💻 computer science

Alternating-Time Temporal Logic with Mean-Payoff Guarantees

This paper introduces ATL*_mp, an extension of Alternating-Time Temporal Logic that combines strategic reasoning with long-run mean-payoff constraints on weighted concurrent game structures, establishing that model checking is 2EXPTIME-complete for one-dimensional and multi-dimensional cases while characterizing the strict hierarchy of memory requirements and the logic's expressiveness for performance-guaranteed synthesis and cooperative rational verification.

Original authors: Muhammad Najib

Published 2026-08-04

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CC BY 4.0

Original authors: Muhammad Najib

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the director of a massive, chaotic theme park with thousands of moving parts: roller coasters, food stalls, and security teams, all controlled by different groups of agents. Your job isn't just to make sure the rides don't crash (a safety check); you also need to ensure the park makes enough money, keeps the lines moving fast, and treats every visitor fairly over the long run. In the world of computer science, this is the challenge of "multi-agent systems." Scientists use special languages called logics to write rules for these digital worlds. One famous language, called ATL, is like a manager asking, "Can my team of robots force the system to stay safe, no matter what the other robots do?" But ATL has a blind spot: it can check if the ride is safe, but it can't check if the ride is profitable or efficient over time. It's like checking if a car has brakes, but not checking how much gas it burns. To fix this, researchers needed a way to mix "safety rules" with "long-term scorekeeping," creating a new kind of logic that can demand both a happy ending and a high score simultaneously.

This paper introduces a new, super-charged logic called ATL∗mp (Alternating-Time Temporal Logic with Mean-Payoff guarantees). Think of it as a new rulebook for our theme park manager. The author shows that you can now ask a very specific, powerful question: "Can my team of robots find one single plan that keeps the park safe forever and guarantees that we earn a specific amount of money per hour, no matter how the other agents try to mess things up?" The big surprise they found is that you can't just check safety and money separately and hope they work together. Sometimes, a team has a plan to be safe and a different plan to be rich, but no single plan that does both. The new logic forces the team to find that "perfect plan" that does everything at once.

The researcher proved that checking if such a perfect plan exists is incredibly hard for computers to solve—so hard that it takes a massive amount of time, even for the smartest algorithms we have (a complexity class called 2Exptime). However, they also discovered some fascinating rules about how much "memory" the robots need. If the robots have perfect memory (remembering every single move ever made), they can achieve the absolute best possible score. If they only have a small, finite memory (like a simple checklist), they can get almost as good as the perfect score, but they might miss the exact top number. The paper shows that to get very close to that perfect score, the robots might need a checklist that grows huge depending on how precise the score target is. For example, if you want a score of 1/3, they need a certain amount of memory; if you want 1/1000, they need a much bigger memory.

The paper also explores what happens when you have multiple goals at once, like maximizing profit for two different food stalls simultaneously. They found that while the logic can handle these complex, multi-goal scenarios, it hits a wall when trying to solve certain "cooperative" problems where the goal depends on comparing the current score to a moving target. In simple terms, the new logic is great at saying, "Make sure we earn at least $100," but it struggles to say, "Make sure we earn more than the other team earned in the last round," because the "last round's score" keeps changing.

In the end, the author provides a complete map of how hard it is to solve these problems, showing exactly where the limits of our current computer power lie. They didn't just invent a new language; they built a rigorous testing ground that tells us exactly what is possible, what is impossible, and how much memory our digital agents need to be truly successful in a complex, competitive world.

Technical Summary: Alternating-Time Temporal Logic with Mean-Payoff Guarantees

Problem Statement
Alternating-time temporal logic (ATL) and its extension ATL* are standard formalisms for reasoning about strategic abilities in multi-agent systems, specifically whether a coalition can enforce a temporal objective regardless of the actions of other agents. However, these logics lack the capacity to express long-run quantitative performance guarantees, such as energy consumption, throughput, or average rewards. Conversely, mean-payoff games capture long-run quantitative objectives but do not inherently combine them with complex temporal requirements.

This paper addresses the gap by investigating whether a coalition possesses a single strategy that simultaneously enforces a temporal objective and guarantees specific long-run mean-payoff thresholds against every counter-strategy. The author emphasizes that this combined requirement is strictly stronger than requiring the temporal and quantitative objectives to be enforceable separately; a strategy satisfying the temporal goal might fail the payoff constraint, and vice versa.

Methodology and Logic Definition
The author introduces ATL* $_{mp}$ , a conservative extension of ATL* interpreted over Weighted Concurrent Game Structures (WCGS). In this framework:

Syntax: Strategic modalities are augmented with mean-payoff constraints: $\langle\langle C \rangle\rangle_{\Lambda} \psi$ . Here, $\Lambda$ is a conjunction of constraints of the form $mp_j \geq q$ (mean payoff in dimension $j$ is at least $q$ ), and $\psi$ is a temporal path formula (LTL).
Semantics: The logic requires the existence of a single coalition strategy that ensures the temporal property $\psi$ and the quantitative constraints $\Lambda$ hold for all plays generated against any strategy of the opposing agents.
Memory Models: The paper analyzes the logic under three strategy classes: memoryless (positional), finite-memory, and perfect-recall.

Key Technical Contributions

Non-Decomposability: The paper proves that the combined modality $\langle\langle C \rangle\rangle_{\Lambda} \psi$ is not equivalent to the conjunction of separate qualitative and quantitative abilities ( $\langle\langle C \rangle\rangle \psi \land \langle\langle C \rangle\rangle_{\Lambda} \top$ ). A single strategy must satisfy both conditions simultaneously.
Round-Preserving Sequentialisation: To reduce the verification problem to known game-theoretic problems, the author proposes a specific sequentialisation of concurrent games. Unlike standard sequentialisations that insert intermediate states (which disrupt "next" operators in LTL), this construction preserves the "round" structure. It couples the game with a deterministic parity automaton (DPA) for the temporal objective, ensuring the automaton advances exactly once per concurrent round. This establishes a correspondence between strategies in the original concurrent game and the resulting turn-based mean-payoff parity game.
Model Checking Algorithms:
- One-Dimensional Constraints: For constraints involving a single weight dimension, the model checking problem is shown to be 2Exptime-complete under both perfect-recall and finite-memory semantics. This matches the complexity of standard ATL*. The upper bound is driven by the determinisation of the temporal objective (LTL to DPA), while the game solving (mean-payoff parity) is in NP $\cap$ coNP.
- Multi-Dimensional Constraints: For arbitrary conjunctive constraints over multiple dimensions, model checking under finite-memory semantics remains 2Exptime-complete. The proof relies on reducing the problem to the arbitrary-initial-credit problem for multi-dimensional energy parity games.
- Memoryless Semantics: Under memoryless semantics, the problem is PSpace-complete, even with arbitrary conjunctive constraints.
Memory Hierarchy and Bounds:
- The paper establishes a strict hierarchy of strategic abilities: memoryless $\subset$ finite-memory $\subset$ perfect-recall. There exist formulas satisfiable only by perfect-recall strategies and others satisfiable by finite-memory but not memoryless strategies.
- However, for one-dimensional constraints, finite-memory strategies can approximate perfect-recall abilities arbitrarily closely. Specifically, if a threshold $q$ is enforceable by perfect recall, any threshold $q' < q$ is enforceable by a finite-memory strategy.
- The author provides tight bounds on memory requirements, showing that the size of a finite-memory witness may grow linearly with the denominator of the threshold (and thus exponentially with the binary encoding of the threshold), even for fixed games and temporal objectives.
Expressiveness and Applications:
- The logic supports temporal synthesis with performance guarantees, allowing for the specification of reactive controllers that must satisfy safety/liveness properties while maintaining long-run resource bounds.
- It supports aggregate and multi-criteria objectives, such as utilitarian (sum of utilities) or egalitarian (minimum of utilities) guarantees.
- Rational Verification: The paper relates ATL* $_{mp}$ to cooperative rational verification (specifically the "core"). It demonstrates that while the logic can express deviations from a fixed payoff baseline, it cannot directly encode the standard core definition for mean-payoff preferences because the deviation threshold depends on the payoff of the candidate profile, which the logic cannot name or compare dynamically.

Significance and Open Problems
The paper demonstrates that combining strategic temporal reasoning with mean-payoff constraints is decidable and retains the same worst-case complexity as ATL* for one-dimensional constraints, despite the added quantitative dimension. The introduction of round-preserving sequentialisation is a key methodological advance for handling concurrent games with temporal objectives.

The primary open question identified is the decidability and complexity of multi-dimensional mean-payoff parity games under perfect-recall strategies. While finite-memory semantics are fully characterized, the perfect-recall case for multi-dimensional constraints remains unresolved. Additionally, the author notes that their strategy correspondence relies on perfect information and deterministic transitions, leaving the extension to imperfect-information or stochastic models as future work.

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