Revisiting the Institutional Approach to Herbrands Theorem Ionu uu - PowerPoint PPT Presentation

Revisiting the Institutional Approach to Herbrand’s Theorem Ionuţ Ţuţu 1,2 José Luiz Fiadeiro 1 1 Department of Computer Science, Royal Holloway University of London 2 Simion Stoilow Institute of Mathematics of the Romanian Academy 6 th Conference on Algebra and Coalgebra in Computer Science Nijmegen, 2015

Herbrand’s Fundamental Theorem • central result in proof theory • deals with the reduction of provability in first-order logic to provability in propositional logic ∃ { x 1 , . . . , x n } · ρ ( x 1 , . . . , x n ) is valid if and only if there is a sequence of terms t 1 , . . . , t n such that ρ ( t 1 , . . . , t n ) is valid 1929

Herbrand’s Fundamental Theorem • central result in proof theory • deals with the reduction of provability in first-order logic to provability in propositional logic ∃ { x 1 , . . . , x n } · ρ ( x 1 , . . . , x n ) is valid if and only if there is a sequence of terms t 1 , . . . , t n such that ρ ( t 1 , . . . , t n ) is valid 1929

Herbrand’s Fundamental Theorem • central result in proof theory • deals with the reduction of provability in first-order logic to provability in propositional logic • difficulties in following the proof and errors reported by Bernays and Gödel Herbrand 1929 1940

Herbrand’s Fundamental Theorem • central result in proof theory • deals with the reduction of provability in first-order logic to provability in propositional logic • gaps and counterexamples found by Dreben, Andrews, and Aanderaa • the publication of the first emended (and detailed) proof of the result Herbrand 1929 1963

The resolution inference rule • introduced by Robinson • well-suited for automation ∃ X · Q ∧ g ∀ Y · c ← H θ ∃ X ′ · θ ( Q ) ∧ θ ( H ) • led to the development of logic programming – prolog (Kowalski & Colmerauer) Herbrand 1929 1965

The resolution inference rule • introduced by Robinson • well-suited for automation ∃ X · Q ∧ g ∀ Y · c ← H θ ∃ X ′ · θ ( Q ) ∧ θ ( H ) • led to the development of logic programming – prolog (Kowalski & Colmerauer) Herbrand Robinson 1929 1973 1965

Foundations of logic programming Given a logic program Γ , the answers to an existential query can be found simply by examining a term model – the least Herbrand model – instead ∃ { x } · “ x is a number” of all the models that satisfy Γ . ∧ “ x Prolog programmers 1. Γ � Σ ∃ X · ρ can change a lightbulb” 2. 0 Σ , Γ � Σ ∃ X · ρ 3. There exists ψ : X → Y such that Γ � Σ ∀ Y · ψ ( ρ ) . Herbrand Robinson 1929 1965 1984

Foundations of logic programming Given a logic program Γ , the answers to an existential query can be found simply by examining a term model – the least Herbrand model – instead ∃ { x } · “ x is a number” of all the models that satisfy Γ . ∧ “ x Prolog programmers 1. Γ � Σ ∃ X · ρ can change a lightbulb” 2. 0 Σ , Γ � Σ ∃ X · ρ 3. There exists ψ : X → Y such that Γ � Σ ∀ Y · ψ ( ρ ) . Herbrand Robinson 1929 1965 1984

Foundations of logic programming Given a logic program Γ , the answers to an existential query can be found simply by examining a term model – the least Herbrand model – instead ∃ { x } · “ x is a number” of all the models that satisfy Γ . ∧ “ x Prolog programmers 1. Γ � Σ ∃ X · ρ can change a lightbulb” 2. 0 Σ , Γ � Σ ∃ X · ρ 3. There exists ψ : X → Y such that Γ � Σ ∀ Y · ψ ( ρ ) . Herbrand Robinson 1929 1965 1984

Foundations of logic programming Given a logic program Γ , the answers to an existential query can be found simply by examining a term model – the least Herbrand model – instead ∃ { x } · “ x is a number” of all the models that satisfy Γ . ∧ “ x Prolog programmers 1. Γ � Σ ∃ X · ρ can change a lightbulb” 2. 0 Σ , Γ � Σ ∃ X · ρ 3. There exists ψ : X → Y such that Γ � Σ ∀ Y · ψ ( ρ ) . Herbrand Robinson 1929 1965 1984

A multitude of variants • relational first-order logic • many-sorted equational logic • higher-order logic • hidden algebra ∃ { x , y } · sorted ( 2, 3, x , y , 5 ) • institution-independent • service-oriented • abstract logic programming Herbrand Robinson 1929 1965 1984

A multitude of variants • relational first-order logic • many-sorted equational logic • higher-order logic • hidden algebra ∃ { x : Num } · sorted ( 2, 3, x ) = T • institution-independent • service-oriented • abstract logic programming Herbrand Robinson Lloyd 1929 1965 1984

A multitude of variants • relational first-order logic • many-sorted equational logic • higher-order logic • hidden algebra ∃ { s : List → B } · s [ 2, 3, 5 ] = T • institution-independent • service-oriented • abstract logic programming Herbrand Robinson Lloyd 1929 1965 1984

A multitude of variants • relational first-order logic • many-sorted equational logic • higher-order logic • hidden algebra ∃ { s : Stream } · s ∼ tail ( s ) • institution-independent • service-oriented • abstract logic programming Herbrand Robinson Lloyd 1929 2002 1965 1984

� � A multitude of variants • relational first-order logic • many-sorted equational logic � S ig, Sen, Mod, � � • higher-order logic Sen ( Σ ) • hidden algebra Sen ✶ • institution-independent Σ � Σ ✌ • service-oriented Mod Mod ( Σ ) • abstract logic programming Herbrand Robinson Lloyd 1929 2004 1965 1984

A multitude of variants • relational first-order logic • many-sorted equational logic � S ig, Sen, Mod, � � • higher-order logic subject to a • hidden algebra satisfaction condition: for every ϕ : Σ → Ω , • institution-independent M ∈ | Mod ( Ω ) | , ρ ∈ Sen ( Σ ) • service-oriented iff M ↾ ϕ � Σ ρ M � Ω ϕ ( ρ ) • abstract logic programming Herbrand Robinson Lloyd 1929 2004 1965 1984

A multitude of variants • relational first-order logic ∃ o · { r 1 , r 2 } • many-sorted equational logic • higher-order logic • hidden algebra • institution-independent • service-oriented • abstract logic programming Herbrand Robinson Lloyd 1929 2004 2013 1965 1984

A multitude of variants • relational first-order logic • many-sorted equational logic • higher-order logic • hidden algebra ∃ { x , y } · sorted ( 2, 3, x , y , 5 ) • institution-independent χ : � F , P � ֒ →� F ∪ { x , y } , P � • service-oriented ∃ χ · sorted ( 2, 3, x , y , 5 ) • abstract logic programming Herbrand Robinson Lloyd 1929 2004 2013 1965 1984

� � � � � � � A multitude of variants • relational first-order logic Sen Σ ( X ) • many-sorted equational logic ✴ � • higher-order logic Σ X ✖ ❴ Mod Σ ( X ) • hidden algebra ϕ Sen Ω ( X ϕ ) • institution-independent X ϕ ✴ � � Ω ✖ • service-oriented Mod Ω ( X ϕ ) • abstract logic programming Herbrand Robinson Lloyd 1929 2004 2014 1965 1984

� � A multitude of variants • relational first-order logic Sen Σ ( X ) • many-sorted equational logic ✴ � • higher-order logic Σ X ✖ Mod Σ ( X ) • hidden algebra • institution-independent ∃ X · ρ • service-oriented • abstract logic programming Herbrand Robinson Lloyd 1929 2004 2014 1965 1984

� � Institutions as functors • each institution I = � S ig , Sen , Mod , � � can be identified with a functor Rooms and corridors I : S ig → R oom α � S , M , � � � S ′ , M ′ , where I ( Σ ) = � Sen ( Σ ) , Mod ( Σ ) , � Σ � � ′ � β • similarly, substitution systems can be defined as functors S : S ubst → G / R oom Herbrand Robinson Lloyd 1929 2004 2014 1965 1984

� � � From institutions to substitution systems • let Q be a class of signature morphisms of an institution I : S ig → R oom I ( Σ ) For every I -signature Σ we obtain a substi- I ( χ 2 ) I ( χ 1 ) tution system SI Q Σ : S ubst Q Σ → I ( Σ ) / R oom : I ( Σ ( χ 1 )) I ( Σ ( χ 2 )) • the objects of S ubst Q Σ are signature morphisms χ : Σ → Σ ( χ ) belonging to Q � Sen Σ ( ψ ) ,Mod Σ ( ψ ) � • a Σ -substitution ψ : χ 1 → χ 2 is a corridor � Sen Σ ( ψ ) , Mod Σ ( ψ ) � : I ( Σ ( χ 1 )) → I ( Σ ( χ 2 )) Herbrand Robinson Lloyd 1929 2004 2014 1965 1984

� � � Quantification spaces • for every subcategory Q ⊆ S ig � , the functor ϕ Σ ′ Σ dom : Q → S ig gives rise to a natural transformation ι Q : ( _ / Q ) ⇒ dom op ; ( _ / C ) χ ′ χ � Ω ′ Ω φ Definition. Q is said to be a quantification space for an institution I : S ig → R oom if �→ ι Q , χ 1. every arrow in Q forms a pushout in S ig , and ϕ � Σ ′ Σ 2. ι Q is a natural isomorphism. Herbrand Robinson Lloyd 1929 2004 2014 1965 1984

� � � Quantification spaces • for every subcategory Q ⊆ S ig � , the functor ϕ Σ ′ Σ dom : Q → S ig gives rise to a natural transformation ι Q : ( _ / Q ) ⇒ dom op ; ( _ / C ) χ ϕ χ Σ ( χ ) ϕ χ � Σ ′ ( χ ϕ ) Definition. Q is said to be a quantification space for an institution I : S ig → R oom if �→ ι Q , χ 1. every arrow in Q forms a pushout in S ig , and ϕ � Σ ′ Σ 2. ι Q is a natural isomorphism. Herbrand Robinson Lloyd 1929 2004 2014 1965 1984