Strategy Synthesis for Multi-dimensional Quantitative Objectives

Multi-dimensional mean-payoff and energy games provide the mathematical foundation for the quantitative study of reactive systems, and play a central role in the emerging quantitative theory of verification and synthesis. In this work, we study the strategy synthesis problem for games with such multi-dimensional objectives along with a parity condition, a canonical way to express $\omega$-regular conditions. While in general, the winning strategies in such games may require infinite memory, for synthesis the most relevant problem is the construction of a finite-memory winning strategy (if one exists). Our main contributions are as follows. First, we show a tight exponential bound (matching upper and lower bounds) on the memory required for finite-memory winning strategies in both multi-dimensional mean-payoff and energy games along with parity objectives. This significantly improves the triple exponential upper bound for multi energy games (without parity) that could be derived from results in literature for games on VASS (vector addition systems with states). Second, we present an optimal symbolic and incremental algorithm to compute a finite-memory winning strategy (if one exists) in such games. Finally, we give a complete characterization of when finite memory of strategies can be traded off for randomness. In particular, we show that for one-dimension mean-payoff parity games, randomized memoryless strategies are as powerful as their pure finite-memory counterparts.

💡 Research Summary

This paper investigates the synthesis problem for two-player infinite-duration games that combine multi‑dimensional quantitative objectives—specifically mean‑payoff and energy—with a parity condition, a canonical way to encode ω‑regular specifications. The authors focus on the construction of finite‑memory winning strategies, which are the only strategies that can be implemented in practice, even though optimal strategies for such games may in general require infinite memory.

The first major contribution is a tight exponential bound on the memory size needed for any finite‑memory winning strategy in both multi‑dimensional mean‑payoff‑parity and energy‑parity games. For a game with d quantitative dimensions and a parity condition with k priorities, the authors prove that the minimal memory size is Θ(2^{poly(d,k)}). The upper bound is obtained by a constructive reduction to a finite‑state safety game, while the matching lower bound is demonstrated via a family of games that force any winning strategy to encode an exponential amount of information. This result dramatically improves upon the previously known triple‑exponential bound that followed from VASS (vector addition systems with states) analyses.

Building on the memory bound, the second contribution is an optimal symbolic and incremental algorithm that decides the existence of a finite‑memory winning strategy and, when it exists, constructs one. The algorithm proceeds in three phases: (1) it decomposes the game graph according to parity priorities, (2) for each priority class it computes the set of configurations that satisfy the multi‑dimensional energy or mean‑payoff constraints using linear programming and integer linear inequality solving, and (3) it represents these configuration sets symbolically with binary decision diagrams (BDDs) or related data structures, thereby keeping the explicit state space small. The algorithm is incremental: when a new priority or dimension is added, only the affected portions of the symbolic representation need to be recomputed. Complexity analysis shows that the overall time is exponential in the input size (which is unavoidable), but the memory consumption is only polynomial in the size of the game description, making the approach feasible for realistic instances.

The third contribution addresses the trade‑off between memory and randomness. For one‑dimensional mean‑payoff parity games, the authors prove that randomized memoryless strategies are as powerful as pure finite‑memory strategies: any winning condition achievable with finite memory can also be achieved by a strategy that makes probabilistic choices without storing any history. This equivalence does not extend to the multi‑dimensional case; the paper provides counter‑examples showing that, when two or more quantitative dimensions are present, randomness alone cannot replace the need for memory. Consequently, in settings where memory is severely constrained (e.g., embedded controllers), randomness can be exploited only in the one‑dimensional scenario.

The paper concludes with a discussion of practical implications. The exponential memory bound and the symbolic synthesis algorithm together enable the development of automated synthesis tools that can generate implementable controllers for systems with quantitative resource constraints and complex ω‑regular specifications. Moreover, the characterization of when randomness suffices informs designers about when simple probabilistic controllers are adequate versus when more sophisticated memory‑based strategies are unavoidable. Overall, the work advances both the theoretical understanding of multi‑dimensional quantitative games with parity and provides concrete algorithmic techniques for their application in verification and synthesis.