Alternating-Time Temporal Logic with Mean-Payoff Guarantees

2026-08-02

AI Summary

This work investigates whether a coalition in multi-agent systems admits a single strategy that simultaneously satisfies temporal objectives and guarantees that the long-run average payoff meets or exceeds a given threshold across multiple dimensions. To this end, we extend the ATL* logic over weighted concurrent game structures by introducing ATL*_mp, which tightly integrates strategic modalities with temporal goals and conjunctive multi-dimensional mean-payoff constraints, requiring correctness against all possible behaviors of the remaining agents. Our main contributions include the first formal integration of mean-payoff guarantees with ATL* temporal specifications; a characterization of the memory requirements for such strategies, establishing tight linear upper and lower bounds; a proof that model checking under one-dimensional constraints is 2EXPTIME-complete; a demonstration that finite-memory strategies can approximate the optimal values achievable under perfect recall; and illustrative case studies highlighting the framework’s expressiveness in strategy synthesis and multi-criteria optimization.

Abstract

Alternating-time temporal logic and its extensions provide several ways of combining strategic and quantitative reasoning. We study a particular combination: whether a coalition has a single strategy that enforces a temporal objective while guaranteeing given long-run mean-payoff thresholds. We introduce ATL*_mp, an extension of ATL* over weighted concurrent game structures in which each strategic modality carries a conjunctive mean-payoff constraint. The temporal and quantitative requirements must hold against every behaviour of the remaining agents, and the existence of such a strategy cannot in general be reduced to the two requirements considered separately. For one-dimensional constraints, model checking is 2EXPTIME-complete under both perfect-recall and finite-memory semantics, matching ATL*. For the pure quantitative fragment and fragments restricted to ATL or GR(1) temporal objectives, model checking has lower complexity. With multi-dimensional conjunctive constraints, model checking under finite-memory semantics remains 2EXPTIME-complete. We show that memoryless, finite-memory, and perfect-recall abilities form a strict hierarchy, while finite-memory strategies still achieve every threshold strictly below the perfect-recall supremum. We give tight linear upper and lower bounds on the required memory as a function of the denominator of the threshold, even when the game and temporal monitor are fixed. We give several examples of properties expressible in the logic, including temporal synthesis with performance guarantees and aggregate and multi-criteria objectives. We also relate the logic to cooperative rational verification, showing that it can express beneficial deviations from fixed payoff baselines, but not directly reproduce the standard ATL* encoding of the core for dichotomous preferences.

Problem

• Alternating-Time Temporal Logic
• Mean-Payoff
• Strategic Reasoning
• Quantitative Verification
• Concurrent Game Structures

Innovation

• Alternating-Time Temporal Logic
• Mean-Payoff Constraints
• Weighted Concurrent Games
• Finite-Memory Strategies
• Model Checking Complexity