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Revisiting the Institutional Approach to Herbrands Theorem Ionu uu 1,2 Jos 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

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SLIDE 1

Revisiting the Institutional Approach to Herbrand’s Theorem

Ionuţ Ţuţu1,2 José Luiz Fiadeiro1

1Department of Computer Science, Royal Holloway University of London 2Simion Stoilow Institute of Mathematics of the Romanian Academy

6th Conference on Algebra and Coalgebra in Computer Science Nijmegen, 2015

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SLIDE 2

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

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SLIDE 3

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

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SLIDE 4

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

  • difficulties in following the proof and

errors reported by Bernays and Gödel

1929 Herbrand

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SLIDE 5

1963

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

1929 Herbrand

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SLIDE 6

1929 Herbrand 1965

The resolution inference rule

  • introduced by Robinson
  • well-suited for automation

∃X· Q ∧ g ∀Y · c ← H ∃X′· θ(Q) ∧ θ(H) θ

  • led to the development of logic

programming – prolog (Kowalski & Colmerauer)

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SLIDE 7

1929 Herbrand 1973

The resolution inference rule

  • introduced by Robinson
  • well-suited for automation

∃X· Q ∧ g ∀Y · c ← H ∃X′· θ(Q) ∧ θ(H) θ

  • led to the development of logic

programming – prolog (Kowalski & Colmerauer)

1965 Robinson

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SLIDE 8

1929 Herbrand 1965 Robinson 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

  • f all the models that satisfy Γ.
  • 1. Γ Σ ∃X· ρ
  • 2. 0Σ,Γ Σ ∃X· ρ
  • 3. There exists ψ: X → Y such that

Γ Σ ∀Y · ψ(ρ). ∃{x}· “x is a number” ∧ “x Prolog programmers

can change a lightbulb”

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SLIDE 9

1929 Herbrand 1965 Robinson 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

  • f all the models that satisfy Γ.
  • 1. Γ Σ ∃X· ρ
  • 2. 0Σ,Γ Σ ∃X· ρ
  • 3. There exists ψ: X → Y such that

Γ Σ ∀Y · ψ(ρ). ∃{x}· “x is a number” ∧ “x Prolog programmers

can change a lightbulb”

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SLIDE 10

1929 Herbrand 1965 Robinson 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

  • f all the models that satisfy Γ.
  • 1. Γ Σ ∃X· ρ
  • 2. 0Σ,Γ Σ ∃X· ρ
  • 3. There exists ψ: X → Y such that

Γ Σ ∀Y · ψ(ρ). ∃{x}· “x is a number” ∧ “x Prolog programmers

can change a lightbulb”

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SLIDE 11

1929 Herbrand 1965 Robinson 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

  • f all the models that satisfy Γ.
  • 1. Γ Σ ∃X· ρ
  • 2. 0Σ,Γ Σ ∃X· ρ
  • 3. There exists ψ: X → Y such that

Γ Σ ∀Y · ψ(ρ). ∃{x}· “x is a number” ∧ “x Prolog programmers

can change a lightbulb”

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SLIDE 12

1929 Herbrand 1965 Robinson 1984

A multitude of variants

  • relational first-order logic
  • many-sorted equational logic
  • higher-order logic
  • hidden algebra
  • institution-independent
  • service-oriented
  • abstract logic programming

∃{x, y}· sorted(2, 3, x, y, 5)

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SLIDE 13

1929 Herbrand 1965 Robinson 1984 Lloyd

A multitude of variants

  • relational first-order logic
  • many-sorted equational logic
  • higher-order logic
  • hidden algebra
  • institution-independent
  • service-oriented
  • abstract logic programming

∃{x: Num}·sorted(2, 3, x) = T

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SLIDE 14

1929 Herbrand 1965 Robinson 1984 Lloyd

A multitude of variants

  • relational first-order logic
  • many-sorted equational logic
  • higher-order logic
  • hidden algebra
  • institution-independent
  • service-oriented
  • abstract logic programming

∃{s: List → B}·s [2, 3, 5] = T

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SLIDE 15

1929 Herbrand 1965 Robinson 1984 Lloyd 2002

A multitude of variants

  • relational first-order logic
  • many-sorted equational logic
  • higher-order logic
  • hidden algebra
  • institution-independent
  • service-oriented
  • abstract logic programming

∃{s: Stream}· s ∼ tail(s)

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SLIDE 16

1929 Herbrand 1965 Robinson 1984 Lloyd 2004

A multitude of variants

  • relational first-order logic
  • many-sorted equational logic
  • higher-order logic
  • hidden algebra
  • institution-independent
  • service-oriented
  • abstract logic programming

Sig, Sen, Mod, Sen(Σ) Σ

✶

Sen

  • ✌

Mod

  • Mod(Σ)

Σ

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SLIDE 17

1929 Herbrand 1965 Robinson 1984 Lloyd 2004

A multitude of variants

  • relational first-order logic
  • many-sorted equational logic
  • higher-order logic
  • hidden algebra
  • institution-independent
  • service-oriented
  • abstract logic programming

Sig, Sen, Mod,

subject to a satisfaction condition: for every ϕ: Σ → Ω, M ∈ |Mod(Ω)|, ρ ∈ Sen(Σ) M↾ϕ Σ ρ iff M Ω ϕ(ρ)

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SLIDE 18

1929 Herbrand 1965 Robinson 1984 Lloyd 2004 2013

A multitude of variants

  • relational first-order logic
  • many-sorted equational logic
  • higher-order logic
  • hidden algebra
  • institution-independent
  • service-oriented
  • abstract logic programming

∃o· {r1, r2}

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SLIDE 19

1929 Herbrand 1965 Robinson 1984 Lloyd 2004 2013

A multitude of variants

  • relational first-order logic
  • many-sorted equational logic
  • higher-order logic
  • hidden algebra
  • institution-independent
  • service-oriented
  • abstract logic programming

∃{x, y}

χ: F,P֒ →F∪{x,y},P

· sorted(2, 3, x, y, 5)

∃χ· sorted(2, 3, x, y, 5)

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SLIDE 20

1929 Herbrand 1965 Robinson 1984 Lloyd 2004 2014

A multitude of variants

  • relational first-order logic
  • many-sorted equational logic
  • higher-order logic
  • hidden algebra
  • institution-independent
  • service-oriented
  • abstract logic programming

SenΣ(X)

  • Σ

ϕ

  • X

✴

  • ✖
  • ❴
  • ModΣ(X)
  • SenΩ(Xϕ)

Ω Xϕ ✴

✖

  • ModΩ(Xϕ)
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SLIDE 21

1929 Herbrand 1965 Robinson 1984 Lloyd 2004 2014

A multitude of variants

  • relational first-order logic
  • many-sorted equational logic
  • higher-order logic
  • hidden algebra
  • institution-independent
  • service-oriented
  • abstract logic programming

SenΣ(X) Σ X

✴

  • ✖
  • ModΣ(X)
  • ∃X· ρ
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SLIDE 22

1929 Herbrand 1965 Robinson 1984 Lloyd 2004 2014

Institutions as functors

  • each institution I = Sig, Sen, Mod,

can be identified with a functor I: Sig → Room where I(Σ) = Sen(Σ), Mod(Σ), Σ

  • similarly, substitution systems can be

defined as functors S: Subst → G / Room Rooms and corridors S,

α

  • M,

S′, M′,

β

  • ′
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SLIDE 23

1929 Herbrand 1965 Robinson 1984 Lloyd 2004 2014

From institutions to substitution systems

  • let Q be a class of signature morphisms of

an institution I: Sig → Room For every I-signature Σ we obtain a substi- tution system SIQ

Σ : SubstQ Σ → 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

SenΣ(ψ), ModΣ(ψ): I(Σ(χ1)) → I(Σ(χ2)) I(Σ)

I(χ1)

  • I(χ2)
  • I(Σ(χ1))

SenΣ(ψ),ModΣ(ψ)

  • I(Σ(χ2))
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SLIDE 24

1929 Herbrand 1965 Robinson 1984 Lloyd 2004 2014

Quantification spaces

  • 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.

Σ

ϕ

  • χ
  • Σ′

χ′

  • Ω

φ

Ω′

→

ιQ,χ

Σ

ϕ Σ′

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SLIDE 25

1929 Herbrand 1965 Robinson 1984 Lloyd 2004 2014

Quantification spaces

  • 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.

Σ

ϕ

  • χ
  • Σ′

χϕ

  • Σ(χ) ϕχ Σ′(χϕ)

→

ιQ,χ

Σ

ϕ Σ′

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SLIDE 26

1929 Herbrand 1965 Robinson 1984 Lloyd 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) Mod(Σ(χ))

_↾χ

  • iχ

Mχ / Mod(Σ)

forgetful

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SLIDE 27

1929 Herbrand 1965 Robinson 1984 Lloyd 2004 2014

Representable signature extensions

  • Proposition. The representation of

signature extensions generalizes to a functor RQ

Σ : SubstQ Σ → Mod(Σ), where

  • for every χ: Σ → Σ(χ) in |Q|, RQ

Σ(χ) = Mχ,

  • for every substitution ψ: χ1 → χ2,

RQ

Σ(ψ) = (i−1 χ2 ; ModΣ(ψ) ; iχ1)(1Mχ2).

Moreover, for every Σ-substitution ψ, ModΣ(ψ) is uniquely determined by RQ

Σ(ψ).

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1929 Herbrand 1965 Robinson 1984 Lloyd 2004 2014

Representable signature extensions

  • Proposition. Every morphism of signatures

ϕ: Σ → Σ′ gives rise to a functor Ψϕ : SubstΣ → SubstΣ′ defined as follows: I(Σ)

ϕ

  • I(χ1)
  • I(χ2)
  • I(Σ′)

I(χϕ

1 )

  • I(χϕ

2 )

  • I(Σ(χ1))

I(ϕχ1)

  • ψ
  • I(Σ′(χϕ

1 ))

  • RQ

Σ′

−1

RQ

Σ(ψ)ϕ

  • I(Σ(χ2))

I(ϕχ2)

I(Σ′(χϕ

2 ))

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SLIDE 29

1929 Herbrand 1965 Robinson 1984 Lloyd 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.

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SLIDE 30

1929 Herbrand 1965 Robinson 1984 Lloyd 2004 2014

Herbrand’s theorem revisited Let Σ, Γ be a lp and ∃χ· ρ a query such that

  • Σ and Σ, Γ have initial models 0Σ and 0Σ,Γ,
  • Mχ is projective with respect to the unique

homomorphism !Γ : 0Σ → 0Σ,Γ, and

  • the sentence ρ is basic.

The following statements are equivalent:

  • 1. Γ Σ ∃χ· ρ.
  • 2. 0Σ,Γ Σ ∃χ· ρ.
  • 3. ∃χ· ρ admits a Γ-solution.

Mχ

f

  • h

0Σ

!Γ

  • 0Σ,Γ

M ρ iff Mρ → M

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SLIDE 31

Thank you!

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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.