We consider two-player turn-based games with zero-reachability and zero-safety objectives generated by extended vector addition systems with states. Although the problem of deciding the winner in such games is undecidable in general, we identify several decidable and even tractable subcases of this problem obtained by restricting the number of counters and/or the sets of target configurations.
Vector addition systems with states (VASS) are an abstract computational model equivalent to Petri nets (see, e.g., [27,29]) which is well suited for modelling and analysis of distributed concurrent systems. Roughly speaking, a k-dimensional VASS, where k ≥ 1, is an automaton with a finite control and k unbounded counters which can store non-negative integers. Depending on its current control state, a VASS can choose and perform one of the available transitions. A given transition changes the control state and updates the vector of current counter values by adding a fixed vector of integers which labels the transition. For simplicity, we assume that transition labels can increase/decrease each counter at most by one. Since the counters cannot become negative, transitions which attempt to decrease a zero counter are disabled. Configurations of a given VASS are written as pairs pv, where p is a control state and v ∈ N k a vector of counter values.
In this paper, we consider extended VASS games which enrich the modelling power of VASS in two orthogonal ways.
(1) Transition labels can contain symbolic components (denoted by ω) whose intuitive meaning is “add an arbitrarily large non-negative integer to a given counter”. For example, a single transition p -→ q labeled by (1, ω) represents an infinite number of “ordinary” transitions labeled by (1,0), (1,1), (1,2), . . . A natural source of motivation for introducing symbolic labels are systems with multiple resources that can be consumed and produced simultaneously by performing a transition. The ω components can then be conveniently used to model “resource reloading” (see also the example below).
(2) To model the interaction between a system and its environment, the set of control states is split into two disjoint subsets of controllable and environmental states. Transitions from the controllable and environmental states then correspond to the events generated by the system and its environment, respectively.
Hence, the semantics of a given extended VASS game M is a possibly infinitely-branching turn-based game G M with infinitely many vertices which correspond to the configurations of M. The game G M is initiated by putting a token on some configuration pv. The token is then moved from vertex to vertex by two players, and , who select transitions in the controllable and environmental configurations according to some strategies. Thus, they produce an infinite sequence of configurations called a play. Desired properties of M can be formalized as objectives, i.e., admissible plays. The central problem is the question whether player (the system) has a winning strategy which ensures that the objective is satisfied for every strategy of player (the environment). We refer to, e.g., [32,13,35] for more comprehensive expositions of results related to games in formal verification. In this paper, we are mainly interested in zero-safety objectives (or, dually, zero-reachability objectives), consisting of plays where no counter is decreased to zero, i.e., a given system never reaches a situation when some of its resources are insufficient. As a simple example, consider a workshop which “consumes” wooden sticks, screws, wires, etc., and produces puppets of various kinds which are then sold at the door. From time to time, the manager may decide to issue an order for screws or other supplies, and thus increase their number by a finite but essentially unbounded amount (the manager certainly aims at choosing the “right” number of screws which are needed to produce all puppets that can be sold in next few days). Controllable states can be used to model the actions taken by workshop employees, and environmental states model the behaviour of unpredictable customers. We wonder whether the workshop manager has a strategy which ensures that at least one puppet of each kind is always available for sell, regardless what the unpredictable customers do (the model can of course reflect only selected aspects of customers’ behaviour). Note that a winning strategy for the manager must also resolve the symbolic ω value used to model the order of screws by specifying a concrete number of screws that should be ordered.
Technically, we consider extended VASS games with non-selective and selective zeroreachability objectives, where the set of target configurations that should be reached by player and avoided by player is either Z and Z C , respectively. Here, -the set Z consists of all pv such that v ℓ = 0 for some ℓ (i.e., some counter is zero); -the set Z C , where C is a subset of control states, consists of all pv ∈ Z such that p ∈ C.
Our main results can be summarized as follows:
(a) The problem of deciding the winner in k-dimensional extended VASS games (where k ≥ 2) with Z-reachability objectives is in (k-1)-EXPTIME. (b) A finite description of the winning region for each player (i.e., the set of all vertices where the player wins) is computable in (k-1)-exponential time. (c) Winning
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