## Finite equational bases in process algebra: Results and open questions (2005)

### Cached

### Download Links

- [www.cs.vu.nl]
- [www.brics.dk]
- [www.brics.dk]
- DBLP

### Other Repositories/Bibliography

Venue: | Processes, Terms and Cycles: Steps on the Road to Infinity, LNCS 3838 |

Citations: | 28 - 19 self |

### BibTeX

@INPROCEEDINGS{Aceto05finiteequational,

author = {Luca Aceto and Wan Fokkink and Anna Ingolfsdottir and Bas Luttik},

title = {Finite equational bases in process algebra: Results and open questions},

booktitle = {Processes, Terms and Cycles: Steps on the Road to Infinity, LNCS 3838},

year = {2005},

pages = {338--367},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. Van Glabbeek (1990) presented the linear time/branching time spectrum of behavioral equivalences for finitely branching, concrete, sequential processes. He studied these semantics in the setting of the basic process algebra BCCSP, and tried to give finite complete axiomatizations for them. Obtaining such axiomatizations in concurrency theory often turns out to be difficult, even in the setting of simple languages like BCCSP. This has raised a host of open questions that have been the subject of intensive research in recent years. Most of these questions have been settled over BCCSP, either positively by giving a finite complete axiomatization, or negatively by proving that such an axiomatization does not exist. Still some open questions remain. This paper reports on these results, and on the state-of-the-art in axiomatizations for richer process algebras with constructs like sequential and parallel composition. 1

### Citations

3409 | Communicating Sequential Processes
- Hoare
(Show Context)
Citation Context ... behavioural semantics in van Glabbeek’s spectrum is offered in [24]. The equational axiomatizations offered ibidem are over the language BCCSP, a common fragment of Milner’s CCS [36] and Hoare’s CSP =-=[31]-=- suitable for describing finite synchronization trees, and characterize the differences between behavioural semantics in terms of a few revealing axioms. If the answer to the second basic question men... |

3218 |
Communication and Concurrency
- Milner
- 1989
(Show Context)
Citation Context ...on, which is widely used in light of its flexibility and applicability. In particular, they underlie Plotkin’s Structural Operational Semantics [41, 42] and, following Milner’s pioneering work on CCS =-=[36]-=-, are by now the standard formalism for describing the semantics of various process description languages. LTSs model processes by explicitly describing their states and their transitions from state t... |

1289 | A Structured Approach to Operational Semantics
- Plotkin
- 1981
(Show Context)
Citation Context ...al formalism for the description of concurrent computation, which is widely used in light of its flexibility and applicability. In particular, they underlie Plotkin’s Structural Operational Semantics =-=[41, 42]-=- and, following Milner’s pioneering work on CCS [36], are by now the standard formalism for describing the semantics of various process description languages. LTSs model processes by explicitly descri... |

495 | R.: Algebraic laws for nondeterminism and concurrency
- Hennessy, Milner
- 1985
(Show Context)
Citation Context ...ly by proving that such an axiomatization does not exist. Still some open questions remain—especially for process description languages and behavioural equivalences that, like observation equivalence =-=[29, 36]-=-, abstract, in some formal sense, from events in process behaviours that are deemed to be unobservable. In this paper, we report on positive and negative results pertaining to the existence of (finite... |

393 |
An Algebraic Theory of Processes
- Hennessy
- 1988
(Show Context)
Citation Context ...and that much more on it may be found in, e.g., the classic reference [15]. A self-contained presentation from a computer science perspective of the topics we now proceed to introduce may be found in =-=[27]-=-. 2.1 Preliminaries Σ-Algebras We start from a countably infinite set V of variables with typical elements x, y, w, z. A signature Σ consists of a set of operation symbols, disjoint from V , together ... |

377 |
Representation of events in nerve nets and finite automata, Automata studies, Annals of mathematics studies
- Kleene
- 1956
(Show Context)
Citation Context ...opment of modern algebra rests, to concurrency theory.sAn example of a typical axiom that holds for all of the classic algebras in the ACP family, and is familiar from the theory of regular languages =-=[16, 33]-=-, is (x + y) · z ≈ (x · z) + (y · z) . In the above equation, the operation symbols + and · stand for “alternative composition” (or nondeterministic choice) and “sequencing”, respectively. Intuitively... |

360 | Process algebra for synchronous communication
- Bergstra, Klop
(Show Context)
Citation Context ...heory of the expansion has a finite axiomatization. A classic example of this line of research, which can again be traced back to Jan Willem Klop’s work in concurrency theory, is offered by the paper =-=[10]-=-. There Bergstra and Klop showed how to give a finite axiomatization of the language ACP using the auxiliary left and communication merge operators to characterize parallel composition. As shown by Mo... |

290 | The Linear Time – Branching Time Spectrum II; The semantics of sequential systems with silent moves (extended abstract
- Glabbeek
- 1993
(Show Context)
Citation Context ...n its applications in process algebra, is the study of results pertaining to the existence or non-existence of finite bases for algebras. In the realm of concurrency theory, van Glabbeek presented in =-=[23, 24]-=- the linear time/branching time spectrum of behavioral equivalences for finitely branching, concrete, sequential processes. He studied these semantics in the setting of the basic process algebra BCCSP... |

277 |
Regular Algebra and Finite Machines
- Conway
- 1971
(Show Context)
Citation Context ...opment of modern algebra rests, to concurrency theory.sAn example of a typical axiom that holds for all of the classic algebras in the ACP family, and is familiar from the theory of regular languages =-=[16, 33]-=-, is (x + y) · z ≈ (x · z) + (y · z) . In the above equation, the operation symbols + and · stand for “alternative composition” (or nondeterministic choice) and “sequencing”, respectively. Intuitively... |

256 | Branching Time and Abstraction in Bisimulation Semantics - Glabbeek, Weijland - 1996 |

218 | Algebra of communicating processes with abstraction - Bergstra, Klop - 1985 |

176 |
On the structure of abstract algebras
- Birkhoff
- 1935
(Show Context)
Citation Context ...e of these two main methodologies. Model-theoretic Techniques If a set of equations E is sound in an algebra A, we say that A is a model for E. By Birkhoff’s completeness theorem for equational logic =-=[13]-=-, each equation that is derived from E holds in A, if A is a model for E. Thus, to prove that an equation t ≈ u is not derivable from E it is sufficient to find an algebra that is a model for E but no... |

159 |
Formal verification of parallel programs
- KELLER
- 1976
(Show Context)
Citation Context ...y, one may define intuitively appealing models of its axiom system as quotients of the collection of labelled transition systems modulo some behavioural congruence. Labelled transition systems (LTSs) =-=[32]-=- are a fundamental formalism for the description of concurrent computation, which is widely used in light of its flexibility and applicability. In particular, they underlie Plotkin’s Structural Operat... |

122 |
A complete inference system for a class of regular behaviours
- Milner
- 1984
(Show Context)
Citation Context ...t algebra of closed terms T(Σ) modulo ∼. Since each element of the algebra T(Σ)/∼ is denotable, it follows that E is also ω-complete. To the best of our knowledge, this technique was first applied in =-=[35]-=- by Milner to show completeness of his inference system for bisimulation equivalence over the regular fragment of the Calculus of Communicating Systems (CCS) [36]. 11sCover Equations This technique fr... |

106 |
Process Algebra. Cambridge Tracts in Theoretical Computer Science 18
- Baeten, Weijland
- 1990
(Show Context)
Citation Context ...Willem Klop’s main contributions to the theory of concurrency is the development of the ACP family of process algebras in collaboration with Jan Bergstra—see the original papers [8–12], the textbooks =-=[6, 18]-=- and the historical paper [5]. Process algebras in the ACP style are defined, following the tradition of the algebraic specification of abstract data types, relying on tools from universal algebra and... |

103 | Introduction to Process Algebra
- Fokkink
- 1999
(Show Context)
Citation Context ...Willem Klop’s main contributions to the theory of concurrency is the development of the ACP family of process algebras in collaboration with Jan Bergstra—see the original papers [8–12], the textbooks =-=[6, 18]-=- and the historical paper [5]. Process algebras in the ACP style are defined, following the tradition of the algebraic specification of abstract data types, relying on tools from universal algebra and... |

95 | The linear time – branching time spectrum I; the semantics of concrete, sequential processes
- Glabbeek
- 2001
(Show Context)
Citation Context ...sus on what constitutes an appropriate notion of observable behaviour for reactive systems has led to a large number of proposals for behavioural equivalences for concurrent processes. (See the study =-=[24]-=-, where van Glabbeek presents the linear time/branching time spectrum—a lattice of known behavioural equivalences and preorders over LTSs, ordered by inclusion.) Having defined a model of an axiom sys... |

60 | The algebra of recursively defined processes and the algebra of regular processes - Bergstra, Klop - 1984 |

57 | A Brief History of Process Algebra
- Baeten
(Show Context)
Citation Context ...to the theory of concurrency is the development of the ACP family of process algebras in collaboration with Jan Bergstra—see the original papers [8–12], the textbooks [6, 18] and the historical paper =-=[5]-=-. Process algebras in the ACP style are defined, following the tradition of the algebraic specification of abstract data types, relying on tools from universal algebra and equational logic. More speci... |

50 |
Axioms for Concurrency
- Moller
- 1989
(Show Context)
Citation Context ...et, occurrences of action names in axioms are interpreted as variables (or action schemes). Bisimulation The core axioms in Table 1 are sound and ground complete for BCCSP modulo bisimulation. Moller =-=[37]-=- proved using normal forms that this axiomatization is ω-complete; Groote provided an alternative proof of this result in [25] using inverted substitutions. A1 x + y ≈ y + x A2 (x + y) + z ≈ x + (y + ... |

47 | The equational theory of pomsets
- GISCHER
- 1988
(Show Context)
Citation Context ...finite subset E of the equational theory, construct an algebra AE that is a model of E, but in which some of the equations en fail. Examples of the application of this strategy may be found in, e.g., =-=[1, 2, 16, 22]-=-. 12sProof-theoretic Techniques Recall that an equation t ≈ u is derivable from a set of equations E if there is a sequence ti ≈ ui (1 ≤ i ≤ n) of equations such that – tn = t and un = u, and – for ea... |

45 | Algebra of communicating processes - Bergstra, Klop - 1986 |

45 |
A Course in Universal Algebra. Graduate Texts in Mathematics
- Burris, Sankappanavar
- 1981
(Show Context)
Citation Context ...paper. We state at the outset that we shall not need very deep results or 4sconstructions from universal algebra in what follows, and that much more on it may be found in, e.g., the classic reference =-=[15]-=-. A self-contained presentation from a computer science perspective of the topics we now proceed to introduce may be found in [27]. 2.1 Preliminaries Σ-Algebras We start from a countably infinite set ... |

42 |
Fixed point semantics in process algebra
- Bergstra, Klop
- 1982
(Show Context)
Citation Context ...mon to a collection of algebraic structures—namely, their models. Early examples of models of the axiom systems for ACP style process algebras were the “projective limit” model— as employed in, e.g., =-=[8]-=-—, and the “graph model” adopted in [11]. Given a language in the ACP family, one may define intuitively appealing models of its axiom system as quotients of the collection of labelled transition syst... |

32 |
The importance of the left merge operator in process algebras
- Moller
(Show Context)
Citation Context ...re Bergstra and Klop showed how to give a finite axiomatization of the language ACP using the auxiliary left and communication merge operators to characterize parallel composition. As shown by Moller =-=[38, 39]-=-, auxiliary operators are needed to obtain a finite basis for that language because the process algebras CCS and ACP without the auxiliary left merge operator from [8] do not have a finite equational ... |

28 |
Axiomatising finite concurrent processes
- Hennessy
- 1988
(Show Context)
Citation Context ...s are indispensable. Three such auxiliary operators have been proposed in the literature: Bergstra and Klop introduced the left merge (� ) in [8] and the communication merge (|) in [10], and Hennessy =-=[28]-=- introduced an operation that we call Hennessy’s merge (| / ). In the remainder of this section we discuss these auxiliary operators in the context of BCCSP. In the next section we examine the consequ... |

28 | Bisimulation equivalence is decidable for normed process algebra
- Hirshfeld, Jerrum
- 1999
(Show Context)
Citation Context ...ther axioms). It is an example of a so-called mixed equation, equating a parallel composition and a sequential composition. There is a deep theory of mixed equations developed by Hirshfeld and Jerrum =-=[30]-=- for the benefit of their proof that bisimulation equivalence is decidable for normed PA. The proof in [19] that the presented axiomatization is ω-complete partly relies on their theory. Incorporation... |

25 |
A new strategy for proving ω–completeness with applications in process algebra
- GROOTE
- 1990
(Show Context)
Citation Context ...ble 1 are sound and ground complete for BCCSP modulo bisimulation. Moller [37] proved using normal forms that this axiomatization is ω-complete; Groote provided an alternative proof of this result in =-=[25]-=- using inverted substitutions. A1 x + y ≈ y + x A2 (x + y) + z ≈ x + (y + z) A3 x + x ≈ x A6 x + 0 ≈ x Table 1. The axioms for bisimulation. 2-Nested Simulation and Possible Futures Aceto, Fokkink, va... |

24 | A menagerie of non-finitely based process semantics over BPA*—from ready simulation to completed traces
- Aceto, Fokkink, et al.
- 1998
(Show Context)
Citation Context ...finite subset E of the equational theory, construct an algebra AE that is a model of E, but in which some of the equations en fail. Examples of the application of this strategy may be found in, e.g., =-=[1, 2, 16, 22]-=-. 12sProof-theoretic Techniques Recall that an equation t ≈ u is derivable from a set of equations E if there is a sequence ti ≈ ui (1 ≤ i ≤ n) of equations such that – tn = t and un = u, and – for ea... |

19 | CCS with Hennessy’s merge has no finite equational axiomatization
- Aceto, Fokkink, et al.
(Show Context)
Citation Context ...ra and Klop in [10], it turns out that the operation itself cannot be finitely axiomatized. Assuming the CCS synchronization mechanism (see the beginning of Section 4), the authors recently proved in =-=[3]-=- that there does not exist a finite set of sound BCCSP �,| /-equations from which all equations of the form a0 | / ψn ≈ aψn + are equationally derivable. 4.4 Overview n� i=0 τa i (with ψn = n� āa i , ... |

17 | On the Axiomatizability of Ready Traces, Ready Simulation and Failure Traces
- Blom, Fokkink, et al.
- 2003
(Show Context)
Citation Context ...n. However, they cannot be derived from E. Ready Simulation Van Glabbeek presented a conditional axiom for ready simulation equivalence: I(x) = I(y) ⇒ a(x + y) ≈ a(x + y) + ay. Blom, Fokkink and Nain =-=[14]-=- showed that a sound and ground complete finite equational axiomatization for BCCSP modulo ready simulation exists. It is obtained by extending the four core axioms with a(bx + by + z) ≈ a(bx + by + z... |

15 | Nested semantics over finite trees are equationally hard
- Aceto, Fokkink, et al.
(Show Context)
Citation Context .... A1 x + y ≈ y + x A2 (x + y) + z ≈ x + (y + z) A3 x + x ≈ x A6 x + 0 ≈ x Table 1. The axioms for bisimulation. 2-Nested Simulation and Possible Futures Aceto, Fokkink, van Glabbeek and Ingolfsdottir =-=[4]-=- proved that BCCSP modulo any semantics no coarser than possible futures and no finer than 2-nested simulation does not possess a finite sound and ground complete axiomatization. The infinite family o... |

15 |
The nonexistence of finite axiomatisations for CCS congruences
- Moller
- 1990
(Show Context)
Citation Context ...re Bergstra and Klop showed how to give a finite axiomatization of the language ACP using the auxiliary left and communication merge operators to characterize parallel composition. As shown by Moller =-=[38, 39]-=-, auxiliary operators are needed to obtain a finite basis for that language because the process algebras CCS and ACP without the auxiliary left merge operator from [8] do not have a finite equational ... |

13 |
An ω-complete equational specification of interleaving
- Fokkink, Luttik
- 2000
(Show Context)
Citation Context ...zation of BPA. Moreover, the axiomatization is ω-complete. As far as we know, this latter result does not explicitly appear in print, but a proof can be extracted from the ω-completeness proof for PA =-=[19]-=- that we discuss below. In [38], Moller adapted his proof that BCCSP � is not finitely based to the setting with sequential composition. The infinite family of equations he uses to establish this resu... |

10 | The max-plus algebra of the natural numbers has no finite equational basis, Theoretical Comput
- Aceto, Ésik, et al.
- 2003
(Show Context)
Citation Context ...finite subset E of the equational theory, construct an algebra AE that is a model of E, but in which some of the equations en fail. Examples of the application of this strategy may be found in, e.g., =-=[1, 2, 16, 22]-=-. 12sProof-theoretic Techniques Recall that an equation t ≈ u is derivable from a set of equations E if there is a sequence ti ≈ ui (1 ≤ i ≤ n) of equations such that – tn = t and un = u, and – for ea... |

10 |
Partial evaluation and ω-completeness of algebraic specifications
- Heering
- 1986
(Show Context)
Citation Context ...matizations that are ω-complete. In fact, using an ω-complete axiomatization one can avoid proofs by (structural) induction in favour of purely equational reasoning. Moreover, as argued by Heering in =-=[26]-=-, ωcompleteness of an axiom system is desirable in the partial evaluation of programs. A classic example of an axiom system that is not ω-complete is that for the lambdacalculus—see [40]. Many of the ... |

10 |
The λ-calculus is ω-incomplete
- Plotkin
- 1974
(Show Context)
Citation Context ...by Heering in [26], ωcompleteness of an axiom system is desirable in the partial evaluation of programs. A classic example of an axiom system that is not ω-complete is that for the lambdacalculus—see =-=[40]-=-. Many of the existing axiomatizations of behavioural equivalences over expressive process description languages studied in concurrency theory are powerful enough to prove all of the valid equalities ... |

9 | On finite alphabets and infinite bases: From ready pairs to possible worlds
- Fokkink, Nain
- 2004
(Show Context)
Citation Context ...ompleteness of his inference system for bisimulation equivalence over the regular fragment of the Calculus of Communicating Systems (CCS) [36]. 11sCover Equations This technique from Fokkink and Nain =-=[20]-=- is tailored to BCCSP. The aim is to obtain an explicit description of the equational theory for a particular semantics. The central idea is that if an equation t ≈ u is sound for BCCSP modulo some se... |

9 | A finite basis for failure semantics
- Fokkink, Nain
- 2005
(Show Context)
Citation Context ...xiomatization for BCCSP modulo failure pairs is obtained by extending the four core axioms with a(bx + by + z) ≈ a(bx + by + z) + a(bx + z) ax + a(y + z) ≈ ax + a(y + z) + a(x + y) . Fokkink and Nain =-=[21]-=- proved using cover equations that when A is infinite, this axiomatization is ω-complete. They also proved that when A is finite, one extra axiom is needed to obtain an ω-complete axiomatization: |A| ... |

8 | An omega-complete equational specification of interleaving - FOKKINK, LUTTIK - 2000 |

7 |
An interactive proof tool for process algebras
- Lin
- 1992
(Show Context)
Citation Context ...sed to perform system verifications in a purely syntactic way using general purpose theorem provers or proof checkers, and form the basis of purpose-built axiomatic verification tools like, e.g., PAM =-=[34]-=-. A positive answer to the first basic question raised above is therefore not just theoretically pleasing, but has potential practical applications. From the theoretical point of view, complete axioma... |

6 |
Series and parallel operations on pomsets
- ÉSIK, OKAWA
- 1999
(Show Context)
Citation Context ...s an equation in E that is not provable from E ′ . This can be achieved using either of the two general proof strategies described above. Applications of this proof methodology may be found in, e.g., =-=[16, 17]-=-. 3 On Finite Bases for BCCSP 3.1 The Linear Time/Branching Time Spectrum Van Glabbeek presented in [23, 24] the linear time/branching time spectrum of behavioural equivalences for finitely branching,... |

3 | Bas Luttik. CCS with Hennessy’s merge has no finite equational axiomatization - Aceto, Fokkink, et al. - 2003 |

2 | Fokkink and Sumit Nain. On finite alphabets and infinite bases: from ready pairs to possible worlds - Wan |

1 |
Which data types have omega-complete initial algebra specifications
- Bergstra, Heering
- 1994
(Show Context)
Citation Context ...d above is negative, then one may resort to expanding the signature with auxiliary operations, thus adding expressive power for the purpose of axiomatizing the equational theory. Bergstra and Heering =-=[7]-=- have proved that every algebra with a recursively enumerable equational theory has a finite complete equational axiomatization if it may involve a hidden sort and some auxiliary hidden functions. Tha... |