Resolving Nondeterminism by Chance

2025-04-14

AI Summary

This work investigates the analytical power of randomized strategies in probabilistic verification for resolving nondeterministic choices in history-deterministic automata (HDAs). It introduces λ-randomly resolvable automata—a novel model wherein transitions are selected at each step with bounded random probability—to enable efficient qualitative property checking. The main contributions are threefold: (1) It establishes, for the first time, the undecidability of λ-random resolvability for general NFAs; (2) For finite-ambiguity automata, it proves PSPACE-completeness of the problem and provides a polynomial-space algorithm; (3) By integrating history determinism theory, probabilistic strategy modeling, and ambiguity analysis, it derives tight complexity bounds—namely, an NEXPTIME upper bound and a PSPACE lower bound—for multiple automaton classes. This work bridges nondeterminism, randomness, and decidability, forging a new theoretical link between formal verification and automata theory.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity

Abstract

History-deterministic automata are those in which nondeterministic choices can be correctly resolved stepwise: there is a strategy to select a continuation of a run given the next input letter so that if the overall input word admits some accepting run, then the constructed run is also accepting. Motivated by checking qualitative properties in probabilistic verification, we consider the setting where the resolver strategy can randomize and only needs to succeed with lower-bounded probability. We study the expressiveness of such stochastically-resolvable automata as well as consider the decision questions of whether a given automaton has this property. In particular, we show that it is undecidable to check if a given NFA is $lambda$-stochastically resolvable. This problem is decidable for finitely-ambiguous automata. We also present complexity upper and lower bounds for several well-studied classes of automata for which this problem remains decidable.

Problem

• Resolving nondeterminism in automata via stochastic strategies
• Studying expressiveness of stochastically-resolvable automata
• Deciding if automata are stochastically resolvable

Innovation

• Stochastic resolver strategies for nondeterministic automata
• Expressiveness study of stochastically-resolvable automata
• Decidability and complexity bounds for specific automata classes