,thent (cid:14) B. s Proof. Byastraightforward induction onthelengthofQ. Theorem4(SoundnessTheorem) Let A be a closed negation and implication free arithmetical formula. Suppose that u(cid:14)Aand consider the game 1Back(T ). Letw be asin definition 15. Then w isa A recursive winningstrategyfromAforplayerone. Proof. Thetheoremwillbeprovedinthefullversionofthispaper. Theideaistoproveitbycontradiction, assuming there is an infinite w -correct play. Then one can produce an increasing sequence of states. Usingtheorems1and2,onecanshowthatEloise’smoveseventually stabilize andthatthegameresults inawinnningposition forEloise. 4 Examples Minimum Principle for functions over natural numbers. The minimum principle states that every function f over natural numbers has aminimum value, i.e. there exists an f(n)∈N such that for every m∈N f(m)≥ f(n). We can prove this principle in HA+EM , for any f in the language. We assume 1 P(y,x)≡ f(x)<y, but, in order to enhance readability, we will write f(x)<y rather than the obscure P(y,x). Wedefine: Lessef(n):=∃a f(a )≤n F.Aschieri 15 Lessf(n):=∃a f(a )<n Notlessf(n):=∀a f(a )≥n Thenweformulate-inequivalent form-theminimumprinciple as: Hasminf :=∃y.Notlessf(y)∧Lessef(y) The informal argument goes as follows. As base case of the induction, we just observe that f(k)≤0, implies f has a minimum value (i.e. f(k)). Afterwards, if Notlessf(f(0)), we are done, we have find the minimum. Otherwise, Lessf(f(0)), and hence f(a )< f(0) for some a given bythe oracle. Hence f(a )≤ f(0)−1andweconclude that f hasaminimumvaluebyinduction hypothesis. Nowwegivethe formal proofs, which are natural deduction trees, decorated withtermsofT , as Class formalizedin[1]. Wefirstprovethat∀n.(Lessef(n)→Hasminf)→(Lessef(S(n))→Hasminf)holds. [Notlessf(S(n))] [Lessf(S(n))] EP:∀n.Notlessf(S(n))∨Lessf(S(n)) D1 D2 E n:Notlessf(S(n))∨Lessf(S(n)) Hasminf Hasminf P D:Hasminf l w D:Lessef(S(n))→Hasminf 2 l w l w D:(Lessef(n)→Hasminf)→(Lessef(S(n))→Hasminf) 1 2 l nl w l w D:∀n(Lessef(n)→Hasminf)→(Lessef(S(n)→Hasminf) 1 2 wherethetermDislookedatlater,D istheproof 1 v :Notlessf(S(n)) w :Lessef(S(n)) 1 2 hv ,w i:Notlessf(S(n))∧Lessef(S(n)) 1 2 hS(n),hv ,w ii:Hasminf 1 2 andD istheproof 2 [x : f(z)<S(n)] 2 x : f(z)≤n 2 w :[Lessef(n)→Hasminf] hz,x i:Lessef(n) 1 2 v :[Lessf(S(n))] w hz,x i:Hasminf 2 1 2 w hp v ,p v i:Hasminf 1 0 2 1 2 WeprovenowthatLessef(0)→Hasminf x :[f(z)≤0] 1 x : f(z)=0 1 x : f(a )≥ f(z) 0/ : f(z)≤ f(z) 1 la x :Notlessf(f(z)) hz,0/i:Lessef(f(z)) 1 hla x ,hz,0/ii:Notlessf(f(z))∧Lessef(f(z)) 1 w:[Lessef(0)] hf(z),hla x ,hz,0/iii:Hasminf 1 hf(p w),hlap w,hp w,0/iii:Hasminf 0 1 0 F :=l whf(p w),hlap w,hp w,0/iii:Lessef(0)→Hasminf 0 1 0 Thereforewecanconclude withtheinduction rulethat la N RF(l nl w l w D)a :∀x.Lessef(x)→Hasminf 1 2 Andnowthethesis: