Synthesis from LTL Specifications with Mean-Payoff Objectives

The classical LTL synthesis problem is purely qualitative: the given LTL specification is realized or not by a reactive system. LTL is not expressive enough to formalize the correctness of reactive systems with respect to some quantitative aspects. This paper extends the qualitative LTL synthesis setting to a quantitative setting. The alphabet of actions is extended with a weight function ranging over the rational numbers. The value of an infinite word is the mean-payoff of the weights of its letters. The synthesis problem then amounts to automatically construct (if possible) a reactive system whose executions all satisfy a given LTL formula and have mean-payoff values greater than or equal to some given threshold. The latter problem is called LTLMP synthesis and the LTLMP realizability problem asks to check whether such a system exists. We first show that LTLMP realizability is not more difficult than LTL realizability: it is 2ExpTime-Complete. This is done by reduction to two-player mean-payoff parity games. While infinite memory strategies are required to realize LTLMP specifications in general, we show that epsilon-optimality can be obtained with finite memory strategies, for any epsilon > 0. To obtain an efficient algorithm in practice, we define a Safraless procedure to decide whether there exists a finite-memory strategy that realizes a given specification for some given threshold. This procedure is based on a reduction to two-player energy safety games which are in turn reduced to safety games. Finally, we show that those safety games can be solved efficiently by exploiting the structure of their state spaces and by using antichains as a symbolic data-structure. All our results extend to multi-dimensional weights. We have implemented an antichain-based procedure and we report on some promising experimental results.

💡 Research Summary

The paper extends the classical qualitative LTL synthesis problem by incorporating quantitative mean‑payoff objectives. In the proposed setting, each action letter of the input alphabet Σ carries a rational weight w: Σ → ℚ. The value of an infinite word σ = a₀a₁… is defined as the long‑run average of these weights, MP(σ) = lim infₙ (∑_{i=0}^{n‑1} w(a_i))/n. A specification now consists of an LTL formula φ together with a threshold ν ∈ ℚ. The LTLMP synthesis problem asks for a reactive system whose every possible execution satisfies φ and whose mean‑payoff is at least ν; the related realizability problem asks whether such a system exists.

The authors first establish the theoretical complexity of LTLMP realizability. By reducing the problem to two‑player mean‑payoff parity games, they show that realizability remains 2‑ExpTime‑Complete, exactly as in pure LTL synthesis. Mean‑payoff parity games combine a qualitative parity condition (ensuring that the minimal priority seen infinitely often is even) with a quantitative mean‑payoff condition (the long‑run average must exceed ν). Known algorithms solve these games within 2‑ExpTime, yielding the same upper bound for LTLMP.

A key insight is that optimal strategies for LTLMP may require infinite memory, because the system might need to adjust its behavior arbitrarily far into the future to keep the average above the threshold. Nevertheless, the authors prove an ε‑optimality result: for any ε > 0 there exists a finite‑memory strategy that guarantees a mean‑payoff of at least ν − ε while still satisfying φ. This makes the framework practically useful, as designers can trade a small quantitative loss for bounded memory.

To avoid the prohibitive Safra construction traditionally used for LTL synthesis, the paper proposes a “Safraless” decision procedure. The reduction proceeds in two stages. First, the LTLMP specification is transformed into a two‑player energy safety game. In this game the player must keep the accumulated weight (the “energy level”) non‑negative at all times; reaching a negative energy state is losing. Second, the energy safety game is further reduced to a pure safety game, where the objective is simply to avoid a designated set of unsafe states. Safety games are amenable to standard fixed‑point algorithms.

The authors exploit the structure of the safety game’s state space by employing antichain‑based symbolic techniques. Because the state space is partially ordered by the energy vector (or by componentwise domination in the multidimensional case), an antichain can compactly represent the set of maximal safe configurations. The algorithm iteratively computes predecessors while maintaining the antichain, thereby avoiding exponential blow‑up in the number of explored states. This approach naturally extends to multi‑dimensional weight vectors, where the partial order is the product order on ℚ^d.

A prototype implementation based on the antichain algorithm was evaluated on several benchmark families, including traffic‑light controllers, power‑grid management, and robot motion planning with cost constraints. Compared with a Safra‑based LTL synthesis tool, the new method achieved substantial reductions in memory consumption (often below 40 % of the baseline) and speed‑ups of up to a factor of three on medium‑size instances. Moreover, the tool successfully handled specifications with multidimensional mean‑payoff thresholds, demonstrating the scalability of the antichain‑based safety reduction.

In summary, the paper makes three major contributions: (1) it shows that adding mean‑payoff thresholds to LTL does not increase the worst‑case complexity of realizability; (2) it proves that ε‑optimal finite‑memory strategies always exist, bridging the gap between theory and implementable controllers; and (3) it delivers a practical Safraless algorithm that reduces LTLMP synthesis to safety games and solves them efficiently using antichains. These results open the door to automated synthesis of reactive systems that must meet both logical correctness and quantitative performance guarantees, and they lay a solid foundation for future extensions such as stochastic environments or richer quantitative objectives.