## An Elimination Theorem for Regular Behaviours with Integration (1993)

Venue: | LNCS 715 |

Citations: | 6 - 2 self |

### BibTeX

@INPROCEEDINGS{Fokkink93anelimination,

author = {Wan Fokkink},

title = {An Elimination Theorem for Regular Behaviours with Integration},

booktitle = {LNCS 715},

year = {1993},

pages = {432--446},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

This chapter deals with an extension of the process algebra ACP with rational time and integration. We determine a proper subdomain of the regular processes for which an elimination theorem holds, namely, for each pair of processes p0 ; p1 in this class there is a process q in this class such that p0kp1 and q are bisimilar. Some simple examples show that if this subdomain is enlarged, then the elimination result is lost. The subdomain is equivalent to the model of timed automata from Alur and Dill. 1 Introduction In recent years, process algebras such as CCS, CSP and ACP, have been extended with constructs that mean to describe some notion of either discrete or dense time. This chapter is based on the approach of Baeten and Bergstra [3], which extends ACP with real time. They introduced the notion of integration, which expresses the possibility that an action occurs somewhere within a time interval. The construct R v2V p executes the process p, where the behaviour of p may depend on...

### Citations

2176 | A theory of timed automata
- Alur, Dill
- 1994
(Show Context)
Citation Context ...ndicates that neither one can allow variables v i as bounds b kj . 3.5 Orderings on bounds 9 Example 3.5 Define X = R 0!w!1 a[w] \Delta Y (w) Y (v) = R w=1\Gammav a[w] \Delta Y (v): The process Xka 0 =-=[2]-=- (with aja 0 = ffi) is not regular. Each trace of the process X is of the form a[r] \Delta a[1 \Gamma r] \Delta a[1 \Gamma r] \Delta ::: with r 2 h0; 1i. If r 2 h(n \Gamma 2)=(n \Gamma 1); (n \Gamma 1... |

161 | Real Time Process Algebra
- Baeten, Bergstra
- 1991
(Show Context)
Citation Context ... algebras such as CCS, CSP and ACP, have been extended with constructs that mean to describe some notion of either discrete or dense time. This chapter is based on the approach of Baeten and Bergstra =-=[3]-=-, which extends ACP with real time. They introduced the notion of integration, which expresses the possibility that an action occurs somewhere within a time interval. The construct R v2V p executes th... |

128 |
A complete inference system for a class of regular behaviours
- Milner
- 1984
(Show Context)
Citation Context ... r ((b)x) U r (x) U r (y) U r (xky) U r (t X [b 1 =v 1 ; :::; b ar(X) =v ar(X) ]) U r (X(b 1 ; :::; b ar(X) )) Table 1: Action rules for ACPrI 3.2 A counter-example 7 a linear recursive specification =-=[11]-=-. Here, we use this property as the definition of regularity. A recursive specification is linear if its equations are of the form X(v 1 ; :::; v ar(X) ) = X i Z OE i a i [w] \Delta Y i (b i1 ; :::; b... |

114 |
Process Algebra. Cambridge Tracts in Theoretical Computer Science 18
- Baeten, Weijland
- 1990
(Show Context)
Citation Context ...with a process @H (p 1 k \Delta \Delta \Delta kp k ), 1 2 2 THE SYNTAX AND SEMANTICS where p 1 ; :::; p k are regular. Elimination theorems have been deduced for regular processes in untimed ACP (see =-=[4]-=-), and in timed ACP without integration [6]. In this chapter, we set out to deduce an elimination theorem for regular processes in timed ACP with integration. A simple example will show that in genera... |

28 |
Substitution revisited
- Stoughton
- 1988
(Show Context)
Citation Context ...low unrestricted substitution by renaming bound variables. In the sequel, process terms are considered modulo ff-conversion, and when a substitution is applied, bound variables are renamed. Stoughton =-=[12]-=- presented a simple treatment of this technique. If for a substitution oe there are finitely many variables v 1 ; :::; v n such that oe(v i ) 6= v i , then often we will use the standard notation p[oe... |

24 |
On the Ill-Timed but Well-Caused
- Aceto, Murphy
- 1993
(Show Context)
Citation Context ...ith relative time and integration, denoted by ACPrI, together with recursion. For the elimination result it is essential that we restrict the time domain to the rational numbers. For example, if X = a=-=[1]-=- \Delta X and Y = b[ p 2] \Delta Y , then the merge cannot be eliminated from XkY . 2.1 Bounds and conditions In the sequel, we assume a countably infinite set of time variables TVar. 2.2 Process term... |

19 |
Abstraction in real time process algebra, in
- Klusener
- 1991
(Show Context)
Citation Context ...chapter, we restrict to prefix integration, and integration is parametrized by conditions, which consist of inequalities between linear expressions of variables. These notions originate from Klusener =-=[10]-=-. This chapter concerns regular processes. Traditionally, a process is regular if it consists of a finite number of states. However, here such a definition would not work, due to the presence of the i... |

7 |
An e ective axiomatization for real time ACP
- Fokkink, Klusener
- 1995
(Show Context)
Citation Context ... then often we will use the standard notation p[oe(v 1 )=v 1 ; :::; oe(v n )=v n ] for oe(p). 2.4 Operational semantics Table 1 contains an operational semantics for time-closed processes, taken from =-=[7]-=-. We focus on relative time, i.e. we assume that an expression a[r] denotes an action a that is executed exactly r time units after the previous action has been executed. Time starts at zero, which me... |

5 |
The ABP and CABP – a comparison of performances in real time process algebra
- Hillebrand
- 1994
(Show Context)
Citation Context ...e restrictions tend to describe different kinds of irregular behaviour, such as accelerations and oscillations. The subdomain seems to be sufficiently wide for practical purposes, see e.g. Hillebrand =-=[9]-=-. The subdomain of strongly regular processes can be linked to the class of timed automata from Alur and Dill [2]. It is not possible to obtain a precise translation between the processes in our subdo... |

3 |
rule formats for parameterized and statebearing processes. Unpublished manuscript
- SOS
- 1995
(Show Context)
Citation Context ..., then q a[r] \Gamma! p , - if U r (p) and pBq, then U r (q). The rules in Table 1 can be fit into the congruence format for action rules with types, data and variable binding of Bloom and Vaandrager =-=[5]-=-. Strong bisimulation defined by transition rules within this format is always a congruence on the algebra of closed terms, which means that if p $ p 0 and q $ q 0 , then p + q $ p 0 + q 0 and pkq $ p... |

3 |
Regular processes with rational time and silent steps
- Fokkink
- 1992
(Show Context)
Citation Context ...ta kp k ), 1 2 2 THE SYNTAX AND SEMANTICS where p 1 ; :::; p k are regular. Elimination theorems have been deduced for regular processes in untimed ACP (see [4]), and in timed ACP without integration =-=[6]-=-. In this chapter, we set out to deduce an elimination theorem for regular processes in timed ACP with integration. A simple example will show that in general, the merge can not be eliminated from pro... |

3 |
Real time calculi and expansion theorems
- Godskesen, Larsen
- 1992
(Show Context)
Citation Context ...ollection is too small. Because in this class equivalences such as Z 0!v!1 a[v] k Z 1!w!2 a 0 [w] $ Z 0!v!1 a[v] \Delta Z 1\Gammav!w!2\Gammav a 0 [w] cannot be expressed anymore. Godskesen and Larsen =-=[8]-=- provided a rigorous proof that time dependencies are essential in order to obtain an expansion theorem in a 8 3 AN ELIMINATION THEOREM timed setting. (Aceto and Murphy [1] proposed the notion of `ill... |