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

2. General question: For a given (weak) Propositional Proof System investigate the proof complexity (as a function of n) of uniform sequences [η] n of tautologies. For a given (weak) Propositional Proof System investigate the proof complexity (as a function of n) of uniform sequences [η] n of tautologies. Setup for this: Given a large class C of propositions from predicate logic. Each proposition η Є C defines a sequence of propositions [η] n where [η] n is a propositional formula expressing that there is no model of η of size n Setup for this: Given a large class C of propositions from predicate logic. Each proposition η Є C defines a sequence of propositions [η] n where [η] n is a propositional formula expressing that there is no model of η of size n

3. Uniform sequences of Tautologies Often uniform sequences of proposition formulae are usually produced by the Paris-Wilkie translation. Often uniform sequences of proposition formulae are usually produced by the Paris-Wilkie translation. For many nice theorems in weak propositional proof complexity we need a slightly different translation (the Riis- Sitharam or RS-translation). For many nice theorems in weak propositional proof complexity we need a slightly different translation (the Riis- Sitharam or RS-translation).

4. Difference between the PW and RS translations in a nutshell In the PW-translation [A] of A:=Ez z+z=n Λ φ In the PW-translation [A] of A:=Ez z+z=n Λ φ becomes equivalent to 0 if n is odd and [φ] if n is even. becomes equivalent to 0 if n is odd and [φ] if n is even. In the RS-translation all function and relation symbols are treated as uninterpreted symbols. In the RS-translation all function and relation symbols are treated as uninterpreted symbols. (i.e. +, ≤, * etc are treated as general uninterpreted relation symbols). This ensure that the translation [A] n (i.e. +, ≤, * etc are treated as general uninterpreted relation symbols). This ensure that the translation [A] n (essentially) becomes closed under the action of the symmetric group. (essentially) becomes closed under the action of the symmetric group.,

5. Idea For a given (weak) Propositional Proof System P classify the proof complexity behaviour of [η] n for a given class C of formulae η of predicate logic. For a given (weak) Propositional Proof System P classify the proof complexity behaviour of [η] n for a given class C of formulae η of predicate logic. Let c P (n) denote the length of the shortest proof of [η] n. Let c P (n) denote the length of the shortest proof of [η] n. Question: Which complexity functions c P (n) can occur? Question: Which complexity functions c P (n) can occur?

6. Tree resolution proofs Theorem [Riis 99]: Theorem [Riis 99]: The class of complexity functions C tr (n) is included in the set of all functions that either are bounded by a fixed polynomial or has growth rate faster than 2 cn for some c>0. The class of complexity functions C tr (n) is included in the set of all functions that either are bounded by a fixed polynomial or has growth rate faster than 2 cn for some c>0.

7. Tree resolution proofs Conjecture [With Danchev]: Conjecture [With Danchev]: The class of complexity functions C(n) has in three distinct types of behaviour: The class of complexity functions C(n) has in three distinct types of behaviour: (1) C(n) with C(n)<p(n) for some polynomial p. (1) C(n) with C(n)<p(n) for some polynomial p. (2) C(n) for which there exists 0<c<d such that 2 cn <C(n) < 2 dn for all but finitely many values of n. (2) C(n) for which there exists 0<c<d such that 2 cn <C(n) < 2 dn for all but finitely many values of n. (3) C(n)>2 cn log(n) for some constant c>0. (3) C(n)>2 cn log(n) for some constant c>0. New type of problem: In case (1) is C(n) given by a polynomial (for all but finitely many exceptional values of n)? New type of problem: In case (1) is C(n) given by a polynomial (for all but finitely many exceptional values of n)?

8. Resolution proofs (Hilbert style) Theorem [Dantchev, Riis 03]: The class of complexity functions C res (n) that arise from “relativised first order formula” is included in the class of functions that either are polynomial bounded or has growth rate faster than 2 cn for some c>0. The class of complexity functions C res (n) that arise from “relativised first order formula” is included in the class of functions that either are polynomial bounded or has growth rate faster than 2 cn for some c>0.

9. Resolution proofs (Hilbert style) Conjecture [With Dantchev] Conjecture [With Dantchev] The class of complexity functions C res (n) includes all functions that either are polynomially bounded or has growth rate faster than 2 cn for some c>0. The class of complexity functions C res (n) includes all functions that either are polynomially bounded or has growth rate faster than 2 cn for some c>0.

10. New results Let [η] n be a uniform sequence of propositional formula. Let [η] n be a uniform sequence of propositional formula. Let S true :={n : [η] n is true} Let S true :={n : [η] n is true} Let S false :={n : [η] n is false}. Let S false :={n : [η] n is false}. Here n denote (as usual) the size of the underlying model. Here n denote (as usual) the size of the underlying model.

11. BDF proofs Theorem A: Theorem A: If S false and S true both are infinite there exists an infinite subset N’ in S true such that the sequence [η] n for any constant d, and for any infinite N’’ subset of N’ requires exponential size depth d Frege- Proofs. If S false and S true both are infinite there exists an infinite subset N’ in S true such that the sequence [η] n for any constant d, and for any infinite N’’ subset of N’ requires exponential size depth d Frege- Proofs.

12. BDF proofs Theorem B: Theorem B: Assume S false ={s 1,s 2,s 3,…,s n,…} is not very sparse (for each c >0 we have Assume S false ={s 1,s 2,s 3,…,s n,…} is not very sparse (for each c >0 we have (s k+1 -s k ) c <s k for infinitely many values of k). (s k+1 -s k ) c <s k for infinitely many values of k). Then for any d Є {1,2,…} [η] n requires exponential size depth d Frege-Proofs (on any infinite subset N’ of S true )

13. Corollaries: The propositional version of each of the following statements has no subexponential Bound depth Frege proofs: The propositional version of each of the following statements has no subexponential Bound depth Frege proofs: (1) There is no field structure on {1,2, …,n} (when n not is a prime power). (2) The set {1,2,…,n} cannot be organised as a vector space over the field F q (when n not is a power of q). (3) There is no 3-regular graph G on the set {1,2,…,n} (when n is odd) (4) There is no proper rectangular grid on the the set {1,2,..,n} (when n is a prime number).

14. Conjecture (for Resolution as well as for BDF): The presence of a hard instance where C(n) is large, force C(m) to be large for values of m close to n The presence of a hard instance where C(n) is large, force C(m) to be large for values of m close to n (Justification: hard instances ought to have similar consequences as impossible instances) (Justification: hard instances ought to have similar consequences as impossible instances) In general C(n) is smooth without big (local) jumps when n increase. In general C(n) is smooth without big (local) jumps when n increase.

15. Nullstellensatz-Proofs (over Field of characteristic q). Theorem [Riis 06]: Theorem [Riis 06]: If P n is a uniform generated sequence of polynomial equations. Then there are 4 distinct behaviours of the NS degree complexity D(n) If P n is a uniform generated sequence of polynomial equations. Then there are 4 distinct behaviours of the NS degree complexity D(n) (1) D(n) is bound by a constant (2) D(n) ≥ l(n) for all sufficiently large values of n (3) There exists a constant c such that the residue class of n modulo q c determine if case (1) or case (2) apply (4) The function D(n) has fluctuating complexity that fluctuates in a very specific pattern such that (essentially) all complexities between constant and l(n) occur on some infinite set. Here log(n) ≤ l(n) ≤ n/2 is a universal function that can be chosen independently of the actual sequence P n. Here log(n) ≤ l(n) ≤ n/2 is a universal function that can be chosen independently of the actual sequence P n.

16. Example of equations that have fluctuating NS-complexity ∑ j x ij + ∑ j y ij -1 =0 for i Є {1,2,…,n} ∑ j x ij + ∑ j y ij -1 =0 for i Є {1,2,…,n} ∑ j x ij -1=0 for j Є {1,2,…,n} ∑ j x ij -1=0 for j Є {1,2,…,n} ∑ i y ij -1=0 for j Є {1,2,…,n} ∑ i y ij -1=0 for j Є {1,2,…,n} x ij x ik = 0 for i,j,k Є {1,2,…,n}, j ≠ k x ij x ik = 0 for i,j,k Є {1,2,…,n}, j ≠ k y ij y ik =0 for i,j,k Є {1,2,…,n}, j ≠ k y ij y ik =0 for i,j,k Є {1,2,…,n}, j ≠ k x ij y ik =0 for i,j,k Є {1,2,…,n} x ij y ik =0 for i,j,k Є {1,2,…,n} y ji y ki =0 for i,j,k Є {1,2,…,n}, j ≠ k y ji y ki =0 for i,j,k Є {1,2,…,n}, j ≠ k This system of equations (that formalise the statement that there is no bijection form n to 2n) has essentially NS degree complexity D(n)=q a(q,n) where a(q,n) denote the power of q in the prime factor decomposition of n. This system of equations (that formalise the statement that there is no bijection form n to 2n) has essentially NS degree complexity D(n)=q a(q,n) where a(q,n) denote the power of q in the prime factor decomposition of n. Thus case (4) in the classification is non-empty. Thus case (4) in the classification is non-empty.

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

18. Is something similar valid for strong propositional systems? Is it possible that the full Frege proof system only allows certain special complexity functions? Is it possible that the full Frege proof system only allows certain special complexity functions? Hard to say since each C(n) might (for each uniform sequence of tautologies) be bound by a polynomial. Hard to say since each C(n) might (for each uniform sequence of tautologies) be bound by a polynomial.

19. Sporadic Proofs: Consider a uniform sequences [η] n of tautologies. A proof R m is sporadic if there is no uniform sequence P n of proofs of [η] n where the proof complexity of P m is as low as that of R m. Consider a uniform sequences [η] n of tautologies. A proof R m is sporadic if there is no uniform sequence P n of proofs of [η] n where the proof complexity of P m is as low as that of R m. For many proof systems a high density of sporadic proofs is not possible! If fact all the proof systems I have analysed only allowed a finite number of sporadic proofs. For many proof systems a high density of sporadic proofs is not possible! If fact all the proof systems I have analysed only allowed a finite number of sporadic proofs.

20. Sporadic Proofs: Something vaguely like Kreisels Conjecture: Short proof of A(n) in PA for each n implies that the universal sentence Something vaguely like Kreisels Conjecture: Short proof of A(n) in PA for each n implies that the universal sentence Forall x A(n) has a proof in PA. Forall x A(n) has a proof in PA. Kreisel like Conjecture: Kreisel like Conjecture: If we for an uniform sequences [η] n of tautologies can find short (size bound by a fixed polynomial p(n) ) proofs of [η] n for each n, then the corresponding system of bounded arithmetic proves η. If we for an uniform sequences [η] n of tautologies can find short (size bound by a fixed polynomial p(n) ) proofs of [η] n for each n, then the corresponding system of bounded arithmetic proves η. If an uniform sequence [η] n of tautologies has a high density of short proofs, (i.e. has short proofs for many values of n) this force in fact the sequence [η] n to have a sequence of fairly uniformly given proofs. If an uniform sequence [η] n of tautologies has a high density of short proofs, (i.e. has short proofs for many values of n) this force in fact the sequence [η] n to have a sequence of fairly uniformly given proofs.

21. How to construct hard tautologies based on predicate logic? Is there a method to turn a first order formulae into a hard sequence of tautologies? Relativation was one such method, but for stronger systems we need harder tautologies. Is there a method to turn a first order formulae into a hard sequence of tautologies? Relativation was one such method, but for stronger systems we need harder tautologies.

22. Mathematical approaches to (weak) Propositional Complexity Few examples: Ajtai: M. Ajtai, The independence of the modulo p counting principles (1994) Krajicek, Scanlon: Combinatorics with definable sets: Euler Characteristics and Grothendieck rings. (2000) Krajicek: On degree of ideal membership proofs from uniform families of polynomials over a finite field. (2001)