Infinite lexicographic products of positional objectives

TL;DR

This paper extends the understanding of positional determinacy in infinite games on graphs by introducing infinite lexicographic products and analyzing their impact on various parity objectives and complexity classes.

Contribution

It introduces two notions of infinite lexicographic products indexed by ordinals and proves they preserve positionality, extending prior results to more complex game structures.

Findings

Infinite lexicographic products preserve positionality.

Max-Parity objectives over countable ordinals are complete for the infinite difference hierarchy levels.

Min-Parity objectives are complete for the class Σ₀₃.

Abstract

This paper contributes to the study of positional determinacy of infinite duration games played on potentially infinite graphs with neutral transitions. 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.