Simplified lower bounds for propositional proofs (original) (raw)

On the Proof Complexity of Paris-Harrington and Off-Diagonal Ramsey Tautologies

ACM Transactions on Computational Logic, 2016

We study the proof complexity of Paris-Harrington’s Large Ramsey Theorem for bi-colorings of graphs and of off-diagonal Ramsey’s Theorem. For Paris-Harrington, we prove a non-trivial conditional lower bound in Resolution and a non-trivial upper bound in bounded-depth Frege. The lower bound is conditional on a (very reasonable) hardness assumption for a weak (quasi-polynomial) Pigeonhole principle in R es (2). We show that under such an assumption, there is no refutation of the Paris-Harrington formulas of size quasi-polynomial in the number of propositional variables. The proof technique for the lower bound extends the idea of using a combinatorial principle to blow up a counterexample for another combinatorial principle beyond the threshold of inconsistency. A strong link with the proof complexity of an unbalanced off-diagonal Ramsey principle is established. This is obtained by adapting some constructions due to Erdős and Mills. We prove a non-trivial Resolution lower bound for a ...

Hardness amplification in proof complexity

Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10, 2010

We present a general method for converting any family of unsatisfiable CNF formulas that is hard for one of the simplest proof systems, tree resolution (ordinary backtracking search), into formulas that require large rank in very strong proof systems, which include any proof system that manipulates polynomials or polynomial threshold functions of degree at most k (known as Th(k) proofs). These include high degree versions of Lovász-Schrijver systems, (LS(k), LS + (k)), high degree versions of Cutting Planes proofs (CP(k)), Sherali-Adams, and Lasserre proofs.

A Generic Time-Space Lower Bound for Proof Complexity

2008

We show that for all reals c and d such that c 2 d < 4 there exists a positive real e such that tautologies cannot be decided by both a nondeterministic algorithm that runs in time n c , and a nondeterministic algorithm that runs in time n d and space n e. In particular, for every d < 3 √ 4 there exists a positive e such that tautologies cannot be decided by a nondeterministic algorithm that runs in time n d and space n e .

Complexity of proofs in classical propositional logic

Logic from computer science, 1992

This paper surveys the work of the last two decades on the complexity of proofs in classical propositional logic, and its connection with open problems in theoretical computer science. The paper ends with a short list of open problems.

On proving circuit lower bounds against the polynomial-time hierarchy: positive and negative results

We consider the problem of proving circuit lower bounds against the polynomialtime hierarchy. We give both positive and negative results. For the positive side, for any fixed integer k > 0, we give an explicit Σ p 2 language, acceptable by a Σ p 2 -machine with running time O(n k 2 +k ), that requires circuit size > n k . This provides a constructive version of an existence theorem of Kannan [Kan82]. Our main theorem is on the negative side. We give evidence that it is infeasible to give relativizable proofs that any single language in the polynomialtime hierarchy requires super polynomial circuit size. Our proof techniques are based on the decision tree version of the Switching Lemma for constant depth circuits and Nisan-Wigderson pseudorandom generator.

Some elementary proofs of lower-bounds in complexity theory:(prepublication)

We introduce a new and easily applicable criterion called rank immunity for estimating the minimal number of multiplications needed to compute a set of bilinear forms in commuting variables. The result is obtained by an elimination argument after canonically embedding computations in a quotient ring R/Z, where Z is an appropriately chosen ideal that is left invariant under the eliminations. The criterion combines the well-known arguments based on elimination and on row rank, but in contrast to (for instance) colnrnn-and mixed-rank arguments it normally leads to better elementary estimates than were derivable in a uniform manner before.

On proving circuit lower bounds against the polynomial-time hierarchy

We consider the problem of proving circuit lower bounds against the polynomialtime hierarchy. We give both positive and negative results. For the positive side, for any fixed integer k > 0, we give an explicit Σ p 2 language, acceptable by a Σ p 2 -machine with running time O(n k 2 +k ), that requires circuit size > n k . This provides a constructive version of an existence theorem of Kannan [Kan82]. Our main theorem is on the negative side. We give evidence that it is infeasible to give relativizable proofs that any single language in the polynomialtime hierarchy requires super polynomial circuit size. Our proof techniques are based on the decision tree version of the Switching Lemma for constant depth circuits and Nisan-Wigderson pseudorandom generator.

On the bounds for the main proof measures in some propositional proof systems

2014

Various proof complexity characteristics are investigated in three propositional proof systems, based on determinative disjunctive normal forms. The comparative analysis for size, time, space, width of proofs is given. For some formula family we obtain in our systems simultaneously bounds for different proof complexity measures (asymptotically the same upper and lower bounds for each measures). These results can be generalized for the other formulas and for the other systems also..

Lower bounds and the hardness of counting properties

2002

Rice's Theorem states that all nontrivial language properties of recursively enumerable sets are undecidable. Borchert and Stephan (BSOO) started the search for complexity-theoretic analogs of Rice's Theorem, and proved that every nontrivial counting property of boolean circuits is UP-hard. Hemaspaandra and Rothe (HROO] improved the UP-hardness lower bound to UPocwhardness. The present paper raises the lower bound for nontrivial counting properties from UP O(l)-hardness to FewPhardness, i.e., from constant-ambiguity nondeterminism to polynomialambiguity nondeterminism. Furthermore, we prove that this lower bound is rather tight with respect to relativizable techniques, i.e., no relativizable technique can raise this lower bound to FewP-:s;f-tt-hardness. We also prove a Rice-style theorem for NP, namely that every nontrivial language property of NP sets is NP-hard.