## Partial-Order Process Algebra (and its Relation to Petri Nets) (2000)

Venue: | Handbook of Process Algebra. Elsevier Science |

Citations: | 5 - 1 self |

### BibTeX

@INPROCEEDINGS{Baeten00partial-orderprocess,

author = {J.C.M. Baeten and T. Basten},

title = {Partial-Order Process Algebra (and its Relation to Petri Nets)},

booktitle = {Handbook of Process Algebra. Elsevier Science},

year = {2000},

publisher = {Elsevier, North-Holland}

}

### OpenURL

### Abstract

To date, many different formalisms exist for describing and analyzing the behavior of concurrent systems. Petri nets and process algebras are two well-known classes of such formalisms. Petri-net theory is well suited for reasoning about concurrent systems in a partial-order framework; it handles causal relationships between actions of concurrent systems in an explicit way. Process algebras, on the other hand, often provide a total-order framework, which means that information about causalities is not always accurate. This chapter illustrates how to develop a partial-order process algebra in the style of ACP. It is shown how to extend such an algebraic theory with a causality mechanism inspired by Petri-net theory. In addition, the chapter clarifies the concepts of interleaving and non-interleaving process algebra; total-order semantics for concurrent systems are often incorrectly referred to as interleaving semantics. Key words: process algebra -- Petri nets -- concurrency -- ...

### Citations

3430 |
Communication and Concurrency
- Milner
- 1989
(Show Context)
Citation Context ...tal-order and a partial-order setting. The theory ACP originates from [13]. Good introductions to ACP-style process algebra can be found in [5, 6]. Other well-known process-algebraic theories are CCS =-=[43, 44]-=- and CSP [35]. Specifications and verifications in ACP-style process algebra are based on an equational style of reasoning, whereas CCS and CSP emphasize model-based reasoning. In the latter case, the... |

1639 |
Petri nets: properties, analysis, and applications
- Murata
- 1989
(Show Context)
Citation Context ...cs for P/T nets than the one chosen in this section. However, the framework of this section is sufficient for the remainder of this chapter. Good starting points for further reading on Petri nets are =-=[17, 46, 51, 55, 56, 57]-=-. In the remainder of this chapter, we introduce a process-algebraic theory with both a total-order and a step semantics. In addition, we study several concepts from Petri-net theory in the algebraic ... |

1418 |
A Calculus of Communicating Systems
- Milner
- 1980
(Show Context)
Citation Context ...tal-order and a partial-order setting. The theory ACP originates from [13]. Good introductions to ACP-style process algebra can be found in [5, 6]. Other well-known process-algebraic theories are CCS =-=[43, 44]-=- and CSP [35]. Specifications and verifications in ACP-style process algebra are based on an equational style of reasoning, whereas CCS and CSP emphasize model-based reasoning. In the latter case, the... |

858 |
Coloured Petri Nets, Basic Concepts, Analysis Methods and Practical Use
- Jensen
- 1997
(Show Context)
Citation Context ...1 Place fusion The most commonly used approach to modularizing Petri nets is the one where interfaces between modular nets are based on places. It appears in the Petri-net frameworks of, for example, =-=[15, 16, 34, 38]-=- and [8, Chapter 3]. It is also supported by tools as Design/CPN [39], ExSpect [7], and PEP [18]. Figure 8.1(a) shows a labeled P/T net of which the set of places is partitioned into two sets by means... |

700 |
Concurrency and automata on infinite sequences
- Park
- 1981
(Show Context)
Citation Context ...antics. The starting point is an algebraic theory in the style of the Algebra of Communicating Processes (ACP) [13]. The basic semantic equivalence that is used throughout the chapter is bisimilarity =-=[50]-=-. Bisimilarity is often used to provide process-algebraic theories with a semantics that, in the terminology of this chapter, can be characterized as a branching-time, interleaving, total-order semant... |

394 |
Representation of events in nerve nets and finite automata
- Kleene
- 1956
(Show Context)
Citation Context ...KS4 Table 5.4: The equational theory (ACP # + RN)(AC, # ). The binary Kleene star # was introduced in ACP-style process algebra in [11]. It is adapted from the original star operator as introduced in =-=[40]-=-. In [11], also the Axioms BKS1 through BKS4 appear, although BKS4 is formulated only for the specific renaming operators encapsulation and abstraction. Abstraction in 15 process algebra is not covere... |

385 |
Kommunikation mit Automaten
- Petri
- 1962
(Show Context)
Citation Context ...ic theories in the linear-time/branching-time spectrum. The interested reader is referred to [26, 27, 28]. Another important theme of this chapter is the study of concepts known from Petri-net theory =-=[52]-=- in the process-algebraic framework developed in this chapter. The Petri-net formalism is a well-known theory for describing and analyzing concurrent systems. Petri nets have been used both as a langu... |

384 | Process algebra for synchronous communication
- Bergstra, Klop
- 1984
(Show Context)
Citation Context ...pic is to illustrate a way to develop a process-algebraic theory with a partial-order semantics. The starting point is an algebraic theory in the style of the Algebra of Communicating Processes (ACP) =-=[13]-=-. The basic semantic equivalence that is used throughout the chapter is bisimilarity [50]. Bisimilarity is often used to provide process-algebraic theories with a semantics that, in the terminology of... |

311 | The linear time - branching time spectrum ii
- Glabbeek
- 1993
(Show Context)
Citation Context ...interleaving variant of such a process algebra. This chapter does not discuss variations of process-algebraic theories in the linear-time/branching-time spectrum. The interested reader is referred to =-=[26, 27, 28]-=-. Another important theme of this chapter is the study of concepts known from Petri-net theory [52] in the process-algebraic framework developed in this chapter. The Petri-net formalism is a well-know... |

267 |
Event structures
- Winskel
- 1987
(Show Context)
Citation Context ...ders. Examples of such frameworks can be found in [25, 31, 37, 54]. Other well-known partial-order based theories are trace theory as described in [41] and the theory of event structures presented in =-=[47, 60]-=-. Future research Several topics for future study have already been mentioned earlier in this chapter. At this point, we mention two other general directions for future research. First, it is interest... |

240 | Modelling concurrency with partial orders
- Pratt
- 1986
(Show Context)
Citation Context ...aken an approach to modeling and analyzing concurrent systems based on an explicit representation of concurrent-system behavior in terms of partial orders. Examples of such frameworks can be found in =-=[25, 31, 37, 54]-=-. Other well-known partial-order based theories are trace theory as described in [41] and the theory of event structures presented in [47, 60]. Future research Several topics for future study have alr... |

239 |
Predicate Calculus and Program Semantics
- Dijkstra, Scholten
- 1990
(Show Context)
Citation Context ...nce. The notion of subbags is defined as expected: Bag X is a subbag of Y , denoted X # Y , if and only if for all a # A, X (a) # Y (a). Quantifiers For quantifiers, we use the convenient notation of =-=[23]-=- in which the bound variables are represented explicitly. A quantifier is a commutative and associative operator with a unit element. Examples of well-known quantifiers are the logical quantifiers # a... |

238 |
Algebra of communicating processes with abstraction
- Bergstra, Klop
- 1985
(Show Context)
Citation Context ...ively expected from a two-place buffer. In the current algebraic framework, it is not possible to hide actions. The extension of ACP-style process algebra with an abstraction operator is described in =-=[6, 14]-=-. It goes beyond the scope of this chapter to treat abstraction in detail. c 2 s 3 r 1 r 1 s 3 s 3 s 3 r 1 r 1 (a) (b) Figure 5.30: A two-place buffer in a total-order semantics. 5.4 A step semantics ... |

217 |
Process Algebra
- Baeten, Weijland
- 1990
(Show Context)
Citation Context ... m # (# i : i # N : c i e i #)) The next step is to show the following result. (ACP c # + RN + SC)(AC(C, A), #) # # ca(I ) (# I # m # (# i : i # N : c e i #)) = # ca(I ) (# I m (# i : i # N : e i #)) =-=(5)-=- That is, the extra input causes in the merge quantification in term (4) do not prevent the occurrence of any action of this quantification after the execution of the action corresponding to Petri ele... |

169 |
Net Theory and the Modeling of Systems
- Petri
(Show Context)
Citation Context ...cs for P/T nets than the one chosen in this section. However, the framework of this section is sufficient for the remainder of this chapter. Good starting points for further reading on Petri nets are =-=[17, 46, 51, 55, 56, 57]-=-. In the remainder of this chapter, we introduce a process-algebraic theory with both a total-order and a step semantics. In addition, we study several concepts from Petri-net theory in the algebraic ... |

129 | Structural operational semantics
- Aceto, Fokkink, et al.
- 2001
(Show Context)
Citation Context ... [y W (y,x) | yFx] and ox = [y W (x,y) | x Fy]. 9 Example 4.4. Consider again the P/T net in Figure 4.2. The bag of input places of transition r is the singleton bag containing place 1; that is, ir = =-=[1]-=-. The bag of output places of r is [2, 3]; or = [2, 3]. The bag of output transitions of place 6 is empty; o6 = 0. A labeled P/T net as defined above is a static structure. Labeled P/T nets also have ... |

128 |
Trace theory
- Mazurkiewicz
- 1986
(Show Context)
Citation Context ... of concurrent-system behavior in terms of partial orders. Examples of such frameworks can be found in [25, 31, 37, 54]. Other well-known partial-order based theories are trace theory as described in =-=[41]-=- and the theory of event structures presented in [47, 60]. Future research Several topics for future study have already been mentioned earlier in this chapter. At this point, we mention two other gene... |

117 |
The Box Calculus: A New Causal Algebra with Multi-label Communication
- Best, Devilllers, et al.
- 1992
(Show Context)
Citation Context ...1 Place fusion The most commonly used approach to modularizing Petri nets is the one where interfaces between modular nets are based on places. It appears in the Petri-net frameworks of, for example, =-=[15, 16, 34, 38]-=- and [8, Chapter 3]. It is also supported by tools as Design/CPN [39], ExSpect [7], and PEP [18]. Figure 8.1(a) shows a labeled P/T net of which the set of places is partitioned into two sets by means... |

105 | The syntax and semantics of CRL - Groote, Ponse - 1995 |

102 | The linear time – branching time spectrum I; the semantics of concrete, sequential processes
- Glabbeek
- 2001
(Show Context)
Citation Context ...interleaving variant of such a process algebra. This chapter does not discuss variations of process-algebraic theories in the linear-time/branching-time spectrum. The interested reader is referred to =-=[26, 27, 28]-=-. Another important theme of this chapter is the study of concepts known from Petri-net theory [52] in the process-algebraic framework developed in this chapter. The Petri-net formalism is a well-know... |

97 |
Process algebra, volume 18 of Cambridge tracts
- Baeten, Weijland
- 1990
(Show Context)
Citation Context ...h is an immediate corollary of property (3) above. For all J # N with J # = #, (ACP c # + RN + SC)(AC(C, A), #) # # ca(I ) (# I # m # (| j : j # J : c j e j )) = # ca(I ) (# I m (| j : j # J : e j )) =-=(6)-=- Assume that n > 0. To prove property (5) in this case, the following auxiliary notations are needed. Based on Definition 10.8 (Petri element), it is possible to lift the consumption and production fu... |

92 | The syntax and semantics of μCRL - Groote, Ponse - 1994 |

79 |
Petri net models for algebraic theories of concurrency (extended abstract
- Glabbeek, Vaandrager
- 1987
(Show Context)
Citation Context ...ebraic theory via a Petri-net semantics with a partial-order semantics. The most well-known example of this approach is the Petri Box Calculus of [15, 16]. Other examples of this line of research are =-=[19, 21, 29, 30, 45, 49, 59]-=-. In this chapter, the converse approach is pursued. In particular, in Section 6.3, a translation from labeled P/T nets into algebraic terms is given. Other examples of this approach are [20, 22]. The... |

68 | Algebraic process verification - Groote, Reniers - 2001 |

51 |
Process algebra with iteration and nesting
- Bergstra, Bethke, et al.
- 1994
(Show Context)
Citation Context ...x + y) # z) + z) = (x + y) # z BKS3 # f (x # y) = # f (x) # # f (y) BKS4 Table 5.4: The equational theory (ACP # + RN)(AC, # ). The binary Kleene star # was introduced in ACP-style process algebra in =-=[11]-=-. It is adapted from the original star operator as introduced in [40]. In [11], also the Axioms BKS1 through BKS4 appear, although BKS4 is formulated only for the specific renaming operators encapsula... |

47 | The equational theory of pomsets
- Gischer
- 1988
(Show Context)
Citation Context ...aken an approach to modeling and analyzing concurrent systems based on an explicit representation of concurrent-system behavior in terms of partial orders. Examples of such frameworks can be found in =-=[25, 31, 37, 54]-=-. Other well-known partial-order based theories are trace theory as described in [41] and the theory of event structures presented in [47, 60]. Future research Several topics for future study have alr... |

47 |
On partial languages
- Grabowski
- 1981
(Show Context)
Citation Context ...aken an approach to modeling and analyzing concurrent systems based on an explicit representation of concurrent-system behavior in terms of partial orders. Examples of such frameworks can be found in =-=[25, 31, 37, 54]-=-. Other well-known partial-order based theories are trace theory as described in [41] and the theory of event structures presented in [47, 60]. Future research Several topics for future study have alr... |

43 |
Informations Systems Engineering - A formal approach
- Hee
- 1994
(Show Context)
Citation Context ...1 Place fusion The most commonly used approach to modularizing Petri nets is the one where interfaces between modular nets are based on places. It appears in the Petri-net frameworks of, for example, =-=[15, 16, 34, 38]-=- and [8, Chapter 3]. It is also supported by tools as Design/CPN [39], ExSpect [7], and PEP [18]. Figure 8.1(a) shows a labeled P/T net of which the set of places is partitioned into two sets by means... |

43 |
Petri nets: an introduction, volume 4
- Reisig
- 1985
(Show Context)
Citation Context ...cs for P/T nets than the one chosen in this section. However, the framework of this section is sufficient for the remainder of this chapter. Good starting points for further reading on Petri nets are =-=[17, 46, 51, 55, 56, 57]-=-. In the remainder of this chapter, we introduce a process-algebraic theory with both a total-order and a step semantics. In addition, we study several concepts from Petri-net theory in the algebraic ... |

42 |
Global renaming operators in concrete process algebra
- Baeten, Bergstra
- 1988
(Show Context)
Citation Context ... | x Fy]. 9 Example 4.4. Consider again the P/T net in Figure 4.2. The bag of input places of transition r is the singleton bag containing place 1; that is, ir = [1]. The bag of output places of r is =-=[2, 3]-=-; or = [2, 3]. The bag of output transitions of place 6 is empty; o6 = 0. A labeled P/T net as defined above is a static structure. Labeled P/T nets also have a behavior. The behavior of a net is dete... |

42 |
Fernández C. Nonsequential processes – A Petri net view
- Best, C
- 1988
(Show Context)
Citation Context |

37 |
A Survey of Equivalence Notions of Net Based Systems
- Pomello, Rozenberg, et al.
- 1992
(Show Context)
Citation Context ... in a framework based on some other partial-order semantics. There are partial-order semantics that capture causalities more accurately than the step semantics used in this chapter (see, for example, =-=[53]-=-, for more details). Second, it would be interesting to investigate the application of the concepts studied in this chapter in the development and analysis of complex concurrent systems. Such applicat... |

36 |
A distributed operational semantics for CCS based on condition/event systems
- Degano, Nicola, et al.
- 1988
(Show Context)
Citation Context ...ebraic theory via a Petri-net semantics with a partial-order semantics. The most well-known example of this approach is the Petri Box Calculus of [15, 16]. Other examples of this line of research are =-=[19, 21, 29, 30, 45, 49, 59]-=-. In this chapter, the converse approach is pursued. In particular, in Section 6.3, a translation from labeled P/T nets into algebraic terms is given. Other examples of this approach are [20, 22]. The... |

33 |
Flow models of distributed computations: Three equivalent semantics for CCS
- BOUDOL, CASTELLANI
- 1994
(Show Context)
Citation Context ...ebraic theory via a Petri-net semantics with a partial-order semantics. The most well-known example of this approach is the Petri Box Calculus of [15, 16]. Other examples of this line of research are =-=[19, 21, 29, 30, 45, 49, 59]-=-. In this chapter, the converse approach is pursued. In particular, in Section 6.3, a translation from labeled P/T nets into algebraic terms is given. Other examples of this approach are [20, 22]. The... |

25 |
Terms of Nets: Systems Design with Petri Nets and Process Algebra
- In
- 1998
(Show Context)
Citation Context ...learly, property (5) conforms to (ACP c # + RN + SC)(AC(C, A), #) # E(#) (7) The basis of the inductive proof is the following property. For all i # N , (ACP c # + RN + SC)(AC(C, A), #) # E(N \ {i }) =-=(8)-=- 71 Property (8) is easily proven, because in each case the quantifications in the desired equality reduce to single action-prefix constants. The induction hypothesis is as follows. For all J with # #... |

21 | A complete equational axiomatization for prefix iteration
- Fokkink
- 1994
(Show Context)
Citation Context ...lems can be solved by replacing the sequential-composition operator by a set of actionprefix operators in the style of CCS [43, 44] and the binary Kleene star by a class of prefix-iteration operators =-=[12, 24]-=-. It is important to define a suitable class of actions that can form the basis for these prefix operators. The following definition is inspired by the notion of a transition in a P/T net. A Petri ele... |

18 |
On the relationship of CCS and Petri nets
- Goltz, Mycroft
(Show Context)
Citation Context |

16 |
Nonaxiomatisability of equivalences over finite state processes
- Sewell
- 1997
(Show Context)
Citation Context ...he best of our knowledge, it is still an open problem whether the equational theory (ACP # +RN+ RSP # +SC)(AC, # ) is complete for model M(AC, # ). (Note, however, that it follows from the results of =-=[58]-=- that the theory (ACP # + RN + SC)(AC, # ), which is obtained by removing the recursion principle RSP # , is not complete for M(AC, # ).) Interleaving process algebra In the previous subsection, it ha... |

11 | Process algebra with recursive operations
- Bergstra, Fokkink, et al.
(Show Context)
Citation Context ...ecific renaming operators encapsulation and abstraction. Abstraction in 15 process algebra is not covered in this chapter. An extensive study of iteration in ACP-style process algebra can be found in =-=[12]-=-. In Table 5.4, function f ranges over the set of renaming functions RF as introduced above. Axiom BKS1 is characteristic for iteration. It states that behavior x is iterated an arbitrary number of ti... |

11 |
Layered Design of Parallel Systems
- Janssen
- 1994
(Show Context)
Citation Context |

10 |
PEP: Documentation and User Guide. Universitat Hildesheim. Available together with the tool via: http://www.informatik.uni-hildesheim.de/pep
- Best, Grahlmann
(Show Context)
Citation Context ...n modular nets are based on places. It appears in the Petri-net frameworks of, for example, [15, 16, 34, 38] and [8, Chapter 3]. It is also supported by tools as Design/CPN [39], ExSpect [7], and PEP =-=[18]-=-. Figure 8.1(a) shows a labeled P/T net of which the set of places is partitioned into two sets by means of a dashed box. Places inside the box are internal places, whereas places outside the box are ... |

9 |
A Consistent Model for Nets and Process Algebras
- Best, Devillers, et al.
- 2001
(Show Context)
Citation Context |

8 |
The Total Order Assumption
- Baeten
- 1992
(Show Context)
Citation Context ...e operator. The specification follows the same approach as the one illustrated in Examples 6.3 and 6.4. Let I be the set of causes {1, 2, 3, 4, 5}. Consider the following specification. # ca(I ) (# I =-=[1,2]-=- (([1, 2], [r 1 ], [3]) # # # ([3], [c 2 ], [4, 5]) # # # ([4], [r 1 ], [1]) # # # ([5], [s 3 ], [2]) # #)) (3) Calculations similar to the calculations in Example 6.3 show that also this specificatio... |

8 |
Non interleaving process algebra
- Baeten, Bergstra
- 1993
(Show Context)
Citation Context ...fferent initial actions. However, in the initial state of process X , only one of these fifteen actions can actually occur. # I 1 (X) = (1, r, [2, 3]) # I [2,3] (X) = (1, r, [2, 3]) ((2, p 1 , 4) # I =-=[3,4]-=- (X) + (3, p 2 , 5) # I [2,5] (X) + ([2, 3], [ p 1 , p 2 ], [4, 5]) # I [4,5] (X)) = (1, r, [2, 3]) ((2, p 1 , 4) (3, p 2 , 5) + (3, p 2 , 5) (2, p 1 , 4) + ([2, 3], [ p 1 , p 2 ], [4, 5])) # I [4,5] ... |

6 |
Nets, Terms and Formulas, volume 23 of Cambridge Tracts
- Olderog
- 1991
(Show Context)
Citation Context |

5 |
Design/CPN. A reference manual
- Jensen, Christensen, et al.
- 1996
(Show Context)
Citation Context ...one where interfaces between modular nets are based on places. It appears in the Petri-net frameworks of, for example, [15, 16, 34, 38] and [8, Chapter 3]. It is also supported by tools as Design/CPN =-=[39]-=-, ExSpect [7], and PEP [18]. Figure 8.1(a) shows a labeled P/T net of which the set of places is partitioned into two sets by means of a dashed box. Places inside the box are internal places, whereas ... |

5 |
Degrees of non-determinism and concurrency: A Petri net view
- Nielsen, Thiagarajan
- 1984
(Show Context)
Citation Context ...ter, can be characterized as a branching-time, interleaving, total-order semantics. An interesting aspect of bisimilarity is that it can be turned into a semantic equivalence called step bisimilarity =-=[48]-=- that provides the basis for a partial-order view on the behavior of concurrent systems. By means of step bisimilarity, it is possible to obtain a processalgebraic theory with a branching-time, interl... |

5 |
Representing CCS Programs by Finite Predicate/Transition Nets
- Taubner
- 1989
(Show Context)
Citation Context |

4 |
Petri nets and algebraic calculi of processes
- Boudol, Roucairol, et al.
- 1985
(Show Context)
Citation Context ..., 45, 49, 59]. In this chapter, the converse approach is pursued. In particular, in Section 6.3, a translation from labeled P/T nets into algebraic terms is given. Other examples of this approach are =-=[20, 22]-=-. The main difference between the approach taken in this chapter and other approaches is that this chapter emphasizes equational reasoning, whereas other approaches often emphasize the semantic framew... |

4 |
Combining CCS and Petri Nets via Structural Axioms
- Montanari, Yankelevich
- 1994
(Show Context)
Citation Context |