Data vectors generalise finite multisets: they are finitely supported functions into a commutative monoid. We study the question if a given data vector can be expressed as a finite sum of others, only assuming that 1) the domain is countable and 2) the given set of base vectors is finite up to permutations of the domain.Based on a succinct representation of the involved permutations as integer linear constraints, we derive that positive instances can be witnessed in a bounded subset of the domain.For data vectors over a group we moreover study when a data vector is reversible, that is, if its inverse is expressible using only nonnegative coefficients. We show that if all base vectors are reversible then the expressibility problem reduces to checking membership in finitely generated subgroups. Moreover, checking reversibility also reduces to such membership tests.These questions naturally appear in the analysis of counter machines extended with unordered data: namely, for data vectors over (Z d , +) expressibility directly corresponds to checking state equations for Coloured Petri nets where tokens can only be tested for equality. We derive that in this case, expressibility is in NP, and in P for reversible instances. These upper bounds are tight: they match the lower bounds for standard integer vectors (over singleton domains). ACM Subject Classification F.1.1 Models of Computation Keywords and phrases Computation with atoms, vector addition systemsDigital Object Identifier 10.4230/LIPIcs.xxx.yyy.p IntroductionFinite collections of named values are basic structures used in many areas of theoretical computer science. They can be used for instance to model databases snapshots or define the operational semantics of programming languages. We can formalize these as functions v : D → X from some countable domain D of names or data, into some value space X, and call such functions (X-valued) data vectors. Often the actual names used are not relevant and instead one is interested in data vectors up to renaming, i.e., one wants to consider vectors v and w equivalent if v = w • θ for some permutation θ : D → D of the domain. We consider the case where the value space X has additional algebraic structure. Namely, we focus on data vectors where the values are from some commutative monoid (M, +, 0) and where all but finitely many names are mapped to the neutral element. A natural question then asks if a given data vector is expressible as a sum of vectors from a given set, where the monoid operation is lifted to data vectors pointwise.
Abstract. We consider average-energy games, where the goal is to minimize the long-run average of the accumulated energy. While several results have been obtained on these games recently, decidability of average-energy games with a lower-bound constraint on the energy level (but no upper bound) remained open; in particular, so far there was no known upper bound on the memory that is required for winning strategies. By reducing average-energy games with lower-bounded energy to infinite-state mean-payoff games and analyzing the density of low-energy configurations, we show an almost tight doubly-exponential upper bound on the necessary memory, and that the winner of average-energy games with lower-bounded energy can be determined in doubly-exponential time. We also prove EXPSPACE-hardness of this problem. Finally, we consider multi-dimensional extensions of all types of average-energy games: without bounds, with only a lower bound, and with both a lower and an upper bound on the energy. We show that the fully-bounded version is the only case to remain decidable in multiple dimensions.
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