Mean-Payoff Pushdown Games

Krishnendu Chatterjee, Yaron Velner
2012 2012 27th Annual IEEE Symposium on Logic in Computer Science  
Two-player games on graphs are central in many problems in formal verification and program analysis such as synthesis and verification of open systems. In this work we consider solving recursive game graphs (or pushdown game graphs) that can model the control flow of sequential programs with recursion. While pushdown games have been studied before with qualitative objectives, such as reachability and ω-regular objectives, in this work we study for the first time such games with the most
more » ... died quantitative objective, namely, mean-payoff objectives. In pushdown games two types of strategies are relevant: (1) global strategies, that depend on the entire global history; and (2) modular strategies, that have only local memory and thus do not depend on the context of invocation, but only on the history of the current invocation of the module. Our main results are as follows: (1) One-player pushdown games with mean-payoff objectives under global strategies are decidable in polynomial time. (2) Twoplayer pushdown games with mean-payoff objectives under global strategies are undecidable. (3) One-player pushdown games with mean-payoff objectives under modular strategies are NPhard. (4) Two-player pushdown games with mean-payoff objectives under modular strategies can be solved in NP (i.e., both one-player and two-player pushdown games with mean-payoff objectives under modular strategies are NP-complete). We also establish the optimal strategy complexity showing that global strategies for mean-payoff objectives require infinite memory even in one-player pushdown games; and memoryless modular strategies are sufficient in twoplayer pushdown games. Finally we also show that all the problems have the same complexity if the stack boundedness condition is added, where along with the mean-payoff objective the player must also ensure that the stack height is bounded. Microsoft faculty fellows award, and the RICH Model Toolkit (ICT COST Action IC0901) with every transition and the goal of one of the players is to maximize the long-run average of the rewards (and the goal of the opponent is to minimize). Two-player finite-state games with meanpayoff objectives have been studied in [16, 36, 27] , and more recently applied in synthesis of reactive systems with quality guarantee [6] and robustness [7] . Moreover recently many quantitative logics and automata theoretic formalisms have been proposed with mean-payoff objectives in their heart to express properties such as reliability requirements, and resource bounds of reactive systems [12, 8, 15] . Thus pushdown games with mean-payoff objectives would be a central theoretical question for model checking of quantitative logics (specifying reliability and resource bounds) on reactive systems with recursion feature. Pushdown mean-payoff games. In this work we study for the first time pushdown games with mean-payoff objectives (to the best of our knowledge mean-payoff objectives have not been studied in the context of pushdown games). In pushdown games two types of strategies are relevant and studied in literature. The first is the global strategies, where a global strategy can choose the successor state depending on the entire global history of the play (where history is the finite sequence of configurations of the current prefix of a play). The second is the modular strategies, and modular strategies are understood more intuitively in the model of games on recursive state machines. A recursive state machine (RSM) consists of a set of component machines (or modules). Each module has a set of nodes (atomic states) and boxes (each of which is mapped to a module), a well-defined interface consisting of entry and exit nodes, and edges connecting nodes/boxes. An edge entering a box models the invocation of the module associated with the box and an edge leaving the box represents return from the module. In the game version the nodes are partitioned into player-1 nodes and player-2 nodes. Due to recursion the underlying global state-space is infinite and isomorphic to pushdown games. The equivalence of pushdown games and recursive games has been established in [5] . A modular strategy is a strategy that has only local memory, and thus, the strategy does not depend on the context of invocation of the module, but only on the history within the current invocation of the module. In other words, modular strategies are appealing because they are stackless strategies, decomposable into one for each module. In this work we will study pushdown games with mean-payoff objectives for both global and modular strategies. Previous results. Pushdown games with qualitative objectives were studied in [33, 32] . It was shown in [33] that solving pushdown games (i.e., determining the winner in pushdown games) with reachability objectives under global strategies is EXPTIME-hard, and pushdown games with parity objectives under global strategies can be solved in EXPTIME. Thus it follows that pushdown games with reachability and parity objectives under global strategies are EXPTIME-complete. The notion of modular strategies in games on recursive state machines was introduced in [5, 4]. It was shown that the modular strategies problem is NP-complete in pushdown games with reachability and parity objectives in general [5, 4] . The results of [5] also presents more refined complexity results in terms of the number of exit nodes, showing that if every module has single exit, then the problem is polynomial for reachability objectives [5] and in NP ∩ coNP for parity objectives [4] . Our contributions. In this work we present a complete characterization of the computational and strategy complexity of pushdown games and pushdown systems (one-player pushdown games or pushdown automata) with mean-payoff objectives. Solving a pushdown system (resp. pushdown game) with respect to a mean-payoff objective is to decide whether there exists a path that (resp. a winning strategy to ensure that every path possible given the strategy) satisfies the mean-payoff objective. Our main results for computational complexity are as follows. 1. Global strategies. We show that pushdown systems (one-player pushdown games) with mean-payoff objectives under global strategies can be solved in polynomial time, whereas solving pushdown games with mean-payoff objectives under global strategies is undecidable. 2. Modular strategies. Solving pushdown systems with single exit nodes with mean-payoff objectives under modular strategies is NP-hard, and pushdown games with mean-payoff objectives under modular strategies can be solved in NP. Thus both pushdown systems and pushdown games with mean-payoff objectives under modular strategies are NP-complete. Our results are shown in Table 1 . First observe that our hardness result for modular strategies is different from the NP-hardness of [5] because the hardness result of [5] shows hardness for games with reachability objectives and require that the number of modules with multiple exit nodes are not bounded (in fact if every module of the recursive game has a single exit, then the problem is in PTIME Global strategies Modular strategies Pushdown systmes PTIME NP-complete (NP-hard for single exit) Pushdown games Undecidable NP-complete
doi:10.1109/lics.2012.30 dblp:conf/lics/ChatterjeeV12 fatcat:np3mdikzubgn3jnqjtqfa7gip4