On the Design of a Galculator

1. On the Design of a Galculator Paulo Silva Departamento de Informática Universidade do Minho Braga, Portugal November 16, 2009 Guimarães

2. Outline Introduction Motivation Objectives Ingredients Indirect equality Galois connections Relation and fork algebras Point-free transform Galois and Galculator Concluding remarks Contributions Future work

4. Software correctness Current approaches Software correctness is an ambitious challenge Sometimes proofs are hindered by the theory Tool support seems to be important Alternatives Sometimes algebraic approaches are possible Algebras “abstract” the underlying logic Proofs become more syntactic Galois connections can play an important role

5. Software correctness Current approaches Software correctness is an ambitious challenge Sometimes proofs are hindered by the theory Tool support seems to be important Alternatives Sometimes algebraic approaches are possible Algebras “abstract” the underlying logic Proofs become more syntactic Galois connections can play an important role

6. Whole division implementation Haskell code x ‘div ‘ y | x < y = 0 | x y = (x − y ) ‘div ‘ y + 1 for non-negative x and positive y . This is the code. Where is the speciﬁcation?

7. Whole division implementation Haskell code x ‘div ‘ y | x < y = 0 | x y = (x − y ) ‘div ‘ y + 1 for non-negative x and positive y . This is the code. Where is the speciﬁcation?

8. Whole division specification Implicit definition c =x ÷y ⇔ ∃r : 0 r <y : x =c×y +r Explicit definition x ÷y = z :: z × y x Galois connection z ×y x ⇔ z x ÷y (y > 0)

11. Specification vs. Implementation We can verify if the implementation meets the specification. We can calculate the implementation from the specification.

12. Whole division From speciﬁcation to implementation We want to calculate the implementation x ÷ y = (x − y ) ÷ y + 1 if x x ÷y =0 if x < y from speciﬁcation z ×y x ⇔ z x ÷y (y > 0) Some useful Galois connections a−b =c ⇔ a=c+b a−b c ⇔ a c+b y

13. Proof when x z y x÷y ⇔ { z×y x ⇔ z x÷y assuming x 0, y > 0 } x z×y ⇔ { cancellation, thanks to a − b z × y −y ⇔ c⇔a c+b } x−y { distributivity } (z − 1)×y ⇔ { z×y z−1 ⇔ x ⇔ z x÷y assuming x (x − y )÷y { a−b z x −y c⇔a (x − y ) ÷ y +1 c+b } y}

19. Proof when x < y z ⇔ x÷y { z×y z×y ⇔ x ⇔ z x÷y } x { transitivity, since x < y } z ×y ⇔ x ∧ z ×y <y { since y = 0 } z ×y ⇔ { z z 0 x ∧ z 0 0 entails z × y x, since 0 x }

24. Objectives Exploit equational reasoning Use Galois connections in equational proofs Integrate Galois connections, fork algebras and indirect equality Galois language Language for mathematical reasoning Equivalent to ﬁrst-order logic Strongly typed Front-end for the Galculator

25. Objectives Galculator = Galois connections + calculator Proof assistant based on Galois connections, their algebra and associated tactics Exploit the state-of-the-art Haskell technology in the development of a proof assistant prototype

27. Indirect equality Deﬁnition (Indirect inequality) a b ⇔ ∀ x :: x a⇒x b a b ⇔ ∀ x :: b x ⇒a x Deﬁnition (Indirect equality) a=b ⇔ ∀ x :: x a⇔x b a=b ⇔ ∀ x :: a x ⇔b x

28. Galois connections Deﬁnition (Galois connection) Given two preordered sets (A, A ) and (B, B ) and two functions B o f A and A o g 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 ,

29. B B or (A, A) o (f ,g) (B, B)

30. Galois connections f a B ⊤A b ⇔ g a A gb ⊤B f A B f⊤A g B g f B A

31. Algebra Identity connection. (A, A) o (id,id) (A, A) Composition. if (A, ) o (f ,g) (h,k ) (B, ) and (B, ) o (C, ) then (h◦f ,g ◦k ) (A, ) o (C, ) Converse. if (A, ) o (f ,g) (B, ) then (B, ) o (g,f ) (A, ) Relator. For every relator F, if (A, ) o (f ,g) (B, ) then (FA, F (F f ,F g) )o (FB, F )

32. Relation and fork algebras Relation algebras Extension of Boolean algebras Single inference rule: substitution of equals by equals Amenable for syntactic manipulation Equivalent to a three-variable fragment of ﬁrst-order logic Fork algebras Extend relation algebras with a pairing operator Equivalent in expressive and deductive power to ﬁrst-order logic Equational reasoning

33. Relation and fork algebras Relation algebras Extension of Boolean algebras Single inference rule: substitution of equals by equals Amenable for syntactic manipulation Equivalent to a three-variable fragment of ﬁrst-order logic Fork algebras Extend relation algebras with a pairing operator Equivalent in expressive and deductive power to ﬁrst-order logic Equational reasoning

34. Point-free definitions Definition (Galois connection) f◦ ◦ B = A ◦ g Definition (Indirect equality) f =g f =g ⇔ ⇔ ◦ f ◦ ◦ f = =g ◦ ◦ g ◦

36. Galois language Module Deﬁnition GC Deﬁnition Axiom Type Theorem Strategy Proof Step Fork Formula Rewriting Combinator Term Derivation Order Function Galois connection

37. Galculator prototype Combine GC Derive Laws Relation algebra Derive Properties Derive Rules TRS Strategies Combine Theory domain Derive

39. Contributions Study about Galois connections Survey of the most important theoretical results Comprehensive study of different approaches to combine Galois connections Relation with category theory Survey of applications Innovative approach Fork algebras used together with Galois connections Use of indirect equality Amenable for either pencil-and-paper or computer assisted proofs

40. Contributions Galois language Follows from the theoretical concepts Strongly typed Galois connections introduce some semantic support while reasoning in a syntactic level Galculator prototype Proof assistant prototype based on Galois connections First proof engine to calculate directly with point-free Galois connections Application of advanced and innovative implementation techniques

41. Future work Integration with host theorem provers (e.g., Coq) Mechanization of point-free transform Automated proofs Free-theorems Extension of the type system Evaluation of the language Application to abstract interpretation

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