Class Of Infinite-state Systems Research Articles

On the Decidability of Non Interference over Unbounded Petri Nets

Non-interference, in transitive or intransitive form, is defined here over unbounded (Place/Transition) Petri nets. The definitions are adaptations of similar, well-accepted definitions introduced earlier in the framework of labelled transition systems. The interpretation of intransitive non-interference which we propose for Petri nets is as follows. A Petri net represents the composition of a controlled and a controller systems, possibly sharing places and transitions. Low transitions represent local actions of the controlled system, high transitions represent local decisions of the controller, and downgrading transitions represent synchronized actions of both components. Intransitive non-interference means the impossibility for the controlled system to follow any local strategy that would force or dodge synchronized actions depending upon the decisions taken by the controller after the last synchronized action. The fact that both language equivalence and bisimulation equivalence are undecidable for unbounded labelled Petri nets might be seen as an indication that non-interference properties based on these equivalences cannot be decided. We prove the opposite, providing results of decidability of non-interference over a representative class of infinite state systems.

On-Line Monitoring of Large Petri Net Models Under Partial Observation

We consider a Petri Net model of the plant. The observation is given by a subset of transitions whose occurrence is always and immediately sensed by a monitoring agent. Other transitions not in this subset are silent (unobservable). Classical on-line monitoring techniques, which are based on the estimation of the current state of the plant and the detection of the occurrence of undesirable events (faults), are not suitable for models of large systems due to high spatial complexity (exponential in the size of the entire model). In this paper we propose a method based on the explanation of plant observation. A legal trace minimally explains the observation if it includes all unobservable transitions whose firing is needed to enable the observed transitions. To do so, starting from an observable transition, using backward search techniques, a set of minimal explanations is derived, which are sufficient for detecting whether a fault event must have occurred for sure in the plant or not. The technique also allows production of a set of basis markings for the estimation of the current state of the plant. The set of all possible current markings can then be characterized as the unobservable reach of these basis markings. The computational complexity of the algorithm depends on the size of the largest connected subnet which includes only unobservable transitions. This allows monitoring of plants of any size in which there is no large unobservable subnet. We also illustrate the applicability of the method for the monitoring of a class of infinite state systems, unbounded Petri Nets with unobservable trap circuits, and we show how this can be useful for distributed implementations.

On Computation of State Avoidance Control for Infinite State Systems in Assignment Program Framework

We study supervisory control of discrete event systems with potentially infinite state-space using state variables for representation and specification. An assignment program model consisting of state variables and a finite set of conditional assignment statements is used for representing a discrete event system, and a predicate over state variables is used for representing a state avoidance control specification. The contribution of this paper is to show how to perform supervisory control computations symbolically. In the case of a Petri net (vector addition system) with the set of forbidden states being a right-closed set, we present a finitely terminating algorithm for maximally permissive supervision. Discrete-event systems are systems with discrete states that evolve in response to discrete events. The state space of such systems can be finite or infinite. The latter case occurs when the state-variables can take infinitely many values such as integers. Certain states in a given system may be "bad", such as a deadlocking state. Then, controllers must be designed to restrict the system behavior by dynamically disabling events occurring in the system so that system never reaches the bad states. Also, certain events can be uncontrollable and cannot be disabled. State-avoidance control of potentially infinite-state discrete-event systems is studied in this paper. A compact program-like modeling formalism has been adopted. Although the control problem for infinite-state systems is in general not solvable in an automated fashion owing to its undecidability established We develop a symbolic technique, that is iterative in nature and can be automated, for computing a control strategy. If the iteration terminates (there is no guarantee though), a controller is computed. We illustrate through several examples where the iterative computation does terminate. Finally, we show that for a certain class of infinite-state systems that can be modeled as Petri nets, the iterative computation is guaranteed to terminate whenever the state-avoidance set is lower-bounded (every state that "dominates" a bad state is itself bad).

Constraint-Based Verification of Parameterized Cache Coherence Protocols

We propose a new method for the parameterized verification of formal specifications of cache coherence protocols. The goal of parameterized verification is to establish system properties for an arbitrary number of caches. In order to achieve this purpose we define abstractions that allow us to reduce the original parameterized verification problem to a control state reachability problem for a system with integer data variables. Specifically, the methodology we propose consists of the following steps. We first define an abstraction in which we only keep track of the number of caches in a given state during the execution of a protocol. Then, we use linear arithmetic constraints to symbolically represent infinite sets of global states of the resulting abstract protocol. For reasons of efficiency, we relax the constraint operations by interpreting constraints over real numbers. Finally, we check parameterized safety properties of abstract protocols using symbolic backward reachability, a strategy that allows us to obtain sufficient conditions for termination for an interesting class of protocols. The latter problem can be solved by using the infinite-state model checker HyTech: Henzinger, Ho, and Wong-Toi, “A model checker for hybrid systems,” Proc. of the 9th International Conference on Computer Aided Verification (CAV'97), Lecture Notes in Computer Science, Springer, Haifa, Israel, 1997, Vol. 1254, pp. 460–463. HyTech handles linear arithmetic constraints using the polyhedra library of Halbwachs and Proy, “Verification of real-time systems using linear relation analysis,” Formal Methods in System Design, Vol. 11, No. 2, pp. 157–185, 1997. By using this methodology, we have automatically validated parameterized versions of widely implemented write-invalidate and write-update cache coherence protocols like Synapse, MESI, MOESI, Berkeley, Illinois, Firefly and Dragon (Handy, The Cache Memory Book, Academic Press, 1993). With this application, we have shown that symbolic model checking tools like HyTech, originally designed for the verification of hybrid systems, can be applied successfully to new classes of infinite-state systems of practical interest.

Simulation Preorder over Simple Process Algebras

We consider the problem of simulation preorder/equivalence between infinite-state processes and finite-state ones. First, we describe a general method how to utilize the decidability of bisimulation problems to solve (certain instances of) the corresponding simulation problems. For certain process classes, the method allows us to design effective reductions of simulation problems to their bisimulation counterparts and some new decidability results for simulation have already been obtained in this way. Then we establish the decidability border for the problem of simulation preorder/equivalence between infinite-state processes and finite-state ones w.r.t. the hierarchy of process rewrite systems. In particular, we show that simulation preorder (in both directions) and simulation equivalence are decidable in EXPTIME between pushdown processes and finite-state ones. On the other hand, simulation preorder is undecidable between PA and finite-state processes in both directions. These results also hold for those PA and finite-state processes which are deterministic and normed, and thus immediately extend to trace preorder. Regularity (finiteness) w.r.t. simulation and trace equivalence is also shown to be undecidable for PA. Finally, we prove that simulation preorder (in both directions) and simulation equivalence are intractable between all classes of infinite-state systems (in the hierarchy of process rewrite systems) and finite-state ones. This result is obtained by showing that the problem whether a BPA (or BPP) process simulates a finite-state one is PSPACE -hard and the other direction is co NP -hard; consequently, simulation equivalence between BPA (or BPP) and finite-state processes is also co NP -hard.

Weak bisimilarity between finite-state systems and BPA or normed BPP is decidable in polynomial time

We prove that weak bisimilarity is decidable in polynomial time between finite-state systems and several classes of infinite-state systems: context-free processes and normed basic parallel processes (normed BPP). To the best of our knowledge, these are the first polynomial algorithms for weak bisimilarity problems involving infinite-state systems.

Deciding bisimulation-like equivalences with finite-state processes

We show that characteristic formulae for finite-state systems up to bisimulation-like equivalences (e.g., strong and weak bisimilarity) can be given in the simple branching-time temporal logic EF. Since EF is a very weak fragment of the modal μ-calculus, model checking with EF is decidable for many more classes of infinite state systems. This yields a general method for proving decidability of bisimulation-like equivalences between infinite-state processes and finite-state ones. We apply this method to the class of PAD processes, which strictly subsumes PA and pushdown (PDA) processes, showing that a large class of bisimulation-like equivalences (including, e.g., strong and weak bisimilarity) is decidable between PAD and finite-state processes. On the other hand, we also demonstrate that no ‘reasonable’ bisimulation-like equivalence is decidable between state-extended PA processes and finite-state ones. Furthermore, weak bisimilarity with finite-state processes is shown to be undecidable even for state-extended BPP (which are also known as ‘parallel pushdown processes’).

Preface: Volume 5

This volume contains a selection of papers from the First International Workshop on Verification of Infinite State Systems (Infinity'96), which took place at the University of Pisa (Italy), 30—31 August, 1996, as a satellite of CONCUR'96, the 7th International Conference on Concurrency Theory.The aim of the workshop was to provide a forum for researchers interested in the development of mathematical techniques for the analysis of infinite state systems, a topic which has received a concerted effort within the Concurrency Theory community over the past few years. The basis of this effort has been the realization that an understanding of infinite state systems is necessary in order to have a complete picture of general process algebras, Petri nets, or other formalisms incorporating value-passing, real-time, hybrid, and/orprobabilistic aspects. Its importance has grown however by the furtherrealization that techniques which are developed for infinite state systems—-particularly structural techniques—-can potentially provide elegant solutions to the state-space explosion problem in the analysis of finite state systems, as well as to classical problems in language theory. Possible topics for inclusion are: decidablility issues for equivalence and model checking over various classes of infinite state systems; complexity results for decidability results; connections and applications to questions in classical automata and formal language theory; and tools and case studies involving nontrivial applications of methods for the analysis of infinite state systems.In addition to the regular sessions, Infinity featured three invited talks, given by Yoram Hirshfeld (Tel Aviv University, Israel), Thomas A. Henzinger (University of California at Berkeley, USA), and Javier Esparza (Technische Universität München, Munich, Germany).In addition to the Editors, the Program Committee includedJulian Bradfield (Edinburgh) and Faron Moller (Stockholm). We also would like to thank the referees who assisted in theselection of the regular contributions among the 24 submissions received.Finally, warm thanks are due to Tiziana Margaria, who has provided invaluable assistancein the preparation of the text of this volume. Without her you wouldnot hold this volume in your hands, nor could you browse it on your screen.March 1997Bernhard SteffenDidier Caucal

Verifying Programs with Unreliable Channels

We consider the verification of a particular class of infinite-state systems, namely systems consisting of finite-state processes that communicate via unbounded lossy FIFO channels. This class is able to model, e.g., link protocols such as the Alternating Bit Protocol and HDLC. For this class of systems, we show that several interesting verification problems are decidable by giving algorithms for verifying (1) thereachability problem—is a finite set of global states reachable from some other global state of the system ? (2)safety properties over tracesformulated as regular sets of allowed finite traces, and (3)eventuality properties—do all computations of a system eventually reach a given set of states? We have used the algorithms to verify some idealized sliding-window protocols with reasonable time and space resources. Our results should be contrasted with the well-known fact that these problems are undecidable for systems with unboundedperfectFIFO channels.

Decidable Subsets of CCS

CCS is a universal formalism: any computable function is computed by some CCS agent. Moreover, one can reduce the halting problem for Turing machines to the problem of deciding bisimilarity of two CCS agents, thus demonstrating the undecidability of the equivalence checking problem. In this paper, we demonstrate the limits of decidability of CCS. In particular, we show that by simply disallowing either of communication or both restriction and relabelling, we arrive at a sub-language which still describes a rich class of infinite state systems but for which bisimulation is decidable. We also demonstrate complete axiomatisations for these sublanguages. We compare these results with the undecidability of all other common equivalences