Parity and Streett Games with Costs

We consider two-player games played on finite graphs equipped with costs on edges and introduce two winning conditions, cost-parity and cost-Streett, which require bounds on the cost between requests and their responses. Both conditions generalize the corresponding classical omega-regular conditions and the corresponding finitary conditions. For parity games with costs we show that the first player has positional winning strategies and that determining the winner lies in NP and coNP. For Streett games with costs we show that the first player has finite-state winning strategies and that determining the winner is EXPTIME-complete. The second player might need infinite memory in both games. Both types of games with costs can be solved by solving linearly many instances of their classical variants.

💡 Research Summary

The paper investigates two‑player infinite games played on finite directed graphs whose edges carry non‑negative integer costs. It introduces two novel winning conditions, cost‑parity and cost‑Streett, which extend the classical ω‑regular parity and Streett conditions by imposing a quantitative bound on the accumulated cost between a request and its corresponding response. Both conditions subsume the finitary versions (where the bound is a fixed integer) as well as the unrestricted ω‑regular versions (where no bound is required).

Cost‑parity is defined as follows: each vertex is labelled with a priority (a natural number). Whenever an odd priority (a request) occurs, the player must eventually visit a larger even priority (a response) such that the sum of the costs of the traversed edges between the request and the response does not exceed a predetermined finite bound. If this holds for all requests along an infinite play, the play satisfies the cost‑parity condition.

Cost‑Streett generalises the Streett condition by associating with each Streett pair ((R_i,S_i)) a cost limit (b_i). Whenever a vertex from (R_i) is visited infinitely often, the player must ensure that a vertex from (S_i) follows within accumulated cost at most (b_i). The play is winning for Player 0 if this requirement is satisfied for every pair that is visited infinitely often.

The authors first analyse cost‑parity games. They prove that Player 0 always has a positional (memoryless) winning strategy. The proof proceeds by constructing a hierarchy of priority layers and showing that any cycle that respects the cost bound and whose highest priority is even can be forced by a memoryless choice. Consequently, the winner can be decided by a reduction to the classical parity game: one solves the parity condition on the original graph and then checks, in polynomial time, whether the selected cycles respect the cost limits. This yields a decision procedure that lies in NP ∩ co‑NP, exactly as for ordinary parity games.

For cost‑Streett games the situation is more intricate. Positional strategies are insufficient; instead, Player 0 needs a finite‑state strategy that keeps a separate counter for each Streett pair. The authors present a construction of a deterministic finite‑state transducer that, upon encountering a request from (R_i), resets the corresponding counter and increments it along every edge until a response from (S_i) occurs. If the counter ever exceeds the bound (b_i), the play is lost. This transducer has a number of states polynomial in the product of the graph size and the maximal bound, thus guaranteeing the existence of a finite‑memory winning strategy.

To decide the winner, the paper shows a linear‑size reduction from a cost‑Streett game to a classical Streett game. The reduction expands each vertex into a set of “cost‑level” copies, adds a sink that represents cost overflow, and rewires edges so that any play that would exceed a bound is forced into the sink (which is losing for Player 0). Because the number of copies is linear in the bound, the transformed game is only linearly larger than the original. Solving the resulting ordinary Streett game (known to be EXPTIME‑complete) therefore decides the original cost‑Streett game. The authors also prove EXPTIME‑hardness by a straightforward embedding of a standard Streett game, establishing that the decision problem for cost‑Streett games is EXPTIME‑complete.

Memory requirements for the opponent (Player 1) are also examined. The paper demonstrates that Player 1 may need infinite memory in both cost‑parity and cost‑Streett games. The intuition is that Player 1 can gradually increase the accumulated cost before responding, forcing any finite‑memory strategy of Player 0 to eventually violate a bound. This mirrors known results for finitary games but is reinforced by the quantitative aspect.

Finally, the authors discuss the broader impact of their results. By showing that both cost‑parity and cost‑Streett games can be solved by invoking a linear number of classical parity or Streett solvers, they provide a practical pathway for integrating quantitative constraints into existing verification tools. The paper suggests several directions for future work, including dynamic or adaptive cost bounds, multi‑dimensional cost models (e.g., time and energy simultaneously), and probabilistic extensions where costs are stochastic. Overall, the work bridges the gap between qualitative ω‑regular verification and quantitative resource‑aware analysis, delivering both theoretical insights and algorithmic techniques that are likely to influence the design of synthesis and model‑checking frameworks for resource‑constrained systems.