Infinite lexicographic products of positional objectives

2025-06-17

AI Summary

This work addresses positional determinacy of infinite games on infinite graphs, focusing on prefix-independent objectives under infinite lexicographic products and positional extensions of Max/Min-Parity objectives (with arbitrary countable ordinal priorities) in edge-labeled one-player games. Methodologically, it integrates game theory, ordinal induction, descriptive set theory (particularly Σ⁰₂/Σ⁰₃ classes and the Δ²(Σ⁰₂) difference hierarchy), and Parity semantics analysis. The contributions include: (i) the first construction of two ordinal-indexed infinite lexicographic products and a proof of their positional preservation for prefix-independent objectives; (ii) establishing Δ²(Σ⁰₂)-completeness for Max-Parity and Σ⁰₃-completeness for Min-Parity; and (iii) uncovering new closure properties of positional languages under union and neutral-letter projection. These results advance the theoretical understanding of determinacy, definability, and structural robustness in infinite games with complex winning conditions.

Technology Category

• Game Theory and Economic Paradigms: Imperfect Information
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

This paper contributes to the study of positional determinacy of infinite duration games played on potentially infinite graphs. Recently, [Ohlmann, TheoretiCS 2023] established that positionality of prefix-independent objectives is preserved by finite lexicographic products. We propose two different notions of infinite lexicographic products indexed by arbitrary ordinals, and extend Ohlmann's result by proving that they also preserve positionality. In the context of one-player positionality, this extends positional determinacy results of [Gr""adel and Walukiewicz, Logical Methods in Computer Science 2006] to edge-labelled games and arbitrarily many priorities for both Max-Parity and Min-Parity. Moreover, we show that the Max-Parity objectives over countable ordinals are complete for the infinite levels of the difference hierarchy over $Sigma^0_2$ and that Min-Parity is complete for the class $Sigma^0_3$. We obtain therefore positional languages that are complete for all those levels, as well as new insights about closure under unions and neutral letters.

Problem

• Extends positional determinacy to infinite lexicographic products in games
• Generalizes one-player positionality to edge-labelled games with many priorities
• Proves completeness of Max-Parity and Min-Parity objectives for hierarchy levels

Innovation

• Infinite lexicographic products indexed by ordinals
• Extends positionality to edge-labelled games
• Max-Parity and Min-Parity objectives completeness results