On Simon’s Congruence Closure of a String

Abstract

AbstractTwo strings are Simon’s \(\sim _k\)-congruent if they have the same set of subsequences of length at most k. We study the Simon’s congruence closure of a string, which is regular by definition. Given a string w over an alphabet \(\varSigma \), we present an efficient DFA construction that accepts all \(\sim _k\)-congruent strings with respect to w. We also present lower bounds for the state complexity of the Simon’s congruence closure. Finally, we design a polynomial-time algorithm that answers the following open problem: “given a string w over a fixed-sized alphabet, an integer k and a (regular or context-free) language L, decide whether there exists a string \(v \in L\) such that \(w \sim _k v\).” The problem is NP-complete for a variable-sized alphabet.KeywordsSimon’s congruenceState complexityFinite automataShortlex normal forms