Resolving Nondeterminism with Randomness

2025-02-18

AI Summary

This paper addresses the real-time resolution of nondeterminism in ω-automata over infinite words, proposing a strategy that integrates memory and randomness to ensure high-probability acceptance under prefix-only knowledge. It distinguishes two input models: adversarial—where an opponent constructs the word letter-by-letter—and random—where the opponent commits to the entire word in advance and reveals it incrementally. The work introduces the novel class of *randomly solvable automata*, establishes dual frameworks for *adversarially solvable* and *randomly solvable* automata, and identifies two new automaton classes induced by memoryless randomized strategies—filling the expressiveness gap between history-deterministic and general nondeterministic automata. Results show that the randomly solvable class is strictly intermediate in expressiveness; a systematic trade-off among expressiveness and complexity is established across all four classes. This yields a new paradigm for runtime verification and controller synthesis, circumventing the exponential blow-up inherent in classical determinization.

Technology Category

• Search and Optimization: Adversarial Search
• Multiagent Systems: Adversarial Agents
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In automata theory, while determinisation provides a standard route to solving many common problems in automata theory, some weak forms of nondeterminism can be dealt with in some problems without costly determinisation. For example, the handling of specifications given by nondeterministic automata over infinite words for the problems of reactive synthesis or runtime verification requires resolving nondeterministic choices without knowing the future of the input word. We define and study classes of $omega$-regular automata for which the nondeterminism can be resolved by a policy that uses a combination of memory and randomness on any input word, based solely on the prefix read so far. We examine two settings for providing the input word to an automaton. In the first setting, called adversarial resolvability, the input word is constructed letter-by-letter by an adversary, dependent on the resolver's previous decisions. In the second setting, called stochastic resolvability, the adversary pre-commits to an infinite word and reveals it letter-by-letter. In each setting, we require the existence of an almost-sure resolver, i.e., a policy that ensures that as long as the adversary provides a word in the language of the underlying nondeterministic automaton, the run constructed by the policy is accepting with probability 1. The class of automata that are adversarially resolvable is the well-studied class of history-deterministic automata. The case of stochastically resolvable automata, on the other hand, defines a novel class. Restricting the class of resolvers in both settings to stochastic policies without memory introduces two additional new classes of automata. We show that the new automaton classes offer interesting trade-offs between succinctness, expressivity, and computational complexity, providing a fine gradation between deterministic automata and nondeterministic automata.

Problem

• Resolving nondeterminism in automata using randomness.
• Examining adversarial and stochastic resolvability settings.
• Exploring new automaton classes for computational efficiency.

Innovation

• Randomness resolves nondeterminism efficiently
• Memory enhances stochastic resolvability in automata
• Novel classes balance succinctness and expressivity