Small witnesses, accepting lassos and winning strategies in omega-automata and games

Obtaining accepting lassos, witnesses and winning strategies in omega-automata and games with omega-regular winning conditions is an integral part of many formal methods commonly found in practice today. Despite the fact that in most applications, the lassos, witnesses and strategies found should be as small as possible, little is known about the hardness of obtaining small such certificates. In this paper, we survey the known hardness results and complete the complexity landscape for the cases not considered in the literature so far. We pay particular attention to the approximation hardness of the problems as approximate small solutions usually suffice in practice.

💡 Research Summary

The paper conducts a comprehensive study of the computational complexity and approximability of finding small certificates—accepting lassos, witnesses (finite words of the form uv^ω), and winning strategies—in ω‑automata and two‑player games with ω‑regular winning conditions. The authors begin by motivating the importance of small certificates in formal verification, synthesis, and model checking, noting that smaller witnesses are easier to understand and more efficient to use in subsequent steps such as circuit synthesis or error tracing.

A key contribution is a unified definition of certificate size. For arena‑based strategies (positional or finite‑memory), size is measured by the number of reachable (position, memory) pairs along some execution; for stand‑alone strategies, size is the number of states in the corresponding Mealy or Moore machine. The size of a lasso is the length of its cycle plus the length of the prefix leading to the cycle, and the size of a witness uv^ω is |u| + |v|. These definitions allow the authors to treat automata (one‑player games) and two‑player games within the same framework.

The paper first surveys known results: shortest accepting lassos for Büchi automata can be computed in polynomial time, while for generalized Büchi automata the problem is NP‑complete; finding shortest witnesses is NP‑complete even for Büchi automata; and determining a minimum positional strategy in safety games is NP‑complete with no PTAS. Building on this foundation, the authors fill the gaps in the literature by proving NP‑completeness (or polynomial‑time solvability) for all remaining acceptance conditions—co‑Büchi, Rabin, Streett, Muller, parity, and generalized Büchi—both for automata and for games where memoryless determinacy does not hold.

All the decision problems are shown to lie in NP, and the authors present a novel reduction that establishes approximation hardness: for many of the problems, any polynomial‑factor approximation would imply P = NP. In particular, they prove that approximating the minimal witness size within any polynomial factor is NP‑hard even for safety automata, and that approximating the smallest arena‑based strategy in safety games suffers the same hardness. Consequently, in practice, one cannot hope for efficient algorithms that guarantee a bounded approximation ratio for these tasks.

Despite these negative results, the paper also offers a positive contribution: an exponential‑approximation scheme that, for a wide class of problems, produces a certificate whose size is at most exponential in the optimum. This scheme provides a theoretical fallback when exact solutions are infeasible.

An important structural insight is that for acceptance conditions that guarantee memoryless determinacy (e.g., parity, Büchi, co‑Büchi), the minimal sizes of arena‑based and stand‑alone strategies coincide. This equivalence means that designers can freely choose the representation that best fits downstream processing (e.g., converting a strategy into a hardware circuit).

The authors summarize all results in a comprehensive table (Table 1, p. 8), clearly delineating the boundary between tractable and intractable cases, as well as between problems that admit any polynomial approximation and those that do not. This complete landscape serves both theoreticians—who can now identify open directions such as tighter approximation algorithms for the few remaining cases—and practitioners, who can better assess the feasibility of obtaining small certificates for their specific verification or synthesis tasks.