[Paper Review] Extending Finite-Memory Determinacy by Boolean Combination of Winning Conditions

Stéphane Le Roux, Arno Pauly+1 more|arXiv (Cornell University)|Jan 1, 2018

Computability, Logic, AI Algorithms4 references4 citations

TL;DR

This paper establishes a general framework for finite-memory (FM) determinacy in two-player games on finite graphs by introducing the concept of regularly-predictable winning conditions and hypothetical subgame-perfect equilibria (hSPE). It proves that Boolean combinations of such conditions with regular languages preserve FM determinacy, enabling unified analysis of complex multi-objective games like energy-parity or energy-mean-payoff combinations, with memory bounds derived from automata constructions and subgame-perfect strategy synthesis.

ABSTRACT

We study finite-memory (FM) determinacy in games on finite graphs, a central question for applications in controller synthesis, as FM strategies correspond to implementable controllers. We establish general conditions under which FM strategies suffice to play optimally, even in a broad multi-objective setting. We show that our framework encompasses important classes of games from the literature, and permits to go further, using a unified approach. While such an approach cannot match ad-hoc proofs with regard to tightness of memory bounds, it has two advantages: first, it gives a widely-applicable criterion for FM determinacy; second, it helps to understand the cornerstones of FM determinacy, which are often hidden but common in proofs for specific (combinations of) winning conditions.

Motivation & Objective

• To provide a general, abstract criterion for finite-memory determinacy in two-player games on finite graphs, applicable across diverse winning conditions.
• To unify and extend existing results on FM determinacy for complex objectives such as energy, parity, mean-payoff, and their combinations.
• To identify core structural properties—regular predictability and existence of FM hSPE—that underlie FM determinacy, making them transparent and reusable.
• To enable the analysis of new or composite objectives without ad-hoc proofs, by reducing them to known classes via Boolean combinations.
• To derive concrete memory bounds for FM strategies in combined objectives, even when individual components require exponential or infinite memory.

Proposed method

• Introduce the notion of regularly-predictable winning conditions: those for which a finite automaton can recognize histories from which Player 1 has a winning strategy.
• Define hypothetical subgame-perfect equilibrium (hSPE) as a technical tool to ensure that FM strategies exist even in complex subgames.
• Prove that if a class of winning conditions is closed under Boolean combinations and admits FM hSPE, then so does any Boolean combination with regular languages.
• Construct game expansions using product constructions with finite automata to simulate history-dependent conditions, enabling memory-bound analysis.
• Use automaton product constructions to simulate the interaction between winning conditions and memory requirements, particularly for disjunctions and conjunctions.
• Apply subgame-perfect strategy synthesis to derive memory bounds for both players, especially for Player 2, by combining automata and FM strategy representations.

Experimental results

Research questions

• RQ1 Under what general conditions does finite-memory determinacy persist when combining multiple winning conditions via Boolean operations?
• RQ2 Can the core structural features enabling FM determinacy be abstracted and formalized to unify existing results across different game types?
• RQ3 How can memory requirements for FM strategies be bounded in games with complex objectives formed by Boolean combinations of simpler ones?
• RQ4 To what extent can the existence of FM strategies be guaranteed for disjunctions of winning conditions, especially when individual components require infinite memory?
• RQ5 Can the framework be applied to new or composite objectives (e.g., energy-parity, energy-mean-payoff) without re-proving determinacy from scratch?

Key findings

• Boolean combinations of regularly-predictable winning conditions and regular languages preserve finite-memory determinacy, provided the base class is closed under Boolean combinations and admits FM hSPE.
• For battery-like and spill-over-like energy parity games, the paper derives an O(Emax) memory bound, significantly improving upon the |V|·d·W bound for unbounded-energy parity games.
• Player 1 can win disjunctive games (W ∨ LA) using finite memory of size |Q|·|Qg|·m(|V|·|Q|·|Qg|), where Q and Qg are automaton states and m(n) is the memory bound for the subgame.
• Player 2’s winning strategies in disjunctive games require memory size l|A| + m, where l is the size of an automaton witnessing regular-predictability and m is the FM strategy size for the base game.
• The framework reveals that regular-predictability and FM hSPE are sufficient conditions for preserving FM determinacy under Boolean combinations, even when individual components require infinite memory.
• The method provides a systematic way to derive memory bounds for new composite objectives without ad-hoc analysis, at the cost of potentially looser bounds than specialized proofs.