Computational and Proof Complexity of Partial String Avoidability

Dmitry Itsykson,Vsevolod Oparin,Alexander Okhotin

doi:10.1145/3442365

Abstract

The partial string avoidability problem is stated as follows: given a finite set of strings with possible “holes” (wildcard symbols), determine whether there exists a two-sided infinite string containing no substrings from this set, assuming that a hole matches every symbol. The problem is known to be NP-hard and in PSPACE, and this article establishes its PSPACE-completeness. Next, string avoidability over the binary alphabet is interpreted as a version of conjunctive normal form satisfiability problem, where each clause has infinitely many shifted variants. Non-satisfiability of these formulas can be proved using variants of classical propositional proof systems, augmented with derivation rules for shifting proof lines (such as clauses, inequalities, polynomials, etc.). First, it is proved that there is a particular formula that has a short refutation in Resolution with a shift rule but requires classical proofs of exponential size. At the same time, it is shown that exponential lower bounds for classical proof systems can be translated for their shifted versions. Finally, it is shown that superpolynomial lower bounds on the size of shifted proofs would separate NP from PSPACE; a connection to lower bounds on circuit complexity is also established.

Highlights

The field of proof complexity is concerned with the size of proofs for different kinds of logical formulas, under various measures of size
The most common subject, motivated by satisfiability problem (SAT)-solvers, are Boolean formulas in conjunctive normal form (CNF), and there is a substantial body of literature on lower bounds on the size of a proof that a given CNF formula is unsatisfiable
This paper investigates the complexity issues for a variant of CNF formulae, in which every clause exists in countably many variants, with variable numbers shifted by any constant

Summary

Introduction

The field of proof complexity is concerned with the size of proofs for different kinds of logical formulas, under various measures of size. The resulting Shift-CNF depends on countably many variables, and represents uniformly defined constraints applied to all blocks of variables It can be alternatively written as a finite formula, with each clause using a universal quantifier on position numbers, such as in (∀i ∈ Z)(xi+1 ∨ ¬xi+4). The Shift-SAT problem is attractive for being similar to the classical SAT problem, to the point that all proof systems for UNSAT, such as Resolution, Cutting Plane, Polynomial Calculus, etc., can be directly applied to Shift-SAT formulas For every such proof system Π, there is its shifted version, Shift-Π, with an additional derivation rule for adding an arbitrary integer to the numbers of all variables in a constraint.

The partial string avoidability problem

The PSPACE-hardness proof

Proof systems

Lower bounds on the size of shifted proofs

Separation of Resolution with and without shift

Conclusion