The Almost Equivalence by Asymptotic Probabilities for Regular Languages and Its Computational Complexities

TL;DR

This paper introduces p-equivalence based on asymptotic probabilities for regular languages, explores its logical characterization, and analyzes its computational complexity, showing it aligns with standard equivalence problems.

Contribution

It defines p-equivalence using asymptotic probabilities, provides a logical characterization, and establishes that its computational complexity matches that of standard language equivalence.

Findings

P-equivalence is robust and logically characterized.

Computational complexities of p-equivalence match those of standard equivalence.

Proof techniques for p-equivalence extend to generalized equivalences.

Abstract

We introduce p-equivalence by asymptotic probabilities, which is a weak almost-equivalence based on zero-one laws in finite model theory. In this paper, we consider the computational complexities of p-equivalence problems for regular languages and provide the following details. First, we give an robustness of p-equivalence and a logical characterization for p-equivalence. The characterization is useful to generate some algorithms for p-equivalence problems by coupling with standard results from descriptive complexity. Second, we give the computational complexities for the p-equivalence problems by the logical characterization. The computational complexities are the same as for the (fully) equivalence problems. Finally, we apply the proofs for p-equivalence to some generalized equivalences.