Institutions - Part 1 Institution Recap Closure Systems - PowerPoint PPT Presentation

Institutions - Part 1 Liam O’Reilly Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Institutions - Part 1 Institution Recap Closure Systems Π -Institutions Liam O’Reilly Definition of Π -Institutions Relating Institutions and Π -Institutions Summary 09.05.07

Institutions - Part 1 Outline Liam O’Reilly Why Do We Need Institutions? Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Institutions Example - The CASL Institution Definition of Institutions Recap Examples - EL Closure Systems Π -Institutions Example - The CASL Institution Definition of Π -Institutions Relating Institutions and Π -Institutions Summary Recap Closure Systems Π -Institutions Definition of Π -Institutions Relating Institutions and Π -Institutions

Institutions - Part 1 Why Do We Need Institutions? Liam O’Reilly Why Do We Need ◮ There are many different logics in the world, for Institutions? instance: EL, FOL, HOL, SubPCFOL = , temporal Institutions Definition of Institutions logic, Horn clause logic, etc. Examples - EL Example - The CASL ◮ Each program / prover tends to use its own logic. Institution Recap ◮ Many general results are actually completely Closure Systems Π -Institutions independent of what logic system is used. Definition of Π -Institutions Relating Institutions and Institutions allow: Π -Institutions Summary ◮ 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.

Institutions - Part 1 Outline Liam O’Reilly Why Do We Need Institutions? Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Institutions Example - The CASL Institution Definition of Institutions Recap Examples - EL Closure Systems Π -Institutions Example - The CASL Institution Definition of Π -Institutions Relating Institutions and Π -Institutions Summary Recap Closure Systems Π -Institutions Definition of Π -Institutions Relating Institutions and Π -Institutions

Institutions - Part 1 Definition - Institutions Liam O’Reilly Why Do We Need Institutions? Institutions Definition Definition of Institutions Examples - EL An institution is a quadruple � SIGN , gram , mod , | = � Example - The CASL Institution where: Recap Closure Systems ◮ SIGN is a category. Π -Institutions ◮ gram : SIGN → SET is a functor. Definition of Π -Institutions Relating Institutions and Π -Institutions ◮ mod : SIGN op → CAT is a functor. Summary ◮ 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 ′ ) , = Σ p iff M ′ | mod ( σ )( M ′ ) | = Σ ′ gram ( σ )( p ) .

Institutions - Part 1 Outline Liam O’Reilly Why Do We Need Institutions? Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Institutions Example - The CASL Institution Definition of Institutions Recap Examples - EL Closure Systems Π -Institutions Example - The CASL Institution Definition of Π -Institutions Relating Institutions and Π -Institutions Summary Recap Closure Systems Π -Institutions Definition of Π -Institutions Relating Institutions and Π -Institutions

Institutions - Part 1 Signatures for Many-Sorted Equational Logic Liam O’Reilly Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Institution Recap A signature Σ = ( S , Ω) is a pair of sets, where Closure Systems ◮ S is a set of sorts Π -Institutions Definition of Π -Institutions ◮ Ω is a set of total functions symbols, of the form Relating Institutions and Π -Institutions n : s 1 × . . . × s k → s with s 1 , . . . , s k , s ∈ S and k � 0. Summary

Institutions - Part 1 Signature Morphisms for Many-Sorted Liam O’Reilly Equational Logic Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Given two signatures Σ = ( S , Ω) and Σ ′ = ( S ′ , Ω ′ ) , a Institution Recap signature morphism σ : Σ → Σ ′ is a pair ( σ s , σ Ω ) where Closure Systems Π -Institutions ◮ σ s : S → S ′ Definition of Π -Institutions Relating Institutions and ◮ σ Ω : Ω → Ω ′ Π -Institutions Summary such that for each function symbol n : s 1 × . . . s k → s ∈ Ω , k ≥ 0, there exists a function name m with σ Ω ( n : s 1 × . . . s k → s ) = ( m : σ s ( s 1 ) × . . . σ s ( s k ) → σ s ( s )) .

Institutions - Part 1 Models for Many-Sorted Equational Logic Liam O’Reilly Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Institution Given a signature Σ = ( S , Ω) a total algebra(model) for Σ Recap assigns : Closure Systems Π -Institutions ◮ A carrier set A ( s ) to each sort s ∈ S . Definition of Π -Institutions Relating Institutions and ◮ A total function Π -Institutions Summary A ( n : s 1 × s k → s ) : A ( s 1 ) × . . . × A ( s k ) → A ( s ) to each operation ( n : s 1 × . . . × s k → s ) ∈ Ω , k ≥ 0.

Institutions - Part 1 Models Morphisms for Many-Sorted Liam O’Reilly Equational Logic Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Institution Given two models A , B ∈ mod (Σ) , a model morphism Recap h : A → B is a family ( h s ) s ∈ S of functions Closure Systems h s : A ( s ) → B ( s ) such that for any function f ∈ Ω say Π -Institutions Definition of Π -Institutions f = ( n : s 1 × . . . × s k → s ) , k ≥ 0, the following condition Relating Institutions and Π -Institutions holds: Summary h s ( A ( f )( a 1 , . . . , a k )) = B ( f )( h s 1 ( a 1 ) , . . . , h sk ( a k )) for all ( a 1 , . . . , a k ) ∈ A ( s 1 ) × . . . × A ( s k ) .

Institutions - Part 1 Reducts Liam O’Reilly Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Institution Given two signatures Σ = ( S , Ω) and Σ ′ = ( S ′ , Ω ′ ) , and a Recap signature morphism σ : Σ → Σ ′ then for a Σ ′ -algebra A ′ Closure Systems the σ -reduct of A ′ is defined by: Π -Institutions Definition of Π -Institutions ◮ ( A ′ | σ )( s ) = A ′ ( σ ( s )) for all s ∈ S Relating Institutions and Π -Institutions Summary ◮ ( A ′ | σ )( f ) = A ′ ( σ ( f )) for all f ∈ Ω

Institutions - Part 1 Terms in Many-Sorted Equational Logic Liam O’Reilly Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Given a signature Σ = ( S , Ω) and a family of variables Example - The CASL Institution X = ( X s ) s ∈ S of disjoint infinite sets, then T Σ( X ) , s is Recap defined by Closure Systems Π -Institutions 1. X s ⊆ T Σ( X ) , s , Definition of Π -Institutions Relating Institutions and 2. if n : → s is an operation of Ω then n ∈ T Σ( X ) , s , Π -Institutions Summary 3. if n : s 1 × . . . × s k → s , k � 1 is an operation of Ω and if t i ∈ T Σ( X ) , s i , for 1 � i � k , then n ( t 1 , . . . , t k ) ∈ T Σ( X ) , s .

Institutions - Part 1 Sentences Many-Sorted Equational Logic Liam O’Reilly Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Institution Recap For each signature Σ the set of formulae of EL is Closure Systems Π -Institutions Definition of Π -Institutions gram (Σ) = {∀ X . t = u | t , u ∈ T Σ( X ) , s } Relating Institutions and Π -Institutions Summary

Institutions - Part 1 Translation of Sentences Liam O’Reilly Translation of Variables Given σ : Σ → Σ ′ the variable translation is defined as Why Do We Need Institutions? Institutions � σ (( X s ) s ∈ S ) = (( X s ) s ′ ∈ S ′ ) Definition of Institutions Examples - EL Example - The CASL σ ( s )= s ′ Institution Recap Closure Systems Translation of Terms Π -Institutions Given σ : Σ → Σ ′ the term translation is defined as Definition of Π -Institutions Relating Institutions and Π -Institutions Summary σ ( x : s ) = x : σ s ( s ) σ ( f ( t 1 , . . . , t k )) = σ Ω ( f )( σ s ( t 1 ) , . . . , σ s ( t k ) Translation of Sentences Given σ : Σ → Σ ′ the sentence translation is defined as σ ( ∀ X . t = u ) = ∀ σ ( X ) .σ ( t ) = σ ( u )

Institutions - Part 1 Assignments Liam O’Reilly Why Do We Need Institutions? Institutions Definition of Institutions Examples - EL Example - The CASL Institution Recap Given Σ = ( S , Ω) then an assignment of X for A is a Closure Systems Π -Institutions family α = ( α s ) s ∈ S of functions α s : X s → A ( s ) . Definition of Π -Institutions Relating Institutions and Π -Institutions We can just write α : X → A . Summary