On the Design of a Galculator

Paulo Silva

Departamento de InformáticaUniversidade do Minho

Braga, Portugal

November 16, 2009Guimarães

Outline

IntroductionMotivationObjectives

IngredientsIndirect equalityGalois connectionsRelation and fork algebrasPoint-free transform

Galois and Galculator

Concluding remarksContributionsFuture work

Outline

Software correctness

Current approaches

I Software correctness is an ambitious challengeI Sometimes proofs are hindered by the theoryI Tool support seems to be important

AlternativesI Sometimes algebraic approaches are possibleI Algebras “abstract” the underlying logicI Proofs become more syntactic

Galois connections can play an important role

Software correctness

Current approaches

I Software correctness is an ambitious challengeI Sometimes proofs are hindered by the theoryI Tool support seems to be important

AlternativesI Sometimes algebraic approaches are possibleI Algebras “abstract” the underlying logicI Proofs become more syntactic

Galois connections can play an important role

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?

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?

Whole division specification

Implicit definition

c = x ÷ y ⇔ 〈∃ r : 0 6 r < y : x = c × y + r〉

Explicit definition

x ÷ y = 〈∨

z :: z × y 6 x〉

Galois connection

z × y 6 x ⇔ z 6 x ÷ y (y > 0)

Implicit definition

c = x ÷ y ⇔ 〈∃ r : 0 6 r < y : x = c × y + r〉

Explicit definition

x ÷ y = 〈∨

z :: z × y 6 x〉

Galois connection

z × y 6 x ⇔ z 6 x ÷ y (y > 0)

Implicit definition

c = x ÷ y ⇔ 〈∃ r : 0 6 r < y : x = c × y + r〉

Explicit definition

x ÷ y = 〈∨

z :: z × y 6 x〉

Galois connection

z × y 6 x ⇔ z 6 x ÷ y (y > 0)

Specification vs. Implementation

I We can verify if the implementation meets thespecification.

I We can calculate the implementation from thespecification.

Whole divisionFrom specification to implementation

We want to calculate the implementation

x ÷ y = (x − y)÷ y + 1 if x > yx ÷ y = 0 if x < y

from specification

z × y 6 x ⇔ z 6 x ÷ y (y > 0)

Some useful Galois connections

a− b = c ⇔ a = c + ba− b 6 c ⇔ a 6 c + b

Proof when x > y

z 6 x÷y

⇔ { z×y 6 x ⇔ z 6 x÷y assuming x > 0, y > 0 }z×y 6 x

⇔ { cancellation, thanks to a− b 6 c ⇔ a 6 c + b }z × y−y 6 x−y

⇔ { distributivity }

(z − 1)×y 6 x − y

⇔ { z×y 6 x ⇔ z 6 x÷y assuming x > y }

z−1 6 (x − y)÷y

⇔ { a−b 6 c ⇔ a 6 c+b }

z 6 (x − y)÷ y +1

Proof when x > y

z 6 x÷y

(z − 1)×y 6 x − y

z−1 6 (x − y)÷y

⇔ { a−b 6 c ⇔ a 6 c+b }

z 6 (x − y)÷ y +1

Proof when x > y

z 6 x÷y

(z − 1)×y 6 x − y

z−1 6 (x − y)÷y

⇔ { a−b 6 c ⇔ a 6 c+b }

z 6 (x − y)÷ y +1

Proof when x > y

z 6 x÷y

(z − 1)×y 6 x − y

z−1 6 (x − y)÷y

⇔ { a−b 6 c ⇔ a 6 c+b }

z 6 (x − y)÷ y +1

Proof when x > y

z 6 x÷y

(z − 1)×y 6 x − y

z−1 6 (x − y)÷y

⇔ { a−b 6 c ⇔ a 6 c+b }

z 6 (x − y)÷ y +1

Proof when x > y

z 6 x÷y

(z − 1)×y 6 x − y

z−1 6 (x − y)÷y

⇔ { a−b 6 c ⇔ a 6 c+b }

z 6 (x − y)÷ y +1

Proof when x < y

z 6 x÷y

⇔ { z×y 6 x ⇔ z 6 x÷y }z×y 6 x

⇔ { transitivity, since x < y }z × y 6 x ∧ z × y < y

⇔ { since y 6= 0 }

z × y 6 x ∧ z 6 0

⇔ { z 6 0 entails z × y 6 x , since 0 6 x }

Proof when x < y

z 6 x÷y

⇔ { z×y 6 x ⇔ z 6 x÷y }z×y 6 x

⇔ { since y 6= 0 }

z × y 6 x ∧ z 6 0

Proof when x < y

z 6 x÷y

⇔ { z×y 6 x ⇔ z 6 x÷y }z×y 6 x

⇔ { since y 6= 0 }

z × y 6 x ∧ z 6 0

Proof when x < y

z 6 x÷y

⇔ { z×y 6 x ⇔ z 6 x÷y }z×y 6 x

⇔ { since y 6= 0 }

z × y 6 x ∧ z 6 0

Proof when x < y

z 6 x÷y

⇔ { z×y 6 x ⇔ z 6 x÷y }z×y 6 x

⇔ { since y 6= 0 }

z × y 6 x ∧ z 6 0

Objectives

Exploit equational reasoning

I Use Galois connections in equational proofsI Integrate Galois connections, fork algebras and indirect

equality

Galois language

I Language for mathematical reasoningI Equivalent to first-order logicI Strongly typedI Front-end for the Galculator

Objectives

Galculator = Galois connections + calculator

I Proof assistant based on Galois connections, their algebraand associated tactics

I Exploit the state-of-the-art Haskell technology in thedevelopment of a proof assistant prototype

Outline

Indirect equality

Definition (Indirect inequality)

a v b ⇔ 〈∀ x :: x v a⇒ x v b〉a v b ⇔ 〈∀ x :: b v x ⇒ a v x〉

Definition (Indirect equality)

a = b ⇔ 〈∀ x :: x v a ⇔ x v b〉a = b ⇔ 〈∀ x :: a v x ⇔ b v x〉

Galois connections

Definition (Galois connection)Given two preordered sets (A,vA) and (B,vB) and twofunctions B Afoo and A B

goo , the pair (f , g) is a Galoisconnection if and only if, for all a ∈ A and b ∈ B:

f a vB b ⇔ a vA g b

Graphical notation

vA��

vB��

or (A,vA) (B,vB)(f ,g)oo

Galois connections

f a vB b ⇔ a vA g b

⊤A⊤B

⊥A⊥B

Algebra

I Identity connection.

(A,vA) (A,vA)(id ,id)oo

I Composition.

if (A,v) (B,�)(f ,g)oo and (B,�) (C,6)

(h,k)oo then

(A,v) (C,6)(h◦f ,g◦k)oo

I Converse.

if (A,v) (B,�)(f ,g)oo then (B,�) (A,w)

(g,f )oo

I Relator. For every relator F ,

if (A,v) (B,�)(f ,g)oo then (FA,F v) (FB,F �)

(F f ,Fg)oo

Relation and fork algebras

Relation algebras

I Extension of Boolean algebrasI Single inference rule: substitution of equals by equalsI Amenable for syntactic manipulationI Equivalent to a three-variable fragment of first-order logic

Fork algebras

I Extend relation algebras with a pairing operatorI Equivalent in expressive and deductive power to first-order

Equational reasoning

Relation and fork algebras

Relation algebras

I Extension of Boolean algebrasI Single inference rule: substitution of equals by equalsI Amenable for syntactic manipulationI Equivalent to a three-variable fragment of first-order logic

Fork algebras

I Extend relation algebras with a pairing operatorI Equivalent in expressive and deductive power to first-order

Equational reasoning

Point-free definitions

Definition (Galois connection)

f ◦ ◦vB = vA ◦ g

Definition (Indirect equality)

f = g ⇔ � ◦ f = � ◦ gf = g ⇔ f ◦ ◦� = g◦◦ �

Outline

Galois language

Formula

Module

TheoremGC DefinitionDefinition Axiom Strategy

Galois connection

Rewriting Combinator

Derivation

Proof Step

FunctionOrder

Galculator prototype

Combine

Strategies

Combine

Properties Theory domain

Relation algebra

Derive

Derive DeriveDerive

Outline

Contributions

Study about Galois connections

I Survey of the most important theoretical resultsI Comprehensive study of different approaches to combine

Galois connectionsI Relation with category theoryI Survey of applications

Innovative approach

I Fork algebras used together with Galois connectionsI Use of indirect equalityI Amenable for either pencil-and-paper or computer assisted

proofs

Contributions

Galois language

I Follows from the theoretical conceptsI Strongly typedI Galois connections introduce some semantic support while

reasoning in a syntactic level

Galculator prototype

I Proof assistant prototype based on Galois connectionsI First proof engine to calculate directly with point-free

Galois connectionsI Application of advanced and innovative implementation

techniques

Future work

I Integration with host theorem provers (e.g., Coq)I Mechanization of point-free transformI Automated proofsI Free-theoremsI Extension of the type systemI Evaluation of the languageI Application to abstract interpretation

On the Design of a Galculator

Technology

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