Tree Automata Acceptance up to Measurable Defect

2026-05-26

AI Summary

Traditional tree automata employ a rigid acceptance mechanism that cannot tolerate quantifiable structural deviations, limiting their applicability in domains such as software verification. This work proposes ε-acceptance games, introducing bisimulation distance into the acceptance condition of tree automata for the first time, thereby allowing accepted trees to deviate from an ideal model by at most ε. The framework integrates game semantics, bisimulation metrics, tree automata theory, and measure theory to establish a defect-tolerant acceptance semantics. It is shown that a tree T is ε-accepted if and only if there exists a tree T′ accepted in the classical sense such that the bisimulation distance between T and T′ is at most ε. The approach naturally extends to probabilistic branching, revealing a deep connection between automata acceptance and measure-theoretic concepts.

Abstract

Automata acceptance can, in several situations of interest, be captured game-theoretically via acceptance games. The existence of a winning strategy for Verifier then captures the existence of a winning run-tree of a given automaton over a model. However, such acceptance is rigid, in that it does not allow a measurable defect budget, which can be a challenge in software verification. In this paper, we draw inspiration from how bisimulation distance can be defined as an extension of bisimilarity to define epsilon-acceptance games. Our main theorem shows that a tree T is epsilon-accepted iff there is a tree T' that is accepted in the traditional (rigid) sense and the bisimulation distance of T' and T is at most epsilon. Our work also suggests a strong connection with measure theory, of which we give a preliminary exploration via appropriate examples. Our framework is defined over binary trees with leaves and infinite branches, and strictly contains the case in which binary nodes are seen as probabilistic choice and the defect measures the probability of the set of rejected branches.

Problem

• tree automata
• measurable defect
• epsilon-acceptance
• bisimulation distance
• software verification

Innovation

• epsilon-acceptance
• tree automata
• bisimulation distance
• acceptance games
• measure theory