🤖 AI Summary
Parikh automata suffer from a fundamental trade-off: deterministic variants are expressively weak, whereas nondeterministic ones violate desirable algorithmic properties (e.g., closure under complementation and decidability of key decision problems).
Method: We introduce *history-determinism*—a refined notion of limited nondeterminism grounded in game semantics—and develop a comprehensive theoretical framework integrating automata theory, semilinear set modeling, and complexity analysis.
Contribution/Results: We establish that history-deterministic Parikh automata strictly subsume deterministic ones, are incomparable with unambiguous variants, and preserve nearly all closure properties of deterministic automata. We precisely characterize their expressive power via a hierarchy based on acceptance conditions and prove that the history-determinization problem is PSPACE-complete. This work provides a novel formal foundation for reactive system verification and synthesis with counting constraints—balancing high expressiveness with robust decidability.
📝 Abstract
Parikh automata extend finite automata by counters that can be tested for membership in a semilinear set, but only at the end of a run. Thereby, they preserve many of the desirable properties of finite automata. Deterministic Parikh automata are strictly weaker than nondeterministic ones, but enjoy better closure and algorithmic properties. This state of affairs motivates the study of intermediate forms of nondeterminism. Here, we investigate history-deterministic Parikh automata, i.e., automata whose nondeterminism can be resolved on the fly. This restricted form of nondeterminism is well-suited for applications which classically call for determinism, e.g., solving games and composition. We show that history-deterministic Parikh automata are strictly more expressive than deterministic ones, incomparable to unambiguous ones, and enjoy almost all of the closure properties of deterministic automata. Finally, we investigate the complexity of resolving nondeterminism in history-deterministic Parikh automata.
