[Paper Review] Prompt Delay

Felix Klein, Martín Zimmermann|arXiv (Cornell University)|Feb 16, 2016

Formal Methods in Verification23 references4 citations

TL;DR

This paper introduces Prompt-LTL as a well-behaved quantitative winning condition for delay games, showing that solving such games is triply-exponential time complete. It establishes tight triply-exponential bounds on both lookahead and the scope of the parameterized eventually operator, resolving key limitations of prior approaches.

ABSTRACT

Delay games are two-player games of infinite duration in which one player may delay her moves to obtain a lookahead on her opponent's moves. Recently, such games with quantitative winning conditions in weak MSO with the unbounding quantifier were studied, but their properties turned out to be unsatisfactory. In particular, unbounded lookahead is in general necessary. Here, we study delay games with winning conditions given by Prompt-LTL, Linear Temporal Logic equipped with a parameterized eventually operator whose scope is bounded. Our main result shows that solving Prompt-LTL delay games is complete for triply-exponential time. Furthermore, we give tight triply-exponential bounds on the necessary lookahead and on the scope of the parameterized eventually operator. Thus, we identify Prompt-LTL as the first known class of well-behaved quantitative winning conditions for delay games. Finally, we show that applying our techniques to delay games with \omega-regular winning conditions answers open questions in the cases where the winning conditions are given by non-deterministic, universal, or alternating automata.

Motivation & Objective

• To address the unsatisfactory properties of prior delay games with weak MSO winning conditions, particularly the need for unbounded lookahead.
• To identify a class of quantitative winning conditions that yield well-behaved delay games with bounded lookahead.
• To establish tight complexity bounds for solving delay games with Prompt-LTL conditions.
• To answer open questions about delay games with ω-regular winning conditions using the proposed techniques.

Proposed method

• Formalizing delay games with Prompt-LTL, where the eventually operator has a bounded scope parameter.
• Reducing the problem of solving Prompt-LTL delay games to automata-based decision procedures using alternating tree automata.
• Applying game-theoretic techniques to analyze the necessary lookahead and bound it tightly.
• Using a construction based on bounded delay strategies to derive triply-exponential upper bounds on lookahead and operator scope.
• Proving completeness via reductions to known complexity classes, establishing tightness of bounds.
• Extending the framework to handle ω-regular winning conditions via non-deterministic, universal, and alternating automata.

Experimental results

Research questions

• RQ1 Can Prompt-LTL serve as a well-behaved quantitative winning condition for delay games, avoiding the need for unbounded lookahead?
• RQ2 What is the precise complexity of solving delay games with Prompt-LTL winning conditions?
• RQ3 What are the tight bounds on the required lookahead and the scope of the parameterized eventually operator in such games?
• RQ4 Can the proposed techniques resolve open problems in delay games with ω-regular winning conditions?
• RQ5 Do the bounds on lookahead and operator scope remain tight when extending to automata-based winning conditions?

Key findings

• Solving Prompt-LTL delay games is complete for triply-exponential time, establishing the exact complexity class.
• Tight triply-exponential bounds are established on the necessary lookahead for winning strategies.
• Tight triply-exponential bounds are also derived for the scope of the parameterized eventually operator in Prompt-LTL.
• Prompt-LTL is identified as the first known class of well-behaved quantitative winning conditions for delay games.
• The techniques resolve open questions regarding delay games with non-deterministic, universal, and alternating automata as winning conditions.
• The results show that bounded lookahead and bounded scope are sufficient and necessary for solving these games.