Playing with Repetitions in Data Words Using Energy Games

Abstract

We introduce two-player games which build words over infinite alphabets, and we study the problem of checking the existence of winning strategies. These games are played by two players, who take turns in choosing valuations for variables ranging over an infinite data domain, thus generating multi-attributed data words. The winner of the game is specified by formulas in the Logic of Repeating Values, which can reason about repetitions of data values in infinite data words. We prove that it is undecidable to check if one of the players has a winning strategy, even in very restrictive settings. However, we prove that if one of the players is restricted to choose valuations ranging over the Boolean domain, the games are effectively equivalent to single-sided games on vector addition systems with states (in which one of the players can change control states but cannot change counter values), known to be decidable and effectively equivalent to energy games.Previous works have shown that the satisfiability problem for various variants of the logic of repeating values is equivalent to the reachability and coverability problems in vector addition systems. Our results raise this connection to the level of games, augmenting further the associations between logics on data words and counter systems.We denote by Z the set of integers and by N the set of non-negative integers. We let N + denote the set of integers that are strictly greater than 0. For any set S, we denote by S * (resp. S ω ) the set of all finite (resp. countably infinite) sequences of elements in S. For a sequence σ ∈ S * , we denote its length by |σ|. We denote by P(S) (resp. P + (S)) the set of all subsets (resp. non-empty subsets) of S. Logic of repeating valuesWe recall the syntax and semantics of the logic of repeating values from [10,11]. This logic extends the usual propositional linear temporal logic with the ability to reason about repetitions of data values from an infinite domain. We let this logic use both Boolean variables (i.e., propositions) and data variables ranging over an infinite data domain D. The Boolean variables can be simulated by data variables. However, we need to consider fragments of the logic, for which explicitly having Boolean variables is convenient. Let BVARS = {q, t, . . .} be a countably infinite set of Boolean variables ranging over { , ⊥}, and let DVARS = {x, y, . . .} be a countably infinite set of 'data' variables ranging over D. We denote by LRV the logic whose formulas are defined as follows: 1A valuation is the union of a mapping from BVARS to { , ⊥} and a mapping from DVARS to D. A model is a finite or infinite sequence of valuations. We use σ to denote models and σ(i) denotes the i th valuation in σ, where i ∈ N + . For any model σ and position i ∈ N + , the satisfaction relation |= is defined inductively as follows. The semantics of temporal operators next (X), previous (X −1 ), until (U), since (S) and the Boolean connectives are defined in the usual way, but for the sake of completeness we provide their 1 In a pr...

Parameterized Synthesis for Fragments of First-Order Logic Over Data Words

“…Moreover, in the decidable cases, it will be worthwhile to provide tight bounds on cutoffs and the algorithmic complexity of the decision problem. Like in [7,15,16,30,31], our strategies allow the system to have a global view of the whole program run executed so far. However, it is also perfectly natural to consider uniform local strategies where each process only sees its own actions and possibly those that are revealed according to some causal dependencies.…”

“…Unlike in the case of finite alphabets, there is no canonical definition of regular languages. In fact, the synthesis problem has been studied for N-memory automata [7], the Logic of Repeating Values [16], and register automata [15,30,31]. Though there is no agreement on a "regular" automata model, first-order (FO) logic over data words can be considered as a canonical logic, and this is the specification language we consider here.…”

“…One can distinguish two cases here. In [17], a potentially infinite number of data values are involved in an infinite program run (e.g. by dynamic process generation).…”

“…Previous generalizations of Church's synthesis problem to infinite alphabets were generally synchronous in the sense that the system and the environment perform their actions in strictly alternating order. This assumption was made, e.g., in the above-mentioned recent papers [8,15,17,30,31]. If there are several processes, however, it is realistic to relax this condition, which leads us to an asynchronous setting in which the system has no influence on when the environment acts.…”

Round-Bounded Control of Parameterized Systems

“…Unfortunately, the latter do not allow us to require that all local processes terminate in a final state. Interestingly, tight links between VASS/energy games and games played on infinite domains have recently been established [16].…”