Revisiting the Institutional Approach
to Herbrand’s Theorem
Ionuţ Ţuţu1,2
1 Department
2 Simion
José Luiz Fiadeiro1
of Computer Science, Royal Holloway University of London
Stoilow Institute of Mathematics of the Romanian Academy
6th 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
∃{x1 , . . . , xn } · ρ(x1 , . . . , xn ) is valid
if and only if
there is a sequence of terms t1 , . . . , tn
such that ρ(t1 , . . . , tn ) 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
∃{x1 , . . . , xn } · ρ(x1 , . . . , xn ) is valid
if and only if
there is a sequence of terms t1 , . . . , tn
such that ρ(t1 , . . . , tn ) 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 0
· 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 0
· θ(Q) ∧ θ(H)
• led to the development of logic
programming – prolog
(Kowalski & Colmerauer)
Herbrand
1929
Robinson
1965 1973
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
of all the models that satisfy Γ .
∃{x} · “x is a number”
∧ “x Prolog programmers
can change a lightbulb”
1. Γ Σ ∃X · ρ
2. 0Σ,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
of all the models that satisfy Γ .
∃{x} · “x is a number”
∧ “x Prolog programmers
can change a lightbulb”
1. Γ Σ ∃X · ρ
2. 0Σ,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
of all the models that satisfy Γ .
∃{x} · “x is a number”
∧ “x Prolog programmers
can change a lightbulb”
1. Γ Σ ∃X · ρ
2. 0Σ,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
of all the models that satisfy Γ .
∃{x} · “x is a number”
∧ “x Prolog programmers
can change a lightbulb”
1. Γ Σ ∃X · ρ
2. 0Σ,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
∃{x, y} · sorted(2, 3, x, y, 5)
• hidden algebra
• 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
∃{x : Num} · sorted(2, 3, x) = T
• hidden algebra
• 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
∃{s : List → B} · s [2, 3, 5] = T
• hidden algebra
• 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
∃{s : Stream} · s ∼ tail(s)
• hidden algebra
• institution-independent
• service-oriented
• abstract logic programming
Herbrand
Robinson
Lloyd
1929
1965
1984
2002
A multitude of variants
• relational first-order logic
• many-sorted equational logic
hSig, Sen, Mod, i
• higher-order logic
Sen(Σ)
1
Σ
Sen 8
• hidden algebra
• institution-independent
Σ
Mod
&
• service-oriented
Mod(Σ)
• abstract logic programming
Herbrand
Robinson
Lloyd
1929
1965
1984
2004
A multitude of variants
• relational first-order logic
• many-sorted equational logic
hSig, Sen, Mod, i
• higher-order logic
subject to a
satisfaction condition:
• hidden algebra
for every ϕ : Σ → Ω,
M ∈ |Mod(Ω)|, ρ ∈ Sen(Σ)
• institution-independent
Mϕ Σ ρ iff M Ω ϕ(ρ)
• service-oriented
• abstract logic programming
Herbrand
Robinson
Lloyd
1929
1965
1984
2004
A multitude of variants
• relational first-order logic
∃o · {r1 , r2 }
• many-sorted equational logic
• higher-order logic
• hidden algebra
• institution-independent
• service-oriented
• abstract logic programming
Herbrand
Robinson
Lloyd
1929
1965
1984
2004 2013
A multitude of variants
• relational first-order logic
• many-sorted equational logic
• higher-order logic
∃{x, y} · sorted(2, 3, x, y, 5)
• hidden algebra
• institution-independent
χ : hF,Pi,→hF∪{x,y},Pi
∃χ · sorted(2, 3, x, y, 5)
• service-oriented
• abstract logic programming
Herbrand
Robinson
Lloyd
1929
1965
1984
2004 2013
A multitude of variants
• relational first-order logic
SenΣ (X)
Σ
/ 7
X
_
+
ϕ
Ω
SenΩ
/ 7
Xϕ
• many-sorted equational logic
+
• higher-order logic
ModΣ (X)
(Xϕ )
• hidden algebra
O
• institution-independent
ModΩ (Xϕ )
• service-oriented
• abstract logic programming
Herbrand
Robinson
Lloyd
1929
1965
1984
2004 2014
A multitude of variants
• relational first-order logic
SenΣ (X)
Σ
/ 7
X
• many-sorted equational logic
+
• higher-order logic
ModΣ (X)
• hidden algebra
∃X · ρ
• institution-independent
• service-oriented
• abstract logic programming
Herbrand
Robinson
Lloyd
1929
1965
1984
2004 2014
Institutions as functors
• each institution I = hSig, Sen, Mod, i
Rooms and corridors
can be identified with a functor
I : Sig → Room
α
hS, M,
O i
hS 0, M 0, 0 i
where I(Σ) = hSen(Σ), Mod(Σ), Σ i
β
• similarly, substitution systems can be
defined as functors
S : Subst → G / Room
Herbrand
Robinson
Lloyd
1929
1965
1984
2004 2014
From institutions to substitution systems
• let Q be a class of signature morphisms of
an institution I : Sig → Room
I(Σ)
I(χ1 )
I(Σ(χ1 ))
I(χ2 )
I(Σ(χ2 ))
O
hSenΣ (ψ),ModΣ (ψ)i
For every I-signature Σ we obtain a substiQ
tution system SIQ
Σ : SubstΣ → I(Σ) / Room:
• the objects of SubstQ
Σ are signature
morphisms χ : Σ → Σ(χ) belonging to Q
• a Σ-substitution ψ : χ1 → χ2 is a corridor
hSenΣ (ψ), ModΣ (ψ)i : I(Σ(χ1 )) → I(Σ(χ2 ))
Herbrand
Robinson
Lloyd
1929
1965
1984
2004 2014
Quantification spaces
Σ
χ
ϕ
χ0
Ω
φ
7→
Σ
/ Σ0
ϕ
/ Ω0
ιQ,χ
/ Σ0
• for every subcategory Q ⊆ Sig~ , the functor
dom : Q → Sig gives rise to a natural
transformation ιQ : (_ / Q) ⇒ domop ; (_ / C)
Definition. Q is said to be a quantification
space for an institution I : Sig → Room if
1. every arrow in Q forms a pushout in Sig, and
2. ιQ is a natural isomorphism.
Herbrand
Robinson
Lloyd
1929
1965
1984
2004 2014
Quantification spaces
χ
/ Σ0
ϕ
Σ
χϕ
Σ(χ)
ϕχ
/ Σ 0 (χϕ )
7→
Σ
ϕ
ιQ,χ
/ Σ0
• for every subcategory Q ⊆ Sig~ , the functor
dom : Q → Sig gives rise to a natural
transformation ιQ : (_ / Q) ⇒ domop ; (_ / C)
Definition. Q is said to be a quantification
space for an institution I : Sig → Room if
1. every arrow in Q forms a pushout in Sig, and
2. ιQ is a natural isomorphism.
Herbrand
Robinson
Lloyd
1929
1965
1984
2004 2014
Representable signature extensions
Definition. An extension χ : Σ → Σ(χ) is
representable if there exist
• a Σ-model Mχ and
• an isomorphism of categories iχ
such that the following diagram commutes:
Mod(Σ)
O
h
forgetful
_χ
Mod(Σ(χ))
iχ
Herbrand
Robinson
Lloyd
1929
1965
1984
/ Mχ / Mod(Σ)
2004 2014
Representable signature extensions
Proposition. The representation of
signature extensions generalizes to a functor
Q
RQ
Σ : SubstΣ → Mod(Σ), where
• for every χ : Σ → Σ(χ) in |Q|, RQ
Σ (χ) = Mχ ,
• for every substitution ψ : χ1 → χ2 ,
−1
RQ
Σ (ψ) = (iχ2 ; ModΣ (ψ) ; iχ1 )(1Mχ2 ).
Moreover, for every Σ-substitution ψ,
ModΣ (ψ) is uniquely determined by RQ
Σ (ψ).
Herbrand
Robinson
Lloyd
1929
1965
1984
2004 2014
Representable signature extensions
Proposition. Every morphism of signatures
ϕ : Σ → Σ 0 gives rise to a functor
Ψϕ : SubstΣ → SubstΣ 0 defined as follows:
/ I(Σ 0 )
ϕ
I(Σ)
I(χϕ
1 )
I(χ1 )
I(χϕ
2 )
/ I(Σ 0 (χϕ ))
1
I(χ2 )
I(Σ(χ1 ))
ψ
" I(ϕχ1 )
−1
ϕ
RQ
RQ
Σ (ψ)
Σ0
I(Σ(χ2 ))
I(ϕχ2 )
Herbrand
Robinson
Lloyd
1929
1965
1984
" / I(Σ 0 (χϕ ))
2
2004 2014
Deriving generalized substitution systems
Theorem. Every institution I : Sig → Room
equipped with
• an adequate quantification space Q of
representable signature extensions and
• compatible categories SubstΣ of
Q-representable Σ-substitutions,
determines a generalized substitution
system that has model amalgamation.
Herbrand
Robinson
Lloyd
1929
1965
1984
2004 2014
Herbrand’s theorem revisited
Let hΣ, Γ i be a lp and ∃χ · ρ a query such that
h
Mχ
f
0Σ,Γ
/ 0Σ
• Σ and hΣ, Γ i have initial models 0Σ and 0Σ,Γ ,
!Γ
• Mχ is projective with respect to the unique
homomorphism !Γ : 0Σ → 0Σ,Γ , and
• the sentence ρ is basic.
Mρ
iff
Mρ → M
The following statements are equivalent:
1. Γ Σ ∃χ · ρ.
2. 0Σ,Γ Σ ∃χ · ρ.
3. ∃χ · ρ admits a Γ -solution.
Herbrand
Robinson
Lloyd
1929
1965
1984
2004 2014
Thank you!
Further Reading
J. Herbrand.
Investigations in proof theory.
In: From Frege to Gödel: A Source Book in Mathematical Logic,
HUP, 1967.
J. W. Lloyd.
Foundations of Logic Programming, 2nd Edition.
Springer, 1987.
R. Diaconescu.
Herbrand theorems in arbitrary institutions.
Information Processing Letters, Elsevier, 2004.
I. Ţuţu and J. L. Fiadeiro.
From conventional to institution-independent logic programming.
Journal of Logic and Computation, OUP, in press.