## Abstract

Two strings are Simon's ∼_{k}-congruent if they have the same set of subsequences of length at most k. We study Simon's congruence closure of a string, which is regular by definition. Given a string w over an alphabet Σ, we present two efficient DFA constructions that accept all ∼_{k}-congruent strings with respect to w. We also present lower bounds for the state complexity of Simon's congruence closure. Then, 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 or not there exists a string v∈L such that w∼_{k}v.” In addition, for a variable-sized alphabet, we prove that the problem is NP-complete.

Original language | English |
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Article number | 114078 |

Journal | Theoretical Computer Science |

Volume | 972 |

DOIs | |

State | Published - 13 Sep 2023 |

## Keywords

- Finite automata
- Shortlex normal forms
- Simon's congruence
- State complexity