Solving Parity Games on Integer Vectors

Abstract

Abstract. We consider parity games on infinite graphs where configurations are represented by control-states and integer vectors. This framework subsumes two classic game problems: parity games on vector addition systems with states (VASS) and multidimensional energy parity games. We show that the multidimensional energy parity game problem is inter-reducible with a subclass of singlesided parity games on VASS where just one player can modify the integer counters and the opponent can only change control-states. Our main result is that the minimal elements of the upward-closed winning set of these single-sided parity games on VASS are computable. This implies that the Pareto frontier of the minimal initial credit needed to win multidimensional energy parity games is also computable, solving an open question from the literature. Moreover, our main result implies the decidability of weak simulation preorder/equivalence between finite-state systems and VASS, and the decidability of model checking VASS with a large fragment of the modal µ-calculus.

Alternating Vector Addition Systems with States

“…can then be envisioned, and correspond to conditions that must be satisfied by all the branches of a deduction tree. As shown by Abdulla et al [2], such asymmetric games are closely related to multi-dimensional energy games [8,6], see Section 5.…”

“…The asymmetric game semantics described in § 2.1.2 is easily seen to be equivalent to one-sided VASS games as defined in [23,2]. Such a game is played on a VASS with a partitioned state space Q = Q ♦ Q , where Controller owns the states in Q ♦ and can freely manipulate the current vector value, while Environment owns the states in Q and can only change the current state: if q u − → q is a rule in T u and q ∈ Q , then u = 0; these restricted Environment rules correspond to AVASS fork rules.…”

“…Multi-dimensional Energy Games. Abdulla et al [2] have shown the equivalence of AVASS games with the (multi-dimensional) energy games of Brázdil et al [6] and Chatterjee et al [8], where the asymmetry between Controller and Environment is not enforced in the structure of the AVASS or in restricted unary rules for Environment: in such a game, Environment can use arbitrary unary rules. This would lead to an undecidable state reachability game when played on the Q × N d arena [1], but energy games are played instead over Q×Z d -which means that unary rules can be applied even if they yield some negative vector components.…”

“…Surprisingly, this is not really the case: if there is indeed a large number of denominations (e.g. B-VASS games [23], Z-reachability games [6], multi-dimensional energy games [8]), Abdulla, Mayr, Sangnier, and Sproston [2] noted last year that they all defined essentially the same asymmetric class of games, where one player is restricted and cannot update the vector values.…”

“…Indeed, we relate it to energy games in Section 5 (following [2]), to Horn fragments of affine linear logic in Section 6 (following [15]), and to regular simulation problems for VASS in Section 7. Our lower bound improves on all the published bounds for those problems, including the ExpSpace-hardness of simulations between basic parallel processes and finite-state processes due to Lasota [18].…”