Some progress on finding short proofs quickly

Some progress on finding short proofs quickly

Anupam Das

University of Bath

Bath, April 21st 2011

This talk is available athttp://people.bath.ac.uk/ad402/FindingShortProofs.pdf

Outline

I Propositional proof systems

I Proof Size vs. Proof Search

I Gentzen vs. Deep inference systems

I A result for deep inference systems

Propositional proof systems

I A (sound and complete) proof system is a polynomial-time function whose range is the set of propositional tautologies.

I An object in the domain of such a function is called a “proof”.

I Examples:

Propositional proof systems

I A (sound and complete) proof system is a polynomial-time function whose range is the set of propositional tautologies.

I An object in the domain of such a function is called a “proof”.

I Examples:

Propositional proof systems

I A (sound and complete) proof system is a polynomial-time function whose range is the set of propositional tautologies.

I An object in the domain of such a function is called a “proof”.

I Examples:

Proof size

QUESTION: What is the relationship between the size of a tautology and the size of its proof?

I How small can we make proofs?

I Can we find a proof system where all proofs have size polynomial in the size of the tautologies they prove? Such a proof system is called super.

Theorem (Cook, Reckhow)

A super proof system exists iff N P =co-N P.

So the existence of a super proof system seems unlikely.

Proof size

QUESTION: What is the relationship between the size of a tautology and the size of its proof?

I How small can we make proofs?

I Can we find a proof system where all proofs have size polynomial in the size of the tautologies they prove? Such a proof system is called super.

Theorem (Cook, Reckhow)

A super proof system exists iff N P =co-N P.

So the existence of a super proof system seems unlikely.

Proof size

QUESTION: What is the relationship between the size of a tautology and the size of its proof?

I How small can we make proofs?

I Can we find a proof system where all proofs have size polynomial in the size of the tautologies they prove? Such a proof system is called super.

Theorem (Cook, Reckhow)

A super proof system exists iff N P =co-N P.

So the existence of a super proof system seems unlikely.

Proof size

QUESTION: What is the relationship between the size of a tautology and the size of its proof?

I How small can we make proofs?

I Can we find a proof system where all proofs have size polynomial in the size of the tautologies they prove? Such a proof system is called super.

Theorem (Cook, Reckhow)

A super proof system exists iff N P =co-N P.

Proof search

I PROBLEM: Given a tautology, how do we find a proof of it?

I Proof systems tend to be “infinitary”, and there is often no terminating algorithm to find a proof of a tautology.

I Example. Frege: A A→B

−−−−−−−−−−−−−

B . Acould be any formula, from which there are infinitely many to choose.

I Rules like modus ponens are instances of cut, and pose a big problem for proof search.

Proof search

I PROBLEM: Given a tautology, how do we find a proof of it?

I Proof systems tend to be “infinitary”, and there is often no terminating algorithm to find a proof of a tautology.

I Example. Frege: A A→B

−−−−−−−−−−−−−

B . Acould be any formula, from which there are infinitely many to choose.

I Rules like modus ponens are instances of cut, and pose a big problem for proof search.

Proof search

I PROBLEM: Given a tautology, how do we find a proof of it?

I Proof systems tend to be “infinitary”, and there is often no terminating algorithm to find a proof of a tautology.

I Example. Frege: A A→B

−−−−−−−−−−−−−

B . Acould be any formula, from which there are infinitely many to choose.

I Rules like modus ponens are instances of cut, and pose a big problem for proof search.

Proof search

SOLUTION! Gentzen’s Hauptsatz shows that we can eliminate cuts.

Results:

I Finitary proof search.

I Consistency of proof system.

I Subformula property.

However cut-free Gentzen systems can only at best exponentiallysimulate Gentzen systems with cut.

Proof search

SOLUTION! Gentzen’s Hauptsatz shows that we can eliminate cuts.

Results:

I Finitary proof search.

I Consistency of proof system.

I Subformula property.

However cut-free Gentzen systems can only at best exponentiallysimulate Gentzen systems with cut.

Proof search

SOLUTION! Gentzen’s Hauptsatz shows that we can eliminate cuts.

Results:

I Finitary proof search.

I Consistency of proof system.

I Subformula property.

However cut-free Gentzen systems can only at best exponentiallysimulate Gentzen systems with cut.

Statman tautologies

S1 ≡(c1∧d1)^∨[¯c1∨d¯1] ,

S₂ ≡(c₂^∧d₂)^∨[(([¯c₂^∨d¯₂]^∧c₁)^∧([¯c₂^∨d¯₂]^∧d₁))^∨[¯c₁^∨d¯₁]] , S₃ ≡(c₃^∧d₃)^∨(([¯c₃^∨d¯₃]^∧c₂)^∧([¯c₃^∨d¯₃]^∧d₂))^∨

((([¯c₃^∨d¯₃]^∧[¯c₂^∨d¯₂])^∧c₁)^∧(([¯c₃^∨d¯₃]^∧[¯c₂^∨d¯₂])^∧d₁))^∨ [¯c1∨d¯1] .

. . .

Cut-free sequent calculus proofs are exponentialin the size of the tautology.

Statman tautologies

S1 ≡(c1∧d1)^∨[¯c1∨d¯1] ,

S₂ ≡(c₂^∧d₂)^∨[(([¯c₂^∨d¯₂]^∧c₁)^∧([¯c₂^∨d¯₂]^∧d₁))^∨[¯c₁^∨d¯₁]] , S₃ ≡(c₃^∧d₃)^∨(([¯c₃^∨d¯₃]^∧c₂)^∧([¯c₃^∨d¯₃]^∧d₂))^∨

((([¯c₃^∨d¯₃]^∧[¯c₂^∨d¯₂])^∧c₁)^∧(([¯c₃^∨d¯₃]^∧[¯c₂^∨d¯₂])^∧d₁))^∨ [¯c1∨d¯1] .

. . .

Cut-free sequent calculus proofs are exponentialin the size of the

Question

How well can we optimise a system for both proof search and proof size?

Question

How well can we optimise a system for both proof search and proof size?

Deep inference

1 Inference rules operatearbitrarily deep within formulae. ξ (A

−−

B )

2 The distinction between object and meta level is abolished. Proofs are top-down symmetric.

Properties:

I Systems with cut are as efficient as their “shallow” counterparts.

I Exponential speedup for cut-free systems.

I There is a quasipolynomial cut-elimination procedure.

I Only has a restricted version of the subformula property.

Conjecture

All cuts can be eliminated in polynomial-time.

Deep inference

1 Inference rules operatearbitrarily deep within formulae. ξ (A

−−

B )

2 The distinction between object and meta level is abolished. Proofs are top-down symmetric.

Properties:

I Systems with cut are as efficient as their “shallow” counterparts.

I Exponential speedup for cut-free systems.

I There is a quasipolynomial cut-elimination procedure.

I Only has a restricted version of the subformula property.

Conjecture

All cuts can be eliminated in polynomial-time.

Deep inference

1 Inference rules operatearbitrarily deep within formulae. ξ (A

−−

B )

2 The distinction between object and meta level is abolished. Proofs are top-down symmetric.

Properties:

I Systems with cut are as efficient as their “shallow” counterparts.

I Exponential speedup for cut-free systems.

I There is a quasipolynomial cut-elimination procedure.

I Only has a restricted version of the subformula property.

Conjecture

All cuts can be eliminated in polynomial-time.

Deep inference

1 Inference rules operatearbitrarily deep within formulae. ξ (A

−−

B )

2 The distinction between object and meta level is abolished. Proofs are top-down symmetric.

Properties:

I Systems with cut are as efficient as their “shallow” counterparts.

I Exponential speedup for cut-free systems.

I There is a quasipolynomial cut-elimination procedure.

I Only has a restricted version of the subformula property.

Deep inference

I If conjecture is true then we have “finitary proof-search” with as short proofs as Frege/Gentzen with cut.

I While proof-search is now finitely branching, the “branching degree” is still unbounded, since we can now operate on any connective. Can we do better?

I QUESTION: How deep do we need to go?

Deep inference

I If conjecture is true then we have “finitary proof-search” with as short proofs as Frege/Gentzen with cut.

I While proof-search is now finitely branching, the “branching degree”

is still unbounded, since we can now operate on any connective. Can we do better?

I QUESTION: How deep do we need to go?

Deep inference

I If conjecture is true then we have “finitary proof-search” with as short proofs as Frege/Gentzen with cut.

I While proof-search is now finitely branching, the “branching degree”

is still unbounded, since we can now operate on any connective. Can we do better?

I QUESTION: How deep do we need to go?

Surprise!

Symmetry alone is enough! For propositional logic, the same speedup in proof size can be attained just from top-down symmetry in the system.

ξ{D}

ρ(2)−−−−−−−

ξ{D⁰} ρ⁰ :

ξ{D}

(A^∧[B^∨(C^∧D)])

−−−−−−−−−−−−−−−−−−−−−−−−−−

(A^∧[B^∨C]^∧[B^∨D])

ρ(1)−−−−−−−−−−−−−−−−−−−−−−−−−−−

(A^∧[B^∨C]^∧[B^∨D⁰])

−−−−−−−−−−−−−−−−−−−−−−−−−−−

(A^∧[B^∨(C^∧D⁰)])

ξ{D⁰}

Surprise!

Symmetry alone is enough! For propositional logic, the same speedup in proof size can be attained just from top-down symmetry in the system.

ξ{D}

ρ(2)−−−−−−−

ξ{D⁰} ρ⁰ :

ξ{D}

(A^∧[B^∨(C^∧D)])

−−−−−−−−−−−−−−−−−−−−−−−−−−

(A^∧[B^∨C]^∧[B^∨D])

ρ(1)−−−−−−−−−−−−−−−−−−−−−−−−−−−

(A^∧[B^∨C]^∧[B^∨D⁰])

−−−−−−−−−−−−−−−−−−−−−−−−−−−

(A^∧[B^∨(C^∧D⁰)])

Technicalities

I What about the equations?

Turn them into inference rules and bound their depth too.

I Can we derive thedistributivity laws? Yes, in the presence of cocontraction.

I Can we still admit structural rules? Yes, and with no (significant) effect on proof complexity.

Theorem

Cut-free deep inference systems containing cocontraction can be restricted to allow just shallow inferences, with only polynomial increase in proof size.

Technicalities

I What about the equations? Turn them into inference rules and bound their depth too.

I Can we derive thedistributivity laws? Yes, in the presence of cocontraction.

I Can we still admit structural rules? Yes, and with no (significant) effect on proof complexity.

Theorem

Cut-free deep inference systems containing cocontraction can be restricted to allow just shallow inferences, with only polynomial increase in proof size.

Technicalities

I What about the equations? Turn them into inference rules and bound their depth too.

I Can we derive thedistributivity laws?

Yes, in the presence of cocontraction.

I Can we still admit structural rules? Yes, and with no (significant) effect on proof complexity.

Theorem

Cut-free deep inference systems containing cocontraction can be restricted to allow just shallow inferences, with only polynomial increase in proof size.

Technicalities

I What about the equations? Turn them into inference rules and bound their depth too.

I Can we derive thedistributivity laws? Yes, in the presence of cocontraction.

I Can we still admit structural rules? Yes, and with no (significant) effect on proof complexity.

Theorem

Cut-free deep inference systems containing cocontraction can be restricted to allow just shallow inferences, with only polynomial increase in proof size.

Technicalities

I What about the equations? Turn them into inference rules and bound their depth too.

I Can we derive thedistributivity laws? Yes, in the presence of cocontraction.

I Can we still admit structural rules?

Yes, and with no (significant) effect on proof complexity.

Theorem

Cut-free deep inference systems containing cocontraction can be restricted to allow just shallow inferences, with only polynomial increase in proof size.

Technicalities

I What about the equations? Turn them into inference rules and bound their depth too.

I Can we derive thedistributivity laws? Yes, in the presence of cocontraction.

I Can we still admit structural rules? Yes, and with no (significant) effect on proof complexity.

Theorem

Cut-free deep inference systems containing cocontraction can be restricted to allow just shallow inferences, with only polynomial increase in proof size.

Technicalities

I What about the equations? Turn them into inference rules and bound their depth too.

I Can we derive thedistributivity laws? Yes, in the presence of cocontraction.

I Can we still admit structural rules? Yes, and with no (significant) effect on proof complexity.

Theorem

Cut-free deep inference systems containing cocontraction can be restricted to allow just shallow inferences, with only polynomial increase in proof size.

So this means...

I Equates to augmenting a cut-free sequent system with the following two rules.

Γ, A∨B

−−−−−−−−−−

Γ, A, B

Γ, A∧B

−−−−−−−−−−−−

Γ, A Γ, B

I No longer a sequent calculus; it breaks the asymmetry induced by the tree-like structure of sequent calculus proofs.

I Less restricted version of subformula property than that for regular deep inference.

I Worse for proof search than Gentzen systems, but much better than regular deep inference while still giving access to short proofs.

So this means...

I Equates to augmenting a cut-free sequent system with the following two rules.

Γ, A∨B

−−−−−−−−−−

Γ, A, B

Γ, A∧B

−−−−−−−−−−−−

Γ, A Γ, B

I No longer a sequent calculus; it breaks the asymmetry induced by the tree-like structure of sequent calculus proofs.

I Less restricted version of subformula property than that for regular deep inference.

I Worse for proof search than Gentzen systems, but much better than regular deep inference while still giving access to short proofs.

Example

Thanks!