Edit Distance for Pushdown Automata

TL;DR

This paper investigates the computational complexity and decidability of calculating the edit distance from pushdown automata to finite automata, revealing which problems are solvable and under what conditions.

Contribution

It provides a comprehensive analysis of the decidability and complexity of edit distance problems between pushdown automata and finite automata.

Findings

Deciding if the edit distance is at most a threshold is decidable.

Deciding if the edit distance is finite is decidable.

The problem of computing exact edit distance is undecidable.

Abstract

The edit distance between two words $w_1, w_2$ is the minimal number of word operations (letter insertions, deletions, and substitutions) necessary to transform $w_1$ to $w_2$. The edit distance generalizes to languages $\mathcal{L}_1, \mathcal{L}_2$, where the edit distance from $\mathcal{L}_1$ to $\mathcal{L}_2$ is the minimal number $k$ such that for every word from $\mathcal{L}_1$ there exists a word in $\mathcal{L}_2$ with edit distance at most $k$. We study the edit distance computation problem between pushdown automata and their subclasses. The problem of computing edit distance to a pushdown automaton is undecidable, and in practice, the interesting question is to compute the edit distance from a pushdown automaton (the implementation, a standard model for programs with recursion) to a regular language (the specification). In this work, we present a complete picture of decidability and complexity for the following problems: (1)~deciding whether, for a given threshold $k$, the edit distance from a pushdown automaton to a finite automaton is at most $k$, and (2)~deciding whether the edit distance from a pushdown automaton to a finite automaton is finite.