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Institutions - Part 1 Liam OReilly Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Institutions - Part 1 Institution Recap Closure Systems -Institutions Liam OReilly

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SLIDE 1

Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Institutions - Part 1

Liam O’Reilly 09.05.07

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Outline

Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Institution Recap Closure Systems Π-Institutions Definition of Π-Institutions Relating Institutions and Π-Institutions

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Why Do We Need Institutions?

◮ There are many different logics in the world, for

instance: EL, FOL, HOL, SubPCFOL=, temporal logic, Horn clause logic, etc.

◮ Each program / prover tends to use its own logic. ◮ Many general results are actually completely

independent of what logic system is used. Institutions allow:

◮ Translation of sentences from logic to logic whilst

preserving soundness.

◮ Forces us to write down logics in a standard way. ◮ Allows us to use tools from one logic on another

logic. An institution captures how truth can be preserved under change of symbols.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Outline

Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Institution Recap Closure Systems Π-Institutions Definition of Π-Institutions Relating Institutions and Π-Institutions

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Definition - Institutions

Definition

An institution is a quadruple SIGN, gram, mod, | = where:

◮ SIGN is a category. ◮ gram : SIGN → SET is a functor. ◮ mod : SIGNop → CAT is a functor. ◮ For every Σ : SIGN, |

=Σ: mod(Σ) × gram(Σ) which satisfies the satisfaction condition: for every σ : Σ → Σ′, p ∈ gram(Σ) and M′ ∈ mod(Σ′), mod(σ)(M′) | =Σ p iff M′ | =Σ′ gram(σ)(p).

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Outline

Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Institution Recap Closure Systems Π-Institutions Definition of Π-Institutions Relating Institutions and Π-Institutions

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Signatures for Many-Sorted Equational Logic

A signature Σ = (S, Ω) is a pair of sets, where

◮ S is a set of sorts ◮ Ω is a set of total functions symbols, of the form

n : s1 × . . . × sk → s with s1, . . . , sk, s ∈ S and k 0.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Signature Morphisms for Many-Sorted Equational Logic

Given two signatures Σ = (S, Ω) and Σ′ = (S′, Ω′) , a signature morphism σ : Σ → Σ′ is a pair (σs, σΩ) where

◮ σs : S → S′ ◮ σΩ : Ω → Ω′

such that for each function symbol n : s1 × . . . sk → s ∈ Ω, k ≥ 0, there exists a function name m with σΩ(n : s1 × . . . sk → s) = (m : σs(s1) × . . . σs(sk) → σs(s)).

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Models for Many-Sorted Equational Logic

Given a signature Σ = (S, Ω) a total algebra(model) for Σ assigns :

◮ A carrier set A(s) to each sort s ∈ S. ◮ A total function

A(n : s1 × sk → s) : A(s1) × . . . × A(sk) → A(s) to each operation (n : s1 × . . . × sk → s) ∈ Ω, k ≥ 0.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Models Morphisms for Many-Sorted Equational Logic

Given two models A, B ∈ mod(Σ), a model morphism h : A → B is a family (hs)s∈S of functions hs : A(s) → B(s) such that for any function f ∈ Ω say f = (n : s1 × . . . × sk → s), k ≥ 0, the following condition holds: hs(A(f)(a1, . . . , ak)) = B(f)(hs1(a1), . . . , hsk(ak)) for all (a1, . . . , ak) ∈ A(s1) × . . . × A(sk).

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Reducts

Given two signatures Σ = (S, Ω) and Σ′ = (S′, Ω′), and a signature morphism σ : Σ → Σ′ then for a Σ′-algebra A′ the σ-reduct of A′ is defined by:

◮ (A′ |σ)(s) = A′(σ(s)) for all s ∈ S ◮ (A′ |σ)(f) = A′(σ(f)) for all f ∈ Ω

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Terms in Many-Sorted Equational Logic

Given a signature Σ = (S, Ω) and a family of variables X = (Xs)s∈S of disjoint infinite sets, then TΣ(X),s is defined by

  • 1. Xs ⊆ TΣ(X),s,
  • 2. if n :→ s is an operation of Ω then n ∈ TΣ(X),s,
  • 3. if n : s1 × . . . × sk → s, k 1 is an operation of Ω and

if ti ∈ TΣ(X),si, for 1 i k, then n(t1, . . . , tk) ∈ TΣ(X),s.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Sentences Many-Sorted Equational Logic

For each signature Σ the set of formulae of EL is gram(Σ) = {∀X.t = u | t, u ∈ TΣ(X),s}

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Translation of Sentences

Translation of Variables

Given σ : Σ → Σ′ the variable translation is defined as σ((Xs)s∈S) = ((

  • σ(s)=s′

Xs)s′∈S′)

Translation of Terms

Given σ : Σ → Σ′ the term translation is defined as σ(x : s) = x : σs(s) σ(f(t1, . . . , tk)) = σΩ(f)(σs(t1), . . . , σs(tk)

Translation of Sentences

Given σ : Σ → Σ′ the sentence translation is defined as σ(∀X.t = u) = ∀σ(X).σ(t) = σ(u)

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Assignments

Given Σ = (S, Ω) then an assignment of X for A is a family α = (αs)s∈S of functions αs : Xs → A(s). We can just write α : X → A.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Evaluation of Terms

Given a signature Σ = (S, Ω), a Σ-Algebra A, a set of variables X, a term t ∈ TΣ(X) and an assignment α : X → A then A(α)(t) is defined by:

  • 1. A(α)(t) = αs(x) if t = x with x ∈ Xs, s ∈ S,
  • 2. A(α)(t) = A(w) if t = n and w = (n :→ s) ∈ Ω,
  • 3. A(α)(t) = A(w)(A(α)(t1), . . . , A(α)(tk))

if t = n(t1, . . . , tk), w = (n : s1, . . . , sk → s) ∈ Ω, k 1 and ti ∈ TΣ(X),si, 1 i k.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

The Satisfaction Relation for Many-Sorted Equational Logic

Let Σ be a signature. A | =Σ ∀X.t = u :iff for all assignments α : X → A, A(α)(t) = A(α)(u) for each Σ-Algebra A and for each equation ∀X.t = u ∈ EL(Σ).

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Outline

Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Institution Recap Closure Systems Π-Institutions Definition of Π-Institutions Relating Institutions and Π-Institutions

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

CASL Signatures

A many-sorted signature Σ = (S, TF, PF, P) consists of:

◮ A set S of sort symbols. ◮ Sets TFw,s and PFw,s of total function symbols and

partial function symbols such that TFw,s ∩ PFw,s = ∅, for each function profile (w, s) consisting of a sequence or argument sorts w ∈ S∗ and a result sort s ∈ S (constants are treated as functions with no arguments).

◮ A set Pw of predicate symbols for each predicate

profile consisting of a sequence of argument sorts w ∈ S∗.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

CASL Signatures Morphisms

A many-sorted signature morphism σ : (S, TF, PF, P) → (S′, TF ′, PF ′, P′) consists of:

◮ A mapping from S to S′. ◮ For each w ∈ S∗, s ∈ S, a mapping between the

corresponding sets of functions, resp. predicate

  • symbols. A partial function symbol may be mapped

to a total function symbol, but not vice versa.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

CASL Models

For a many sorted signature Σ = (S, TF, PF, P) a many-sorted model M ∈ mod(Σ) is a many-sorted first-order structure consisting of a many-sorted partial algebra:

◮ A non-empty carrier set sM for each sort s ∈ S (let

wM denote the Cartesian product sM

1 × . . . × sM n

where w = s1 . . . sn).

◮ A partial function f M from wM to sM for each function

symbol f ∈ TFw,s or f ∈ PFw,s, the function being required to be total in the former case.

◮ A predicate pM ⊆ wM for each predicate symbol

p ∈ Pw.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

CASL Model Morphisms

A (weak) many-sorted homomorphism h from M1 to M2, with M1, M2 ∈ mod(S, TF, PF, P), consist of a function hs : sM1 → sM2 for each s ∈ S preserving not only the values of functions but also their definedness, and preserving the truth of predicates. Any signature morphism σ : Σ → Σ′ determines the many-sorted reduct of each model M′ ∈ mod(Σ′) to a model M ∈ mod(Σ), defined by interpreting symbols of Σ in M in the same way that their images under σ are interpreted in M′.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

CASL Sentences

The many-sorted terms on a signature Σ = (S, TF, PF, P) and a set of sorted, non-overloaded variables X are build from:

◮ Variables from X. ◮ application of qualified function symbols in TF ∪ PF

to arguments terms of appropriate sorts. gram(Σ) are the usual closed many-sorted first-order logic formulae, built from atomic formulae using quantification (over sorted variables) and logical connectives.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

CASL Satisfaction Relation

The satisfaction of a sentence in a structure M is determined as usual by the holding of its atomic formulae w.r.t. assignments of (defined) values to all the variables that occur in them, the values assigned to variables of sort s being in sM.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Outline

Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Institution Recap Closure Systems Π-Institutions Definition of Π-Institutions Relating Institutions and Π-Institutions

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Closure Systems

Definition of Closure Systems

A closure system s a pair L, c where L is a set and c : 2L → 2L is a total function satisfying the following properties:

◮ Reflexivity: for every Φ ⊆ L, Φ ⊆ c(Φ). ◮ Monotonicity: for every Φ, Γ ⊆ L, Φ ⊆ Γ implies

c(Φ) ⊆ c(Γ).

◮ Idempotence: for every Φ ⊆ L, c(c(Φ)) ⊆ c(Φ).

Category CLOS

The category CLOS has as objects closure systems and morphisms f : L, c → L′, c′ are the maps f : L → L′ such that f(c(Φ)) ⊆ c′(f(Φ)) for all Φ ⊆ L.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Outline

Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Institution Recap Closure Systems Π-Institutions Definition of Π-Institutions Relating Institutions and Π-Institutions

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Definition - Π-Institutions

Definition

A π-institutions consists of a pair SIGN, clos, where SIGN is a category (of signatures) and clos : SIGN → CLOS is a functor.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Alternative Definition - Π-Institutions

Definition

A π-institutions consists of :

◮ A category SIGN. ◮ A functor gram : SIGN → SET. ◮ for every Σ : SIGN, a consequence relation

Σ: 2gram(Σ) × gram(Σ) satisfying the following properties:

◮ For every p ∈ gram(Σ), p Σ p. ◮ For every p ∈ gram(Σ) and Φ1, Φ2 ⊆ gram(Σ),

if Φ1 ⊆ Φ2 and Φ1 Σ p then Φ2 Σ p.

◮ For every p ∈ gram(Σ) and Φ1, Φ2 ⊆ gram(Σ),

if Φ1 Σ p and for every p′ ∈ Φ1, Φ2 Σ p′ then Φ2 Σ p.

◮ For every σ : Σ → Σ′, p ∈ gram(Σ) and

Φ ⊆ gram(Σ), Φ Σ p implies gram(σ)(Φ) Σ′ gram(σ)(p).

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Both Π-Institutions are Equivalent

Definition 1 to Definition 2

◮ The gram functor is the composition of clos with the

forgetful functor that maps closure systems to the underlying languages.

◮ Every closure operator defines a consequence

relation: Φ Σ p iff p ∈ cΣ(Φ).

Definition 2 to Definition 1

◮ The closure operator itself is derived from the

consequence relation: for every Φ ⊆ gram(Σ), cΣ(Φ) = {p ∈ gram(Σ) : Φ Σ p}

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Outline

Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Institution Recap Closure Systems Π-Institutions Definition of Π-Institutions Relating Institutions and Π-Institutions

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Every Institution Presents A Π-Institution

Theorem

Every institution SIGN, gram, mod, | = presents the π-institution SIGN, gram, , where for every signature Σ, p ∈ gram(Σ) and Φ ⊆ gram(Σ), Φ Σ p iff for every M ∈ mod(Σ), M | =Σ Φ implies M | =Σ p.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Proof Outline

In order to prove this we must prove:

  • 1. For every p ∈ gram(Σ), p Σ p.
  • 2. For every p ∈ gram(Σ) and Φ1, Φ2 ⊆ gram(Σ),

if Φ1 ⊆ Φ2 and Φ1 Σ p then Φ2 Σ p.

  • 3. For every p ∈ gram(Σ) and Φ1, Φ2 ⊆ gram(Σ),

if Φ1 Σ p and for every p′ ∈ Φ1, Φ2 Σ p′ then Φ2 Σ p.

  • 4. For every σ : Σ → Σ′, p ∈ gram(Σ) and

Φ ⊆ gram(Σ), Φ Σ p implies gram(σ)(Φ) Σ′ gram(σ)(p).

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Proof - Part 3

To show:

For every p ∈ gram(Σ) and Φ1, Φ2 ⊆ gram(Σ), if Φ1 Σ p and for every p′ ∈ Φ1, Φ2 Σ p′ then Φ2 Σ p.

Definition of

Φ Σ p iff for every M ∈ mod(Σ), M | =Σ Φ implies M | =Σ p.

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Proof - Part 4

To show:

For every σ : Σ → Σ′, p ∈ gram(Σ) and Φ ⊆ gram(Σ), Φ Σ p implies gram(σ)(Φ) Σ′ gram(σ)(p).

Definition of

Φ Σ p iff for every M ∈ mod(Σ), M | =Σ Φ implies M | =Σ p.

Satisfaction Condition

For every σ : Σ → Σ′, p ∈ gram(Σ) and M′ ∈ mod(Σ′), mod(σ)(M′) | =Σ p iff M′ | =Σ′ gram(σ)(p).

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Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions

Definition of Institutions Examples - EL Example - The CASL Institution

Recap

Closure Systems

Π-Institutions

Definition of Π-Institutions Relating Institutions and

Π-Institutions

Summary

Summary

◮ Institutions provide a frame work for dealing with

logics, that capture the notions of translations and models.

◮ Π-Institutions and Institutions are equivalent, but

useful for different purposes.

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Institutions - Part 1 Liam O’Reilly

The Atomic Formulae

The atomic formulae are:

◮ Application of qualified predicate symbols p ∈ P to

argument terms of appropriate sorts.

◮ Assertions about the definedness of fully-qualified

terms.

◮ Existential and strong equations between

fully-quantified terms of the same sort.