We develop and study the complexity of propositional proof systems of varying strength extending resolution by allowing it to operate with disjunctions of linear equations instead of clauses. We demonstrate polynomial-size refutations for hard tautologies like the pigeonhole principle, Tseitin graph tautologies and the clique-coloring tautologies in these proof systems. Using (monotone) interpolation we establish an exponential-size lower bound on refutations in a certain, considerably strong, fragment of resolution over linear equations, as well as a general polynomial upper bound on (non-monotone) interpolants in this fragment. We then apply these results to extend and improve previous results on multilinear proofs (over fields of characteristic 0), as studied in [Ran Raz, Iddo Tzameret, The strength of multilinear proofs. Comput. Complexity (in press)]. Specifically, we show the following: •Proofs operating with depth-3 multilinear formulas polynomially simulate a certain, considerably strong, fragment of resolution over linear equations.•Proofs operating with depth-3 multilinear formulas admit polynomial-size refutations of the pigeonhole principle and Tseitin graph tautologies. The former improve over a previous result that established small multilinear proofs only for the functional pigeonhole principle. The latter are different from previous proofs, and apply to multilinear proofs of Tseitin mod p graph tautologies over any field of characteristic 0. We conclude by connecting resolution over linear equations with extensions of the cutting planes proof system.
All Science Journal Classification (ASJC) codes
- Algebraic proof systems
- Cutting planes
- Feasible monotone interpolation
- Multilinear proofs
- Proof complexity