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Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Institutions - Part 2
Liam O’Reilly
16.05.07
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Outline
From Last TimeRelating Institutions to Π-InstitutionsRecap - Theories in Closure SystemsRelating Π-Institutions to Institutions
Institution Morphisms and MapsMany Notions and NamesRecap - Natural TransformationsInstitution Morphisms
Hets Demo
Propositions for maps and morphisms
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Outline
From Last TimeRelating Institutions to Π-InstitutionsRecap - Theories in Closure SystemsRelating Π-Institutions to Institutions
Institution Morphisms and MapsMany Notions and NamesRecap - Natural TransformationsInstitution Morphisms
Hets Demo
Propositions for maps and morphisms
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Every Institution Presents A Π-Institution
TheoremEvery institution 〈SIGN, gram, mod, |=〉 presents theπ-institution 〈SIGN, gram, 〉,where for every signature Σ, p ∈ gram(Σ) andΦ ⊆ gram(Σ),Φ Σ p iff for every M ∈ mod(Σ), M |=Σ Φimplies M |=Σ p.
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
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 pthen 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).
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
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 impliesM |=Σ p.
Satisfaction ConditionFor every σ : Σ → Σ′, p ∈ gram(Σ) and M ′ ∈ mod(Σ′),mod(σ)(M ′) |=Σ p iff M ′ |=Σ′ gram(σ)(p).
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Outline
From Last TimeRelating Institutions to Π-InstitutionsRecap - Theories in Closure SystemsRelating Π-Institutions to Institutions
Institution Morphisms and MapsMany Notions and NamesRecap - Natural TransformationsInstitution Morphisms
Hets Demo
Propositions for maps and morphisms
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Theories in Closure Systems
Given a closure system 〈L, c〉I We say Φ ⊆ L is closed iff Φ = c(Φ).I We define the category THEO〈L,c〉 whose objects
(theories) are the closed subsets of L andmorphisms are given by inclusions.
I We define the category PRES〈L,c〉 whose objects(theories presentations) are the subsets of L andmorphisms are given by the preorder Φ ≤ Γ iffc(Φ) ⊆ c(Γ).
I We define the category SPRES〈L,c〉 whose objects(strict presentations) are the subsets of L ordered byinclusion.
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Outline
From Last TimeRelating Institutions to Π-InstitutionsRecap - Theories in Closure SystemsRelating Π-Institutions to Institutions
Institution Morphisms and MapsMany Notions and NamesRecap - Natural TransformationsInstitution Morphisms
Hets Demo
Propositions for maps and morphisms
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Every Π-Institution Presents An Institution
TheoremEvery π-institution 〈SIGN, clos〉 presents the institution〈SIGN, gram = clos; forget, mod, |=〉,where for every signature Σ, p ∈ gram(Σ) andΦ ∈ mod(Σ)
Φ |= p iff p ∈ Φ
We have to prove that the satisfaction condition holds.
Satisfaction ConditionFor every σ : Σ → Σ′, p ∈ gram(Σ) and M ′ ∈ mod(Σ′),mod(σ)(M ′) |=Σ p iff M ′ |=Σ′ gram(σ)(p).
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Outline
From Last TimeRelating Institutions to Π-InstitutionsRecap - Theories in Closure SystemsRelating Π-Institutions to Institutions
Institution Morphisms and MapsMany Notions and NamesRecap - Natural TransformationsInstitution Morphisms
Hets Demo
Propositions for maps and morphisms
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Many Notions, Many Names and Confusion
Fiaderio Π-Inst Inst MapGoguen Inst ComorphismMeseguer Entailment System Plain MapMossakowski Plain RepresentationTarlecki Representations
We had originally hoped to survey andsystematise all the distinct notions of morphismthat apply to the close variants of institutions;although we found even this limited goalimpractical.
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Outline
From Last TimeRelating Institutions to Π-InstitutionsRecap - Theories in Closure SystemsRelating Π-Institutions to Institutions
Institution Morphisms and MapsMany Notions and NamesRecap - Natural TransformationsInstitution Morphisms
Hets Demo
Propositions for maps and morphisms
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Definition - Natural Transformations
DefinitionGiven two functors ψ : D → C and ϕ : D → C, a naturaltransformation τ from ψ to ϕ, denoted by ψ τ−→ ϕ orτ : ψ
·−→ ϕ, is a function that assigns to each object d ofD a morphism τ d : ψ(d) → ϕ(d) of C such that, for everymorphism f : d → d ′ of D,
τ d ;ϕ(f ) = ψ(f ); τ d ′
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Outline
From Last TimeRelating Institutions to Π-InstitutionsRecap - Theories in Closure SystemsRelating Π-Institutions to Institutions
Institution Morphisms and MapsMany Notions and NamesRecap - Natural TransformationsInstitution Morphisms
Hets Demo
Propositions for maps and morphisms
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Definition - Institution Morphism
DefinitionLet ι = 〈SIGN, gram, mod, |=〉 andι′ = 〈SIGN′, gram′, mod′, |=′〉 be institutions. Aninstitution morphism ρ : ι → ι′ is a triple 〈Φ,α,β〉 where:
I Φ : SIGN → SIGN′ is a functor.I α : Φ; gram′ → gram is a natural transformation.I β : mod → Φ; mod′ is a natural transformation.
such that the following property(the invariance condition)holds for any signature Σ ∈| SIGN |, m ∈| mod(Σ) | andφ′ ∈ gram′(Φ(Σ)) :
m |=Σ αΣ(φ′) iff βΣ(m) |=′Φ(Σ) φ′
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Definition - Institution Map (Comorphism)
DefinitionLet ι = 〈SIGN, gram, mod, |=〉 andι′ = 〈SIGN′, gram′, mod′, |=′〉 be institutions. Aninstitution morphism ρ : ι → ι′ is a triple 〈Φ,α,β〉 where:
I Φ : SIGN → SIGN′ is a functor.I α : gram → Φ; gram′ is a natural transformation.I β : Φ; mod′ → mod is a natural transformation.
such that the following property(the invariance condition)holds for any signature Σ ∈| SIGN |, m′ ∈| mod′(Φ(Σ)) |and φ ∈ gram(Σ) :
βΣ(m′) |=Σ φ iff m′ |=′Φ(Σ) αΣ(φ)
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Outline
From Last TimeRelating Institutions to Π-InstitutionsRecap - Theories in Closure SystemsRelating Π-Institutions to Institutions
Institution Morphisms and MapsMany Notions and NamesRecap - Natural TransformationsInstitution Morphisms
Hets Demo
Propositions for maps and morphisms
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
CASL Specification subsorts.casl
spec subsorts =sort Nat
ops 0,1 : Nat
sort Pos = { x:Nat . not(x=0)}op pre : Pos -> Nat;
suc : Nat -> Pos
op one:Pos
axiomforall n:Nat . pre(suc(n))=n;1=one;
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Mapping PFOL= to FOL=
A PFOL=-signature Σ = (S, TF , PF , P) is translated to aFOL=-presentation having
SignatureSig(Φ(Σ)) =(S, TF ] PF ] {⊥: s | s ∈ S}, P ] {D : s | s ∈ S})
Set of axioms Ax(Φ(Σ))∃x : s • Ds(x) s ∈ S (1)¬Ds(⊥s) s ∈ S (2)Ds(f (x1, . . . , xn)) ⇔
∧Dsi (xi) f : s1 . . . sn → s ∈ TF (3)
Ds(g(x1, . . . , xn)) ⇒∧
Dsi (xi) g : s1 . . . sn → s ∈ PF (4)p(x1, . . . , xn) ⇒
∧Dsi (xi) p : s1 . . . sn ∈ P (5)
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
CASL Specification partial.casl
spec partial =sort Nat
ops 0,1 : Nat
op pre : Nat ->? Nat;suc : Nat -> Nat
axiomforall n:Nat . pre(suc(n))=n;
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Outline
From Last TimeRelating Institutions to Π-InstitutionsRecap - Theories in Closure SystemsRelating Π-Institutions to Institutions
Institution Morphisms and MapsMany Notions and NamesRecap - Natural TransformationsInstitution Morphisms
Hets Demo
Propositions for maps and morphisms
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Proposition for Institution Morphism
Let ρ = 〈Φ,α,β〉 : ι → ι′ be an institution map. Thefunctor Φ extends to THEOι → THEOι′ by establishingΦ(〈Σ, Γ〉) = 〈Φ(Σ), c(αΣ(Γ))〉.
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Proposition for Institution Map
Let ρ = 〈Ψ,α′,β′〉 : ι → ι′ be an institution morphism.The functor Ψ extends to THEOι′ → THEOι throughΨ(〈Σ′, Γ′〉) = 〈Ψ(Σ′),α′−1
Σ′ (Γ′)〉.
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Proposition
Let ι = 〈SIGN, gram, mod, |=〉 andι′ = 〈SIGN′, gram′, mod′, |=′〉 be institutions,ρ = 〈Φ,α,β〉 : ι → ι′ be an institution map and〈Ψ,α′,β′〉 : ι → ι′ be a morphism such that ψ is a rightadjoint of φ, and, for every Σ ∈| SIGN |,αΣ = gram(ηΣ);α′
Φ(Σ) where η is the unit of theadjunction, then
I The functor υ : THEOι′ → THEOι induced by theinstitution morphism 〈Ψ,α′,β′〉, is a right adjoint ofthe functor THEOι → THEOι′ induced by theinstitution map 〈Φ,α,β〉.
I If each component of β′ is surjective, then the unitsηΣ are conservative.
Institutions - Part 2
Liam O’Reilly
From Last TimeRelating Institutions toΠ-Institutions
Recap - Theories in ClosureSystems
Relating Π-Institutions toInstitutions
InstitutionMorphisms andMapsMany Notions and Names
Recap - NaturalTransformations
Institution Morphisms
Hets Demo
Propositions formaps andmorphisms
Summary
Summary
I Institutions provide a frame work for dealing withlogics, that capture the notions of sentences, modeland satisfaction between models and sentences.
I Institution morphisms and Comorphisms allow us totranslate between institutions, which allow us to useprograms on one logic with another logic.
I They actually have a practical use in the real world.eg. Hets.