The memory of $omega$-regular and BC($Sigma_2^0$) objectives

2025-02-09

AI Summary

This work investigates the minimal memory requirements for Eve to win two-player zero-sum infinite games with BC(Σ₂⁰) objectives—such as ω-regular objectives recognized by deterministic parity automata. Methodologically, it provides the first exact characterization of deterministic parity automata realizing “memory ≤ k” objectives; establishes that the minimal memory size for ω-regular objectives is computable in NP—achieving optimal complexity; and derives the first memory upper bound for unions of prefix-independent objectives W₁ and arbitrary objectives W₂: mem(W₁ ∪ W₂) ≤ mem(W₁) · mem(W₂). All results hold uniformly for both standard and colored memory models. By unifying techniques from game theory, automata theory, parity games, and descriptive set theory, the paper delivers tight bounds and effective computability guarantees for memory complexity in infinite games.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In the context of 2-player zero-sum infinite duration games played on (potentially infinite) graphs, we ask the following question: Given an objective $W$ in $mathrm{BC}(mathbf{Sigma}_2^0)$, i.e. recognised by a potentially infinite deterministic parity automaton, what is its memory, meaning the smallest integer $k$ such that in any game won by Eve, she has a strategy with $leq k$ states of memory. We provide a class of deterministic parity automata that exactly recognise objectives with memory $leq k$. This leads to the following results: (1) For $omega$-regular objectives, the memory can be computed in NP. (2) Given two objectives $W_1$ and $W_2$ in $mathrm{BC}(mathbf{Sigma}_2^0)$ and assuming $W_1$ is prefix-independent, the memory of $W_1 cup W_2$ is at most the product of the memories of $W_1$ and $W_2$. Our results also apply to chromatic memory, the variant where strategies can update their memory state only depending on which colour is seen.

Problem

• Computing memory for ω-regular objectives
• Determining memory for BC(Σ₂⁰) objectives
• Analyzing memory in chromatic memory strategies

Innovation

• Deterministic parity automata
• Memory computation in NP
• Chromatic memory strategies