## Lifting Non-Finite Axiomatizability Results to Extensions of Process Algebras

### Cached

### Download Links

- [www.ru.is]
- [www.win.tue.nl]
- [www.win.tue.nl]
- [www.win.tue.nl]
- [www.cs.vu.nl]
- [www.win.tue.nl]
- [www.cs.vu.nl]
- DBLP

### Other Repositories/Bibliography

Citations: | 3 - 2 self |

### BibTeX

@MISC{Aceto_liftingnon-finite,

author = {Luca Aceto and Wan Fokkink and Anna Ingólfsdóttir and MohammadReza Mousavi},

title = {Lifting Non-Finite Axiomatizability Results to Extensions of Process Algebras },

year = {}

}

### OpenURL

### Abstract

This paper presents a general technique for obtaining new results pertaining to the non-finite axiomatizability of behavioural (pre)congruences over process algebras from old ones. The proposed technique is based on a variation on the classic idea of reduction mappings. In this setting, such reductions are translations between languages that preserve sound (in)equations and (in)equational proofs over the source language, and reflect families of (in)equations responsible for the non-finite axiomatizability of the target language. The proposed technique is applied to obtain a number of new non-finite axiomatizability theorems in process algebra via reduction to Moller’s celebrated non-finite axiomatizability result for CCS. The limitations of the reduction technique are also studied. In particular, it is shown that prebisimilarity is not finitely based over CCS with the divergent process Ω, but that this result cannot be proved by a reduction to the non-finite axiomatizability of CCS modulo bisimilarity.

### Citations

3398 | Communicating sequential processes
- Hoare
- 1985
(Show Context)
Citation Context ...bisimilarity. 1 Introduction Process algebras, such as the Algebra of Communicating Processes (ACP) [12], the Calculus of Communicating Systems (CCS) [28] and Communicating Sequential Processes (CSP) =-=[23]-=-, are prototype languages for the description of reactive systems. Since these languages may be used for describing specifications of process behaviour as well as their implementations, an important i... |

3203 |
Communication and concurrency
- Milner
- 1989
(Show Context)
Citation Context ... to the non-finite axiomatizability of CCS modulo bisimilarity. 1 Introduction Process algebras, such as the Algebra of Communicating Processes (ACP) [12], the Calculus of Communicating Systems (CCS) =-=[28]-=- and Communicating Sequential Processes (CSP) [23], are prototype languages for the description of reactive systems. Since these languages may be used for describing specifications of process behaviou... |

1286 | A structural approach to operational semantics
- Plotkin
- 1981
(Show Context)
Citation Context ...study paves the way for several other meta-theorems, once further assumptions are made regarding the underlying models. For example, we expect that, by committing to SOS rules in the style of Plotkin =-=[36]-=- as means of defining the semantics of the formalism, one may invoke existing meta-theorems from the theory of SOS (see, e.g., [10]) to provide sufficient syntactic conditions guaranteeing that the pr... |

522 |
Introduction to the theory of computation
- Sipser
- 1997
(Show Context)
Citation Context ...on a variation on the classic idea of reduction mappings, which underlies the proofs of many classic undecidability results in computability theory and of lower bounds in complexity theory—see, e.g., =-=[37]-=- for a textbook presentation. The basic idea underlying the reduction-based method we propose in this study is as follows. Assume that we have a language Lo that we know is not finitely axiomatizable ... |

360 | Process algebra for synchronous communication
- Bergstra, Klop
- 1984
(Show Context)
Citation Context ...r study. The addition of a single operator to a non-finitely axiomatizable formalism may make it finitely axiomatizable (e.g., adding the left-merge operator to the synchronization-free subset of CCS =-=[13]-=-). Conversely, the addition of a single operator may ruin the finite axiomatizability of a calculus (e.g., adding parallel composition to the sequential subset of CCS [28, 30]). Also, apparently simpl... |

267 |
Semantics of Programming Languages: Structures and Techniques. Foundations of Computing
- Gunter
- 1992
(Show Context)
Citation Context ...). Structural mappings afford the following crucial property, which describes their interplay with substitutions and is akin to the classic “substitution lemma” from denotational semantics—see, e.g., =-=[19]-=-. In the statement of the subsequent lemma, for each substitution σ over Σe we use �σ to denote the substitution over Σo mapping each variable x to � σ(x). Lemma 1. Let � : T (Σe) → T (Σo) be a struct... |

201 |
Process Algebra
- Baeten, Weijland
- 1990
(Show Context)
Citation Context ...t that this result cannot be proved by a reduction to the non-finite axiomatizability of CCS modulo bisimilarity. 1 Introduction Process algebras, such as the Algebra of Communicating Processes (ACP) =-=[12]-=-, the Calculus of Communicating Systems (CCS) [28] and Communicating Sequential Processes (CSP) [23], are prototype languages for the description of reactive systems. Since these languages may be used... |

172 |
A temporal calculus of communicating systems
- Moller, Tofts
- 1990
(Show Context)
Citation Context ...bility results for the following process algebras modulo their corresponding notions of (pre)congruence: 1. Discrete-time CCS modulo timed bisimilarity [38], 2. Temporal CCS modulo timed bisimilarity =-=[32]-=-, 3. ATP modulo timed bisimilarity [35], 4. TACS UT modulo faster-than preorder [24], 5. TACS LT modulo MT-preorder [25], 2s6. TACS modulo urgent timed bisimilarity [26] and 7. IMC modulo strong Marko... |

121 | Structural operational semantics
- Aceto, Fokkink, et al.
- 1999
(Show Context)
Citation Context ..., we expect that, by committing to SOS rules in the style of Plotkin [36] as means of defining the semantics of the formalism, one may invoke existing meta-theorems from the theory of SOS (see, e.g., =-=[10]-=-) to provide sufficient syntactic conditions guaranteeing that the premises of our algebraic meta-theorem hold. A promising future direction of research is to study whether one can apply our meta-theo... |

120 |
A domain equation for bisimulation
- Abramsky
- 1991
(Show Context)
Citation Context ...t hold that p ↓. So, for instance, a.Ω ↓, but Ω ↑ and a || Ω ↑. The following notion of prebisimilarity is a relevant notion of behavioural preorder in the presence of divergence as adopted in, e.g., =-=[1, 7, 8, 20, 27]-=-. We refer the interested reader to those papers and the references therein for a wealth of results on the semantic theory of CCS Ω modulo prebisimilarity. Definition 11. The relation ⊏ ∼pre is the la... |

105 | The algebra of timed processes ATP: theory and application
- Nicollin, Sifakis, et al.
- 1994
(Show Context)
Citation Context ...s algebras modulo their corresponding notions of (pre)congruence: 1. Discrete-time CCS modulo timed bisimilarity [38], 2. Temporal CCS modulo timed bisimilarity [32], 3. ATP modulo timed bisimilarity =-=[35]-=-, 4. TACS UT modulo faster-than preorder [24], 5. TACS LT modulo MT-preorder [25], 2s6. TACS modulo urgent timed bisimilarity [26] and 7. IMC modulo strong Markovian bisimilarity [22]. Our meta-theore... |

71 |
Interactive Markov Chains and the Quest for Quantified Quality. Volume 2428 of LNCS
- Hermanns
- 2002
(Show Context)
Citation Context ...med bisimilarity [35], 4. TACS UT modulo faster-than preorder [24], 5. TACS LT modulo MT-preorder [25], 2s6. TACS modulo urgent timed bisimilarity [26] and 7. IMC modulo strong Markovian bisimilarity =-=[22]-=-. Our meta-theorem is algebraic in nature and does not rely on any assumption on the specification of the semantics of the languages to which it can be applied. We believe that the general result we p... |

61 |
Real-time behaviour of asynchronous agents
- Yi
- 1990
(Show Context)
Citation Context ...algebras. Namely, we prove non-finite axiomatizability results for the following process algebras modulo their corresponding notions of (pre)congruence: 1. Discrete-time CCS modulo timed bisimilarity =-=[38]-=-, 2. Temporal CCS modulo timed bisimilarity [32], 3. ATP modulo timed bisimilarity [35], 4. TACS UT modulo faster-than preorder [24], 5. TACS LT modulo MT-preorder [25], 2s6. TACS modulo urgent timed ... |

50 |
Axioms for Concurrency
- Moller
- 1989
(Show Context)
Citation Context ...(nr. 070030041) and “New Developments in Operational Semantics” (nr. 080039021) of the Icelandic Research Fund.sfinite axiomatizability of process algebras go back to the Ph.D. thesis of Faron Moller =-=[29]-=-, in which he showed that strong bisimilarity is not finitely based over CCS and over ACP without the left-merge operator. Since then, several other non-finite axiomatizability results have been obtai... |

40 | Termination, deadlock, and divergence
- Aceto, Hennessy
(Show Context)
Citation Context ... algebra, whose non-finite axiomatizability cannot be shown in that fashion. 5.1 CCS Ω and Prebisimilarity Consider the following syntax for the language CCS Ω (a variant of the calculus presented in =-=[7]-=-). P ::= 0 | Ω | a.P | P + P | P || P The operational semantics of CCS Ω is given by two ingredients: a → transitions, which are defined by the same deduction rules as in CCS (thus, Ω has no outgoing ... |

35 | µCRL: A toolset for analysing algebraic specifications
- Blom, Fokkink, et al.
(Show Context)
Citation Context ...ooking at the equations that those equivalences satisfy. Finally, an axiomatization of the relevant notion of equivalence may form the basis for verification tools based on theorem-proving technology =-=[14, 18]-=-. From the theoretical point of view, a fundamental question in the study of algebras of processes is whether they afford a finite (in)equational axiomatization. The first negative results concerning ... |

33 |
A term model for synchronous processes
- Hennessy
- 1981
(Show Context)
Citation Context ...t hold that p ↓. So, for instance, a.Ω ↓, but Ω ↑ and a || Ω ↑. The following notion of prebisimilarity is a relevant notion of behavioural preorder in the presence of divergence as adopted in, e.g., =-=[1, 7, 8, 20, 27]-=-. We refer the interested reader to those papers and the references therein for a wealth of results on the semantic theory of CCS Ω modulo prebisimilarity. Definition 11. The relation ⊏ ∼pre is the la... |

32 |
The importance of the left merge operator in process algebras
- Moller
(Show Context)
Citation Context ...sed over T (Σo). Typically, such a negative result is shown by exhibiting an infinite collection E of sound inequations that cannot be proved from any finite sound axiomatization over Σo. (See, e.g., =-=[1, 3, 4, 6, 8, 11, 12, 20, 22]-=- and the references therein.) In the light of the above theorem, to show that �e is not finitely based over T (Σe) it suffices only to exhibit an E-reflecting reduction from T (Σe) to T (Σo). As the e... |

31 | A conservative look at operational semantics with variable binding
- Fokkink, Verhoef
- 1998
(Show Context)
Citation Context ...ses of our algebraic meta-theorem hold. A promising future direction of research is to study whether one can apply our meta-theorem in conservative and orthogonal language extensions (in the sense of =-=[11, 17]-=- and [34], respectively). The paper is organized as follows. In Section 2, we review some preliminary definitions from universal algebra. Section 3 presents our reduction-based technique for proving n... |

31 |
Relating Processes with Respect to Speed
- Moller, Tofts
- 1991
(Show Context)
Citation Context ...f TACS LT as Σe since we use this language as our source language in applying Theorem 1. 17 asThe notion of preorder that is considered over TACS LT in [25] is the MT-preorder due to Moller and Tofts =-=[33]-=-. Definition 8. The relation ⊐ ∼MT , called MT-preorder, is the largest relation such that, for all p, q ∈ C(Σe), p ⊐ ∼MT q iff for all p ′ ∈ C(Σe), and action µ, 1. if p µ → p ′ , then there exist a ... |

29 | The formal specification language mCRL2
- Groote, Mathijssen, et al.
(Show Context)
Citation Context ...ooking at the equations that those equivalences satisfy. Finally, an axiomatization of the relevant notion of equivalence may form the basis for verification tools based on theorem-proving technology =-=[14, 18]-=-. From the theoretical point of view, a fundamental question in the study of algebras of processes is whether they afford a finite (in)equational axiomatization. The first negative results concerning ... |

28 | Finite equational bases in process algebra: Results and open questions
- Aceto, Fokkink, et al.
(Show Context)
Citation Context ...over CCS and over ACP without the left-merge operator. Since then, several other non-finite axiomatizability results have been obtained for a wide collection of very basic process algebras—see, e.g., =-=[5]-=- for a survey of such results. In general, results concerning (non-)finite axiomatizability are very vulnerable to small changes in, and extensions of, the formalism under study. The addition of a sin... |

19 | CCS with Hennessy’s merge has no finite equational axiomatization
- Aceto, Fokkink, et al.
(Show Context)
Citation Context ... axiomatization modulo ⊏ ∼pre . Proof. Assume, towards a contradiction, that E is a finite collection of sound axioms over CCS Ω modulo ⊏ ∼pre , and that E is ground-complete. It follows from Lemma 11=-=(4)-=- that Moller’s set of equations M is sound modulo ⊏ ∼pre . Thus, we have that E ⊢ M. Moreover, we may assume that each of the proofs of equations in M from E only mentions closed terms. Let E ′ be the... |

15 | The nonexistence of finite axiomatisations for CCS congruences - Moller - 1990 |

14 | On the axiomatizability of priority
- Aceto, Chen, et al.
(Show Context)
Citation Context ... CCS axioms, namely E ′ , which can prove all the equations in M. To prove the above claim, observe that all the process terms involved in a proof of an equation in M are strongly convergent (Lemma 11=-=(2)-=-). It follows that if t ≤ u is contained in E ′ , then Ω does not occur in t and u (Lemma 11(3)). Therefore t and u are CCS terms, and the equation t = u is sound over CCS modulo strong bisimilarity, ... |

13 | Ready to preorder: Get your BCCSP axiomatization for free
- Aceto, Fokkink, et al.
- 2007
(Show Context)
Citation Context ...zability theorems for different formalisms, and spare researchers (some of) the delicate technical analysis needed to adapt the proofs of such results. Despite some initial proposals, like the one in =-=[3]-=-, it is fair to say that such a general theory is missing to date. In this paper, we present a meta-theorem offering a general technique that can be used to prove non-finite axiomatizability results, ... |

11 |
Algebraic Theory of Processes. Foundations of Computing Series
- Hennessy
- 1988
(Show Context)
Citation Context ...tions for future and ongoing research. 2 Preliminaries We begin by recalling some basic notions from universal algebra that will be used throughout the paper. We refer the interested reader to, e.g., =-=[21]-=- for more information. A signature Σ is a set of function symbols f, g, . . . with fixed arities. A function symbol of arity zero is often called a constant (symbol). Given a signature Σ and a set of ... |

10 | Bisimilarity is not finitely based over BPA with interrupt
- Aceto, Fokkink, et al.
(Show Context)
Citation Context ...sed over T (Σo). Typically, such a negative result is shown by exhibiting an infinite collection E of sound inequations that cannot be proved from any finite sound axiomatization over Σo. (See, e.g., =-=[1, 3, 4, 6, 8, 11, 12, 20, 22]-=- and the references therein.) In the light of the above theorem, to show that �e is not finitely based over T (Σe) it suffices only to exhibit an E-reflecting reduction from T (Σe) to T (Σo). As the e... |

10 | Conservative extension in structural operational semantics
- Aceto, Fokkink, et al.
- 2001
(Show Context)
Citation Context ...ses of our algebraic meta-theorem hold. A promising future direction of research is to study whether one can apply our meta-theorem in conservative and orthogonal language extensions (in the sense of =-=[11, 17]-=- and [34], respectively). The paper is organized as follows. In Section 2, we review some preliminary definitions from universal algebra. Section 3 presents our reduction-based technique for proving n... |

8 |
CPO models for compact GSOS languages
- Aceto, Ingolfsdottir
- 1996
(Show Context)
Citation Context ...t hold that p ↓. So, for instance, a.Ω ↓, but Ω ↑ and a || Ω ↑. The following notion of prebisimilarity is a relevant notion of behavioural preorder in the presence of divergence as adopted in, e.g., =-=[1, 7, 8, 20, 27]-=-. We refer the interested reader to those papers and the references therein for a wealth of results on the semantic theory of CCS Ω modulo prebisimilarity. Definition 11. The relation ⊏ ∼pre is the la... |

7 | Nonfinite axiomatizability of the equational theory of shuffle
- Ésik, Bertol
- 1998
(Show Context)
Citation Context ...sed over T (Σo). Typically, such a negative result is shown by exhibiting an infinite collection E of sound inequations that cannot be proved from any finite sound axiomatization over Σo. (See, e.g., =-=[1, 3, 4, 6, 8, 11, 12, 20, 22]-=- and the references therein.) In the light of the above theorem, to show that �e is not finitely based over T (Σe) it suffices only to exhibit an E-reflecting reduction from T (Σe) to T (Σo). As the e... |

7 | Orthogonal extensions in structural operational semantics
- Mousavi, Reniers
- 2005
(Show Context)
Citation Context ...gebraic meta-theorem hold. A promising future direction of research is to study whether one can apply our meta-theorem in conservative and orthogonal language extensions (in the sense of [11, 17] and =-=[34]-=-, respectively). The paper is organized as follows. In Section 2, we review some preliminary definitions from universal algebra. Section 3 presents our reduction-based technique for proving non-finite... |

6 |
Nonfinite axiomatizability of shuffle inequalities
- Bloom, Ésik
- 1995
(Show Context)
Citation Context |

5 | Impossibility results for the equational theory of Timed CCS
- Aceto, Ingolfsdottir, et al.
- 2007
(Show Context)
Citation Context ...ing, may ruin the (non-)finite axiomatizability results and make their proofs obsolete (e.g., adding timing to synchronization-free CCS with left merge makes it nonfinitely axiomatizable, as shown in =-=[9]-=-). Furthermore, proofs of non-finite axiomatizability results in the concurrency-theory literature are extremely delicate and error-prone; they are often rather long, and involve several levels of str... |

5 |
Bisimulation on speed: worst-case efficiency
- Lüttgen, Vogler
- 2004
(Show Context)
Citation Context ... of (pre)congruence: 1. Discrete-time CCS modulo timed bisimilarity [38], 2. Temporal CCS modulo timed bisimilarity [32], 3. ATP modulo timed bisimilarity [35], 4. TACS UT modulo faster-than preorder =-=[24]-=-, 5. TACS LT modulo MT-preorder [25], 2s6. TACS modulo urgent timed bisimilarity [26] and 7. IMC modulo strong Markovian bisimilarity [22]. Our meta-theorem is algebraic in nature and does not rely on... |

5 | Bisimulation on speed: A unified approach
- Lüttgen, Vogler
- 2006
(Show Context)
Citation Context ... CCS modulo timed bisimilarity [32], 3. ATP modulo timed bisimilarity [35], 4. TACS UT modulo faster-than preorder [24], 5. TACS LT modulo MT-preorder [25], 2s6. TACS modulo urgent timed bisimilarity =-=[26]-=- and 7. IMC modulo strong Markovian bisimilarity [22]. Our meta-theorem is algebraic in nature and does not rely on any assumption on the specification of the semantics of the languages to which it ca... |

4 | Ready to preorder: The case of weak process semantics - Chen, Fokkink, et al. |

2 | Bisimulation on speed: Lower time bounds
- Lüttgen, Vogler
- 2004
(Show Context)
Citation Context ...e CCS modulo timed bisimilarity [38], 2. Temporal CCS modulo timed bisimilarity [32], 3. ATP modulo timed bisimilarity [35], 4. TACS UT modulo faster-than preorder [24], 5. TACS LT modulo MT-preorder =-=[25]-=-, 2s6. TACS modulo urgent timed bisimilarity [26] and 7. IMC modulo strong Markovian bisimilarity [22]. Our meta-theorem is algebraic in nature and does not rely on any assumption on the specification... |

2 | Ready to preorder: An algebraic and general proof - Escrig, Rodriguez, et al. |

1 |
A modal characterisation of observable machine
- Milner
- 1981
(Show Context)
Citation Context |

1 | A domain equation for bisimulation. Information and Computation - Abramsky - 1991 |

1 | Current Trends in Theoretical Computer Science Entering the 21st Century - Baeten, Weijland - 1990 |