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.
Does the trace language of a given vector addition system (VAS) intersect with a given context-free language? This question lies at the heart of several verification questions involving recursive programs with integer parameters. In particular, it is equivalent to the coverability problem for VAS that operate on a pushdown stack. We show decidability in dimension one, based on an analysis of a new model called grammarcontrolled vector addition systems. IntroductionPushdown systems are a well-known and natural formalization of recursive programs. Vector addition systems (VAS) are widely used to model concurrent systems and programs with integer variables. Pushdown vector addition systems (pushdown VAS) combine the two: They are VAS extended with a pushdown stack and allow to model, for instance, asynchronous programs [6] and, more generally, programs with recursion and integer variables.Despite the model's relevance for automatic program verification, most classical model-checking problems are so far only partially solved. Termination and boundedness are decidable but their complexity is open [12]. Coverability and reachability are known to be Tower-hard [9], but their decidability is open. In fact, reachability and the seemingly simpler coverability problem are essentially the same for pushdown VAS: there is a simple logarithmic-space reduction from reachability to coverability that only adds one extra dimension.Contributions. Our main result is that coverability is decidable for 1-dimensional pushdown VAS. We work with a new grammar-based model called grammarcontrolled vector addition systems (GVAS), which amounts to VAS restricted to firing sequences defined by a context-free grammar. In dimension one, this model corresponds to two-stack pushdown systems where one of the two stacks uses a single stack symbol. To prove our main result, we show that it is enough to check finitely many potential certificates of coverability. The latter are parse trees of the context-free grammar annotated with counter information from the 1-dimensional VAS. We truncate these annotated parse trees thanks to an This work was partially supported by ANR project ReacHard (ANR-11-BS02-001).
Blondin et al. showed at LICS 2015 that two-dimensional vector addition systems with states have reachability witnesses of length exponential in the number of states and polynomial in the norm of vectors. The resulting guess-and-verify algorithm is optimal (PSPACE), but only if the input vectors are given in binary. We answer positively the main question left open by their work, namely establish that reachability witnesses of pseudo-polynomial length always exist. Hence, when the input vectors are given in unary, the improved guess-and-verify algorithm requires only logarithmic space.
One-counter nets (OCN) are Petri nets with exactly one unbounded place. They are equivalent to a subclass of one-counter automata with only a weak test for zero.We show that weak simulation preorder is decidable for OCN and that weak simulation approximants do not converge at level ω, but only at ω 2 . In contrast, other semantic relations like weak bisimulation are undecidable for OCN [17], and so are weak (and strong) trace inclusion (Sec. 7).
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