Sporadic Propositional Proofs

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Sporadic Propositional Sporadic Propositional Proofs Proofs S S øren Riis øren Riis Queen mary, Queen mary, University of London University of London New Directions in Proof Complexity New Directions in Proof Complexity 11 11 th th of April 2006 at the Newton Instutute of April 2006 at the Newton Instutute Cambridge Cambridge

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Sporadic Propositional Proofs. S øren Riis Queen mary, University of London New Directions in Proof Complexity 11 th of April 2006 at the Newton Instutute Cambridge. General question:. - PowerPoint PPT Presentation

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Page 1: Sporadic Propositional Proofs

Sporadic Propositional Sporadic Propositional ProofsProofs

SSøren Riisøren Riis

Queen mary, Queen mary,

University of LondonUniversity of London

New Directions in Proof Complexity New Directions in Proof Complexity

1111thth of April 2006 at the Newton Instutute Cambridge of April 2006 at the Newton Instutute Cambridge

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General question:General question:

For a given (weak) Propositional Proof System For a given (weak) Propositional Proof System investigate the proof complexity (as a function investigate the proof complexity (as a function of of nn) of uniform sequences ) of uniform sequences [[ηη]]nn of tautologies.of tautologies.

Setup for this: Given a large class Setup for this: Given a large class CC of of propositions from predicate logic. Each propositions from predicate logic. Each proposition proposition ηη ЄЄ C C defines a sequence of defines a sequence of propositions propositions [[ηη]]nn where where [[ηη]]nn is a propositional is a propositional formula expressing that there is no model of formula expressing that there is no model of ηη

of sizeof size nn

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Uniform sequences of Uniform sequences of Tautologies Tautologies

• Often uniform sequences of Often uniform sequences of proposition formulae are usually proposition formulae are usually produced by the Paris-Wilkie produced by the Paris-Wilkie translation. translation.

• For many nice theorems in weak For many nice theorems in weak propositional proof complexity we propositional proof complexity we need a slightly different translation need a slightly different translation (the Riis-Sitharam or RS-translation).(the Riis-Sitharam or RS-translation).

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Difference between the PW Difference between the PW and RS translations in a and RS translations in a nutshellnutshell In the PW-translation In the PW-translation [A][A] of of A:=Ez z+z=n A:=Ez z+z=n ΛΛ φφ becomes equivalent to becomes equivalent to 00 if if nn is odd and is odd and [[φφ]] if if nn is is

even. even. In the RS-translation all function and relation symbols In the RS-translation all function and relation symbols

are treated as uninterpreted symbols. are treated as uninterpreted symbols. (i.e. (i.e. +, ≤, *+, ≤, * etc are treated as general uninterpreted etc are treated as general uninterpreted

relation symbols). This ensure that the translation relation symbols). This ensure that the translation [A][A]nn

(essentially) becomes closed under the action of the (essentially) becomes closed under the action of the symmetric group.symmetric group.

,,

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IdeaIdea

For a given (weak) Propositional Proof For a given (weak) Propositional Proof System System PP classify the proof complexity classify the proof complexity behaviour of behaviour of [[ηη]]nn for a given class for a given class CC of of formulae formulae ηη of predicate logic. of predicate logic.

Let Let ccPP(n)(n) denote the length of the shortest denote the length of the shortest proof of proof of [[ηη]]nn. .

Question: Which complexity functions Question: Which complexity functions ccPP(n)(n) can occur?can occur?

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Tree resolution proofsTree resolution proofs

Theorem [Riis 99]:Theorem [Riis 99]:

The class of complexity functions The class of complexity functions CCtr tr

(n)(n) is included in the set of all is included in the set of all functions that either are bounded by functions that either are bounded by a fixed polynomial or has growth rate a fixed polynomial or has growth rate faster than faster than 22cncn for some for some c>0c>0..

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Tree resolution proofsTree resolution proofs

Conjecture [With Danchev]:Conjecture [With Danchev]: The class of complexity functions The class of complexity functions C(n)C(n) has in three distinct types of has in three distinct types of

behaviour:behaviour: (1) (1) C(n)C(n) with with C(n)<p(n)C(n)<p(n) for some polynomial for some polynomial pp.. (2) (2) C(n)C(n) for which there exists for which there exists 0<c<d0<c<d such that such that 22cn cn <C(n) < 2<C(n) < 2dndn for for

all but finitely many values of all but finitely many values of nn.. (3) (3) C(n)>2C(n)>2cn log(n)cn log(n) for some constant for some constant c>0c>0. .

New type of problem: In case (1) is New type of problem: In case (1) is C(n)C(n) given by a given by a polynomial (for all but finitely many exceptional polynomial (for all but finitely many exceptional values ofvalues of nn)?)?

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Resolution proofs (Hilbert Resolution proofs (Hilbert style)style)

Theorem [Dantchev, Riis 03]:Theorem [Dantchev, Riis 03]:

The class of complexity functions The class of complexity functions CCres res

(n)(n) that arise from “relativised first that arise from “relativised first order formula” is included in the order formula” is included in the class of functions that either are class of functions that either are polynomial bounded or has growth polynomial bounded or has growth rate faster than rate faster than 22cncn for some for some c>0c>0..

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Resolution proofs (Hilbert Resolution proofs (Hilbert style)style)

ConjectureConjecture [With Dantchev] [With Dantchev]

The class of complexity functions The class of complexity functions CCres res

(n)(n) includes all functions that either includes all functions that either are polynomially bounded or has are polynomially bounded or has growth rate faster than growth rate faster than 22cncn for some for some c>0c>0..

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New results New results

Let Let [[ηη]]nn be a uniform sequence of be a uniform sequence of propositional formula. propositional formula.

Let Let SStrue true :={n : [:={n : [ηη]]nn is true} is true}

Let Let SSfalse false :={n : [:={n : [ηη]]nn is false}. is false}.

Here Here nn denote (as usual) the size of denote (as usual) the size of the underlying model.the underlying model.

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BDF proofsBDF proofs

Theorem A:Theorem A:

If If SSfalsefalse and and SStrue true both are infinite there both are infinite there exists an infinite subset exists an infinite subset N’N’ in in SStruetrue

such thatsuch that the sequence the sequence [[ηη]]nn for any for any constant constant dd, and for any infinite , and for any infinite N’’N’’ subset of subset of N’N’ requires exponential size requires exponential size depth depth dd Frege-Proofs. Frege-Proofs.

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BDF proofsBDF proofs

Theorem B:Theorem B:

Assume Assume SSfalse false ={s={s11,s,s22,s,s33,…,s,…,snn,…},…} is not is not very sparse (for each very sparse (for each c >0c >0 we have we have

(s(sk+1k+1-s-skk))cc<s<skk for infinitely many values for infinitely many values of of kk). ).

Then for any Then for any d d ЄЄ {1,2,…} [ {1,2,…} [ηη]]nn requires requires exponential size depth exponential size depth dd Frege-Proofs Frege-Proofs (on any infinite subset (on any infinite subset N’N’ of of SStruetrue))

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Corollaries:Corollaries:

The propositional version of each of the following The propositional version of each of the following statements has no subexponential Bound depth statements has no subexponential Bound depth Frege proofs:Frege proofs:

(1)(1) There is no field structure on There is no field structure on {1,2, …,n}{1,2, …,n} (when (when nn not is a prime not is a prime power).power).

(2)(2) The set The set {1,2,…,n}{1,2,…,n} cannot be organised as a vector space over cannot be organised as a vector space over the field the field FFqq (when n not is a power of (when n not is a power of qq).).

(3)(3) There is no There is no 33-regular graph -regular graph GG on the set on the set {1,2,…,n}{1,2,…,n} (when (when nn is is odd)odd)

(4)(4) There is no proper rectangular grid on the the set There is no proper rectangular grid on the the set {1,2,..,n}{1,2,..,n} (when (when nn is a prime number). is a prime number).

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Conjecture (for Resolution as Conjecture (for Resolution as well as for BDF):well as for BDF):

• The presence of a hard instance The presence of a hard instance where where C(n)C(n) is large, force is large, force C(m)C(m) to be to be large for values of large for values of mm close to close to nn

(Justification: hard instances ought to have similar (Justification: hard instances ought to have similar consequences as impossible instances)consequences as impossible instances)

In general In general C(n)C(n) is smooth without big is smooth without big (local) jumps when (local) jumps when nn increase. increase.

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Nullstellensatz-Proofs (over Nullstellensatz-Proofs (over Field of characteristic q).Field of characteristic q). Theorem [Riis 06]:Theorem [Riis 06]: If PIf Pn n is a uniform generated sequence of polynomial equations. is a uniform generated sequence of polynomial equations.

Then there are Then there are 44 distinct behaviours of the NS degree distinct behaviours of the NS degree complexity complexity D(n)D(n)

(1)(1) D(n)D(n) is bound by a constant is bound by a constant (2)(2) D(n) ≥ l(n)D(n) ≥ l(n) for all sufficiently large values of for all sufficiently large values of nn(3)(3) There exists a constant There exists a constant cc such that the residue class of such that the residue class of nn

modulo modulo qqcc determine if case (1) or case (2) applydetermine if case (1) or case (2) apply(4)(4) The function The function D(n)D(n) has fluctuating complexity that has fluctuating complexity that

fluctuates in a very specific pattern such that (essentially) fluctuates in a very specific pattern such that (essentially) all complexities between constant and all complexities between constant and l(n) l(n) occur on some occur on some infinite setinfinite set..

Here Here log(n) ≤ l(n) ≤ n/2log(n) ≤ l(n) ≤ n/2 is a universal function that can be is a universal function that can be chosen independently of the actual sequence chosen independently of the actual sequence PPnn..

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Example of equations that Example of equations that have fluctuating NS-complexityhave fluctuating NS-complexity• ∑∑j j xxijij + ∑ + ∑jj y yijij -1 =0 for i -1 =0 for i ЄЄ {1,2,…,n} {1,2,…,n}• ∑∑jj x xijij -1=0 for j -1=0 for j ЄЄ {1,2,…,n} {1,2,…,n}• ∑∑ii y yijij -1=0 for j -1=0 for j ЄЄ {1,2,…,n} {1,2,…,n}• xxijijxxikik = 0 for i,j,k = 0 for i,j,k ЄЄ {1,2,…,n}, j ≠ k {1,2,…,n}, j ≠ k• yyijijyyikik=0 for i,j,k =0 for i,j,k ЄЄ {1,2,…,n}, j ≠ k {1,2,…,n}, j ≠ k• xxijijyyikik=0 for i,j,k =0 for i,j,k ЄЄ {1,2,…,n} {1,2,…,n}• yyjijiyykiki=0 for i,j,k =0 for i,j,k ЄЄ {1,2,…,n}, j ≠ k {1,2,…,n}, j ≠ k• yyjijiyykiki=0 for i,j,k =0 for i,j,k ЄЄ {1,2,…,n}, j ≠ k {1,2,…,n}, j ≠ k

This system of equations (that formalise the statement that there is This system of equations (that formalise the statement that there is no bijection form no bijection form nn to to 2n2n) has essentially NS degree complexity ) has essentially NS degree complexity D(n)=qD(n)=qa(q,n)a(q,n) where where a(q,n)a(q,n) denote the power of denote the power of qq in the prime factor in the prime factor decomposition of decomposition of nn..

Thus case (4) in the classification is non-empty.Thus case (4) in the classification is non-empty.

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Summery:Summery:

• For weak propositional proof systems For weak propositional proof systems each uniform sequence of tautologies each uniform sequence of tautologies has proof complexity has proof complexity C(n)C(n) that that behaves in quite regular fashions. behaves in quite regular fashions.

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Is something similar valid for Is something similar valid for strong propositional systems?strong propositional systems?

• Is it possible that the full Frege proof Is it possible that the full Frege proof system only allows certain special system only allows certain special complexity functions?complexity functions?

Hard to say since each Hard to say since each C(n)C(n) might might (for each uniform sequence of (for each uniform sequence of tautologies) be bound by a tautologies) be bound by a polynomial.polynomial.

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Sporadic Proofs:Sporadic Proofs:

Consider a uniform sequences Consider a uniform sequences [[ηη]]nn of of

tautologies. A proof tautologies. A proof RRmm is is sporadicsporadic if there is no if there is no uniform sequence uniform sequence PPnn of proofs of of proofs of [[ηη]]nn where the where the

proof complexity of proof complexity of PPmm is as low as that of is as low as that of RRmm..

For many proof systems For many proof systems a high density of a high density of sporadic proofs is not possible!sporadic proofs is not possible! If fact all the If fact all the proof systems I have analysed only allowed a proof systems I have analysed only allowed a finite number of sporadic proofs. finite number of sporadic proofs.

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Sporadic Proofs:Sporadic Proofs:

Something vaguely like Something vaguely like Kreisels ConjectureKreisels Conjecture: Short proof of : Short proof of A(n)A(n) in PA for each in PA for each nn implies that the universal sentence implies that the universal sentence

Forall x A(n)Forall x A(n) has a proof in PA. has a proof in PA.

Kreisel like ConjectureKreisel like Conjecture:: If we for an uniform sequences If we for an uniform sequences [[ηη]]nn of tautologies can find of tautologies can find

short (size bound by a fixed polynomial short (size bound by a fixed polynomial p(n) p(n) ) proofs of ) proofs of [[ηη]]nn for each for each nn, then the corresponding system of bounded , then the corresponding system of bounded arithmetic proves arithmetic proves ηη. .

If an uniform sequence If an uniform sequence [[ηη]]nn of tautologies has a high density of tautologies has a high density of short proofs, (i.e. has short proofs for many values of of short proofs, (i.e. has short proofs for many values of nn) ) this force in fact the sequence this force in fact the sequence [[ηη]]nn to have a sequence of to have a sequence of fairly uniformly given proofs.fairly uniformly given proofs.

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How to construct hard How to construct hard tautologies based on predicate tautologies based on predicate logic?logic?• Is there a method to turn a first order Is there a method to turn a first order

formulae into a hard sequence of formulae into a hard sequence of tautologies? Relativation was one tautologies? Relativation was one such method, but for stronger such method, but for stronger systems we need harder tautologies.systems we need harder tautologies.

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Mathematical approaches to Mathematical approaches to (weak) Propositional (weak) Propositional ComplexityComplexityFew examples:Few examples:Ajtai: Ajtai: M. Ajtai, M. Ajtai, The independence of the The independence of the

modulo p counting principles (1994)modulo p counting principles (1994) Krajicek, Scanlon: Krajicek, Scanlon: Combinatorics with Combinatorics with

definable sets: Euler Characteristics and definable sets: Euler Characteristics and Grothendieck rings. (2000)Grothendieck rings. (2000)

Krajicek: Krajicek: On degree of ideal membership On degree of ideal membership proofs from uniform families of proofs from uniform families of polynomials over a finite fieldpolynomials over a finite field. (2001). (2001)