Computing Optimal Coverability Costs in Priced Timed Petri Nets

We consider timed Petri nets, i.e., unbounded Petri nets where each token carries a real-valued clock. Transition arcs are labeled with time intervals, which specify constraints on the ages of tokens. Our cost model assigns token storage costs per time unit to places, and firing costs to transitions. We study the cost to reach a given control-state. In general, a cost-optimal run may not exist. However, we show that the infimum of the costs is computable.Comment: 26 pages. Contribution to LICS 201

Timed Basic Parallel Processes

Timed basic parallel processes (TBPP) extend communication-free Petri nets (aka. BPP or commutative context-free grammars) by a global notion of time. TBPP can be seen as an extension of timed automata (TA) with context-free branching rules, and as such may be used to model networks of independent timed automata with process creation. We show that the coverability and reachability problems (with unary encoded target multiplicities) are PSPACE-complete and EXPTIME-complete, respectively. For the special case of 1-clock TBPP, both are NP-complete and hence not more complex than for untimed BPP. This contrasts with known super-Ackermannian-completeness and undecidability results for general timed Petri nets. As a result of independent interest, and basis for our NP upper bounds, we show that the reachability relation of 1-clock TA can be expressed by a formula of polynomial size in the existential fragment of linear arithmetic, which improves on recent results from the literature

Petri Nets with Time and Cost

We consider timed Petri nets, i.e., unbounded Petri nets where each token carries a real-valued clock. Transition arcs are labeled with time intervals, which specify constraints on the ages of tokens. Our cost model assigns token storage costs per time unit to places, and firing costs to transitions. We study the cost to reach a given control-state. In general, a cost-optimal run may not exist. However,we show that the infimum of the costs is computable.Comment: In Proceedings Infinity 2012, arXiv:1302.310

Minimal Cost Reachability/Coverability in Priced Timed Petri Nets

Abstract. We extend discrete-timed Petri nets with a cost model that assigns token storage costs to places and firing costs to transitions, and study the minimal cost reachability/coverability problem. We show that the minimal costs are computable if all storage/transition costs are non-negative, while even the question of zero-cost coverability is undecidable in the case of general integer costs.

Priced Timed Petri Nets

We consider priced timed Petri nets, i.e., unbounded Petri nets where each token carries a real-valued clock. Transition arcs are labeled with time intervals, which specify constraints on the ages of tokens. Furthermore, our cost model assigns token storage costs per time unit to places, and firing costs to transitions. This general model strictly subsumes both priced timed automata and unbounded priced Petri nets. We study the cost of computations that reach a given control-state. In general, a computation with minimal cost may not exist, due to strict inequalities in the time constraints. However, we show that the infimum of the costs to reach a given control-state is computable in the case where all place and transition costs are non-negative. On the other hand, if negative costs are allowed, then the question whether a given control-state is reachable with zero overall cost becomes undecidable. In fact, this negative result holds even in the simpler case of discrete time (i.e., integer-valued clocks).Comment: 51 pages. LMCS journal version of arXiv:1104.061

Solving Parity Games on Integer Vectors

We consider parity games on infinite graphs where configurations are represented by control-states and integer vectors. This framework subsumes two classic game problems: parity games on vector addition systems with states (vass) and multidimensional energy parity games. We show that the multidimensional energy parity game problem is inter-reducible with a subclass of single-sided parity games on vass where just one player can modify the integer counters and the opponent can only change control-states. Our main result is that the minimal elements of the upward-closed winning set of these single-sided parity games on vass are computable. This implies that the Pareto frontier of the minimal initial credit needed to win multidimensional energy parity games is also computable, solving an open question from the literature. Moreover, our main result implies the decidability of weak simulation preorder/equivalence between finite-state systems and vass, and the decidability of model checking vass with a large fragment of the modal mu-calculus.Comment: 30 page

Verification of priced and timed extensions of Petri Nets with multile instances

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Informática, Departamento de Sistemas Informáticos y Computación, leída el 25-01-2016Las redes de Petri son un lenguaje formal muy adecuado para la modelizacíon, ańalisis y verificacíon de sistemas concurrentes con infinitos estados. En particular, son muy apropiadas para estudiar las propiedades de seguridad de dichos sistemas, dadas sus buenas propiedades de decidibilidad. Sin embargo, en muchas ocasiones las redes de Petri carecen de la expresividad necesaria para representar algunas caracteŕısticas fundamentales de los sistemas que se manejan hoy en d́ıa, como el manejo de tiempo real, costes reales, o la presencia de varios procesos con un ńumero no acotado de estados ejecut́andose en paralelo. En la literatura se han definido y estudiado algunas extensiones de las redes de Petri para la representaci ́on de las caracteŕısticas anteriores. Por ejemplo, las “Redes de Petri Temporizadas” [83, 10](TPN) incluyen el manejo de tiempo real y las ν-redes de Petri [78](ν-PN) son capaces de representar un ńumero no acotado de procesos con infinitos estados ejecut́andose concurrentemente. En esta tesis definimos varias extensiones que réunen estas dos caracteŕısticas y estudiamos sus propiedades de decidibilidad. En primer lugar definimos las “ν-Redes de Petri Temporizadas”, que réunen las caracteŕısticas expresivas de las TPN y las ν-PN. Este nuevo modelo es capaz de representar sistemas con un ńumero no acotado de procesos o instancias, donde cada proceso es representado por un nombre diferente, y tiene un ńumero no acotado de relojes reales. En este modelo un reloj de una instancia debe satisfacer ciertas condiciones (pertenecer a un intervalo dado) para formar parte en el disparo de una transicíon. Desafortunadamente, demostramos que la verificacíon de propiedades de seguridad es indecidible para este modelo...The model of Petri nets is a formal modeling language which is very suitable for the analysis and verification of infinite-state concurrent systems. In particular, due to its good decidability properties, it is very appropriate to study safety properties over such systems. However, Petri nets frequently lack the expressiveness to represent several essential characteristics of nowadays systems such as real time, real costs, or the managing of several parallel processes, each with an unbounded number of states. Several extensions of Petri nets have been defined and studied in the literature to fix these shortcomings. For example, Timed Petri nets [83, 10] deal with real time and ν-Petri nets [78] are able to represent an unbounded number of different infinite-state processes running concurrently. In this thesis we define new extensions which encompass these two characteristics, and study their decidability properties. First, we define Timed ν-Petri nets by joining together Timed Petri nets and ν-Petri nets. The new model represents systems in which each process (also called instance) is represented by a different pure name, and it is endowed with an unbounded number of clocks. Then, a clock of an instance must satisfy certain given conditions (belonging to a given interval) in order to take part in the firing of a transition. Unfortunately, we prove that the verification of safety properties is undecidable for this model. In fact, it is undecidable even if we only consider two clocks per process. We restrict this model and define Locally-Synchronous ν-Petri nets by considering only one clock per instance, and successfully prove the decidability of safety properties for this model. Moreover, we study the expressiveness of Locally-Synchronous ν-Petri nets and prove that it is the most expressive non Turing-complete extension of Petri nets with respect to the languages they accept...Depto. de Sistemas Informáticos y ComputaciónFac. de InformáticaTRUEunpu

Verification problems for timed and probabilistic extensions of Petri Nets

In the first part of the thesis, we prove the decidability (and PSPACE-completeness) of the universal safety property on a timed extension of Petri Nets, called Timed Petri Nets. Every token has a real-valued clock (a.k.a. age), and transition firing is constrained by the clock values that have integer bounds (using strict and non-strict inequalities). The newly created tokens can either inherit the age from an input token of the transition or it can be reset to zero. In the second part of the thesis, we refer to systems with controlled behaviour that are probabilistic extensions of VASS and One-Counter Automata. Firstly, we consider infinite state Markov Decision Processes (MDPs) that are induced by probabilistic extensions of VASS, called VASS-MDPs. We show that most of the qualitative problems for general VASS-MDPs are undecidable, and consider a monotone subclass in which only the controller can change the counter values, called 1-VASS-MDPs. In particular, we show that limit-sure control state reachability for 1-VASS-MDPs is decidable, i.e., checking whether one can reach a set of control states with probability arbitrarily close to 1. Unlike for finite state MDPs, the control state reachability property may hold limit surely (i.e. using an infinite family of strategies, each of which achieving the objective with probability ≥ 1-e, for every e > 0), but not almost surely (i.e. with probability 1). Secondly, we consider infinite state MDPs that are induced by probabilistic extensions of One-Counter Automata, called One-Counter Markov Decision Processes (OC-MDPs). We show that the almost-sure {1;2;3}-Parity problem for OC-MDPs is at least as hard as the limit-sure selective termination problem for OC-MDPs, in which one would like to reach a particular set of control states and counter value zero with probability arbitrarily close to 1

2008 Abstracts Collection -- IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science

This volume contains the proceedings of the 28th international conference on the Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2008), organized under the auspices of the Indian Association for Research in Computing Science (IARCS)

An algebraic approach to energy problems II - the algebra of energy functions

Energy and resource management problems are important in areas such as embedded systems or autonomous systems. They are concerned with the question whether a given system admits infinite schedules during which certain tasks can be repeatedly accomplished and the system never runs out of energy (or other resources). In order to develop a general theory of energy problems, we introduce energy automata: finite automata whose transitions are labeled with energy functions which specify how energy values change from one system state to another. We show that energy functions form a *-continuous Kleene ω-algebra, as an application of a general result that finitely additive, locally *-closed and T-continuous functions on complete lattices form *-continuous Kleene ω-algebras. This permits to solve energy problems in energy automata in a generic, algebraic way. In order to put our work in context, we also review extensions of energy problems to higher dimensions and to games