Weak Bisimulation Finiteness of Pushdown Systems With Deterministic $\varepsilon$-Transitions Is 2-ExpTime-Complete

2026-08-11

AI Summary

This study addresses whether pushdown systems with deterministic ε-transitions are weakly bisimilar to some finite-state system. By integrating formal verification, automata theory, and complexity analysis, we precisely characterize the computational complexity of this decision problem as 2-ExpTime-complete: we present a 2-ExpTime decision algorithm and establish matching 2-ExpTime-hardness. Furthermore, we prove that the size of the minimal equivalent finite-state system is at most doubly exponential in the size of the input. This result substantially improves upon the previous gap between an Ackermannian upper bound and an ExpTime lower bound, providing a tight complexity classification for the problem.

Abstract

We consider the problem of deciding whether a given pushdown system all of whose $\varepsilon$-transitions are deterministic is weakly bisimulation finite, that is, whether it is weakly bisimulation equivalent to a finite system. We prove that this problem is 2-ExpTime-complete. This consists of three elements: First, we prove that the smallest finite system that is weakly bisimulation equivalent to a fixed pushdown system, if exists, has size at most doubly exponential in the description size of the pushdown system. Second, we propose a fast algorithm deciding whether a given pushdown system is weakly bisimulation equivalent to a finite system of a given size. Third, we prove 2-ExpTime-hardness of the problem. The problem was known to be decidable, but the previous algorithm had Ackermannian complexity (6-ExpSpace in the easier case of pushdown systems without $\varepsilon$-transitions); concerning lower bounds, only ExpTime-hardness was known.

Problem

• weak bisimulation
• pushdown systems
• deterministic ε-transitions
• finiteness
• equivalence

Innovation

• weak bisimulation finiteness
• pushdown systems
• deterministic ε-transitions
• 2-ExpTime-complete
• complexity characterization