Intuitionistic Proof Transformations and their Application to Constructive Program Synthesis

Transcription

1 Intuitionistic Proof Transformations and their Application to Constructive Program Synthesis Uwe Egly Stephan Schmitt presented by: Christoph Kreitz 1 Motivation: Guiding Proof/Program Development 2 Complexity of Proof Transformations 3 Permutation-based Proof Transformations 4 Conclusion and Future Work

2 Motivation Goal: Guide first-order proofs in program development systems Concept: A) Use ATP (matrix prover) for J (Otten & Kreitz, 1995) B) Transform matrix proofs into LJ mc proofs (Schmitt & Kreitz 1995) C) Transform LJ mc proofs into LJ proofs (Egly & Schmitt, 1998) interactive NuPRL System logic specification proof editor constructive proof extraction mechanism program calulus for Intuitionistic Type Theory concrete, interactive proof process J -goal autom theorem prover separation J -subgoal Γ C integration J -formula Γ C proof plan in LJ T2 matrix prover for J proof reconstruction into LJ mc MJ proof T1 Intuitionistic Proof Transformations 1 CADE-WS 1998

3 Transformation LJ mc LJ cut Differences between calculi: LJ and LJ mc Γ, A A Γ, A C l Γ, A A, Γ, A l Similar for rules l, r, r, r is deleted by r, r, r Straightforward simulation of LJ mc by LJ cut : Γ, A A S Γ, A, A S S Γ, A S cut A S, l where S = A 1 ( A n ) ) LJ cut proofs generate non-intuitive programs (extract terms) Transform LJ mc LJ without search Intuitionistic Proof Transformations 2 CADE-WS 1998

4 Complexity of Proof Transformations LJ mc P-simulates LJ LJ cut P-simulates LJ mc LJ does not P-simulate LJ mc K 1 P-simulates K 2 if for every K 2 -proof for a formula F of length n there is a K 1 -proof of F, whose length is not greater than p(n) proof length: number of axioms in a sequent proof Intuitionistic Proof Transformations 3 CADE-WS 1998

5 A Hard Example for LJ G n A n n 1 i=1 (B i 1 (B i A i ) B i ) ((B }{{} 0 A 0 ) B 0 ) }{{} O i A 0 n 1 i=0 B i A i+1 } {{ } N i O 0 G 0 needs one axiom in LJ mc and LJ there exist an LJ mc proof of G n with 6n 2 axioms G n implies a reduction ordering O j O j+1 N j on LJ proofs each LJ proof of G n requires at least 5(2 n 1) axioms Intuitionistic Proof Transformations 4 CADE-WS 1998

6 Permutation-based Proof Transformations Generate more natural LJ proofs (and programs) 1 Transform LJ mc -proof α mc into normal form α mc N Divide α mc into layers wrt axioms and intuitionistic sequents For each layer L: apply local permutation schemata wrt rule non-permutabilities in LJ mc Normal form: Each layer L is transformed into a set of layers M L such that each (topmost) non-single layer L of M L is l-free 2 Eliminate redundant inferences in α mc N Identify the relevant succedent formula D in l free subproofs LJ proof α Intuitionistic Proof Transformations 5 CADE-WS 1998

7 Layer-based Transformations of LJ mc Proofs generative rules: l, l, r, r (increasing size of succedent) d-generative rules: generative rules + r depending on a generative rule axiom axiom Γ, A D, C Γ, B D, C Γ, A B D, C axiom axiom Γ, A D, C Γ, B D, C [Γ, D, A C] [Γ, D, B C] [Γ, D, A B C] Γ D, C Γ, D C [ ] [ ] Γ, D C Permute d-generative rules above all topmost l inferences Intuitionistic Proof Transformations 6 CADE-WS 1998

8 Rule-Permutability in LJ mc proofs Normal form transformation requires schemata for permuting R { l, l, r, r, r} over R { l, l, l, l, r, l} Local permutation lemma: In LJ mc -proofs, rule R is always permutable over R except for R l l, r l r, r r, r R r l r, r l l A C B, A A B C B, A B, C B B C B, A r l??? A, C B B, C B l A B, C B A B C B, A r Intuitionistic Proof Transformations 7 CADE-WS 1998

9 Local Permutation Schemata for R { l, l, r, r, r} over R { l, l, l, l, r, l} 4 Classes of schemata One-premise rule over one-premise rule One-over-Two, Two-over-One, Two-over-Two Proof Duplication in Two-over-Two Schemata Γ, A C, Γ, B C, Γ, A B C, l Γ, A B C D, Γ, A B D, r r over l Γ, A C, Γ, A D, Γ, A C D, r Γ, A B C D, Γ, B C, Γ, B D, Γ, B C D, l r Branch Modification Rule required Intuitionistic Proof Transformations 8 CADE-WS 1998

10 Building Normalized LJ mc -Proofs Branch b in a layer L of α mc is admissible if it contains a topmost l inference r and a topmost d-generative rule below r such that 1 b is a generative branch of, and 2 no l between r and contains a l in its generative branches If r is the lowmost l between r and on an admissible branch b in L then α mc can be transformed into an LJ mc -proof β mc such that occurs above r Let R I,L be the set of all topmost l rules in L which occur above I If all generative branches of containing an r R,L are admissible, then α mc can be transformed into β mc such that occurs above all r R,L α mc can be transformed into a normal form proof α mc N via permutation of inferences Intuitionistic Proof Transformations 9 CADE-WS 1998

11 Example Ax Ax A C, I 1, I 2 E, F, B, A, C l Ax l A C, I 1, I 2 E, F, B, A A C, A E F, C B } {{ } I 1 } {{ } I 2 Ax Ax A C, E F, I 2 E, F, B l E, F, B l Intuitionistic Proof Transformations 10 CADE-WS 1998

12 Extracting LJ Proofs from Normal LJ mc Proofs Deletion lemma: If α mc is an LJ mc proof of Γ without l rules then for some D there is an LJ proof α of Γ D whose length is polynomial in the length of α mc Proof by induction on the depth of α mc β mc 1 B C, Γ D {}}{ B, β mc 2 C, Γ A {}}{ β mc 1 B C, Γ B β mc 2 C, Γ A B C, Γ B C, Γ A if D = B All topmost, non-single layers L in all layer normal forms M L are by construction l- free Locally apply deletion lemma to L Intuitionistic Proof Transformations 11 CADE-WS 1998

13 Summary Cut-based transformations result in non-intuitive program terms LJ does not P-simulate LJ mc, even in the propositional case Permutation based transformation via layer-oriented construction of normal forms in LJ mc preserve original logical specification for program extraction Future Work Combine Permutation approach with controlled cut-introduction Resulting LJ cut proofs preserve intended specification At most polynomial increase of proof length Investigate computational correspondence between structuring LJ proofs with cut and procedural programming concepts Intuitionistic Proof Transformations 12 CADE-WS 1998

14 Example Aa Aa, xbx Aa Aa, xbx Ba yay, Ba Aa yay, xbx r a Ba yay, xbx r a _ l Aa _ Ba yay, xbx xax _ Bx yay, xbx l a subgoal 1 l xax _ Bx, yay z Az xbx ^ l ( xax _ Bx) ^ ( yay z Az) xbx r ( xax _ Bx) ^ ( yay z Az) xbx subgoal 1: Aa Aa, xbx Ba Aa, Ba Ba Aa, xbx r a Aa _ Ba Aa, xbx xax _ Bx Aa, xbx l a xax _ Bx, Aa xbx l xax _ Bx, z Az xbx l a _ l Aa yay, xbx r a subgoal 1 l Aa, yay z Az xbx subgoal 2 _ l Aa _ Ba, yay z Az xbx xax _ Bx, yay z Az xbx l a ( xax _ Bx) ^ ( yay z Az) xbx ( xax _ Bx) ^ ( yay z Az) xbx subgoal 2: Ba yay, Ba Ba yay, xbx r a subgoal 1 Ba, yay z Az xbx subgoal 1 X = Aa: subgoal 1 X = Ba: X, Ab Ab, xbx X, Ab, Ab xbx l l X, Bb Ab, Bb ^ l X, Bb Ab, xbx r b X, Bb, Ab xbx l X, Ab _ Bb, Ab xbx X, xax _ Bx, Ab xbx l b X, xax _ Bx, z Az xbx l b _ l r Intuitionistic Proof Transformations 13 CADE-WS 1998