Synthesis from Weighted Specifications with Partial Domains over Finite Words

In this paper, we investigate the synthesis problem of terminating reactive systems from quantitative specifications. Such systems are modeled as finite transducers whose executions are represented as finite words in $(I\times O)^$, where $I,O$ are finite sets of input and output symbols, respectively. A weighted specification $S$ assigns a rational value (or $-\infty$) to words in $(I\times O)^$, and we consider three kinds of objectives for synthesis, namely threshold objectives where the system’s executions are required to be above some given threshold, best-value and approximate objectives where the system is required to perform as best as it can by providing output symbols that yield the best value and $\varepsilon$-best value respectively w.r.t. $S$. We establish a landscape of decidability results for these three objectives and weighted specifications with partial domain over finite words given by deterministic weighted automata equipped with sum, discounted-sum and average measures. The resulting objectives are not regular in general and we develop an infinite game framework to solve the corresponding synthesis problems, namely the class of (weighted) critical prefix games.

💡 Research Summary

The paper studies the synthesis of terminating reactive systems from quantitative specifications defined over finite words. A specification S maps each word in (I × O)⁎ to a rational value (or –∞). Unlike classical synthesis, the domain of S may be partial: only inputs that belong to dom(S) are required to be handled, which reflects realistic assumptions about the environment. Three quantitative synthesis objectives are considered. (1) Threshold synthesis: for every input u∈dom(S) the produced output must satisfy S(u⊗f(u)) ⊲ t for a given threshold t (⊲∈{>,≥}). (2) Best‑value synthesis: the system must achieve the maximal possible value bestValS(u)=sup{S(u⊗v) | v∈O⁎} for each input. (3) Approximate synthesis: the produced value must be within a prescribed error r of bestValS(u).

The specifications are given by deterministic weighted automata (DWFA) equipped with one of three payoff functions: Sum, Avg, or Discounted‑Sum (Dsum with discount factor λ∈(0,1)). The paper provides a complete decidability and complexity landscape for all combinations of payoff and objective (see Table 1). For example, threshold synthesis for Sum‑automata is NP∩coNP‑complete (both strict and non‑strict), best‑value synthesis for Sum‑automata is in P, while approximate synthesis for Sum‑automata is EXP‑complete. For Dsum‑automata, strict threshold synthesis lies in NP, best‑value synthesis in NP∩coNP, and approximate synthesis is NEXP‑time when λ=1/n, otherwise EXP‑complete‑c.

To obtain these results the authors introduce a novel infinite‑duration game model called weighted critical‑prefix games. The game is played by Adam (environment) and Eve (system) who alternately choose input and output symbols. Certain vertices are marked as “critical”. When the play reaches a critical vertex, a quantitative requirement derived from S must be satisfied immediately; otherwise Eve loses. By encoding the three synthesis problems as critical‑prefix games, the authors reduce them to known quantitative game frameworks (mean‑payoff, discounted‑sum, energy) or to new variants. For Sum and Avg, critical‑prefix threshold games are reduced to mean‑payoff games, yielding NP∩coNP upper bounds. For Dsum, non‑strict thresholds reduce to discounted‑sum games (polynomial time), while strict thresholds require a new technique: memoryless strategies suffice, and checking a memoryless strategy can be done in polynomial time. This also yields a polynomial‑time algorithm for the non‑emptiness of nondeterministic discounted‑sum max‑automata, a problem previously only known to be in PSPACE.

Best‑value synthesis corresponds to zero‑regret determinization of nondeterministic weighted automata. The authors show that for Sum‑ and Avg‑automata this problem is in P, and for Dsum‑automata it is in NP∩coNP, improving earlier NP results for infinite words. Approximate synthesis maps to r‑regret determinization. For Sum‑automata this is EXP‑complete; for Avg‑automata the authors devise a reduction to a new class of partial‑observation critical‑prefix energy games, proving decidability for the subclass needed in synthesis. For Dsum‑automata, they adapt techniques from existing r‑regret results, obtaining NEXP‑time for λ=1/n and EXP‑complete‑c otherwise.

A technical contribution is the construction of domain‑safe automata. By preprocessing a DWFA in polynomial time, the need to monitor whether the current input belongs to dom(S) is eliminated; the game can be played on the transformed automaton where any deviation from the domain leads to an immediate loss for Eve. This avoids an exponential blow‑up that would arise from naïvely tracking subsets of states.

Finally, the authors extend their results to the Church synthesis problem over infinite words for certain weighted safety specifications, showing that if every prefix must satisfy a quantitative requirement, the problem remains decidable.

In summary, the paper delivers (1) a comprehensive decidability and complexity map for quantitative synthesis with partial domains over finite words, (2) a new game‑theoretic tool—weighted critical‑prefix games—that captures the essence of these synthesis problems, and (3) concrete algorithmic techniques for Sum, Avg, and Discounted‑Sum specifications across threshold, best‑value, and approximate objectives. These contributions advance the state of the art in automated system design by enabling quantitative guarantees under realistic environmental assumptions.