Concurrency Relation Research Articles

Leveraging polyhedral reductions for solving Petri net reachability problems

We propose a new method that takes advantage of structural reductions to accelerate the verification of reachability properties on Petri nets. Our approach relies on a state space abstraction, called polyhedral abstraction, which involves a combination between structural reductions and sets of linear arithmetic constraints between the marking of places. We propose a new data structure, called a Token Flow Graph (TFG), that captures the particular structure of constraints occurring in polyhedral abstractions. We leverage TFGs to efficiently solve two reachability problems: first to check the reachability of a given marking and then to compute the concurrency relation of a net, that is, all pairs of places that can be marked together in some reachable marking. Our algorithms are implemented in a tool, called Kong, that we evaluate on a large collection of models used during the 2020 edition of the Model Checking Contest. Our experiments show that the approach works well, even when a moderate amount of reductions applies.

Off-the-shelf automated analysis of liveness properties for just paths

We enrich the operational semantics of a simple process calculus with ACP-style communication with a concurrency relation, so that for every process expression there exists an associated notion of just path. We then present sufficient conditions on the communication function and the syntax of process expressions that facilitate the formulation of justness on the level of labels rather than on individual transitions, taking a designated set of signals into account. This paves the way for the formulation of liveness properties under justness assumptions in the modal mu -calculus and their verification on process specifications with the mCRL2 toolset.

Local Concurrency Detection in Business Process Event Logs

Process mining techniques aim at analyzing records generated during the execution of a business process in order to provide insights on the actual performance of the process. Detecting concurrency relations between events is a fundamental primitive underpinning a range of process mining techniques. Existing approaches to this problem identify concurrency relations at the level of event types under a global interpretation. If two event types are declared to be concurrent, every occurrence of one event type is deemed to be concurrent to one occurrence of the other. In practice, this interpretation is too coarse-grained and leads to over-generalization. This article proposes a finer-grained approach, whereby two event types may be deemed to be in a concurrency relation relative to one state of the process, but not relative to other states. In other words, the detected concurrency relation holds locally, relative to a set of states. Experimental results both with artificial and real-life logs show that the proposed local concurrency detection approach improves the accuracy of existing concurrency detection techniques.

Closed Sets in Occurrence Nets with Conflicts

The semantics of concurrent processes can be defined in terms of partially ordered sets. Occurrence nets, which belong to the family of Petri nets, model concurrent processes as partially ordered sets of occurrences of local states and local events. On the basis of the associated concurrency relation, a closure operator can be defined, giving rise to a lattice of closed sets. Extending previous results along this line, the present paper studies occurrence nets with forward conflicts, modelling families of processes. It is shown that the lattice of closed sets is orthomodular, and the relations between closed sets and some particular substructures of an occurrence net are studied. In particular, the paper deals with runs, modelling concurrent histories, and trails, corresponding to possible histories of sequential components. A second closure operator is then defined by means of an iterative procedure. The corresponding closed sets, here called 'dynamically closed', are shown to form a complete lattice, which in general is not orthocomplemented. Finally, it is shown that, if an occurrence net satisfies a property called B-density, which essentially says that any antichain meets any trail, then the two notions of closed set coincide, and they form a complete, algebraic orthomodular lattice.

Closure Operators and Lattices Derived from Concurrency in Posets and Occurrence Nets

Partially ordered sets (posets), and among them occurrence nets, are a natural formal tool for studying concurrent processes. In a poset, the concurrency relation between elements is explicit. Starting from this relation, and applying standard techniques of lattice theory, one can build a complete lattice whose elements are subsets of the given poset. We study structural properties of such closed subsets, and of the lattice they form. In particular, we show that, if a poset is N-dense, then the lattice of closed subsets is orthomodular. A characterization of K-density, valid for posets, is given on the basis of a relation between lines, or chains, and closed sets. In the case of occurrence nets, we give a characterization of the closed subsets, and define the related notion of “causally closed subset”; a constructive characterization of such subsets is given, which justifies their interpretation as causally closed subprocesses of the occurrence net. We show that, for K-dense occurrence nets, closed subsets and causally closed subsets coincide. By using causally closed subsets, we give another characterization of K-density, related to the algebraicity of the lattice of closed sets.

Orthomodular Lattices Induced by the Concurrency Relation

We apply to locally finite partially ordered sets a construction which associates a complete lattice to a given poset; the elements of the lattice are the subsets of a closure operator, defined starting from the concurrency relation. We show that, if the partially ordered set satisfies a property of local density, i.e.: N-density, then the associated lattice is also orthomodular. We then consider occurrence nets, introduced by C.A. Petri as models of concurrent computations, and define a family of subsets of the elements of an occurrence net; we call those subsets closed because they can be seen as subprocesses of the whole net which are, intuitively, with respect to the forward and backward local state changes. We show that, when the net is K-dense, the causally sets coincide with the sets induced by the closure operator defined starting from the concurrency relation. K-density is a property of partially ordered sets introduced by Petri, on the basis of former axiomatizations of special relativity theory.

Asynchronous automata versus asynchronous cellular automata

In this paper we compare and study some properties of two mathematical models of concurrent systems, asynchronous automata (Zielonka, 1987) and asynchronous cellular automata (Zielonka, 1989). First, we show that these models are “polynomially” related, exhibiting polynomial-time reductions between them. Subsequently, we prove that, in spite of that, the classes of asynchronous automata and of asynchronous cellular automata recognizing a given trace language are, in general, deeply different. In fact, we exhibit a recognizable trace language T with the following properties: there exists a unique minimum asynchronous automaton accepting T, does not exist a unique minimum asynchronous cellular automaton, but there are infinitely many minimal (i.e., unreducible) nonisomorphic asynchronous cellular automata accepting T. We characterize the class of concurrent alphabets for which every recognizable trace language admits a minimum finite state asynchronous cellular automaton as the class of alphabets with full concurrency relation. Finally, extending a result of (Bruschi et al., 1988), we show that for every concurrent alphabet with nontransitive dependency relation, there exists a trace language accepted by infinitely many minimal nonisomorphic asynchronous automata.

On the Existence of Minimum Asynchronous Automata and on the Equivalence Problem for Unambiguous Regular Trace Languages

Given a recognizable trace language T there always exists a minimum (up to isomorphism) monoid automation which recognizes T . In this paper we consider finite state asynchronous automata, introduced by W. Zielonka ("Notes on Finite Asynchronous Automata," Technical Report, Institute of Mathematics, Warsaw Technical University, 1984) and we show that this result cannot be extended to this model. More precisely, we exhibit a recognizable trace language which does not admit a unique minimum asynchronous automaton. This result solves an open problem posed by Zielonka. We further prove that every recognizable trace language defined on a concurrent alphabet with a transitive dependency relation admits a unique minimum asynchronous automation. In the second part of this paper we investigate the equivalence problem for regular trace languages. This problem has been investigated by many authors: in particular, Aalbersberg and Hoogeboom (1987) have shown that the equivalence problem for regular trace languages defined over a concurrent alphabet 〈 Σ , C 〉 is decidable if and only if the concurrency relation C is transitive. We show that this result does not hold when we restrict our attention to unambiguous regular trace languages we exhibit a concurrent alphabet 〈 Σ ′, C ′〉 with a non transitive concurrency relation such that the equivalence problem for unambiguous regular trace languages on 〈 Σ ′, C ′〉 can be decided in polynomial time. Further we show that the problem of distinguishing whether or not a regular expression on a concurrent alphabet 〈 Σ , C 〉 defines an unambiguous regular trace language is undecidable .

Membership problems for regular and context-free trace languages

Trace languages have been introduced in order to describe the behaviour of concurrent systems in the same way as usual formal languages do for sequential system. They can be defined as subsets of a free partially commutative monoid and a theory of trace languages can be developed, generalizing the usual formal languages theory. In this paper, the time complexity of membership problems for regular and context-free trace languages is investigated. It is proved that the membership problem for context free trace languages can be solved in time O ( BM ( n α )), where α is the dimension of the greatest clique of the concurrency relation C and BM ( n ) is the time required for multiplying two arbitrary n × n boolean matrices. For regular trace languages, our method gives an algorithm which requires O ( n α ) time. Finally, the uniform membership problem is shown to be NP-complete.

A Non-Interleaving Semantics for CCS Based on Proved Transitions

When using labelled transition systems to model languages like CCS or TCSP, one specifies transitions by a set of structural rules. We consider labelling transitions with their proofs – in the given system of rules – instead of simple actions. Then the label of a transition identifies uniquely that transition, and one may use this information to define a concurrency relation on (proved) transitions, and a notion of residual of a (proved) transition by a concurrent one. We apply Berry and Lévy’s notion of equivalence by permutations to sequences of proved transitions for CCS to obtain a partial order semantics for this language.

Finite canonical rewriting systems for congruences generated by concurrency relations

WhenC is a concurrency relation on alphabet Σ, then Σ*/=C is a free partially commutative monoid. Here we show that it is decidable in polynomial time whether or not there exists a finite canonical rewriting systemR on Σ such that the congruences ↔R* generated byR and =C induced byC coincide. Further, in case such a systemR exists, one such system can be determined in polynomial time.

Nets, sequential components and concurrency relations

A lattice of unmarked nets is introduced and studied. It is proved that unmarked nets representing the static structure of sequential systems are atoms of that lattice. Marking classes defined by the decomposition of nets into sequential components are introduced and properties (safeness, fireability, etc.) of nets with those marking classes are investigated. The notion of concurrency relation on the system level is defined and discussed. Two different definitions of that relation are given. The first one starts with a given in advance decomposition of a net into sequential components; the second one is constructed on the basis of a given in advance marking class. Both definitions follow from a general concept of the symmetric and irreflexive relation define by a set covering. Petri's postulate about a common element for every global system state and every sequential component is carried up the system level and its strength is discussed. It turns out that if a net is safe and each transition has a possibility to be fired then that postulate implies that the net is decomposable into finite state machines.