Galois: A Language for Proofs Using Galois Connections and Fork Algebras

1. Galois: A Language for Proofs Using Galois connections and Fork Algebras Paulo Silva1 Joost Visser2 Jos Oliveira1 1 CCTC University of Minho Braga, Portugal 2 Software Improvement Group The Netherlands PLMMS09 August 21, 2009 Munich, Germany Paulo Silva (UMinho) Galois PLMMS09 1 / 28

2. Outline Outline 1 Introduction Motivation Objectives 2 Theoretical background Indirect equality Galois connections Fork algebras Point-free transform 3 Galois and Galculator Galois Galculator 4 Summary Summary Future work Paulo Silva (UMinho) Galois PLMMS09 2 / 28

3. Introduction Outline 1 Introduction Motivation Objectives 2 Theoretical background Indirect equality Galois connections Fork algebras Point-free transform 3 Galois and Galculator Galois Galculator 4 Summary Summary Future work Paulo Silva (UMinho) Galois PLMMS09 3 / 28

4. Introduction Motivation Whole division Prove (a b) c = a (c b) for b and c = 0. Easy if is the real number division Also valid in natural numbers but the proof is not so straightforward Paulo Silva (UMinho) Galois PLMMS09 4 / 28

5. Introduction Motivation Whole division specication Implicit denition c =x y r : 0 r 0) PLMMS09 5 / 28

8. Introduction Motivation Proof. n (a b) c { z y nc (n c) b x z x y } a { multiplication is associative } n (c b) { z y n Paulo Silva (UMinho) x y } ab { z y x z a x z x y } a (c b) Galois PLMMS09 6 / 28

13. Introduction Objectives Objectives Galculator = Galois connection + calculator Build a proof assistant based on Galois connections, their algebra and associated tactics Galois Language for mathematical reasoning Equivalent to rst-order logic Typed language Front-end for the Galculator Paulo Silva (UMinho) Galois PLMMS09 7 / 28

14. Theoretical background Outline 1 Introduction Motivation Objectives 2 Theoretical background Indirect equality Galois connections Fork algebras Point-free transform 3 Galois and Galculator Galois Galculator 4 Summary Summary Future work Paulo Silva (UMinho) Galois PLMMS09 8 / 28

15. Theoretical background Indirect equality Indirect inequality Denition (Indirect inequality) a x :: x ax b a Paulo Silva (UMinho) b b x :: b x a x Galois PLMMS09 9 / 28

16. Theoretical background Indirect equality Proof. a=b { Anti-symmetry } a bb a { Indirect inequality } x :: x ax b x :: x bx a { Rearranging quantiers } x :: x ax bx bx a { Mutual implication } x :: x Paulo Silva (UMinho) ax b Galois PLMMS09 10 / 28

21. Theoretical background Indirect equality Indirect equality Denition (Indirect equality) a=b x :: x ax b a=b Paulo Silva (UMinho) x :: a x b x Galois PLMMS09 11 / 28

22. Theoretical background Galois connections Galois connections Denition (Galois connection) Given two preordered sets (A, A ) and (B, B ) and two functions g f Bo A and A o B , the pair (f , g) is a Galois connection if and only if, for all a A and b B: f a B b a A gb Graphical notation A Al f g Paulo Silva (UMinho) ,

23. B B or (A, Galois A) o (f ,g) (B, B) PLMMS09 12 / 28

24. Theoretical background Galois connections Properties Property f a Bba Agb a A a f a B f a b B b g b A g b a A g (f a) f (g b) B b f (g (f a)) = f a g (f (g b)) = g b g (b B b ) = g b A g b f (a A a ) = f a B f a g B= A f A = B Paulo Silva (UMinho) Description Shunting rule Monotonicity (LA) Monotonicity (UA) Lower cancellation Upper cancellation Semi-inverse Semi-inverse Distributivity (UA over meet) Distributivity (LA over join) Top-preservation (UA) Bottom-preservation (LA) Galois PLMMS09 13 / 28

25. Theoretical background Galois connections Galois connections Algebra Identity connection (A, A) o (id,id) (A, A) Composition if (A, ) o (f ,g) (B, ) and (B, ) o (h,k ) (hf ,g k ) (C, ) then (A, ) o (C, ) Composition is associative and the identity is its unit. Galois connections form a category. Paulo Silva (UMinho) Galois PLMMS09 14 / 28

26. Theoretical background Galois connections Galois connections Algebra Converse if (A, ) o (f ,g) (B, ) then (B, ) o (g,f ) (A, ) Relator For every relator F if (A, ) o (f ,g) Paulo Silva (UMinho) (B, ) then (FA, F Galois (F f ,F g) )o (FB, F ) PLMMS09 15 / 28

27. Theoretical background Fork algebras Logic vs. algebra Logic Propositional logic Intuitionistic propositional logic Predicate logic Paulo Silva (UMinho) Galois Algebra Boolean algebra Heyting algebra ?? PLMMS09 16 / 28

28. Theoretical background Fork algebras Relation algebras Extension of Boolean algebras Original work of De Morgan, Peirce and Schrder Further developed by Tarski in his attempt to formalize set theory without variables Amenable for syntactic manipulation Only one inference rule is needed: substitution of equals by equals Equational reasoning Paulo Silva (UMinho) Galois PLMMS09 17 / 28

29. Theoretical background Fork algebras Relation algebras Extension of Boolean algebras Original work of De Morgan, Peirce and Schrder Further developed by Tarski in his attempt to formalize set theory without variables Amenable for syntactic manipulation Only one inference rule is needed: substitution of equals by equals Equational reasoning Paulo Silva (UMinho) Galois PLMMS09 17 / 28

30. Theoretical background Fork algebras Fork algebras Limitation of relation algebras Relations algebras can express rst-order predicates with at most three variables Fork algebras Extend relation algebras with a pairing operator Equivalent in expressive and deductive power to rst-order logic Paulo Silva (UMinho) Galois PLMMS09 18 / 28

31. Theoretical background Fork algebras Fork algebras Limitation of relation algebras Relations algebras can express rst-order predicates with at most three variables Fork algebras Extend relation algebras with a pairing operator Equivalent in expressive and deductive power to rst-order logic Paulo Silva (UMinho) Galois PLMMS09 18 / 28

32. Theoretical background Point-free transform Point-free transform summary Pointwise (bRa) bRa bSa bSa bSa True False b=a aRb c :: bRc cSa x :: xRb xSa x :: aRx bSx bRa cSa bRa dSc a, b :: bRa bSa a, b :: bRa bSa Paulo Silva (UMinho) Galois Pointfree b(R)a b(R S)a b(R S)a b a ba b id a bR a b(R S)a b(R S)a b(S/R)a (b, c)(R S)a (b, d)(R S)(a, c) RS R=S PLMMS09 19 / 28

33. Theoretical background Point-free transform Point-free denitions Denition (Galois connection) f B = A g Denition (Indirect equality) f =g f =g Paulo Silva (UMinho) f Galois f = =g g PLMMS09 20 / 28

34. Galois and Galculator Outline 1 Introduction Motivation Objectives 2 Theoretical background Indirect equality Galois connections Fork algebras Point-free transform 3 Galois and Galculator Galois Galculator 4 Summary Summary Future work Paulo Silva (UMinho) Galois PLMMS09 21 / 28

35. Galois and Galculator Galois Sub-languages of Galois Module Denition GC Denition Axiom Type Theorem Strategy Proof Step Fork Formula Rewriting Combinator Term Derivation Order Paulo Silva (UMinho) Function Galois Galois connection PLMMS09 22 / 28

36. Galois and Galculator Galculator Architecture of Galculator Combine GC Derive Laws Relation algebra Derive Properties Derive Theory domain Derive Rules TRS Strategies Combine Paulo Silva (UMinho) Galois PLMMS09 23 / 28

37. Summary Outline 1 Introduction Motivation Objectives 2 Theoretical background Indirect equality Galois connections Fork algebras Point-free transform 3 Galois and Galculator Galois Galculator 4 Summary Summary Future work Paulo Silva (UMinho) Galois PLMMS09 24 / 28

38. Summary Summary Summary Fork algebras Equivalent to rst-order logic (same expressive and deductive power) Single inference rule: substitution of equals for equals Equational No variables Integrates Galois connections and indirect equality Galois connections Provide structure Introduce semantic information in syntactic reasoning Paulo Silva (UMinho) Galois PLMMS09 25 / 28

39. Summary Summary Summary Galois Follows the mathematical concepts Alternative to rst-order languages Typed approach Galculator Proof assistance prototype based on Galois connections Innovative approach Uses a point-free equational approach Paulo Silva (UMinho) Galois PLMMS09 26 / 28

40. Summary Future work Future work Mechanization of point-free transform Automated proofs Extension of the type system Free-theorems Evaluation of the language Integration with host theorem provers (e.g., Coq) Paulo Silva (UMinho) Galois PLMMS09 27 / 28

41. The End Download Source code and documentation available from www.di.uminho.pt/research/galculator Contact Questions to [email protected] Paulo Silva (UMinho) Galois PLMMS09 28 / 28

Galois: A Language for Proofs Using Galois Connections and Fork Algebras

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Transcript of Galois: A Language for Proofs Using Galois Connections and Fork Algebras