Interactive Learning Based Realizability and 1-Backtracking Games

Steffen v an Bakel, Stefano Berardi, Ulrich Berg er (Eds.): Classical Logic and Computation 2010 (Cl&C’10) EPTCS 47, 2011, pp. 6–20, doi:10.4204/EPTCS.47.3 c  F . As chieri This work is licensed under the Creativ e Commons Attribution License. Interactive Learning Based Realiz ability and 1-Backtracking Games Federico Aschieri Dipartimento di Informatica Univ ersit ` a di T orino Italy School of Electronic Engineering and Computer Science Queen Mary , University of Lond on UK W e prove that intera cti ve learning based classical re alizability ( introduc ed by Aschieri an d Berardi for first order a rithmetic [ 1]) is so und with respect to Coq uand ga me semantics. In pa rticular, any realizer of an imp lication-an d-negation- free arithmetical formula emb odies a winning recursiv e strategy for the 1-Backtrackin g version of T arski games. W e also gi ve examples of realizer and winning strategy extraction for some classical proofs. W e also sketch some ongo ing work about h ow to extend ou r notion of realizability in order to obtain completene ss with respect to Coquand semantics, when it is restricted to 1-Backtrack ing games. 1 Introd uction In this pape r we sho w that learni ng based realizab ility (see Aschieri and Berard i [1]) rel ates to 1- Backtrac king T arski games as intu itionistic realizabili ty (see Kleene [8 ]) relat es to T arski games . It is well know t hat T arski games (see, definit ion 12 belo w) are just a simple way of re phrasing the con cept of cl assical truth i n terms of a game b etween two player s - the first one tryin g to sho w the truth of a formula, the second its f alsehood - a nd that an intuitionist ic realizer giv es a winning re cursi ve strateg y to the fi rst player . The result is quite e xpected: since a realiz er giv es a way of co mputing all the informati on about the truth of a formula, the play er trying to pro ve the truth of that formu la has a recu rsiv e winning strate gy . Ho w e ver , not at all any classic ally prov able arithmetical formula allows a winning recursi ve strate gy for that player; otherwise, the decida bility of the Halting problem wou ld follo w . In [5], C o- quand introd uced a game semant ics for Peano Arithmetic such that, for an y prov able formula A , the first player has a recursi ve winning strate gy , coming from the proof of A . The key idea of that remarka ble result is to modify T arski games, allo wing players to correct their m istak es and backtrac k to a pre vi- ous position. Here w e sho w that learning based realizers hav e direct interpreta tion as winni ng recur siv e strate gies in 1-Backtra cking T arski games (which are a particular case of Coquand games see [4] and definitio n 11 belo w). T he res ult, again, is ex pected: interacti ve learning based reali zers, by desig n, are similar to strategie s in games with backtrack ing: they improv e their computatio nal ability by learnin g from inte raction and counter examples in a con ver gent way; ev entually , they gather enough information about the truth of a formula to win the game. An interesting s tep to wards our result was the Hayashi r ealizability [7]. Ind eed, a realizer in t he s ense of Hayashi represent s a recurs ive winning strateg y in 1-Backtr acking games. Ho wev er , from the com- putati onal point of view , realizers do not relate to 1-Backtrac king games in a significant way: Hayashi winning strateg ies work by ex hausti ve search and, actually , do not learn from the game and from the inter action w ith the other player . As a result of this issue , construct ive upper bounds on the length of F . Aschier i 7 games cannot be obtained, whereas using our realizabil ity it is possible. For example, in the case of the 1-Backtr acking T arski game for the formula ∃ x ∀ y f ( x ) ≤ f ( y ) , the H ayashi realizer checks all the natu- ral numbers until an n such that ∀ y f ( n ) ≤ f ( y ) is found ; on the contr ary , our realize r yields a strate gy which bounds the number of backtrackin gs by f ( 0 ) , as shown in this paper . In this case, the Hayashi strate gy is the same one suggested by the classical truth of the formula, bu t instead one is interested in the constru ctiv e strate gy suggested by its classical pr oof . Since learnin g based realiz ers are extracte d from proof s in HA + EM 1 (Heyti ng A rithmetic with ex- cluded middle over existen tial sente nces, see [1]), one also has an interpretat ion of classical proofs as learnin g strate gies. Moreo ver , studying learn ing based realiz ers in terms of 1-Backt racking games also sheds light on their beha viour and offers an intere sting case stud y in program ext raction and interpreta- tion in classica l arithmeti c. The plan of the paper is the follo wing. In section § 2, we recall the calculu s of realizers and the main notion of intera ctiv e learning based realizabili ty . In sec tion § 3, w e prove our m ain theorem: a realize r of an arithmet ical formula embodies a winning strate gy in its associated 1-Backtracki ng T arski game. In secti on § 4, we extract real izers from two classical proof s and study their beha vior as learn ing strate gies. In section § 5 , we define an extensio n of our realizabil ity and formulate a conj ecture abou t its complete ness with respe ct to 1-Backtrack ing T arski games. 2 The Calculus T Class and Learnin g-Ba sed Realizability The whole con tent of this sec tion is based on Aschieri and Berardi [1], where the reader may als o fi nd full mo tiv ations an d proofs. W e recall he re the de finitions and the resu lts we n eed in the rest of the paper . The winning strat egies for 1-Backtrackin g T arski games will be represent ed by terms of T Class (see [1]). T Class is a system of type d lambda calc ulus which extends G ¨ odel’ s syst em T by adding symbols for non computa ble functi ons and a new type S (denoting a set of states of kno wledge ) toget her with two basic operat ions ov er it. The terms of T Class are compute d with respe ct to a state of kno wledge, which represents a finite approx imation of the non compu table functions used in the system. For a complete definition of T w e refer to G irard [6]. T is simply typed λ -calcu lus, with atomic types N (representin g the set N of natural number s) and Bo ol (repr esenting the set B = { T rue , False } of boolea ns), product types T × U and arro ws types T → U , con stants 0 : N , S : N → N , True , False : Bool , pairs h ., . i , project ions π 0 , π 1 , condition al if T and primitiv e recursio n R T in all types , and the usua l reduct ion rule s ( β ) , ( π ) , ( if ) , ( R ) for λ , h ., . i , if T , R T . From now on, if t , u are terms of T with t = u we denote prov able equa lity in T . If k ∈ N , the numera l denotin g k is the clos ed normal term S k ( 0 ) of type N . All clos ed normal terms of type N are numer als. Any closed normal term of type Bool in T is T rue or False . W e introduce a notation for ternary projectio ns: if T = A × ( B × C ) , with p 0 , p 1 , p 2 we respecti vely denote the terms π 0 , λ x : T . π 0 ( π 1 ( x )) , λ x : T . π 1 ( π 1 ( x )) . If u = h u 0 , h u 1 , u 2 ii : T , then p i u = u i in T for i = 0 , 1 , 2. W e abbre viate h u 0 , h u 1 , u 2 ii : T with h u 0 , u 1 , u 2 i : T . Definition 1 (States of Knowledge and Consiste n t Union) 1. A k -ary predica te of T is any closed normal term P : N k → B ool of T . 2. An ato m is any triple h P ,~ n , m i , wher e P is a ( k + 1 ) -ary pre dicate of T , and ~ n , m ar e ( k + 1 ) numer als, and P ~ nm = True in T . 3. T wo atoms h P ,~ n , m i , h P ′ , ~ n ′ , m ′ i ar e consistent if P = P ′ and ~ n = ~ n ′ in T imply m = m ′ . 4. A state of knowledg e, shortly a state , is any finite set S of pairwise consistent atoms. 8 Interac tiv e Learning Based Realizabi lity and 1-Back tracking Games 5. T wo state s S 1 , S 2 ar e consis tent if S 1 ∪ S 2 is a state . 6. S is the set of all states of knowledg e. 7. The cons istent uni on S 1 U S 2 of S 1 , S 2 ∈ S is S 1 ∪ S 2 ∈ S minus all atoms of S 2 which ar e inconsis tent with some ato m of S 1 . For each state of kno wledge S w e assume ha ving a unique consta nt S denoti ng it; if there i s no ambiguity , we just assume that state constants are str ings o f the form {h P , ~ n 1 , m 1 i , . . . , h P , ~ n k , m k i} , denoting a sta te of kno wledge. W e define with T S = T + S + { S | S ∈ S } the e xtension of T with one atomic type S denot ing S , and a constant S : S for each S ∈ S , and no new reduction rule. Computatio n on states will be defined by a set of algebr aic reduc tion rules we call “functio nal”. Definition 2 (Functional set of rules) Let C be any set of cons tants, each one of some typ e A 1 → . . . → A n → A, for some A 1 , . . . , A n , A ∈ { Bo ol , N , S } . W e say that R is a fun ctional set of reducti on rules for C if R consists, for all c ∈ C and all closed normal terms a 1 : A 1 , . . . , a n : A n of T S , of e xactly one rule ca 1 . . . a n 7→ a, w her e a : A is a closed normal term of T S . W e define two ex tensions of T S : an ex tension T Class with symbols denoting non-computa ble maps X P : N k → Bool , Φ P : N k → N (for each k -ary predicate P of T ) and no computa ble reduction rules, anothe r ext ension T Learn , with the computable ap proximations χ P , φ P of X P , Φ P , an d a computable set of re duction rules. X P and Φ P are inte nded to repre sent respecti vely the orac le mapping ~ n to th e truth val ue of ∃ xP ~ nx , and a S kol em funct ion mapping ~ n to an element m such that ∃ xP ~ nx hold s iff P ~ nm = T rue . W e use the elements of T Class to represent non-computa ble reali zers, and the elements of T Learn to represen t a computa ble “app roximation” of a re alizer . W e denote terms of type S by ρ , ρ ′ , . . . . Definition 3 Assume P : N k + 1 → Bool is a k + 1 -ary pr edicate of T . W e intr oduce the following con- stants : 1. χ P : S → N k → B ool and ϕ P : S → N k → N . 2. X P : N k → B ool and Φ P : N k → N . 3. ⋒ : S → S → S (we denote ⋒ ρ 1 ρ 2 with ρ 1 ⋒ ρ 2 ). 4. Add P : N k + 1 → S and add P : S → N k + 1 → S . 1. Ξ S is the set of all consta nts χ P , ϕ P , ⋒ , add P . 2. Ξ is the set of all const ants X P , Φ P , ⋒ , Add P . 3. T Class = T S + Ξ . 4. A term t ∈ T Class has state / 0 if it has no state constan t dif fer ent fr om / 0 . Let ~ t = t 1 . . . t k . W e interpret χ P s ~ t and ϕ P s ~ t respecti vel y as a “guess ” for the values of the oracle and the Skole m map X P and Φ P for ∃ y . P ~ t y , guess comput ed w .r .t. the kno wledge state deno ted by the constant s . T here is no set of computa ble redu ction rules for the constant s Φ P , X P ∈ Ξ , and there fore no set of computable reduction rules for T Class . If ρ 1 , ρ 2 denote s the states S 1 , S 2 ∈ S , we interp ret ρ 1 ⋒ ρ 2 as denoti ng the c onsistent union S 1 U S 2 of S 1 , S 2 . Add P denote s the map const antly equal to the empty s tate / 0. add P S ~ nm denotes the empty state / 0 if we canno t ad d th e atom h P ,~ n , m i to S , either becau se h P ,~ n , m ′ i ∈ S for some numeral m ′ , or becau se P ~ nm = False . add P S ~ nm denotes the state {h P ,~ n , m i} otherwise. W e define a system T Learn with reductio n rules ov er Ξ S by a func tional redu ction set R S . Definition 4 (The System T Learn ) Let s , s 1 , s 2 be state co nstants denoti ng the s tates S , S 1 , S 2 . Let h P ,~ n , m i be an atom. R S is the followin g fun ctional set of red uction rules for Ξ S : F . Aschieri 9 1. If h P ,~ n , m i ∈ S, then χ P s ~ n 7→ Tru e and ϕ P s ~ n 7→ m, else χ P s ~ n 7→ Fal se and ϕ P s ~ n 7→ 0 . 2. s 1 ⋒ s 2 7→ S 1 U S 2 3. add P s ~ nm 7→ / 0 if either h P ,~ n , m ′ i ∈ S for some numeral m ′ or P ~ nm = Fa lse , and add P s ~ n m 7→ {h P ,~ n , m i} otherwis e. W e define T Learn = T S + Ξ S + R S . Remark. T Learn is nothing but T S with some “syntactic su gar”. T Learn is strong ly normalizing , has Church- Rosser prope rty for close d term of atomic typ es and: Pro position 1 (Normal Form Pr operty for T Learn ) Assume A is either a n atomic type o r a pr oduct type . Then any closed normal term t ∈ T Learn of type A is: a numeral n : N , or a bool ean True , False : Bool , or a state const ant s : S , or a pa ir h u , v i : B × C . Definition 5 Assume t ∈ T Class and s is a stat e constant. W e call “app r oximation of t at sta te s” the term t [ s ] of T Learn obtain ed fr om t by r eplacing each constant X P with χ P s, each constant Φ P with ϕ P s, each consta nt Add P with add P s. If s , s ′ are state co nstants den oting S , S ′ ∈ S , w e write s ≤ s ′ for S ⊆ S ′ . W e say that a seque nce { s i } i ∈ N of state consta nts is a weakly increa sing chain of states (is w .i. for short) , if s i ≤ s i + 1 for all i ∈ N . Definition 6 (Con ver gence) A ssume that { s i } i ∈ N is a w .i. seque nce of sta te constants, and u , v ∈ T Class . 1. u con ver ges in { s i } i ∈ N if ∃ i ∈ N . ∀ j ≥ i . u [ s j ] = u [ s i ] in T Learn . 2. u con ver ges if u con ver ge s in ev ery w .i. sequen ce of state const ants. Our realiza bility semant ics relies on two propert ies of the non computa ble terms of atomic type in T Class . First, if we repeatedl y increase the kno wledge sta te s , e ventual ly the value of t [ s ] stops changing. Second, if t has typ e S , and contains no s tate const ants but / 0 , th en we may effect iv ely fi nd a way of incre asing the kno wledge state s such that e ventual ly we ha ve t [ s ] = / 0 . Theor em 1 (Stability Theorem) Assume t ∈ T Class is a closed term of atomic type A (A ∈ { Bool , N , S } ). Then t is con ver gent. Theor em 2 (Fixed Point Pr operty) L et t : S be a closed term of T Class of state / 0 , and s = S. Define τ ( S ) = S ′ if t [ S ] = S ′ , and f ( S ) = S ∪ τ ( S ) . 1. F or any n ∈ N , define f 0 ( S ) = S and f n + 1 ( S ) = f ( f n ( S )) . Ther e are h ∈ N , S ′ ∈ S suc h that S ′ = f h ( S ) ⊇ S, f ( S ′ ) = S ′ and τ ( S ′ ) = / 0 . 2. W e may ef fectively find a state constan t s ′ ≥ s suc h that t [ s ′ ] = / 0 . Definition 7 (The language L of Peano Arithmetic) 1. The terms of L a re all t ∈ T , suc h that t : N and F V ( t ) ⊆ { x N 1 , . . . , x N n } for some x 1 , . . . , x n . 2. The atomic formulas of L ar e all Qt 1 . . . t n ∈ T , for some Q : N n → Bool closed term of T , and some terms t 1 , . . . , t n of L . 3. The formulas of L ar e bu ilt fr om atomic formulas of L by the connec tives ∨ , ∧ , → ∀ , ∃ as usual. Definition 8 (T ypes for r ealize rs) F or each arithmetical formu la A we define a type | A | of T by induc- tion on A: | P ( t 1 , . . . , t n ) | = S , | A ∧ B | = | A | × | B | , | A ∨ B | = Bool × ( | A | × | B | ) , | A → B | = | A | → | B | , |∀ xA | = N → | A | , |∃ xA | = N × | A | 10 Interac tiv e Learning Based Realizabi lity and 1-Back tracking Games W e define now our notion of realiz ability , which is relativ ized to a kno wledge state s , and diff ers from Kreisel modified realizabilit y for a single detail: if we realize an atomic formula , the atomic formula does not need to be true, unless the realiz er is equa l to the empty set in s . Definition 9 (Realizabili ty) Assume s is a state constant , t ∈ T Class is a closed term of state / 0 , A ∈ L is a closed formula , and t : | A | . Let ~ t = t 1 , . . . , t n : N . 1. t  s P ( ~ t ) if and only if t [ s ] = / 0 in T Learn implies P ( ~ t ) = True 2. t  s A ∧ B if and only if π 0 t  s A and π 1 t  s B 3. t  s A ∨ B if and only if either p 0 t [ s ] = True in T Learn and p 1 t  s A, o r p 0 t [ s ] = Fals e in T Learn and p 2 t  s B 4. t  s A → B if and only if for all u, if u  s A, then t u  s B 5. t  s ∀ xA if and only if for all numera ls n, t n  s A [ n / x ] 6. t  s ∃ xA if and only for some numer al n, π 0 t [ s ] = n in T Learn and π 1 t  s A [ n / x ] W e define t  A if and only if t  s A for all state constan ts s. Theor em 3 If A is a closed formula pr ovable in HA + EM 1 (see [1 ]), then the re exis ts t ∈ T Class suc h tha t t  A. 3 Games, Lear ning and Realizability In this sectio n, w e define the notion of game, its 1-Backtra cking version andT arski games. W e also prov e our main theorem, connec ting learni ng based realizabi lity and 1-Backtra cking T arski games. Definition 10 (Games) 1. A ga me G between two playe rs is a q uadruple ( V , E 1 , E 2 , W ) , wher e V is a set, E 1 , E 2 ar e subset s of V × V such that Dom ( E 1 ) ∩ Dom ( E 2 ) = / 0 , wher e Dom ( E i ) is th e domain of E i , and W is a se t of sequen ces, poss ibly infinite , of e lements of V . The ele ments of V ar e called positi ons of the game; E 1 , E 2 ar e the trans ition rel ations re spectively for playe r one and playe r two: ( v 1 , v 2 ) ∈ E i means that player i can le gally move fr om the position v 1 to the posit ion v 2 . 2. W e define a play to be a walk, possibly infinite , in the graph ( V , E 1 ∪ E 2 ) , i.e. a sequence , possibl y void, v 1 :: v 2 :: . . . :: v n :: . . . of elements of V such that ( v i , v i + 1 ) ∈ E 1 ∪ E 2 for every i. A play of the form v 1 :: v 2 :: . . . :: v n :: . . . is said to start from v 1 . A play is sai d to be complete if it is either infinit e or is equal to v 1 :: . . . :: v n and v n / ∈ D om ( E 1 ∪ E 2 ) . W is re quir ed to be a set of comple te plays. If p is a compl ete play and p ∈ W , we say that player one wins in p. If p is a complet e play and p / ∈ W , we say that player two wins in p. 3. Let P G be the set of finit e plays. Consider a func tion f : P G → V ; a play v 1 :: . . . :: v n :: . . . is said to be f -corr ect if f ( v 1 , . . . , v i ) = v i + 1 for e very i such that ( v i , v i + 1 ) ∈ E 1 4. A winning strate gy fr om position v for player one is a function ω : P G → V suc h t hat every complete ω -corr ect play v :: v 1 :: . . . :: v n :: . . . belongs to W . Notation. If for i ∈ N , i = 1 , . . . , n we hav e that p i = ( p i ) 0 :: . . . :: ( p i ) n i is a finite sequence of elements of length n i , with p 1 :: . . . :: p n we denote the sequen ce ( p 1 ) 0 :: . . . :: ( p 1 ) n 1 :: . . . :: ( p k ) 0 :: . . . :: ( p k ) n k F . Aschieri 11 where ( p i ) j denote s the j -th element of the seque nce p i . Suppose that a 1 :: a 2 :: . . . :: a n is a play of a game G , represen ting, for some reaso n, a bad situatio n for p layer one (for e xample, in the game of chess, a n might be a configuration of the c hessboard in which player one has just lost his queen ). Then, learn t the lesson, player one might wish to erase some of his mov es and come back to the time the p lay was jus t, say , a 1 , a 2 and cho ose, say , b 1 in pl ace of a 3 ; in other words , player one might w ish to bac ktr ack . Then, the game might go on as a 1 :: a 2 :: b 1 :: . . . :: b m and, once again, player one might wan t to backtrack to, say , a 1 :: a 2 :: b 1 :: . . . :: b i , with i < m , and so on... As there is no learning without rememberi ng, player one must k eep in mind the errors made during the play . This is the idea of 1-Backtracki ng games (for more m oti vat ions, we refer the reader to [4] and [3]) and here is our definition. Definition 11 (1-Backtracking Games) Let G = ( V , E 1 , E 2 , W ) be a game. 1. W e define 1 Bac k ( G ) as the game ( P G , E ′ 1 , E ′ 2 , W ′ ) , wher e: 2. P G is the set of finite pla ys of G 3. E ′ 2 : = { ( p :: a , p :: a :: b ) | p , p :: a ∈ P G , ( a , b ) ∈ E 2 } and E ′ 1 : = { ( p :: a , p :: a :: b ) | p , p :: a ∈ P G , ( a , b ) ∈ E 1 } ∪ { ( p :: a :: q :: d , p :: a ) | p , q ∈ P G , p :: a :: q :: d ∈ P G , a ∈ Dom ( E 1 ) d / ∈ Dom ( E 2 ) , p :: a :: q : : d / ∈ W } ; 4. W ′ is the set of finite complete plays p 1 :: . . . :: p n of ( P G , E ′ 1 , E ′ 2 ) suc h that p n ∈ W . Note. The p air ( p :: a :: q :: d , p :: a ) in the definition abo ve of E ′ 2 codifies a bac ktrac king move by play er one (and we point out that q : : d might be the empty seq uence). Remark. Dif ferently from [4], in which both players are allo wed to backtrac k, we onl y cons ider the case in which only play er one is supposed do that (as in [7]). It is not that our result s wou ld not hold: clearly , the proofs in this pape r would work just as fi ne for the definition of 1-Back tracking T arski games giv en in [4]. Howe ver , as noted in [ 4 ], any player -one recursi ve w inning strate gy in our version of the game c an be ef fecti vely transformed into a winning strat egy for player one in the other v ersion the game. Hence, adding backt racking for the seco nd player does not increase the computatio nal challen ge for player one. Moreo ver , the notion of winner of the game gi ven in [4] is st rictly non constr uctiv e and g ames played by player one with the correct winning strate gy may e ven not terminate. Where as, with our definition, we can formu late our main theo rem as a pro gram termination result: whatev er the strate gy chosen by pla yer two, the game terminate s with the win of player one. This is also the spirit of real izability and hence of this paper: the constructi ve information must be computed in a finite amount of time, not in the limit. In the well known T arski games, the re are two p layers an d a formula on the b oard. T he second pla yer - usually called Abelard - tries to sho w that the formula is fals e, while the first player - usuall y called Eloise - tries to sho w that it is true. Let us see the definition. Definition 12 (T arski Games) Let A be a closed implication and ne gation fr ee arithmet ical formula of L . W e define the T arski game for A as the game T A = ( V , E 1 , E 2 , W ) , wher e: 12 Interac tiv e Learning Based Realizabi lity and 1-Back tracking Games 1. V is the set of all subformula occurr ence s of A; that is, V is the smalles t set of for mulas such that, if either A ∨ B or A ∧ B belongs to V , then A , B ∈ V ; if either ∀ xA ( x ) or ∃ xA ( x ) belon gs to V , then A ( n ) ∈ V for all numer als n. 2. E 1 is the set of pairs ( A 1 , A 2 ) ∈ V × V suc h that A 1 = ∃ xA ( x ) and A 2 = A ( n ) , or A 1 = A ∨ B and either A 2 = A or A 2 = B; 3. E 2 is the set of pairs ( A 1 , A 2 ) ∈ V × V suc h that A 1 = ∀ xA ( x ) and A 2 = A ( n ) , or A 1 = A ∧ B and A 2 = A or A 2 = B; 4. W is the set of finite compl ete plays A 1 :: . . . :: A n suc h that A n = True . Note. W e stress that T arski games are defined only for implicati on and neg ation free formulas. Indeed, 1 Bac k ( T A ) , when A contains implications , would be much more in vo lved and less intuiti ve (for a defini- tion of T arski games for e very arith metical formula see for example Berardi [2]). What we want to sho w is that if t  A , t giv es to player one a recursi ve winning strategy in 1 Bac k ( T A ) . The idea of the proof is the follo wing. Suppo se we play as player one. Our strateg y is relati vized to a kno wledge state and we start the game by fixing the actual state of kno wledge as / 0 . T hen we play in the same way as w e would do in the T arski game. For exa mple, if there is ∀ xA ( x ) on the board and A ( n ) is chos en by player two, w e recursi vely play the strategy giv en by t n ; if there is ∃ xA ( x ) on the board, we calculate π 0 t [ / 0 ] = n and play A ( n ) and recursi vely the strategy giv en by π 1 t . If there is A ∨ B on the board, we calculate p 0 t [ / 0 ] , and accordin g as to whether it equa ls True or False , we play the strate gy recurs ive ly giv en by p 1 t or p 2 t . If there is an ato mic formula on the board, if it is true, we win ; otherwise we exte nd the current state with the state / 0 ⋒ t [ / 0 ] , we backtrack and play with respect to the new state of kno wledge and tr ying to k eep as close a s possible to the pre vious game. Eve ntually , we will reach a state lar ge enough to enable our realizer to giv e always correct answers and we will w in. Let us consider first an exampl e and then the formal definition of the winning strategy for Eloise. Example ( EM 1 ) . Giv en a predicat e P of T , and its boolean negatio n predi cate ¬ P (which is repre- sentab le in T ), the realize r E P of EM 1 : = ∀ x . ∃ y P ( x , y ) ∨ ∀ y ¬ P ( x , y ) is defined as λ α N h X P α , h Φ P α , / 0 i , λ m N Add P α m i Accordin g to the rules of the game 1 Back ( T EM 1 ) , Abelard is the first to move and, for some numeral n , choos es the formula ∃ y P ( n , y ) ∨ ∀ y ¬ P ( n , y ) No w is the turn of Eloise and she plays the strateg y giv en by the term h X P n , h Φ P α , / 0 i , λ m N Add P nm i Hence, she computes X P n [ / 0 ] = χ P / 0 n = Fals e (by definition 4), so she plays the formula ∀ y ¬ P ( n , y ) F . Aschieri 13 and Abelard chooses m and plays ¬ P ( n , m ) If ¬ P ( n , m ) = T rue , Eloise wins. Otherwise, she plays the strate gy giv en by ( λ m N Add P α m ) m [ / 0 ] = add P / 0 nm = {h P , n , m i} So, the ne w knowle dge stat e is now {h P , n , m i} and she backtra cks to the formu la ∃ y P ( n , y ) ∨ ∀ y ¬ P ( n , y ) No w , by definitio n 4, X P n [ {h P , n , m i} ] = True and she plays the formula ∃ y P ( n , y ) calcul ates the term π 0 h Φ P n , / 0 i [ {h P , n , m i} ] = ϕ P {h P , n , m i} n = m plays P ( n , m ) and w ins. Notation. In the follo wing, we shall denote with upper case letters A , B , C closed arithmetical formulas, with lower case letters p , q , r plays of T A and w ith upper case letters P , Q , R plays of 1 Back ( T A ) (and all those letters m ay be inde xed by number s). T o av oid conf usion with the plays of T A , plays of 1Back( T A ) will be denoted as p 1 , . . . , p n rather than p 1 :: . . . :: p n . Moreov er , if P = q 1 , . . . , q m , th en P , p 1 , . . . , p n will denote the sequen ce q 1 , . . . , q m , p 1 , . . . p n . Definition 13 F ix u such that u  A . Let p be a finite play of T A startin g with A. W e define by induction on the length of p a term ρ ( p ) ∈ T Class (r ead as ‘the r ealizer adapt to p’) in the following way: 1. If p = A, then ρ ( p ) = u. 2. If p = ( q :: ∃ xB ( x ) :: B ( n ) ) and ρ ( q :: ∃ xB ( x )) = t , then ρ ( p ) = π 1 t . 3. If p = ( q :: ∀ xB ( x ) :: B ( n ) ) and ρ ( q :: ∀ xB ( x )) = t , then ρ ( p ) = t n. 4. If p = ( q :: B 0 ∧ B 1 :: B i ) an d ρ ( q :: B 0 ∧ B 1 ) = t , then ρ ( p ) = π i t . 5. If p = ( q :: B 1 ∨ B 2 :: B i ) an d ρ ( q :: B 1 ∨ B 2 ) = t , then ρ ( p ) = p i t . Given a play P = Q , q :: B of 1 Back ( T A ) , we set ρ ( P ) = ρ ( q :: B ) . Definition 14 F ix u such that u  A. Let ρ be as in definiti on 13 and P be a finite play of 1 Bac k ( T A ) startin g with A. W e define by induct ion on the length of P a state Σ ( P ) (r ead as ‘the state assoc iated to P’) in the followin g way: 1. If P = A, then Σ ( P ) = ∅ . 2. If P = ( Q , p :: B , p :: B : : C ) and Σ ( Q , p :: B ) = s, then Σ ( P ) = s. 3. If P = ( Q , p :: B :: q , p :: B ) and Σ ( Q , p :: B :: q ) = s and ρ ( Q , p :: B :: q ) = t , then if t : S , then Σ ( P ) = s ⋒ t [ s ] , else Σ ( P ) = s. Definition 15 (Winning s trategy for 1Back( T A )) F ix u suc h that u  A. Let ρ and Σ be r espectively as in definit ion 13 and 14. W e define a functio n ω fr om the set of finite plays of 1 Back ( T A ) to set of finite plays of T A ; ω is intended to be a r ecursiv e w inning stra te gy fr om A for player one in 1 Back ( T A ) . 14 Interac tiv e Learning Based Realizabi lity and 1-Back tracking Games 1. If ρ ( P , q :: ∃ xB ( x )) = t , Σ ( P , q :: ∃ xB ( x )) = s and ( π 0 t )[ s ] = n, then ω ( P , q :: ∃ xB ( x )) = q :: ∃ xB ( x ) :: B ( n ) 2. If ρ ( P , q :: B ∨ C ) = t and Σ ( P , q :: B ∨ C ) = s, then if ( p 0 t )[ s ] = True then ω ( P , q :: B ∨ C ) = q :: B ∨ C :: B else ω ( P , q :: B ∨ C ) = q : : B ∨ C :: C 3. If A n is atomic, A n = False , ρ ( P , A 1 :: · · · :: A n ) = t and Σ ( P , A 1 :: · · · :: A n ) = s, then ω ( P , A 1 :: · · · :: A n ) = A 1 :: · · · :: A i wher e i is equa l to the small est j < n such that ρ ( A 1 :: · · · :: A j ) = w and either A j = ∃ x C ( x ) ∧ A j + 1 = C ( n ) ∧ ( π 0 w )[ s ⋒ t [ s ]] 6 = n or A j = B 1 ∨ B 2 ∧ A j + 1 = B 1 ∧ ( p 0 w )[ s ⋒ t [ s ]] = False or A j = B 1 ∨ B 2 ∧ A j + 1 = B 2 ∧ ( p 0 w )[ s ⋒ t [ s ]] = True If suc h j does not exist , w e set i = n. 4. In the other case s, ω ( P , q ) = q. Lemma 1 Suppos e u  A and ρ , Σ , ω as in definition 15. Let Q be a finite ω -corr ect play of 1 Back ( T A ) startin g with A, ρ ( Q ) = t , Σ ( Q ) = s. If Q = Q ′ , q ′ :: B, then t  s B. Pr oof . By a straightfo rward indu ction on the length of Q . Theor em 4 (Sound ness Theor em) Let A be a closed ne gation and implica tion fr ee arithmetica l for- mula. Suppose that u  A and consider the game 1 Bac k ( T A ) . Let ω be as in defin ition 15. Then ω is a r ecurs ive winnin g strate gy fr om A for player one. Pr oof . T he theorem will be prove d in the full version of this paper . T he idea is to prov e it by contra diction, assuming there is an infinite ω -correc t play . Then one can produce an increasing seque nce of states. Using theore ms 1 and 2, one can sho w that Eloise’ s mo ves e ventua lly stabi lize and that the game re sults in a winnning positio n for Eloise . 4 Examples Minimum Principle for functions ov er natural numbers. The minimum principl e states that e very functi on f ov er natu ral numbers has a minimum v alue, i.e. there exists an f ( n ) ∈ N such that for e very m ∈ N f ( m ) ≥ f ( n ) . W e can prov e this princip le in HA + EM 1 , for any f in the language. W e assume P ( y , x ) ≡ f ( x ) < y , but, in order to enhance readabili ty , w e will write f ( x ) < y rather than the obscure P ( y , x ) . W e define: Lesse f ( n ) : = ∃ α f ( α ) ≤ n F . Aschieri 15 Less f ( n ) : = ∃ α f ( α ) < n N ot l ess f ( n ) : = ∀ α f ( α ) ≥ n Then we formulate - in equi va lent form - the minimum principle as: H asmin f : = ∃ y . N o t l ess f ( y ) ∧ Lesse f ( y ) The informal ar gument goes as follo ws. As base case of the induction , we just observ e that f ( k ) ≤ 0, implies f has a minimum v alue (i.e. f ( k ) ). A fterwa rds, if N ot l ess f ( f ( 0 )) , we are done, we ha ve find the minimum. Otherwise , L ess f ( f ( 0 )) , and hence f ( α ) < f ( 0 ) for some α gi ven by the oracle. Hence f ( α ) ≤ f ( 0 ) − 1 and we conclude that f has a minimum val ue by induction hypothes is. No w we giv e the formal proofs, which are natur al deductio n trees, deco rated with terms of T Class , as formalize d in [1]. W e first pro ve t hat ∀ n . ( Lesse f ( n ) → H asmin f ) → ( Lesse f ( S ( n )) → H asmin f ) holds. E P : ∀ n . N ot l ess f ( S ( n )) ∨ Less f ( S ( n )) E P n : N ot l e ss f ( S ( n )) ∨ Less f ( S ( n )) [ N ot l ess f ( S ( n ))] D 1 H asmin f [ Less f ( S ( n ))] D 2 H asmin f D : H a smin f λ w 2 D : L esse f ( S ( n )) → H asmin f λ w 1 λ w 2 D : ( Lesse f ( n ) → H asmin f ) → ( Lesse f ( S ( n )) → H asmin f ) λ n λ w 1 λ w 2 D : ∀ n ( Lesse f ( n ) → H a smin f ) → ( Lesse f ( S ( n ) → H asmin f ) where the term D is look ed at later , D 1 is the proof v 1 : N ot l ess f ( S ( n )) w 2 : Lesse f ( S ( n )) h v 1 , w 2 i : N ot l ess f ( S ( n )) ∧ Lesse f ( S ( n )) h S ( n ) , h v 1 , w 2 ii : H asmin f and D 2 is the proof v 2 : [ Les s f ( S ( n ))] w 1 : [ Les se f ( n ) → H asmin f ] [ x 2 : f ( z ) < S ( n )] x 2 : f ( z ) ≤ n h z , x 2 i : Lesse f ( n ) w 1 h z , x 2 i : H asmin f w 1 h π 0 v 2 , π 1 v 2 i : H asmin f W e prove no w that Lesse f ( 0 ) → H asmin f w : [ Lesse f ( 0 )] x 1 : [ f ( z ) ≤ 0 ] x 1 : f ( z ) = 0 x 1 : f ( α ) ≥ f ( z ) λ α x 1 : N ot l ess f ( f ( z )) / 0 : f ( z ) ≤ f ( z ) h z , / 0 i : Lesse f ( f ( z )) h λ α x 1 , h z , / 0 ii : N ot l e ss f ( f ( z )) ∧ Lesse f ( f ( z )) h f ( z ) , h λ α x 1 , h z , / 0 iii : H asmin f h f ( π 0 w ) , h λ α π 1 w , h π 0 w , / 0 iii : H asmin f F : = λ w h f ( π 0 w ) , h λ α π 1 w , h π 0 w , / 0 iii : Less e f ( 0 ) → H asmin f Therefore we can conclu de with the indu ction rule that λ α N R F ( λ n λ w 1 λ w 2 D ) α : ∀ x . Lesse f ( x ) → H asmin f And no w the thesis: 16 Interac tiv e Learning Based Realizabi lity and 1-Back tracking Games / 0 : f ( 0 ) ≤ f ( 0 ) h 0 , / 0 i : L esse f ( f ( 0 )) λ α N R F ( λ n λ w 1 λ w 2 D ) α : ∀ x . Les se f ( x ) → H asmin f R F ( λ n λ w 1 λ w 2 D ) f ( 0 ) : Less e f ( f ( 0 )) → H asmin f M : = R F ( λ n λ w 1 λ w 2 D ) f ( 0 ) h 0 , / 0 i : H asmin f Let us no w take a close r look to D . W e hav e defined D : = if X P S ( n ) then w 1 h Φ P S ( n ) , / 0 i el se h S ( n ) , h λ β ( Add P ) S ( n ) β , w 2 ii Let s b e a s tate and let u s consider M , the re alizer of H asmin f , in the bas e case of the recu rsion and after in its genera l form durin g the computatio n: R F ( λ n λ w 1 λ w 2 D ) f ( 0 ) h m , / 0 i [ s ] . If f ( 0 ) = 0, M [ s ] = R F ( λ n λ w 1 λ w 2 D ) f ( 0 ) h 0 , / 0 i [ s ] = = F h 0 , / 0 i = h f ( 0 ) , h λ α / 0 , h 0 , / 0 ii If f ( 0 ) = S ( n ) , we ha ve two other cases. If χ P sS ( n ) = Tr ue , then R F ( λ n λ w 1 λ w 2 D ) S ( n ) h m , / 0 i [ s ] = = ( λ n λ w 1 λ w 2 D ) n ( R F ( λ n λ w 1 λ w 2 D ) n ) h m , / 0 i [ s ] = = R F ( λ n λ w 1 λ w 2 D ) n h Φ P ( S ( n )) , / 0 i [ s ] If χ P sS ( n ) = False , then R F ( λ n λ w 1 λ w 2 D ) S ( n ) h m , / 0 i [ s ] = = ( λ n λ w 1 λ w 2 D ) n ( R F ( λ n λ w 1 λ w 2 D ) n ) h m , / 0 i [ s ] = = h S ( n ) , h λ β ( add P ) sS ( n ) β , h m , / 0 iii In the first case, the minum valu e of f has been found. In the second case, the operator R , starting from S ( n ) , recursi vely calls itself on n ; in the third case, it reduce s to its normal form. F rom these equations , we easily deduc e the beha vior of the realizer of H asmin f . In a pse udo imperati ve programming langu age, for the witness of H asmin f we wou ld write: n : = f ( 0 ) ; whil e ( χ P sn = Tr ue , i . e . ∃ m suc h t ha t f ( m ) < n ∈ s ) d o n : = n − 1; re t ur n n ; Hence, when f ( 0 ) > 0, we ha ve, for some numeral k M [ s ] = h k , h λ β ( add P ) sk β , h ϕ P sk , / 0 iii It is clear that k is the minimum valu e of f , accord ing to the partial informatio n prov ided by s about f , and that f ( ϕ P sk ) ≤ k . If s is suf ficiently complete, then k is the tru e minimum of f . The nor mal form of th e realizer M of H asmin f is so simpl e that we ca n immediately extra ct the winning strate gy ω for the 1-Backt raking ve rsion of the T arski game for H asmin f . Suppose the current state of the game is s . If f ( 0 ) = 0, Eloise choose s the formula N ot l ess f ( 0 ) ∧ Lesse f ( 0 ) and wins. If f ( 0 ) > 0, she chooses F . Aschieri 17 N ot l ess f ( k ) ∧ Lesse f ( k ) = ∀ α f ( α ) ≥ k ∧ ∃ α f ( α ) ≤ k If Abelard chooses ∃ α f ( α ) ≤ k , she wins, because she responds w ith f ( ϕ P sk ) ≤ k , which hold s. Sup- pose hence Abelard choose s ∀ α f ( α ) ≥ k and then f ( β ) ≥ k . If it holds, Eloise wins. O therwise , she adds to the current state s ( λ β ( add P ) sk β ) β = ( add P ) sk β = { f ( β ) < k } and backtr acks to H asmin f and then plays again. T his time, she choose s N ot l ess f ( f ( β )) ∧ Lesse f ( f ( β )) (using f ( β ) , w hich was Abelard’ s countere xample to the m inimality of k and is smaller th an her pre vious choice for the minimum v alue). After at most f ( 0 ) back trackings, she wins. Coquand’s E xample. W e in vest igate now an example - due to Coquand - in our frame work of realiza bility . W e want to prov e that for e very function over nat ural numbers and for ev ery a ∈ N the re exi sts x ∈ N suc h that f ( x ) ≤ f ( x + a ) . Thanks to the minimum principle, we can gi ve a very easy classic al proof : H asmin f [ N ot l ess f ( µ ) ∧ Lesse f ( µ )] Lesse f ( µ ) [ N ot l ess f ( µ ) ∧ Lesse f ( µ )] N ot l ess f ( µ ) f ( z + a ) ≥ µ [ f ( z ) ≤ µ ] f ( z ) ≤ f ( z + a ) ∃ x f ( x ) ≤ f ( x + a ) ∀ a ∃ x f ( x ) ≤ f ( x + a ) ∀ a ∃ x f ( x ) ≤ f ( x + a ) ∀ a ∃ x f ( x ) ≤ f ( x + a ) The ext racted realizer is λ a h π 0 π 1 π 1 M , π 0 π 1 M ( π 0 π 1 π 1 M + a ) ⋒ π 1 π 1 π 1 h i where M is the realizer of H asmin f . m : = π 0 π 1 π 1 M [ s ] is a point the pu rported minimum v alue µ : = π 0 M of f is attained at, accordin gly to the informati on in the stat e s (i.e. f ( m ) ≤ µ ). S o, if Abelard choo ses ∃ x f ( x ) ≤ f ( x + a ) Eloise choos es f ( m ) ≤ f ( m + a ) W e hav e to consider the term U [ s ] : = π 0 π 1 M ( π 0 π 1 π 1 M + a ) ⋒ π 1 π 1 π 1 M [ s ] which update s the curr ent state s . S urely , π 1 π 1 π 1 M [ s ] = / 0. π 0 π 1 M [ s ] is equal eithe r to λ β ( add P ) s µ β or to λ α / 0. So, what does U [ s ] actuall y do? W e ha ve: U [ s ] = π 0 π 1 M ( π 0 π 1 π 1 M + a )[ s ] = π 0 π 1 M ( m + a )[ s ] 18 Interac tiv e Learning Based Realizabi lity and 1-Back tracking Games with either π 0 π 1 M ( m + a )[ s ] = / 0 or π 0 π 1 M ( m + a )[ s ] = { f ( m + a ) < f ( m ) } So U [ s ] tests if f ( m + a ) < f ( m ) ; if it is not the case, Eloise wins, otherwise she enl arges the state s , includ ing the informatio n f ( m + a ) < f ( m ) and backt racks to ∃ x f ( x ) ≤ f ( x + a ) . Starting from the state / 0, after k + 1 backtrac kings, it will be reached a state s ′ , which will be of the form { f (( k + 1 ) a < f ( ka ) , . . . , f ( 2 a ) < f ( a ) , f ( a ) < f ( 0 ) } and Eloise will play f (( k + 1 ) a ) ≤ f (( k + 1 ) a + a ) . Hence, the ext racted algorithm for E loise’ s witness is the follo wing: n : = 0; while f ( n ) > f ( n + a ) do n : = n + a ; return n ; 5 Partial R ecursive Lear ning Ba sed Realizability and Completeness In this section we exte nd o ur notion of realizability and increase the computation al power of our realizers, in order to be able to rep resent any parti al recurs ive function and in particular , we conjectur e, ev ery recurs ive strategie s of 1-Backtr acking T arski games. S o, we choo se to add to our calculus a fixed point combina tor Y , such that for ev ery term u : A → A , Y u = u ( Y u ) . Definition 16 (Systems PCF Class and PCF Learn ) W e define PCF Class and PCF Learn to be, r especti vely , the e xtensions of T Class and T Learn obtain ed by adding for ev ery type A a consta nt Y A of type ( A → A ) → A and a new eq uality axiom Y A u = u ( Y A u ) for every term u : A → A. Since in PCF Class there is a schema for unbounded iterati on, properties lik e con ver gence do not hold any more (think about a term taking a states s and returnin g the larges t n such that χ P sn = True ). S o we hav e to ask our realizers to be con ver gent. Hence, for each type A of PCF Class we define a set k A k of terms u : A which we call the set of stab le terms of type A . W e define stable terms by lifting the notion of con ver gence from atomic types (havin g a special case for the atomic type S , as we said) to arrow and produ ct types. Definition 17 (Con ver gence) Assume that { s i } i ∈ N is a w .i. sequen ce of state constants, and u , v ∈ PCF Class . 1. u con ver ges in { s i } i ∈ N if ther e exist s a nor mal for m v such that ∃ i ∀ j ≥ i . u [ s j ] = v in PCF Learn . 2. u con ver ges if u con ver ge s in ev ery w .i. sequen ce of state const ants. Definition 18 (Stable T erms) Let { s i } i ∈ N be a w .i. chai n of states and s ∈ S . Assume A is a type. W e define a set k A k of terms t ∈ PCF Class of type A, by induc tion on A. 1. k S k = { t : S | t con ver ges } 2. k N k = { t : N | t con ver ges } 3. k Bool k = { t : Bool | t con ver ges } 4. k A × B k = { t : A × B | π 0 t ∈ k A k , π 1 t ∈ k B k} 5. k A → B k = { t : A → B | ∀ u ∈ k A k , t u ∈ k B k} If t ∈ k A k , we say that t is a stable term of type A. No w we exten d the notion of realizabilit y with respect to PCF Class and PCF Learn . F . Aschieri 19 Definition 19 (Realizability ) Assume s is a state constant, t ∈ PCF Class is a c losed term of state / 0 , A ∈ L is a closed formula, and t ∈ k| A |k . Let ~ t = t 1 , . . . , t n : N . 1. t  s P ( ~ t ) if and only if t [ s ] = / 0 in PCF Learn implies P ( ~ t ) = True 2. t  s A ∧ B if and only if π 0 t  s A and π 1 t  s B 3. t  s A ∨ B if and only if either p 0 t [ s ] = T rue in PCF Learn and p 1 t  s A, or p 0 t [ s ] = F alse in PCF Learn and p 2 t  s B 4. t  s A → B if and only if for all u, if u  s A, then t u  s B 5. t  s ∀ xA if and only if for all numera ls n, t n  s A [ n / x ] 6. t  s ∃ xA if and only for some numer al n, π 0 t [ s ] = n in PCF Learn and π 1 t  s A [ n / x ] W e define t  A if and only if t  s A for all state constan ts s. The follo wing conjecture will be addressed in the next ve rsion of this paper: Theor em 5 (Conjecture ) Suppose ther e exis ts a re cursive w inning str ate gy for p layer one in 1 Back ( T A ) . Then ther e exi sts a term t of PCF Class suc h that t  A. 6 Conclusions and Further work The main co ntrib ution of this paper is c onceptual, rather than tec hnical, and it shou ld be useful to u nder - stand the si gnificance and see possi ble uses of learni ng based realizabi lity . W e h av e shown h ow lea rning based realizers may be understo od in terms of backtracking games and that this interpreta tion of fers a way of elicitin g constr uctiv e info rmation from the m. The idea is that playing games represents a way of challe nging realizers; they react to the challenge by learni ng from failure and countere xamples. In the conte xt of games, it is also possibl e to appre ciate the notion of con ver gence, i.e. the fac t that realizers stabili ze their beh aviou r as they increase their kno wledge. Indeed, it looks lik e similar ideas are use ful to underst and othe r clas sical realizabiliti es (see for example , M iquel [9]). A further step will be take n in the full versi on of this paper , where we plan to solve the conjec ture about the completeness of learning based realizabili ty with respect to 1Backtracki ng games. A s pointed out by a referee, the conjectu re could be intere sting with respect to a problem of game semanti cs, i.e. whether all recurs iv e innoce nt strategie s are intepret ation of a term of PCF . Refer ences [1] F . Aschieri, S. Berardi, Interactive Learnin g-Based Realiza bility for Heyting Arithmetic with EM 1 , to appear in Logical Methods in Computer Science, 2010 (prepr int: http:// arxiv .org/abs/1007.1785 ) [2] S. Berardi, Semantics for In tuitionistic A rithmetic Based on T arski Games with Retractable Moves. TLCA 2007 [3] S. Berard i, U. De’ Liguo ro, T oward the interpr etatio n of non- constructive reasoning a s no n-mono tonic learn- ing , Infor mation and Compu tation, v ol 207, issue 1, 2009 [4] S. Berardi, T . Coquan d, S. Hay ashi, Games with 1-Ba ctrac king , to appear in Annals of Pure and Applied Logic, 2010 (see also GALOP 2005). [5] T . Coq uand, A Semantic of E vidence for Classical Arithmetic , Journal of Symbolic Logic 6 0, p ag 325 -337 (1995 ) 20 Interac tiv e Learning Based Realizabi lity and 1-Back tracking Games [6] J.-Y . Girard, Pr oofs and T ypes , Cambrid ge Uni versity Press (1989) [7] S. Hayashi, Can Pr o ofs be Animated by Games? , Fundam enta Infor maticae 77(4), pag 331 -343 (2007) [8] S. C. Kleene, On the Interpretation of Intuitionistic Number Theory , Journal of Symbolic Logic 1 0(4), pag 109-1 24 (1945) [9] A. Miquel, Rela ting c lassical r ealizab ility and negative translation for existential witness e xtraction. In T yp ed Lambda Calculi and Applications (TLCA 2009), pp. 188-2 02, 2009