galculator: functional prototype of a galois-connection based proof assistant

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GalculatorFunctional Prototype of a Galois-connection Based

Proof Assistant

Paulo Silva José Nuno Oliveira

Departamento de InformáticaUniversidade do Minho

Braga, Portugal

Principles and Practice of Declarative ProgrammingJuly 15 – 17, 2008

Valencia

Paulo Silva, José Nuno Oliveira (UMinho) Galculator PPDP’08 1 / 27

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Outline

Outline

1 IntroductionMotivationObjectives

2 Theoretical backgroundGalois connectionsPointfree transform

3 GalculatorPrinciplesRepresentation

4 ConclusionConclusionFuture work


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Introduction

Outline






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Introduction Motivation

Software correctness

Current approachesSoftware correctness is an ambitious challengeLogic based approaches benefit from the help of theorem proversSometimes proofs are hindered by the theoryIt is not always easy to devise the correct strategy

AlternativesSometimes algebraic approaches are possibleAlgebras “abstract” the underlying logicProofs become more syntactic

Galois connections can play an important role


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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?


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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)


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Whole division

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

Definition (Indirect equality)

a = b ⇔ 〈∀ x :: x 6 a⇔ x 6 b〉a = b ⇔ 〈∀ x :: a 6 x ⇔ b 6 x〉

Another useful Galois connection

a− b 6 c ⇔ a 6 c + b


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Proof.

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


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Proof.

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 }

z 6 0


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Introduction Objectives

Objectives

GalculatorBuild a proof assistant based on Galois connections, their algebraand associated tactics


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Theoretical background

Outline






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Theoretical background Galois connections

Galois connections

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

B Afoo and A Bgoo , the pair (f , g) is a Galois connection

if and only if, for all a ∈ A and b ∈ B:

f a vB b ⇔ a vA g b

Graphical notation

Af

,,

vA��

Bg

ll

vB��

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


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Properties

Property Description

f a vB b ⇔ a vA g b “Shunting rule”g (b uB b′) = g b uA g b′ Distributivity (UA over meet)

f (a tA a′) = f a tB f a′ Distributivity (LA over join)a vA g (f a) Lower cancellationf (g b) vB b Upper cancellation

a vA a′⇒ f a vB f a′ Monotonicity (LA)b vB b′⇒ g b vA g b′ Monotonicity (UA)

g >B = >A Top-preservation (UA)f ⊥A = ⊥B Bottom-preservation (LA)


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Galois connections — Algebra

Identity connection

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

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

Composition is associative and the identity is its unit.Galois connections form a category.


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Galois connections — Algebra

Converse

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

(g,f )oo

RelatorFor every relator F that distributes through binary intersections,

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

(F f ,Fg)oo


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Theoretical background Pointfree transform

Pointfree transform

Based on the formalization of set theory without variablesproposed by TarskiAbstracts points from definitionsMore compact and crypticMore amenable for syntactical manipulation


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Pointfree transform summary

Pointwise Pointfree〈∃ c :: bRc ∧ cSa〉 b(R ◦ S)a〈∀ x :: xRb⇒ xSa〉 b(R \ S)a〈∀ x :: bRx ⇒ aSx〉 b(S/R)a

bRa ∧ cSa (b, c)〈R, S〉abRa ∧ dSc (b, d)(R × S)(a, c)bRa ∧ bSa b(R ∩ S)abSa ∨ bSa b(R ∪ S)a(f b)R(g a) b(f ◦ ◦ R ◦ g)a

b = a b id aTrue b > aFalse b ⊥ a

〈∀ a, b :: bRa⇒ bSa〉 R ⊆ S〈∀ a, b :: bRa⇔ bSa〉 R = S

〈∀ a :: aRa〉 id ⊆ R


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Pointfree definitions

Definition (Galois connection)

f ◦ ◦vB = vA ◦ g

Definition (Indirect equality)

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


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Galculator

Outline






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Galculator Principles

Design Principles

TRS

GC

Combine

Laws

Rules

Strategies

Combine

Properties Theory domain

Relation algebra

Derive

Derive DeriveDerive


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Galculator Representation

Algebraic data types

List definition — Algebraic data type

data List a = Nil | Cons a (List a)

List definition — Generalized algebraic data type

data List a whereNil :: List aCons :: a→ List a→ List a


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Type representation

data Type a whereBool :: Type BoolChar :: Type CharInt :: Type IntList :: Type a→ Type [a]Set :: Type a→ Type (Set a)Maybe :: Type a→ Type (Maybe a)· × · :: Type a→ Type b → Type (a, b)

Fun :: Type a→ Type b → Type (a← b)Rel :: Type a→ Type b → Type (a↔ b)GC :: Type a→ Type b → Type (GC a b)


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Relational representation

RepresentationRelationFunctionsOrdersGalois connections

Combinators

data R r where·◦ :: R (b ↔ a)→ R (a↔ b)· ◦· · :: Type b → R (c ↔ b)→ R (b ↔ a)→ R (c ↔ a)· × · :: R (b ↔ a)→ R (d ↔ c)→ R ((b, d)↔ (a, c)). . .


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Conclusion

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Conclusion Conclusion

Conclusion

Proof assistant prototype based on Galois connectionsInnovative approach

Combination of Galois connections and pointfree calculusNon-trivial example of the application of distinctive features offunctional languages

Generalized algebraic data typesExistential data typesCombinatorial approaches (parsing, rewriting)Support for embedded domain specific languagesComputations as monadsHigher-order functionsNew: Polymorphic type representation with unification. . .


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Conclusion Future work

Future work

User-friendly syntaxAutomated proofsFree-theoremsIntegration with host theorem provers


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The End

DownloadSource code and documentation available fromwww.di.uminho.pt/research/galculator

ContactQuestions to [email protected]


galculator: functional prototype of a galois-connection based proof assistant

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