: We survey 25 years of research on decidability issues for Petri nets. We collect results on the decidability of important properties, equivalence notions, and temporal logics. 1. Introduction Petri nets are one of the most popular formal models for the representation and analysis of parallel processes. They are due to C.A. Petri, who introduced them in his doctoral dissertation in 1962. Some years later, and independently from Petri's work, Karp and Miller introduced vector addition systems [47], a simple mathematical structure which they used to analyse the properties of "parallel program schemata', a model for parallel computation. In their seminal paper on parallel program schemata, Karp and Miller studied some decidability issues for vector addition systems, and the topic continued to be investigated by other researchers. When Petri's ideas reached the States around 1970, it was observed that Petri nets and vector addition systems were mathematically equivalent, even though thei...

Abstract. High-level Message Sequence Charts are a well-established formalism to specify scenarios of communications in telecommunication protocols. In order to deal with possibly unbounded specifications, we focus on star-connected HMSCs. We relate this subclass with recognizability and MSO-definability by means of a new connection with Mazurkiewicz traces. Our main result is that we can check effectively whether a star-connected HMSC is realizable by a finite system of communicating automata with possibly unbounded channels. Message Sequence Charts (MSCs) are a popular model often used for the documentation of telecommunication protocols. They profit by a standardized visual and textual presentation (ITU-T recommendation Z.120 [11]) and are related to other formalisms such as sequence diagrams of UML. An MSC gives a graphical description of communications between processes. It usually abstracts away from the values of variables and the actual contents of messages. However, this formalism can be used at a very early stage of design to detect errors in the specification

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We look at 1-region membrane computing systems which only use rules of the form Ca Cv, where C is a catalyst anoncatalW:k and v is a(possiblW:kky string ofnoncatal sts. There are norulk of the form a v. Thus, we can think of these systems as"purelx catalxyMWe consider two types: (1) when theinitial configuration containsonl onecatalxkA and (2) when theinitial configuration contains mulains catalsy-" We show that systems of the first type are equivalyM to communication-free Petri nets, which are aly equivalyM to commutative context-free grammars. They defin eprecisel-kq semilel- sets. ThispartialkyM-:W"k an open question (in: WMC-CdeA'02, Lecture Notes in Computer Science,vol 2597, Springer,Berlge 2003, pp. 400 -- 409; Computational" universal P systems without priorities: two catal"k# are su#cient, availt,y at http://psystems.disco.unimib.it, 2003). Systems of the second type define exactl""# recursivelM-kq""yl sets oftupl# (i.e., Turing machinecomputablWk Weal" studyan extended model where therul- are of the form q :(p; Ca Cv) (where q and p are states), i.e., the appl":xyMk of therul- is guided bya #nite-statecontrol For thisgeneral"yM model type (1) aswel as type (2) with some restriction correspond to vector addition systems. Finally, we briefly investigate the closure properties of catalytic systems.

We examine both the modeling power of normal and sinkless Petri nets and the computational complexities of various classical decision problems with respect to these two classes. We argue that although neither normal nor sinkless Petri nets are strictly more powerful than persistent Petri nets, they nonetheless are both capable of modeling a more interesting class of problems. On the other hand, we give strong evidence that normal and sinkless Petri nets are easier to analyze than persistent Petri nets. In so doing, we apply techniques originally developed for conflict-free Petri nets --- a class defined solely in terms of the structure of the the net --- to sinkless Petri nets --- a class defined in terms of the behavior of the net. As a result, we give the first comprehensive complexity analysis of a class of potentially unbounded Petri nets defined in terms of their behavior. 1 Introduction Many aspects of the fundamental nature of computation are often studied via formal m...

1. For 1-membrane catalytic systems (CS's), the sequential version is strictlyweaker than the parallel version in that the former defines (i.e. generates) exactly the semilinear sets, whereas the latter is known to define nonrecursivesets. 2. For 1-membrane communicating P systems (CPS's), the sequential versioncan only define a proper subclass of the semilinear sets, whereas the parallel version is known to define nonrecursive sets.3. Adding a new type of rule of the form: ab! axbyccomedcome to the CPS(a natural generalization of the rule ab! axbyccome in the original model),where x; y 2 fhere; outg, to the sequential 1-membrane CPS makes itequivalent to a vector addition system.

Abstract. This paper argues that flatness appears as a central notion in the verification of counter automata. A counter automaton is called flat when its control graph can be “replaced”, equivalently w.r.t. reachability, by another one with no nested loops. From a practical view point, we show that flatness is a necessary and sufficient condition for termination of accelerated symbolic model checking, a generic semi-algorithmic technique implemented in successful tools like FAST, LASH or TREX. From a theoretical view point, we prove that many known semilinear subclasses of counter automata are flat: reversal bounded counter machines, lossy vector addition systems with states, reversible Petri nets, persistent and conflict-free Petri nets, etc. Hence, for these subclasses, the semilinear reachability set can be computed using a uniform accelerated symbolic procedure (whereas previous algorithms were specifically designed for each subclass). 1

Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a well-stirred solution according to standard chemical kinetics equations. SCRNs have been widely used for describing naturally occurring (bio)chemical systems, and with the advent of synthetic biology they become a promising language for the design of artificial biochemical circuits. Our interest here is the computational power of SCRNs and how they relate to more conventional models of computation. We survey known connections and give new connections between SCRNs and

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Project, and my advisor Shuki Bruck for supporting me during my studies. I would also like to thank Shuki for being a good advisor and collaborator. I am grateful not only to Shuki but to all the people I have worked with, including Erik Winfree and David Soloveichik, in collaboration with whom the material in section 3.4.2 was produced. My family has supported my adventure of being a student, especially my wife Éva, my children András, Adam, and Emma, my mother Sarah, and my grandfather Howard, and to them I am very grateful. iv Relations are everywhere. In particular, we think and reason in terms of mathematical and English sentences that state relations. However, we teach our students much more about how to manipulate functions than about how to manipulate relations. Consider functions. We know how to combine functions to make new functions, how to evaluate functions efficiently, and how to think about compositions of functions. Especially in the area of boolean functions, we have become experts in the theory and art of designing combinations of functions to yield what we want, and this expertise has led to techniques that enable