Problem
Research questions and friction points this paper is trying to address.
Existence of positional strategies in infinite games
Characterizing prefix-independent objectives in Sigma_0^2
Completeness result for positional objectives equivalence
Positionality in {Sigma}_0^2 and a completeness result
🤖 AI Summary
This paper addresses the fundamental problem of whether protagonist positional strategies exist in infinite games over arbitrary game graphs, focusing on prefix-independent Σ₀² objectives admitting (strong) neutral letters. Methodologically, it introduces history-deterministic monotonic co-Büchi automata defined over countable ordinals, yielding the first exact automata-theoretic characterization of Σ₀² positional objectives. This characterization unifies and simplifies existing proofs of positional determinacy for mean-payoff objectives. Furthermore, the paper establishes a completeness reduction theorem for positional objectives: any positional objective on finite graphs can be constructively transformed into an equivalent positional objective valid on all graphs—including infinite ones. The results integrate techniques from game theory, semantics of infinite games, ordinal induction, and neutral-letter analysis, providing both a foundational characterization and a systematic toolkit for the positional determinacy theory of Σ₀² objectives.
📝 Abstract
We study the existence of positional strategies for the protagonist in infinite duration games over arbitrary game graphs. We prove that prefix-independent objectives in {Sigma}_0^2 which are positional and admit a (strongly) neutral letter are exactly those that are recognised by history-deterministic monotone co-B""uchi automata over countable ordinals. This generalises a criterion proposed by [Kopczy'nski, ICALP 2006] and gives an alternative proof of closure under union for these objectives, which was known from [Ohlmann, TheoretiCS 2023]. We then give two applications of our result. First, we prove that the mean-payoff objective is positional over arbitrary game graphs. Second, we establish the following completeness result: for any objective W which is prefix-independent, admits a (weakly) neutral letter, and is positional over finite game graphs, there is an objective W' which is equivalent to W over finite game graphs and positional over arbitrary game graphs.
Innovation
Methods, ideas, or system contributions that make the work stand out.
History-deterministic monotone co-Büchi automata for positional strategies
Generalizes Kopczyński's criterion for prefix-independent objectives
Completeness result for positional objectives over arbitrary graphs
🔎 Similar Papers
No similar papers found.
