Decidability and complexity of Petri nets with unordered data

We prove several decidability and undecidability results for ν-PN, an extension of P/T nets with pure name creation and name management. We give a simple proof of undecidability of reachability, by reducing reachability in nets with inhibitor arcs to it. Thus, the expressive power of ν-PN strictly surpasses that of P/T nets. We encode ν-PN into Petri Data Nets, so that coverability, termination and boundedness are decidable. Moreover, we obtain Ackermann-hardness results for all our decidable decision problems. Then we consider two properties, width-boundedness and depth-boundedness, that factorize boundedness. Width-boundedness has already been proven to be decidable. Here we prove that its complexity is also non-primitive recursive. Then we prove undecidability of depth-boundedness. Finally, we prove that the corresponding “place version” of all the boundedness problems is undecidable for ν-PN. These results carry over to Petri Data Nets.     

On the complexity of simple arithmetic expressions

Let $\text{E}$ be the set of all simple arithmetic expressions of the form E(x) = xT_l…T_k where x is a nonnegative integer variable and each T_i is a multiplication or integer division by a positive integer constant. We investigate the complexity of the inequivalence and the bounded inequivalence problems for expressions in $\text{E}$. (The bounded inequivalence problem is the problem of deciding for arbitrary expressions E₁(x) and E₂(x) and a positive integer l whether or not E₁(x) ≠ E₂(x) for some nonnegative integer x<l. If l = ∞, i.e., there is no upper bound on x, the problem becomes the inequivalence problem.) We show that the inequivalence problem (or equivalently, the equivalence problem) for a large subclass of $\text{E}$ is decidable in polynomial time. Whether or not the problem is decidable in polynomial time for the full class $\text{E}$ remains open. We also show that the bounded inequivalence problem is NP-complete even if the divisors are restricted to be equal to 2. This last result can be used to sharpen some known NP-completeness results in the literature. Note that if division is rational division, all problems are trivially decidable in polynomial time.     

Lower-bound-constrained runs in weighted timed automata

We investigate a number of problems related to infinite runs of weighted timed automata (with a single weight variable), subject to lower-bound constraints on the accumulated weight. Closing an open problem from Bouyer et al. (2008), we show that the existence of an infinite lower-bound-constrained run is—for us somewhat unexpectedly—undecidable for weighted timed automata with four or more clocks.This undecidability result assumes a fixed and known initial credit. We show that the related problem of existence of an initial credit for which there exists a feasible run is decidable in PSPACE. We also investigate the variant of these problems where only bounded-duration runs are considered, showing that this restriction makes our original problem decidable in  NEXPTIME. We prove that the universal versions of all those problems (i.e, checking that all the considered runs satisfy the lower-bound constraint) are decidable in PSPACE.Finally, we extend this study to multi-weighted timed automata: the existence of a feasible run becomes undecidable even for bounded duration, but the existence of initial credits remains decidable (in PSPACE).     

Verifying parallel programs with dynamic communication structures

We address the verification problem of networks of communicating pushdown systems modeling communicating parallel programs with procedure calls. Processes in such networks can read the control state of the other processes according to a given communication structure (specifying the observability rights between processes). The reachability problem of such models is undecidable in general. First, we define a class of networks that effectively preserves recognizability (hence, its reachability problem is decidable). Then, we consider networks where the communication structure can change dynamically during the execution according to a phase graph. The reachability problem for these dynamic networks being undecidable in general, we define a subclass for which it becomes decidable. Then, we consider reachability when the switches in the communication structures are bounded. We show that this problem is undecidable even for one switch. We define a natural class of models for which this problem is decidable. This class can be used in the definition of an efficient semi-decision procedure for the analysis of the general model of dynamic networks. Our techniques allowed to find bugs in two versions of a Windows NT Bluetooth driver.     

Queries on Xml streams with bounded delay and concurrency

Query answering algorithms on Xml streams check answer candidates on the fly in order to avoid the unnecessary buffering whenever possible. The delay and concurrency of a query are two measures for the degree of their streamability. They count the maximal number of stream elements during the life time for some query answer, and respectively, the maximal number of simultaneously alive answer candidates of a query. We study queries defined by deterministic nested word automata, which subsume large streamable fragments of XPath subject to schema restrictions by DTDs modulo P-time translations. We show that bounded and k-bounded delay and concurrency of such automata-defined queries are all decidable in polynomial time in the size of the automaton. Our results are obtained by P-time reduction to the bounded valuedness problem for recognizable relations between unranked trees, a problem that we show to be decidable in P-time.     

Shrinking of Time Petri nets

The problem of the synthesis of time bounds enforcing good properties for reactive systems has been much studied in the literature. These works mainly rely on dioid algebra theory and require important restrictions on the structure of the model (notably by restricting to timed event graphs). In this paper, we address the problems of existence and synthesis of shrinkings of the bounds of the time intervals of a time Petri net, such that a given property is verified. We show that this problem is decidable for CTL properties on bounded time Petri nets. We then propose a symbolic algorithm based on the state class graph for a fragment of CTL. If the desired property “includes” k-boundedness, the proposed algorithm terminates even if the net is unbounded. A prototype has been implemented in our tool Romeo and the method is illustrated on a small case study from the literature.     

Serial and Parallel Algorithms for (k,2)-Partite Graphs

We introduce a class of layered graphs which we call (k,2)-partite and which we argue are an interesting class because of several important applications. We show that testing for (k,2)-partiteness can be done efficiently both on sequential and parallel machines, by showing that membership is in NSPACE(log n) and in NC². We show that (k,2)-partite graphs have bounded path width. We then show that a particular NP-complete problem, namely Maximum Independent Set, is solvable in linear time on bounded pathwidth graphs if the path decomposition is included in the input. Finally, we show that the Maximum Independent Set problem is in NC² for (k,2)-partite graphs. We note that linear time solutions for certain NP-complete problems have been shown for a wider class of graphs, namely partial k-trees. Our linear time algorithm is somewhat simpler in structure. We conjecture that our techniques can be used on many NP-complete problems to yield efficient algorithms for (k,2)-partite graphs.     

A Decision Procedure for a Sublanguage of Set Theory Involving Monotone, Additive, and Multiplicative Functions

MLSS is a decidable sublanguage of set theory involving the predicates membership, set equality, set inclusion, and the operators union, intersection, set difference, and singleton.In this paper we extend MLSS with constructs for expressing monotonicity, additivity, and multiplicativity properties of set-to-set functions. We prove that the resulting language is decidable by reducing the problem of determining the satisfiability of its sentences to the problem of determining the satisfiability of sentences of MLSS.     

Decidability of boundedness problems for Minsky counter machines

The decidability of boundedness problems for Minksy counter machines is studied. It is proved that, for Minsky machines with two counters, the boundedness problem is partially decidable and the problem of the total boundedness is not even partially decidable. For one-counter Minsky machines, these problems are decidable during a time polynomially depending on the total number of local states of the counter machine.     

Decidability of linearizabilities for relaxed data structures

Many recent implementations of concurrent data structures relaxed their linearizability requirements for better performance and scalability. Quasi-linearizability, k-linearizability and regular-relaxed linearizability are three quantitative relaxation variants of linearizability that have been proposed as correctness conditions of relaxed data structures, yet preserving the intuition of linearizability. Quasi-linearizability has been proved undecidable. In this paper, we first show that k-linearizability is undecidable for a bounded number of processes, by reducing quasi-linearizability into it. We then show that regular-relaxed linearizability is decidable for a bounded number of processes. We also find that the number of the states of a relaxed specification is exponential to the number of the states of the underlying specification automaton (representing its relaxation strategy), and polynomial to the number of the states of the underlying quantitative sequential specification and the number of operations.     

Tree automata with equality constraints modulo equational theories

This paper presents new classes of tree automata combining automata with equality test and automata modulo equational theories. We believe that these classes have a good potential for application in e.g. software verification. These tree automata are obtained by extending the standard Horn clause representations with equational conditions and rewrite systems. We show in particular that a generalized membership problem (extending the emptiness problem) is decidable by proving that the saturation of tree automata presentations with suitable paramodulation strategies terminates. Alternatively our results can be viewed as new decidable classes of first-order formula.     

Default theories over monadic languages

In this paper we compare the semantical and syntactical definitions of extensions for open default theories. We prove that, over monadic languages, these definitions are equivalent and do not depend on the cardinality of the underlying infinite world. We also show that, under the domain closure assumption, one free variable open default theories are decidable.     

Task automata: Schedulability,decidability and undecidability

We present a model, task automata, for real time systems with non-uniformly recurring computation tasks. It is an extended version of timed automata with asynchronous processes that are computation tasks generated (or triggered) by timed events. Compared with classical task models for real time systems, task automata may be used to describe tasks (1) that are generated non-deterministically according to timing constraints in timed automata, (2) that may have interval execution times representing the best case and the worst case execution times, and (3) whose completion times may influence the releases of task instances. We generalize the classical notion of schedulability to task automata. A task automaton is schedulable if there exists a scheduling strategy such that all possible sequences of events generated by the automaton are schedulable in the sense that all associated tasks can be computed within their deadlines. Our first technical result is that the schedulability for a given scheduling strategy can be checked algorithmically for the class of task automata when the best case and the worst case execution times of tasks are equal. The proof is based on a decidable class of suspension automata: timed automata with bounded subtraction in which clocks may be updated by subtractions within a bounded zone. We shall also study the borderline between decidable and undecidable cases. Our second technical result shows that the schedulability checking problem will be undecidable if the following three conditions hold: (1) the execution times of tasks are intervals, (2) the precise finishing time of a task instance may influence new task releases, and (3) a task is allowed to preempt another running task.     

P-optimal heuristics

We show that the decision problem for the class C₀ of all closed universal Horn formulae in prenex conjunctive normal form of extended Skolem arithmetic without equality (i.e. first order formulae built up from the multiplication sign, constants for the natural numbers and free occuring predicate symbols) is exponentially time bounded equivalent to the reachability problem for Petri nets if restricted to the class of formulae with (a) only monadic predicate symbols, with (b) only binary disjunctions in the quantifier free matrix and (c) without terms containing a variable more than once. We show that leaving out one of the restrictions (a) to (c) yields classes of formulae whose decision problem can assume any prescribed recursively enumerable complexity in terms of many-one degrees of unsolvability.     

Boundedness problems for Minsky counter machines

In the paper, the decidability of boundedness problems for counter Minsky machines is studied. It is proved that, for Minsky machines with two counters, the boundedness problem is partially decidable, and the total boundedness problem is not even partially decidable. On the other hand, for the one-counter Minsky machines, the above problems are decidable for time polynomially depending on the total number of local states of the counter machine.     

Distributed synthesis for well-connected architectures

We study the synthesis problem for external linear or branching specifications and distributed, synchronous architectures with arbitrary delays on processes. External means that the specification only relates input and output variables. We introduce the subclass of uniformly well-connected (UWC) architectures for which there exists a routing allowing each output process to get the values of all inputs it is connected to, as soon as possible. We prove that the distributed synthesis problem is decidable on UWC architectures if and only if the output variables are totally ordered by their knowledge of input variables. We also show that if we extend this class by letting the routing depend on the output process, then the previous decidability result fails. Finally, we provide a natural restriction on specifications under which the whole class of UWC architectures is decidable.     

Unification Grammars and Off-Line Parsability

Unification grammars are known to be Turing-equivalent; given a grammar G and a word w, it is undecidable whether w L(G). In order to ensure decidability, several constraints on grammars, commonly known as off-line parsability (OLP), were suggested, such that the recognition problem is decidable for grammars which satisfy OLP. An open question is whether it is decidable if a given grammar satisfies OLP. In this paper we investigate various definitions of OLP and discuss their interrelations, proving that some of the OLP variants are indeed undecidable. We then present a novel, decidable OLP constraint which is more liberal than the existing decidable ones.     

Filtering for bilinear systems with a lipschitz nonlinearity using LPV approach

This paper deals with the H _?? filtering problem for a class of nonlinear systems. This class of nonlinear systems is composed of a bilinear part and of a lipschitzian one. The use of an unbiasedness condition for the bilinear part (called quasi unbiasedness condition) permits to parameterize the filter matrices through a single gain. Two LPV (Linear Parameter Varying) extensions of the bounded real lemma are used to solve the filtering problem. This approach reduces the conservatism inherent to the boundedness of the inputs. Then the filtering solution is expressed in terms of LMI (Linear Matrix Inequality) to be verified at the vertices of a polytope. A numerical example is finally given to illustrate our approach.