Causal Semantics of Algebraic Petri Nets distinguishing Concurrency and Synchronicity (original) (raw)
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Synchronous + Concurrent + Sequential = Earlier than + Not later than
Sixth International Conference on Application of Concurrency to System Design, 2006
In this paper, we show how to obtain causal semantics distinguishing "earlier than" and "not later than" causality between events from algebraic semantics of Petri nets. Janicki and Koutny introduced so called stratified order structures (so-structures) to describe such causal semantics. To obtain algebraic semantics, we redefine our own algebraic approach generating rewrite terms via partial operations of synchronous composition, concurrent composition and sequential composition. These terms are used to produce so-structures which define causal behavior consistent with the (operational) step semantics. For concrete Petri net classes with causal semantics derived from processes minimal so-structures obtained from rewrite terms coincide with minimal so-structures given by processes. This is demonstrated exemplarily for elementary nets with inhibitor arcs.
Logic and Algebra in Unfolded Petri Nets: on a Duality Between Concurrency and Causal Dependence
Fundamenta Informaticae, 2019
An orthogonality space is a set endowed with a symmetric and irreflexive binary relation (an orthogonality relation). In a partially ordered set modelling a concurrent process, two such binary relations can be defined: a causal dependence relation and a concurrency relation, and two distinct orthogonality spaces are consequently obtained. When the condition of N-density holds on both these orthogonality spaces, we study the orthomodular poset formed by closed sets defined according to Dacey. We show that the condition originally imposed by Dacey on the orthogonality spaces for obtaining an orthomodular poset from his closed sets is in fact equivalent to N-density. The requirement of N-density was as well fundamental in a previous work on orthogonality spaces with the concurrency relation. Starting from a partially ordered set modelling a concurrent process, we obtain dual results for orthogonality spaces with the causal dependence relation in respect to orthogonality spaces with the concurrency relation.
Domain and event structure semantics for Petri nets with read and inhibitor arcs
2004
We propose a functorial concurrent semantics for Petri nets extended with read and inhibitor arcs, that we call inhibitor nets. Along the lines of the seminal work by Winskel on safe (ordinary) nets, the truly concurrent semantics is given at a categorical level via a chain of coreflections leading from the category SW-IN of semi-weighted inhibitor nets to the category Dom of finitary prime algebraic domains (equivalent to the category PES of prime event structures).
Process Semantics of Petri Nets over Partial Algebra
Lecture Notes in Computer Science, 2000
Petri nets are monoids" is the title and the central idea of the paper . It provides an algebraic approach to define both nets and their processes as terms. A crucial assumption for this concept is that arbitrary concurrent composition of processes is defined, which holds true for place/transition Petri nets where places can hold arbitrarily many tokens. This paper defines a similar concept for elementary Petri nets, which are elementary net systems with arbitrary initial marking. Since markings of elementary nets cannot be added arbitrarily, some operators are only defined partially; hence we employ concepts of partial algebra. The main result of the paper states that the semantics based on process terms agrees with the classical partial-order process semantics for elementary net systems. More precisely, we provide a syntactic equivalence notion for process terms and a bijection from according equivalence classes of process terms to isomorphism classes of partially ordered processes.
Acta Informatica, 1998
This paper describes a high-level Petri net model called M-nets (for modular multilabelled nets). A distinctive feature of this model is that it allows both: unfolding, as do most other high-level net models; and composition -in particular, synchronisation -in a process algebraic style, turning the set of M-nets into an algebraic domain. It turns out that the composition operations of this domain have various algebraic properties. Moreover, the model is such that composition operations are coherent with unfolding, in the sense that the unfolding of a composite high-level net is the composition of the unfoldings of its components.
Functional concurrent semantics for Petri nets with read and inhibitor arcs
2000
We propose a functorial concurrent semantics for Petri nets extended with read and inhibitor arcs, that we call inhibitor nets. Along the lines of the seminal work of Winskel on safe nets, the truly concurrent semantics is given at a categorical level via a chain of functors leading from the category SW-IN of semi-weighted inhibitor nets to the category Dom of finitary prime algebraic domains.
Complete Process Semantics for Inhibitor Nets
Lecture Notes in Computer Science, 2007
In this paper we complete the semantical framework proposed in [13] for process and causality semantics of Petri nets by an additional aim and develop process and causality semantics of place/transition Petri nets with weighted inhibitor arcs (pti-nets) satisfying the semantical framework including this aim. The aim was firstly mentioned in [8] and states that causality semantics deduced from process nets should be complete w.r.t. step semantics in the sense that each causality structure which is consistent with the step semantics corresponds to some process net. We formulate this aim in terms of enabled causality structures. While it is well known that process semantics of place/transition Petri nets (p/tnets) satisfy the additional aim, we show that the most general process semantics of pti-nets proposed so far [13] does not and develop our process semantics as an appropriate generalization.
Partial Order Semantics of Types of Nets
Lecture Notes in Computer Science, 2009
In this paper we define partial order semantics of types of nets. Types of nets are a parametric definition of Petri nets originally developed for a general presentation of the synthesis of Petri nets from (step) transition systems. Partial order semantics of a concrete net (of a certain type) usually are given by the set of labelled partial orders (LPOs) enabled w.r.t. the net. For classical place/transition nets there are several equivalent characterizations of enabled LPOs. We discuss in which way the general notion of types of nets has to be restricted such that these characterizations can also be formulated for nets of such type. In particular we consider under which requirements enabled LPOs can be defined through token flows, which have been proven to be useful for efficient synthesis and verification of Petri nets. The presented concepts form the basis for a general presentation of the synthesis of Petri nets from sets of LPOs.
An algebraic structure of petri nets
Lecture Notes in Computer Science, 1980
A relational model for non-deterministic programs is presented. Several predicate transformers are introduced and it is shown that one of them satisfies all the healthiness criteria indicated by Dijkstra for a useful total correctness predicate transformer.
An Algebraic Semantics for Hierarchical P/T Nets (Extended Abstract)
1999
The first part of this paper gives an algebraic semantics for Place/Transition nets in terms of an algebra which is based on the process algebra ACP. The algebraic semantics is such that a P/T net and its term representation have the same operational behavior. As opposed to other approaches in the literature, the actions in the algebra do not correspond to the firing of a transition, but to the consumption or production of tokens. Equality of P/T nets can be determined in a purely equational way. The second part of this paper extends the results to hierarchical P/T nets. It gives a compositional algebraic semantics for both their complete operational behavior and their highlevel, observable behavior. By means of a non-trivial example, the Alternating-Bit Protocol, it is shown that the notions of abstraction and verification in the process algebra ACP can be used to verify in an equational way whether a hierarchical P/T net satisfies some algebraic specification of its observable behavior. Thus, the theory in this paper can be used to determine whether two hierarchical P/T nets have the same observable behavior. As an example, it is shown that the Alternating-Bit Protocol behaves as a simple one-place buffer. The theory forms a basis for a modular, top-down design methodology based on Petri nets.