Light Monotone Dialectica

Transcription

1 Light Monotone Dialectica Extraction of moduli of uniform continuity for closed terms from Goedel s T of type (IN IN) (IN IN) Mircea-Dan Hernest Project LogiCal Paris, France and GKLI Munich, Germany LFMTP 06 Talk in Seattle, 16 August 2006 Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

2 Outline of this Talk 1 Introduction A seemingly simple game-problem of discrete mathematics Generalization and set-up for the Proof-theoretic machinery 2 Hereditarily Extensional Equality in computer system MinLog Term system, majorizability and Hereditarily Extensional Equality Weakly extensional monotonic Arithmetic for Göedel functionals 3 From Gödel s Dialectica to Light Monotone Dialectica The pure Göedel s functional Dialectica interpretation The Contraction Problem > Achilles heel for Dialectica! The Light Monotone Dialectica majorant extraction 4 Conclusions and Future work The human/computer outcome for our general game-problem Computer really necessary? Implementing Monotone Dialectica Work to be done > the real Light Monotone Dialectica Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

3 A game-problem of discrete mathematics 1 Let f, g, h : IN IN s.t. h fixed & n IN. f (n) h(n) g(n) h(n) Given k IN find L IN s.t. [ i L. f (i) = g(i)] [ j k. f (j) = g(j)] 2 Dummy answer L : k. Hence try to complicate the demand: f, g IN IN h. [ i L. f (i) = g(i) ] [ j k. f (f (j)) = g(g(j)) ] 3 Simple optimal answer max{k, h(0),..., h(k)}. But what about: f, g h. [ i L. f (i) = g(i) ] [ j k. f (f (f (j))) = g(g(g(j))) ] 4 Temptation max{k, h(0),..., h(k), h(h(0)),..., h(h(k))}. False, since f (j) h(j) f (f (j)) h(h(j)), hence f (f (0)) > h(h(j)) possible. 5 How to solve this? And what about the fully general case? f, g IN IN h. [ i L. f (i) = g(i) ] [ j k. f (m) (j) = g (m) (j) ] Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

4 Set-up for the Proof-theoretic machinery (1/2) 1 Extract moduli of uniform continuity for closed terms t m of Goedel s T of type (IN IN) (IN IN) where t m : λh IN IN, k IN. R IN IN (0)[λp, q. h (m) (p) + q](k + 1) 2 Hence t m (h, k) h (m) (0) + h (m) (1) h (m) (k). How??? Let P : f, g. [ i. f (i) = IN g(i) ] [ j. t m (f, j) = IN t m (g, j) ] 3 The above P is a Minimal Logic proof of (almost) t m t m. We apply on P a Light Monotone Dialectica extraction in MinLog. 4 Gödel s Dialectica would give an exact realizer t [f, g, j ] for i s.t. f, g j. f (t [f, g, j]) = IN g(t [f, g, j ]) t m (f, j) = IN t m (g, j) 5 If t maj λf, g, j. t then for k j, h maj h and h f, g one has (L : t(h, h, k)) t [f, g, j ] and therefore such an L is a solution: h f, g IN IN h k. [ i L. f (i) = g(i) ] [ j k. f (m) (j) = g (m) (j) ] Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

5 Set-up for the Proof-theoretic machinery (2/2) Start from proof of hereditarily extensional equality of t to itself. Hence a proof of t (IN IN) (IN IN) t in system Z 0 of Berger- Buchholz-Schwichtenberg, the base logic of machine system MinLog. Hence a Minimal Logic proof without use of Extensionality Axiom. Two extreme approaches: 1 First extract t by Gödel s Dialectica and then majorize it via Howard s algorithm (Kohlenbach s PhD thesis, JSL paper 92). 2 Directly extract t by producing a majorant for the closed extracted term at each of the Dialectica recursion step (Kohlenbach 93). None of the two efficient on the computer. Solution: use an intermediate approach > Extract partial majorants which are not necessarily closed terms, only simplify treatment of Contraction. Also use a Normalization during Extraction, i.e. NbE-normalize the extracted term of the conclusion of a Modus Ponens. (NdE) Huge impact of such Partial Evaluation. No solution without it!! Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

6 The term system a lambda-variant of Göedel s T 1 All finite types generated from IN and IB by the rule σ, τ (στ) 2 tt IB, ff IB equality = ININIB and inequality ININIB, maximum Max INININ 3 0 IN (zero), Suc ININ τ (INτ τ)inτ (successor) and Gödel s recursor Rτ And IBIBIB : λp, q. If IB p q ff Imp IBIBIB : λp, q. If IB p q tt 4 Combinators at all types are defined in terms of λ-abstraction: Σ : λx, y, z. x z (y z) Π : λx, y. x 5 at IB is the unique predicate symbol of WeZ m one IB argument 6 Extensionally defined equality and inequality (below σ {IB, IN}) s = IN t : at(= s t) s = IB t : at(s) at(t) s IN t : at( s t) s IB t : at(t) at(s) s = σ1...σ n σ t : x σ x n σn (s x 1... x n = σ t x 1... x n ) s σ1...σ n σ t : x σ x n σn (s x 1... x n σ t x 1... x n ) Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

7 Majorizability and Hereditarily Extensional Equality (1) x maj IN y : x IN y : at( x IN y IN ) x στ y z σ (x z τ y z) x maj στ y : z σ 1, zσ 2 (z 1 maj σ z 2 x z 1 maj τ y z 2 ) 0 maj IN 0, Suc maj ININ Suc, Σ maj Σ, Π maj Π and R M maj R WeZ m t maj στ t s maj σ s = t s maj τ t s x IN y : x = IN y : at(= x IN y IN ) x = στ y z σ (x z = τ y z) x στ y : z σ 1, zσ 2 (z 1 σ z 2 x z 1 τ y z 2 ) 0 IN 0, Suc ININ Suc, Σ Σ, Π Π and R R WeZ m t στ t s σ s = t s τ t s Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

8 System WeZ m > Implic. Introd. with Contraction 1 WeZ m - Weakly extensional Minimal Arithmetic with and Max 2 Minimal Arith. Heyting Arith. in all finite types HA ω \ A 3 WeZ m - underlying Logic is Natural Deduction, not Hilbert-style! 4 [u : A]... /B +, particular set of instances of A in the same A B parcel (assumption variable) u get discharged; If at least two A get discharged then one has logical Contraction; If moreover A contains at least one positive universal or a negative existential quantifier then one has a computationally relevant Contraction 5 Comp. Relevance relative to both Gödel and Monotone Dialectica {A D (z; T i (z, x, y))} n+1 i=1, {Ci D (x i; T i (z, x, y))} m i=n+2 B D(T (z, x); y) Same tuple z produced by 2 n + 1 m discharged instances of A If {T i } n+1 i=1 non-null (A is Dialectica relevant) Equalization is a must! Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

9 Extensionality/Compatibility and Induction rules E σ,τ : z στ, x σ, y σ. x = σ y zx = τ zy must be forbidden A 0 COMPAT σ with the restriction that. all undischarged assumptions used s = σ t in the proof of s = σ t (here denoted A 0 ) B(s) B(t) are quantifier-free IR 0 equivalent to IA, IR in WeZ m.. A(tt) A(ff ) p IB A(p) A(0) z (A(z) A(Sucz)) (Boolean Induction Axiom) z A(z) R τ x y 0 = τ x R τ x y (Sucz) = τ y(z, R τ x y z) } : AxR τ Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

10 Göedel s functional Dialectica interpretation 1 A translation of proofs which includes a translation of formulas. 2 A(a) A D x y A D (x; y; a) with a all free vars of formula A 3 A D is quantifier-free for Göedel s Dialectica, since decidability needed > this no longer for Monotone setup Bounded Dialectica 4 Recursive syntactic translation from proofs in Constructive Arithmetic (or Classical Arithmetic, modulo the double-negation translation) to proofs in Intuitionistic Arithmetic such that positive occurrences of and negative occurrences of in the proof s conclusion get actually realized by terms in Gödel s T. 5 Contraction Problem: > choose between a number of realizers according to a boolean term associated to the contraction formula; Diller-Nahm: > postpone all choices to the very end by collecting all candidates and making a single final global choice; Monotone Dialectica: > use a simple common upper bound (maximum majorant) of the candidates = extract majorants Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

11 The Light Dialectica interpretation of formulas A D : (A D : A) for prime formulas A (A B) D : x, u y, v [ (A B) D : A D (x; y; a) B D (u; v; b) ] (A B) D : Y, U x, v [ (A B) D : A D (x; Y (x, v)) B D (U(x); v) ] ( za(z, a)) D : z, x y [ ( za(z, a)) D (z, x; y; a) : A D (x; y; z, a) ] ( za(z, a)) D : x y [ ( za(z, a)) D (x; y; a) : z A D (x; y; z, a) ] ( za(z, a)) D : X z, y [ ( za(z, a)) D (X; z, y; a) : A D (X(z ); y; z, a) ( za(z, a)) D : x y [ ( za(z, a)) D (x; y; a) : z A D (x; y; z, a) ] Here is a mapping which assigns to every given variable z a completely new variable z which has the same type of z. Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

12 Exact realizer synthesis by Dialectica Interpretation Extraction and Soundness Theorem: There exists an algorithm which, given at input a WeZ + proof P : {C i } n i=1 A [hence of the conclusion formula A, from the undischarged assumption formulas {C i } n i=1 ] will produce at output 1) the tuples of terms T and {T i} n i=1 2) the tuples of variables {x i } n i=1 and y 3) the verifying proof P D : {C i D (x i; T i (x, y))} n i=1 A D(T (x); y) where x : x 1,..., x n. Moreover, 1 variables x and y are all completely new (not occur in P) 2 the free variables of T and {T i } n i=1 are among the free variables of A and {C i } n i=1 (this one names the free variable condition (FVC) for programs extracted by the Dialectica Interpretation ) [ x, y not occur free in the extracted terms {T i } n i=1 and T ] Notice that: Terms T and {T i } n i=1 are not necessarily closed!!! Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

13 Problem > Implication Introduction with Contraction [u : A]... /B A B + n 1, z n+1 {}}{ z,..., z and x x n+2,..., x m : {A D (z; T i (z, x, y))} n+1 i=1, {Ci D (x i; T i (z, x, y))} m i=n+2 B D(T (z, x); y) 1) Same tuple z produced by n + 1 m discharged instances of A 2) Case: tuples {T i } n+1 i=1 are non-null! Recall that A D is quantifier-free 3) Since {T i } n+1 i=1 non-null = their equalization is a must : S : λx, z, y. If n τ (t D A [z; T 1 ],..., t D A [z; T n ], T n+1 (z, x, y), T n,..., T 1 ) one can now cancell all {A D } n+1 i=1 by a single + in the verifying proof {A D (z; S(x, z, y))} n+1 i=1, {Ci D (x i; S i (x, z, y))} m i=n+2 B D(S(x, z); y) {C i D (x i; S i (x, z, y))} m i=n+2 A D(z; S(x, z, y)) B D (S(x, z); y) Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

14 The Light Monotone Dialectica program extraction Majorant realizer synthesis by Light Monotone Dialectica Theorem: There ex. an algorithm which, given at input a WeZm + proof P : {C i (a i )} n i=1 A(a ) [hence of the conclusion formula A, whose free variables form the tuple a, from the undischarged assumption formulas {C i } n i=1 ] it will produce at output the following (a : a 1,..., a n, a ): 1 tuples of terms {T i [a]} n i=1 and T [a], with free variables among a 2 the tuples of variables {x i } n i=1 and y, all together with 3 the following verifying proof in WeZ m (below let x : x 1,..., x n ): Y 1,... Y n, X [ n i=1 (λa. T i) maj Y i (λa. T ) maj X a, x, y ( { n i=1 Ci D(x i ; Y i (a, x, y); a i )} A D (X(a, x); y; a) ) ] Variables x and y do not occur in P (they are all completely new) = x and y do not occur free in the extracted terms {T i } n i=1 and T. Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

15 Majorizability and Hereditarily Extensional Equality (2) x maj IN y : x IN y : at( x IN y IN ) x στ y z σ (x z τ y z) x maj στ y : z σ 1, zσ 2 (z 1 maj σ z 2 x z 1 maj τ y z 2 ) 0 maj IN 0, Suc maj ININ Suc, Σ maj Σ, Π maj Π and R M maj R WeZ m t maj στ t s maj σ s = t s maj τ t s x IN y : x = IN y : at(= x IN y IN ) x = στ y z σ (x z = τ y z) x στ y : z σ 1, zσ 2 (z 1 σ z 2 x z 1 τ y z 2 ) 0 IN 0, Suc ININ Suc, Σ Σ, Π Π and R R WeZ m t στ t s σ s = t s τ t s Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

16 The WeZ m proof at input & post-extraction ops. 1 Let t ρ be a closed term of Gödel s T. Then WeZ m t ρ t. 2 Let t (IN IN) (IN IN) be a closed T-term. Since WeZ m x IN IN, y IN IN [ x = IN IN y x IN IN y ] (due to weak extensionality + reflexivity) it immediately follows that WeZ m x IN IN, y IN IN [ x = IN IN y t(x) = IN IN t(y) ] 3 Let t[a] be a T-term with free vars a. There exists a corresponding T-term t [a] such that WeZ m λa. t maj λa. t. Very simple t construction: just replace each R in t with the corresponding R M. 4 If the type of a is IN ρ then a M maj a, hence t [a M ] maj t[a]. 5 For a of type IN ρ define a M (k) : Max ρ (a(0),..., a(k)) Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

17 MinLog computer output for our game-problem, m = 3 1 For t 3 : λh IN IN, k IN. R IN IN (0)[λp, q. h (3) (p) + q](k + 1) want f, g h. [ i L. f (i) = g(i) ] [ j k. f (f (f (j))) = g(g(g(j))) ] 2 The MinLog machine outputs in less than one minute: λh, k. max{k, h(0)..., h(k), max{h(0)... h(max{h(0)... h(k)})}} which immediately rewrites more humanly readable as L 3 : λh, k. max{k, h(0), h(1),..., h(max{k, h(0), h(1),..., h(k)})} 3 Recall that for m = 2 and m = 1 the (human) outcomes were L 2 : λh, k. max{k, h(0), h(1),..., h(k)} L 1 : λh, k. k Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

18 Final human solution for our general game-problem Pattern can be noticed (by the human!) in the solution of our problem for terms t m : λh, k. h (m) (0) h (m) (k), with h (m) (i) : h(h... (h(i))) s.t. h appears m times on the right side. t 1 (h, k) k t 2 (h, k) max{k, h(0),..., h( t 1 (h, k))} t 3 (h, k) max{k, h(0),..., h( t 2 (h, k))} Immediate inference of the generic (recursive) solution for m IN: t m+1 (h, k) max{k, h(0),..., h( t m (h, k))} Verify that t m is the optimal modulus of uniform continuity for t m! Now an easy exercise for the human! = See my PhD thesis :) Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

19 Was the Computer really necessary? 1 Maybe not, but what if the problem were more complex/tedious? 2 Certainly helpful for preventing the human error! Effectively! Implementing Monotone Dialectica 1 The light variant of Monotone Dialectica is the result of our implementation effort! Many operations which are easy for the human (mathematician) are not really that easy for the machine! 2 On the computer, the Goal is to produce programs in normal form! 3 Hence improve the Nbe-normalization by its own Partial Evaluation = Normalization during Extraction (NdE) NbE-normalize the term extracted for the conclusion of each Modus Ponens. 4 Only majorize at Contraction = produce a partial majorant which is transformed at the end by replacing each R with its corresp. R M. 5 Why? Well, some of the R may be eliminated during the partial NbE-normalization... Also use the more clever R M, with just 1 R. Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

20 A lot of Work to be done... 1 Completely formalize and explore the limits of Normalization during Extraction (NdE) generic optimization for (t n..(t 2 (t 1 t 0 ))..) 2 Completely formalize these ad-hoc optimizations of the computer implementation of Monotone Dialectica and combine with the light optimization brought by the use of quantifiers without comp. content 3 We suspect that the use of these ncm quantifiers may eliminate some of the comput. contractions in the Hered. Ext. Eq. extraction! This game-problem is already solved for a very particular case only! 4 Find other more interesting T-terms t m, for which the modulus of uniform continuity is far more difficult to find! 5 Find other more interesting examples for the Proof Mining by the Light (monotone) Dialectica on the Computer! 6 Improve the human-interaction side of our Dialectica extraction modules in MinLog, in order to render MinLog for Dialectica as an indispensable computer tool even for the more pure mathematically oriented Proof Mining! Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

21 Short List of related Papers I U. Kohlenbach. Proof Interpretations and the Computational Content of Proofs. Lecture Course, latest version in the author s web page. U. Kohlenbach and P. Oliva. Proof Mining: a systematic way of analysing proofs in Mathematics. Proc. of the Steklov Inst. of Mathem., 242: , U. Kohlenbach. Pointwise hereditary majorization and some applications. Arch. Math. Logic, 31: , U. Kohlenbach. Analysing proofs in Analysis. In Logic: from Foundations to Applications, Keele, 1993, European Logic Colloquium, pages Oxford University Press, Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22

22 Short List of related Papers II M.-D. Hernest. Light Dialectica program extraction from a classical Fibonacci proof Proceedings of DCM@ICALP 06, ENTCS (2007), 10pp. M.-D. Hernest. Light Functional Interpretation. CSL In LNCS 3634 pp , July M.-D. Hernest and U. Kohlenbach. A complexity analysis of functional interpretations. Theoretical Computer Science, 338(1-3): , U. Berger. Uniform Heyting Arithmetic. Annals of Pure and Applied Logic, 133(1-3): , U. Berger, W. Buchholz, and H. Schwichtenberg. Refined program extraction from classical proofs. Annals of Pure and Applied Logic, 114:3 25, Mircea-Dan Hernest (LogiCal) Light Monotone Dialectica FLoC 06, Seattle, 16 Aug / 22