## Completeness in Discrete-Time Process Algebra (1995)

Citations: | 5 - 3 self |

### BibTeX

@TECHREPORT{Reniers95completenessin,

author = {Michel Reniers and Jan Joris Vereijken},

title = {Completeness in Discrete-Time Process Algebra},

institution = {},

year = {1995}

}

### Years of Citing Articles

### OpenURL

### Abstract

We prove soundness and completeness for some ACP-style concrete, relative-time, discrete-time process algebras. We treat non-delayable actions, delayable actions, and immediate deadlock. Basic process algebras are examined extensively, and also some concurrent process algebras are considered. We conclude with ACP drt , which combines all described features in one theory. 1991 Mathematics Subject Classification: 68Q10, 68Q22, 68Q55. 1991 CR Categories: D.1.3, D.3.1, F.1.2, F.3.2. Keywords: ACP, process algebra, discrete time, relative time, elimination, soundness, completeness, BPA, BPA # , BPA - drt --#, BPA - drt --ID, BPA drt --ID, BPA - drt , BPA drt , PA - drt --ID, ACP - drt --ID, PA drt , ACP drt , RSP(USD). Note: The investigations of the first author were supported by Philips Research Laboratories Eindhoven, The Netherlands. The investigations of the second author were supported by the Netherlands Computer Science Research Foundation (SION) with financial support from the N...

### Citations

565 | Term Rewriting Systems
- Klop
- 1992
(Show Context)
Citation Context ...ection 2.2.3 of [13]. In the proofs of elimination theorems we make abundant use of term rewriting analysis. For further details on these techniques, especially the lexicographical path ordering, see =-=[19]-=-. In proving completeness we sometimes make use of Verhoef's General Completeness Theorem. For more information on this, see [25]. As the definitions and notations used for discrete-time process algeb... |

201 |
Process Algebra
- Baeten, Weijland
- 1990
(Show Context)
Citation Context ...P framework of process algebras has been extended with, among many other things, discrete-time extensions. Papers describing such extensions are for example [8, 10, 11]. With the recent appearance of =-=[10, 13]-=- discrete-time process algebra seems to have reached a decent state of maturity, and we felt now was the time to write a paper about the soundness and completeness issues involved with discrete time. ... |

155 | Real time process algebra
- Baeten, Bergstra
- 1991
(Show Context)
Citation Context ...or the undelayable action "a", and "ats(a)" ("any time slice") for the delayable action "a" [8, 10]. . We find the notation "a[1]" for the undelayable=-= action "a", and "a" for the delayable action "a" [5, 6]. . It has been sugg-=-ested to refer to the delayable deadlock constant "#" as livelock, and to the immediate deadlock constant " #" as the immediate time stop, full time stop, or catastrophic deadlock ... |

108 | A congruence theorem for structured operational semantics with predicates and negative premises. Nordic Journal of Computing 2 (2), 274–302. A. Proof of Theorem 9 We show that the problem of deciding whether a universal two-counter machine diverges on inp
- Verhoef
- 1995
(Show Context)
Citation Context ...on Model for BPA) Using bisimulation, we can now construct a model of the axioms of BPA. In order to do this, we first need to know that bisimulation is a congruence with respect to all operators. In =-=[24]-=- it is proven that a sufficient condition for this is that the deduction rules satisfy the so called panth format. It is easy to check that this is indeed the case. We then construct the bisimulation ... |

105 | The algebra of timed processes ATP: theory and application
- Nicollin, Sifakis, et al.
- 1994
(Show Context)
Citation Context ...BPA drt --ID # # rel (a) = a Remark 2.6.10 (Properties of BPA + drt --ID, Part I) The equalities of Proposition 2.6.9 on the page before are not new, but have been described before. See for example [=-=20]. Pr-=-oposition 2.6.11 (Properties of BPA + drt --ID, Part II) For any BPA drt --ID term x we have the following equality: BPA + drt --ID # # �� x = # Proof Using Proposition 2.6.9(ii) we derive: BPA + ... |

36 | A general conservative extension theorem in process algebra
- Verhoef
- 1994
(Show Context)
Citation Context ...on these techniques, especially the lexicographical path ordering, see [19]. In proving completeness we sometimes make use of Verhoef's General Completeness Theorem. For more information on this, see =-=[25]-=-. As the definitions and notations used for discrete-time process algebras have been subject to vehement revision over the past few years, we have compiled an appendix that tries to shed some light on... |

33 | Branching bisimilarity is an equivalence indeed
- Basten
- 1996
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Citation Context ...R # is a bisimulation relation between y and # rel (x) +# rel (y). We must prove that y # BPA - drt --ID #x# # . Then take the symmetric, transitive closure of R # (which is again a bisimulation, see =-=[15-=-]), denoted R # ST , and extend it to a bisimulation relation R between y and #x# # as follows: R = R # ST # (s, #x# # )|s # C(BPA drt --ID) # y # =#s Now, let s be any closed term such that y # =#s.... |

24 |
Process algebra with propositional signals
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- 1997
(Show Context)
Citation Context ...om DRTM2 would be just an instance of Axiom DRTM3 if the x in the latter could take the value #. Similarly, Axioms DRTCM2--DRTCM4 could also be collapsed to just one axiom with the help of # (and see =-=[7]-=- for a theory in which it is hard to find any axiom that could not be formulated better with the help of the empty step!). The absence of # is even felt worse when doing calculations. If we look for e... |

16 |
Discrete time process algebra with abstraction
- Baeten, Bergstra
- 1995
(Show Context)
Citation Context ...rocess Algebras Since [14] appeared, the ACP framework of process algebras has been extended with, among many other things, discrete-time extensions. Papers describing such extensions are for example =-=[8, 10, 11]-=-. With the recent appearance of [10, 13] discrete-time process algebra seems to have reached a decent state of maturity, and we felt now was the time to write a paper about the soundness and completen... |

7 | Fischer’s protocol in timed process algebra
- Vereijken
- 1994
(Show Context)
Citation Context ...rtant differences: . We find the notation "BPA dt " for "BPA - drt --#", "BPA #dt " for "BPA - drt --ID", and "PA #dt " for "PA - drt --ID" =-=[13]. . We find the notation "ACP dt " for "ACP - drt --ID" [6, 13, 23], . We find the notations "# d " and "# rel(1) " for "#-=- rel " [6, 13, 23]. . We find the notation "cts(a)" ("current time slice") for the undelayable action "a", and "ats(a)" ("any time slice") for th... |

4 | Discrete time process algebra with relative timing. Unpublished lecture notes of Eindhoven University of Technology Course 2L920 “Process Algebra
- Baeten, Reniers
(Show Context)
Citation Context ... that all axioms of BPA - drt --ID are also contained in PA - drt --ID. Remark 3.2.10 (Elimination for PA - drt --ID) Elimination for PA - drt --ID is also claimed (without proof) in Theorem 3.2 of [=-=11]-=-. Theorem 3.2.11 (Soundness of PA - drt --ID) The set of closed PA - drt --ID terms modulo bisimulation equivalence is a model of PA - drt --ID. Proof Soundness is proven following the same lines as i... |

3 | Discrete-time process algebra with empty step
- Baeten, Vereijken
- 1996
(Show Context)
Citation Context ...heoretical viewpoint it's troublesome: it does not lend itself well to term rewriting system analysis, and worse, it does not seem to be compatible with the addition of the empty step, as is shown in =-=[12]-=-. On the other hand, the classic style is not ideal either. It appears less intuitive, and it needs more axioms. Compare for example Axioms DRTM5-- DRTM6 of Table 25 on page 70 with Axioms DRTM7--DRTM... |

3 |
The I C-bus in discrete-time process algebra
- Bos, Reniers
- 1996
(Show Context)
Citation Context ...Table 10 on the next page. Therefore we could replace Axioms DRT4--DRT5 in BPA - drt --ID by DRT4A without affecting the soundness or completeness of the resulting theory. This is for example done in =-=[16]. # -=-�� x = # DRT3 a + # = a DRT4 # rel (x) + # = # rel (x) DRT5 Table 9: Axioms for #. One such reason to do so, could be the fact that DRT4A is a straightforward reformulation of Axiom A6 from BPA # ... |

3 | The I 2 C-bus in discrete-time process algebra
- BOS, RENIERS
- 1996
(Show Context)
Citation Context ...n Table 10 on the next page. Therefore we could replace Axioms DRT4–DRT5 in BPA − drt –ID by DRT4A without affecting the soundness or completeness of the resulting theory. This is for example done in =-=[16]-=-. 22sδ·x=δ DRT3 a+δ=a DRT4 σrel(x)+δ=σrel(x) DRT5 Table 9: Axioms forδ. One such reason to do so, could be the fact that DRT4A is a straightforward reformulation of Axiom A6 from BPAδ, in a setting wi... |

2 | editor.Programming Concepts, Methods and Calculi (Procomet ’94), volume A-56 - OLDEROG - 1994 |

1 |
et al., editors. Simplex Sigillum Veri
- AARTS
- 1995
(Show Context)
Citation Context ...el " [6, 13, 23]. . We find the notation "cts(a)" ("current time slice") for the undelayable action "a", and "ats(a)" ("any time slice") for the =-=delayable action "a" [8, 10]. . We find the notation "a[1]" for the undelayable action "a&quo-=-t;, and "a" for the delayable action "a" [5, 6]. . It has been suggested to refer to the delayable deadlock constant "#" as livelock, and to the immediate deadlock consta... |

1 |
Some simple calculations in discrete time process algebra
- BAETEN, BERGSTRA
- 1995
(Show Context)
Citation Context ...ect to axioms, see [14]. Remark 2.6.4 (Notation BPA + drt --ID) By superscripting a theory with a "+" (e.g. BPA + drt --ID), we indicate the presence of the recursion principle RSP(USD). Not=-=e that in [9, 11]-=- the notation BPA drt --ID + RSP(USD) is used instead of BPA + drt --ID. However, we find that notation cumbersome, as it clutters up the formulae. y = # rel (x) +# rel (y) =# y = #x# # RSP(USD) Table... |

1 |
editors.Simplex Sigillum Veri
- AARTS
- 1995
(Show Context)
Citation Context ...3]. –ID”, and “PAδdt” • We find the notation “cts(a)”(“current time slice”) for the undelayable action “a”, and “ats(a)”(“any time slice”) for the delayable action “a”[8,10]. • We find the notation “a=-=[1]-=-” for the undelayable action “a”, and “a” for the delayable action “a” [5, 6]. • It has been suggested to refer to the delayable deadlock constant “δ” aslivelock, and to the immediate deadlock constan... |