On Proof Compression Mechanisms and Deep Inference

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On Proof Compression Mechanisms and Deep Inference

Anupam Das

University of Bath

LAC-GeoCal, 24/11/2011

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Outline of Talk

1 Motivation

Compression Mechanisms in Proof Systems Proof Complexity

Deep Inference Questions

2 The Setting

Calculus of Structures

Embedding Compression Mechanisms

3 Some Theory Algebra

Measuring Proofs

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Motivation

Anupam Das (University of Bath) Proof Compression Mechanisms LAC-GeoCal, 24/11/2011 3 / 20

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Compression Mechanisms

Proof systems often containcompression mechanisms that allow proofs to have smaller representations

, e.g.

I Dag - Reusing an already proved formula/subproof

I Cut - Invoking a lemma

I Substitution - Substituting variables in a proof

I Extension - Naming a theorem or proof Observation

Given a formula A∧B, compression mechanisms candetect similarities between the conjunctsA andB.

E.g. from a proof of A,

I a dag-like system can conclude A^∧A

I a system with substitution can concludeA∧Aσ, whereσ is some substitution of variables.

PROBLEM: How can we capture this property of ‘detecting similarities’ formally?

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Compression Mechanisms

Proof systems often containcompression mechanisms that allow proofs to have smaller representations, e.g.

I Extension - Naming a theorem or proof

Observation

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Compression Mechanisms

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Compression Mechanisms

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Compression Mechanisms

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Proof Complexity

I Types of question: ‘How short are proofs of formulae in a system, up to a polynomial factor?’

I An important relation: polynomial simulation. Definition

A systemS polynomially simulates a systemT if for every proof inS there is a proof in T of the same formula whose size is bounded above by a polynomial in the size of the original proof.

I A class of studied questions: When does a tree-like system polynomially simulate its dag-like counterpart? E.g.

I Tree-like Frege polynomially simulates dag-like Frege. [Krajicek]

I Tree-like cut-free Gentzen does not polynomially simulate dag-like cut-free Gentzen. [Statman]

I More generally the effects of general proof compression mechanisms on proof complexity are studied.

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Proof Complexity

I An important relation: polynomial simulation. Definition

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Proof Complexity

I An important relation: polynomial simulation.

Definition

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Proof Complexity

Definition

I A class of studied questions: When does a tree-like system polynomially simulate its dag-like counterpart?

E.g.

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Proof Complexity

Definition

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Proof Complexity

Definition

I Tree-like cut-free Gentzen does not polynomially simulate dag-like

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Deep Inference

1 Inference rules operatearbitrarily deep within formulae. A^∨(B^∧C)

−−−−−−−−−−−−−−

A^∨(B^∧C⁰)

2 Derivations are top-down symmetric. A`B ⇒B¯ `A.¯

The basic deep inference system, without compression mechanisms, is called KS. Deep inference systems have their own versions of cut and dag:

A^∧A¯

i↑ −−−−−−

⊥

c↑ −−−−−−A A^∧A

but these can be applied anywhere inside a formula. They can be considered generalisations of the usual formulations of cut and dag. Question

Does KS polynomially simulate KS∪ {c↑}? Conjecture: No. [Straßburger]

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Deep Inference

−−−−−−−−−−−−−−

A^∨(B^∧C⁰)

A^∧A¯

i↑ −−−−−−

⊥

c↑ −−−−−−A A^∧A

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Deep Inference

−−−−−−−−−−−−−−

A^∨(B^∧C⁰)

A^∧A¯

i↑ −−−−−−

⊥

c↑ −−−−−−A A^∧A

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Deep Inference

−−−−−−−−−−−−−−

A^∨(B^∧C⁰)

The basic deep inference system, without compression mechanisms, is calledKS.

Deep inference systems have their own versions of cut and dag: A^∧A¯

i↑ −−−−−−

⊥

c↑ −−−−−−A A^∧A

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Deep Inference

−−−−−−−−−−−−−−

A^∨(B^∧C⁰)

The basic deep inference system, without compression mechanisms, is calledKS. Deep inference systems have their own versions of cut and dag:

A^∧A¯

i↑ −−−−−−

⊥

A

c↑ −−−−−−

A^∧A

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Deep Inference

−−−−−−−−−−−−−−

A^∨(B^∧C⁰)

A^∧A¯

i↑ −−−−−−

⊥

A

c↑ −−−−−−

A^∧A

but these can be applied anywhere inside a formula. They can be considered generalisations of the usual formulations of cut and dag.

Question

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Deep Inference

−−−−−−−−−−−−−−

A^∨(B^∧C⁰)

A^∧A¯

i↑ −−−−−−

⊥

A

c↑ −−−−−−

A^∧A

Question

Does KS polynomially simulate KS∪ {c↑}?

Conjecture: No. [Straßburger]

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Deep Inference

−−−−−−−−−−−−−−

A^∨(B^∧C⁰)

A^∧A¯

i↑ −−−−−−

⊥

A

c↑ −−−−−−

A^∧A

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Questions

1 How can we capture compression mechanisms in a way that is independent of any particular proof system?

2 Can we give a definition of ‘compression mechanism’, or what it means to be an uncompressed system.

3 Can we give conditions for when a tree-like system is able to polynomially simulate its dag-like counterpart?

4 More generally, can we give conditions for when any system is able to polynomially simulate that system augmented with some compression mechanism?

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Questions

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Questions

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Questions

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The Setting

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Calculus of Structures

I We will consider those proof systems that can be embedded in the deep inference formalism Calculus of Structures.

I CoS proofs are just sequences of formulae chained together by inference rules from a proof system.

I There is no meta/object level distinction.

I We can embed a wide variety of studied proof systems in this formalism, while preserving proof complexity properties. E.g.

[A^∨C]¯ ^∧[C^∨B]

mp−−−−−−−−−−−−−−−−−−−

A^∨B

A^∧B

mix−−−−−−

A^∨B

A

shr−−−−−−

A^∧A modus ponens mix dag

>

id−−−−−−

A^∨A¯

A

wk−−−−−−

A^∨B

A^∨A

con−−−−−−

A

[A^∨B]^∧[C^∨D]

∧ −−−−−−−−−−−−−−−−−−−

A^∨(B^∧C)^∨D identity weakening contraction

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Calculus of Structures

[A^∨C]¯ ^∧[C^∨B]

mp−−−−−−−−−−−−−−−−−−−

A^∨B

A^∧B

mix−−−−−−

A^∨B

A

shr−−−−−−

>

id−−−−−−

A^∨A¯

A

wk−−−−−−

A^∨B

A^∨A

con−−−−−−

A

[A^∨B]^∧[C^∨D]

∧ −−−−−−−−−−−−−−−−−−−

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Calculus of Structures

I We can embed a wide variety of studied proof systems in this formalism, while preserving proof complexity properties.

E.g. [A^∨C]¯ ^∧[C^∨B]

mp−−−−−−−−−−−−−−−−−−−

A^∨B

A^∧B

mix−−−−−−

A^∨B

A

shr−−−−−−

>

id−−−−−−

A^∨A¯

A

wk−−−−−−

A^∨B

A^∨A

con−−−−−−

A

[A^∨B]^∧[C^∨D]

∧ −−−−−−−−−−−−−−−−−−−

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Calculus of Structures

[A^∨C]¯ ^∧[C^∨B]

mp−−−−−−−−−−−−−−−−−−−

A^∨B

A^∧B

mix−−−−−−

A^∨B

A

shr−−−−−−

>

id−−−−−−

A^∨A¯

A

wk−−−−−−

A^∨B

A^∨A

con−−−−−−

A

[A^∨B]^∧[C^∨D]

∧ −−−−−−−−−−−−−−−−−−−

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Embedding Compression Mechanisms in CoS

We can define compression mechanisms as follows: A^∨(B^∧B)¯

cut−−−−−−−−−−−−−

A

dag−−−−−−

A^∧A

Definition

We say that a systemS “distributes over∧” if, for?∈ {∧,∨}, there is a polynomial-size derivation (A ? B)^∧(A ? C)`A ?(B^∧C).

Theorem

Whenever a system distributes over ∧and has basic equations, cut is equivalent tomp anddag is equivalent toshr. If the system is a deep inference system then cutis equivalent to i↑ anddag is equivalent to c↑.

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Embedding Compression Mechanisms in CoS

We can define compression mechanisms as follows:

A^∨(B^∧B)¯

cut−−−−−−−−−−−−−

A

dag−−−−−−

A^∧A

Definition

Theorem

Whenever a system distributes over ∧and has basic equations, cut is equivalent tomp anddag is equivalent toshr. If the system is a deep inference system then cutis equivalent to i↑ anddag is equivalent to c↑.

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Embedding Compression Mechanisms in CoS

We can define compression mechanisms as follows:

A^∨(B^∧B)¯

cut−−−−−−−−−−−−−

A

dag−−−−−−

A^∧A

Definition

Theorem

Whenever a system distributes over ∧and has basic equations, cut is

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Some Theory

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Rings and Modules

I Asemiring Ris a tuple(R,+,×,0,1)such that:

1 (R,+,0)is a commutative monoid.

2 (R,×,1)is a commmutative monoid.

3 ×distributes over +.

I Amodule V over a semiringR= (R,+,×,0,1)is a tuple (V,+_V,0_V,•) such that:

1 (V,+_V,0_V)is a commutative monoid.

2 •:R×V →V. •left-distributes over+_V and right-distributes over+.

3 ×is associative over•. 0•v= 0_V and1•v=v.

Observation

(TAUT,∧,>,·) forms a module over N, wheren·A≡

n

z }| {

A^∧· · ·^∧A.

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Rings and Modules

I Amodule V over a semiring R= (R,+,×,0,1)is a tuple (V,+_V,0_V,•) such that:

Observation

(TAUT,∧,>,·) forms a module over N, wheren·A≡

n

z }| {

A^∧· · ·^∧A.

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Rings and Modules

I Amodule V over a semiring R= (R,+,×,0,1)is a tuple (V,+_V,0_V,•) such that:

Observation

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Measures and Norms

For a proof system S:

I Ameasuring is a functionM :PRFS→R such thatM ∼ | · | andM is monotone with respect to subproofs.

I Ameasure is a compositionM ◦P whereM is a measuring and P is an assignment of tautologies to proofs that are minimal, up to a polynomial, and closed under subproofs.

I Anorm on a moduleV over a semiring R⊆Ris function µ:V →R such that:

1 µ(a•v) =|a|µ(v). (Homogeneity)

2 µ(v+w)≤µ(v) +µ(w). (Triangle inequality, orM)

3 µ(v) = 0 impliesv= 0_V.

If, in addition we have µ(v+w) =µ(v) +µ(w) then we say thatµ is an L1 norm.

We will be interested in measures that form norms over (TAUT,∧,>,·).

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Measures and Norms

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Measures and Norms

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Measures and Norms

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Measures and Norms

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Measures and Norms

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Main Result

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The Characterisation

Theorem

Let S be a system with an L1 norm. The following are equivalent:

1 dag-S has a norm.

2 dag-S has an L1 norm.

3 dag-S is polynomially equivalent toS.

Proof Idea.

I (2) implies (1): Trivial.

I (1) implies (3): Ifµ=M◦P is a norm for S then it is also a measure for dag-S, and so a norm for dag-S.

I (3) implies (2): Transform proofs by just expanding out dag-steps. Evaluating the size of the resulting proof in a clever way gives only a polynomial blow-up; this relies crucially on homogeneity of the norm.

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The Characterisation

Theorem

Let S be a system with an L1 norm. The following are equivalent:

1 dag-S has a norm.

2 dag-S has an L1 norm.

3 dag-S is polynomially equivalent toS.

Proof Idea.

I (2) implies (1): Trivial.

I (1) implies (3): Ifµ=M◦P is a norm for S then it is also a measure for dag-S, and so a norm for dag-S.

I (3) implies (2): Transform proofs by just expanding out dag-steps.

Evaluating the size of the resulting proof in a clever way gives only a polynomial blow-up; this relies crucially on homogeneity of the norm.

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Further Results

Proposition

Cut-free tree-like Gentzen and deep inference systems have L1 norms.

Corollary

Dag-like cut-free Gentzen has no L1 norm. A cut-free deep inference systems with c↑ has an L1 norm just if it is polynomially simulated by the system without c↑.

Proposition

Tree-like Frege systems have an L1 norm just if dag-like Frege systems do. Conjecture

Neither tree-like nor dag-like Frege systems have an L1 norm.

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Further Results

Proposition

Corollary

Proposition

Tree-like Frege systems have an L1 norm just if dag-like Frege systems do.

Conjecture

Neither tree-like nor dag-like Frege systems have an L1 norm.

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Further Results

Proposition

Corollary

Proposition

Tree-like Frege systems have an L1 norm just if dag-like Frege systems do.

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Conclusions

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Summary

I We can propose a definition of compression mechanism, or rather of being an uncompressed system:

Definition

A system is uncompressed just if it has an L1 norm.

Under this definition, deep inference would not be regarded as a compression mechanism, as without dag, cut and substitution the basic system still has an L1 norm.

I We formulated compression mechanisms dag and cut in a way that is independent of any particular system, but nonetheless does the same job from the point of view of proof complexity.

I We characterised the situations when an uncompressed system p-simulates its dag-like counterpart for a wide variety of systems.

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Summary

Definition

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Summary

Definition

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Summary

Definition

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Summary

Definition

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Further Directions

I Can we extend the main result to account for other compression mechanisms?

Conjecture

A system with an L1 norm cannot polynomially simulate any extension of it that does not have an L1 norm.

I A specific application of this result might be a robustnesstheorem for cut-free deep inference systems.

Conjecture (Idea)

All regular analytic deep inference systems with L1 norms are polynomially equivalent.

for some definitions of ‘regular’ and ‘analytic’.

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Further Directions

Conjecture

Conjecture (Idea)

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Further Directions

Conjecture

Conjecture (Idea)

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Further Directions

Conjecture

Conjecture (Idea)

All regular analytic deep inference systems with L1 norms are polynomially

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Further Directions

Conjecture

Conjecture (Idea)

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On Proof Compression Mechanisms and Deep Inference