No-counterexample Interpretations of Logic and the Geometry of Interaction. Masaru Shirahata Keio University, Hiyoshi Campus

Transcription

1 No-counterexample Interpretations of Logic and the Geometry of Interaction Masaru Shirahata Keio University, Hiyoshi Campus

2 The Plan of the Talk I will first explain the motivation. Then I will mostly explain the nocounterexample interpretation (NCI) according to Tait s work. Finally I will add a small observation of mine and present NCI in a trace-like graphical representation.

3 Introduction The functional interpretations of logic have a flavor of game. The values for existential quantiers are positive and those for universal quantifiers are negative. In the negation the positive and the negative change the roles. I want to relate them to GoI.

4 GoI and Cut-elimination GoI is supposed to model the dynamics of cut-elimination. For the consistency proof the cutelimination of propositional logic is not so interesting. All techniques of the consistency proof is to handle the alternating quantifiers.

5 The Consistency Proof of PA The epsilon substitution method by Hilbert and Ackermann. The Cut-elimination method by Gentzen The Dialectica interpretation by Goedel No-counterexample interpretation by Kreisel.

6 The Pre-history Gentzen s first version of the consistency proof is in terms of reduction. Goedel described Gentzen s idea in his Zilsel lecture, essentially as a nocounter example interpretation. It can be stated in terms of game, recently revived by Coquand

7 Our Convention Consider the sentences in a classical first-order logic. Quantified sentences are regarded as infinitary disjunctions and conjunctions. Negations are pushed inside by the De Morgan duality. In the games, the player s moves are blue and the opponents are red.

8 The Henkin Hintikka Game Start with a sentence. The Player and the Opponent form a new sentence from the sentence in the previous stage. Ends with an atomic sentence. The Player wins if the atomic sentence is true. The opponent wins otherwise.

9 The Moves in the Henkin Hintikka Game φ j Done pφ φ A k

10 The Gentzen Game Start with a list of sentences in the prenex normal form. The Player and the Opponent form a new list of sentences from the list in the previous stage. The player wins if the list contains a true prime (atomic) sentence.

11 The Moves in the Gentzen A,..., p, j A p,..., A,..., Game φ, ( ),... p,... p,... φ k φ,...,... A i,... k

12 Some Restriction The Player does not repeat the same instantiation, in other words, always chooses a different disjunct from the given disjunction. We regard quantifier free sentences as prime (atomic). This restriction is not crucial with respect to the expressive power.

13 The counter-strategy as a function The counter-strategy (trying to falsify) of the Opponent may be seen as a function of the previous moves of the Player. The re-instantiation of the existential sentence may be seen as the change of mind.

14 The counter-strategy as a tree 0... p... 3 o p o o 0... p p...

15 The winning strategy as a path finder True! 0... p... 3 o p o o 0... p p...

16 The no-counterexample Interpretation (NCI) The universally quantified (negative) variables are replaced by the functions of the preceding existentially quantified (positive) variables. For a provable sentence one can find the functionals of those negative functions, yielding the witnesses for the positive variables.

17 A Brief Histroy of NCI NCI was introduced by G. Kreisel,using Herbrand s theorem for FOL and the epsilon substitution for PA. The direct proofs are given by Kohlenbach and Tait. Tait s work dates back to early 960 s, which had been unpublished since then.

18 The NCI and the Gentzen Game u x u x A ( u, x, u, x ) Consider a counter-strategy. A F f, A ( f u, f u u ) u, f F f where u, The winning strategy finds a path., F f, f f f f F f F f

19 Modus Ponens x u x u A ( x, u, x, u ) x u x u va ( x y, u, Ax, ( x u, ) u, x v, u ) y B v,, y v y B v, y

20 Modus Ponens in NCI A F f, g, A g, Gg g,, f F f g gg g, F f,, G, g G g f F f F f G g B HB g H, g g G, g HG g 3 3 H g with g, g ( ) ( ( )() ) B JH h g, hg h J, h H g g h such that A ( g, g ( g h ), g h ) G g g h, G g G g

21 Finding the Counter Strategies A F f A f, g, g ˆ = F f f f ˆ ( ), f F G g F ˆ f f g h, f f = g ˆ G ˆ G ˆ F f f, g G g g g ˆ g h ( )( ) f F ( ), g h ( g h ) = f ˆ F ˆ F ˆ f f ˆ f f F ˆ f f G g f f ˆ g h = F G ˆ g ˆ f f g h

22 The General Pattern Find the solution h for h ( L ( h )) = K ( h ) Solved with h K ( h ) u x L h h ( ) ( ) h L ( h )

23 The Approximation h ( m ) = 0 0 hn + ( m )= K hn h m n ( ) if m = L ( h ) n ( ) otherwise h h n n + L ( hn), K ( hn)

24 The System of α-recursive Functionals Tait introduced the system of recursively definable functionals, allowing the recursion along a primitive recursively definable well-founded partial order α. One can keep track of how much of the initial segment of input functions is necessary to compute the value of the functional, along α.

25 The Solution in the System of α-recursive Functionals Let h ˆ be h such that n + L ( h )= L ( h ) n + n We have L ( h )= L ( h ) K ( h )= K h Hence h ( L ( h ))= n + n + n + ( )= K hn h L ( h ) n ( ) ( )= K ( h ) n +

26 The Solution as the Fixpoint of the Update Operator Assume L ( h )= L ( h ) n + n For m = L ( h ) n + ( ( )) = K hn + h L h n + n + Otherwise h ( m ) = h n + n + ( ) m ( ) = K hn ( ) = n + ( ) h L ( h ) n ( ( ))= n + h L h n + n + ( )= K ( h ) n + h L ( h ) n +

27 The Solution is the Fixpoint of the Update Operator Assume h = U ( h ) Then h ( L ( h )) =U h ( ) = K h ( ) L ( h ) ( ) Assume h ( L ( h ))= K ( h ) For m = L ( h ) U ( h )( m ) = K ( h )= h ( m ) Otherwise U ( h )( m )= h m ( )

28 The Similarity between NCI and GoI The morphisms in GoI and the interpretations in NCI are functions from negatives to positives. The composition in GoI and Modus Ponens of NCI are formed by taking trace and fixpoint, connecting the corresponding negatives and positives. The simple duality is lost in NCI.

29 Trace-like Operation Given Φ, i F j with i Φ, F j: Take the fixpoint of Φk X,... and substitute it in Φ i, F j X, l... X X l X l Φk X l

30 A F f A The Counter Strategies in f, Cyclic Graphs ( ), f F f f g, G g A x, u, g h, x, u F f g G f, ( )( ) f F ( ), g g h B v, y u F u F x x f f G g g h F f f x x y G G G 3 u u v

31 g ˆ = F f f Stage u F u x F F x x x x y G G G 3 u u v

32 Stage f ˆ F ˆ f f = G ˆ g h g u u F F Pr x x u x F F x U u x x y u G G u u G G 3 v

33 ( g h ) = g ˆ G ˆ g ˆ F ˆ f f Stage 3 u u x F F x F u x Pr x F u F x U U x x x y G u G u Pr x u G u G G 3 v

34 f ˆ F ˆ f f ˆ F ˆ f f ˆ = G ˆ g ˆ g h Stage 4 u u F F F F Pr x x x u F Pr x F u F x U U x x u x U u x y x G u G u Pr x G G u u G G 3 v u

35 The General Pattern K x x L Pr u u n x U x

36 The Categorical NCI? Adding the propositional structure is just straightforward. The trace here is partial. We need to find a suitable framework. Study the formal properties of our trace.

37 The Dialectica Interpretation and NCI In Dialectica we have the duality at the cost of higher-types. In Dialectica the cut is composition while in NCI the cut is taking the trace. NCI is a germ of Dialectica?

38 Conclusion We have seen the similarity between NCI and GoI. The predicate logic is quite relevant. The unified framework?