## Undecidable Equivalences for Basic Parallel Processes (1993)

Venue: | 13th Conference on Foundations of Software Technology and Theoretical Computer Science |

Citations: | 25 - 2 self |

### BibTeX

@INPROCEEDINGS{Hüttel93undecidableequivalences,

author = {Hans Hüttel},

title = {Undecidable Equivalences for Basic Parallel Processes},

booktitle = {13th Conference on Foundations of Software Technology and Theoretical Computer Science},

year = {1993},

pages = {454--464},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

. Recent results show that strong bisimilarity is decidable for the class of Basic Parallel Processes (BPP), which corresponds to the subset of CCS definable using recursion, action prefixing, nondeterminism and the full merge operator. In this paper we examine all other equivalences in the linear/branching time hierarchy [12] and show that none of them are decidable for BPP. 1 Introduction Much attention has been devoted to the study of process calculi and in particular to behavioural semantics for these calculi. In order to capture the behavioural aspects of processes, a variety of equivalences have been proposed. Various criteria exist for comparing the merits and deficiencies of these equivalences. A systematic approach consists of classifying the equivalences according to their coarseness. For this purpose van Glabbeek proposed the linear/branching time spectrum which is illustrated in Figure 1 [12]. The least discriminating equivalences are at the bottom of the diagram. Arrows i...

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Citation Context ...iterion for evaluating behavioural equivalences is decidability. For finite-state processes, i.e. finite automata, all equivalences in Figure 1 are decidable [21]. However, it is well known (see e.g. =-=[17]-=-) that completed trace equivalence becomes undecidable when one moves beyond finite automata to context-free grammars, which correspond to processes specified in Basic Process Algebra (BPA) [1]. A rec... |

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Citation Context ...m we have the testing/failures equivalence investigated by De Nicola and Hennessy (see e.g. [15]). At the top of the diagram is bisimulation equivalence (or bisimilarity), a notion introduced by Park =-=[28]-=- and subsequently widely used in process theories. As all equivalences save bisimilarity are defined as the symmetric closure of a preorder, there is a similar hierarchy for the behavioural preorders,... |

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Citation Context ... e.g. [17]) that completed trace equivalence becomes undecidable when one moves beyond finite automata to context-free grammars, which correspond to processes specified in Basic Process Algebra (BPA) =-=[1]-=-. A recent theorem [2, 6, 18] shows that strong bisimilarity is decidable for normed BPA processes. For deterministic processes, the equivalence hierarchy collapses and consequently all equivalences a... |

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Citation Context ...ds to the subset of CCS definable using recursion, action prefixing, nondeterminism and the full merge operator. In this paper we examine all other equivalences in the linear/branching time hierarchy =-=[12]-=- and show that none of them are decidable for BPP. 1 Introduction Much attention has been devoted to the study of process calculi and in particular to behavioural semantics for these calculi. In order... |

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Citation Context ...ble of \Delta. 3 Undecidability of the ready-simulation preorder In the following we shall consider the notion of ready simulation (or 2/3-bisimulation) originated in work by Bloom, Istrail and Meyer =-=[4]-=- and Larsen and Skou [23]. It is the trace congruence induced by the GSOS-format [4]. Our reasons for considering this particular preorder are twofold: firstly, and most importantly, this preorder is ... |

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Citation Context ..., illustrated in Figure 2. One relevant criterion for evaluating behavioural equivalences is decidability. For finite-state processes, i.e. finite automata, all equivalences in Figure 1 are decidable =-=[21]-=-. However, it is well known (see e.g. [17]) that completed trace equivalence becomes undecidable when one moves beyond finite automata to context-free grammars, which correspond to processes specified... |

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Citation Context ...eted trace equivalence becomes undecidable when one moves beyond finite automata to context-free grammars, which correspond to processes specified in Basic Process Algebra (BPA) [1]. A recent theorem =-=[2, 6, 18]-=- shows that strong bisimilarity is decidable for normed BPA processes. For deterministic processes, the equivalence hierarchy collapses and consequently all equivalences are decidable in the determini... |

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Citation Context ... equivalence are undecidable for BPA processes and, using this and Friedman's result, it was shown in [13] that in fact none of the equivalences in Figure 1 are decidable for normed BPA processes. In =-=[10]-=-, Christensen, Huttel and Stirling showed that bisimilarity is in fact decidable for the full BPA calculus. Attention has since focused on the process calculus BPP where a non-communicating parallel (... |

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Citation Context ...ta \Delta \Delta k En as a multiset fjE 1 ; : : : ; En jg. The collection of all multisets over V ar is denoted by V ar\Omega ; multisets will be denoted by Greek letters ff; fi; fl; : : :. Following =-=[7]-=-, a BPP process definition \Delta is said to be in normal form if its defining equations are all of the form X i = P n i j=1 a ij ff ij where the ff ij 2 V ar\Omega . If furthermore all ff ij have car... |

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Citation Context ...eted trace equivalence becomes undecidable when one moves beyond finite automata to context-free grammars, which correspond to processes specified in Basic Process Algebra (BPA) [1]. A recent theorem =-=[2, 6, 18]-=- shows that strong bisimilarity is decidable for normed BPA processes. For deterministic processes, the equivalence hierarchy collapses and consequently all equivalences are decidable in the determini... |

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Citation Context ...calculus. Attention has since focused on the process calculus BPP where a non-communicating parallel (full merge) operator takes the place of sequencing. Christensen, Hirshfeld and Moller have proved =-=[8, 9]-=- that bisimulation equivalence is decidable for BPP by using a tableau technique similar to that used in Bisimulation equivalence ? 2-nested simulation equivalence i i i) ? Ready simulation equivalenc... |

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Citation Context ...ble for deterministic BPA processes. In [20] Huynh and Tian have shown that readiness and failures equivalence are undecidable for BPA processes and, using this and Friedman's result, it was shown in =-=[13]-=- that in fact none of the equivalences in Figure 1 are decidable for normed BPA processes. In [10], Christensen, Huttel and Stirling showed that bisimilarity is in fact decidable for the full BPA calc... |

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Citation Context ...calculus. Attention has since focused on the process calculus BPP where a non-communicating parallel (full merge) operator takes the place of sequencing. Christensen, Hirshfeld and Moller have proved =-=[8, 9]-=- that bisimulation equivalence is decidable for BPP by using a tableau technique similar to that used in Bisimulation equivalence ? 2-nested simulation equivalence i i i) ? Ready simulation equivalenc... |

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Citation Context ...f trace equivalence and possible futures equivalence, in the sense that 1-bounded-tr-bisimulation corresponds to trace equivalence and 2-bounded-tr-bisimulation is the possible futures equivalence of =-=[30]. Definition-=-15. n-bounded-tr-bisimulation, written �� n tr , is defined inductively as follows. -- E �� 0 tr F for all processes E and F , -- E �� n+1 tr F iff ffl if E w \Gamma!E 0 then 9F 0 such tha... |

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Citation Context ...eted trace equivalence becomes undecidable when one moves beyond finite automata to context-free grammars, which correspond to processes specified in Basic Process Algebra (BPA) [1]. A recent theorem =-=[2, 6, 18]-=- shows that strong bisimilarity is decidable for normed BPA processes. For deterministic processes, the equivalence hierarchy collapses and consequently all equivalences are decidable in the determini... |

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Citation Context ...ministic case. However, as early as in [11] it was shown by Friedman (in the setting of simple grammars) that the completed trace inclusion preorder is undecidable for deterministic BPA processes. In =-=[20]-=- Huynh and Tian have shown that readiness and failures equivalence are undecidable for BPA processes and, using this and Friedman's result, it was shown in [13] that in fact none of the equivalences i... |

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Citation Context ...decidable for normed BPA processes. For deterministic processes, the equivalence hierarchy collapses and consequently all equivalences are decidable in the deterministic case. However, as early as in =-=[11]-=- it was shown by Friedman (in the setting of simple grammars) that the completed trace inclusion preorder is undecidable for deterministic BPA processes. In [20] Huynh and Tian have shown that readine... |

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Citation Context ... any Minsky machine we construct two processes that are in the ready simulation preorder if and only if the machine halts with zero-valued counters. The reduction is based on an idea due to Hirshfeld =-=[16]-=-. The rest of the paper is organized as follows. Section 2 deals with the necessary definitions of BPP processes. In Section 3 we show that the ready simulation preorder is undecidable. In the remaind... |

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Citation Context ...r F 0 and ffl if F w \Gamma!F 0 then 9E 0 such that E w \Gamma!E 0 and E 0 �� n tr F 0 . This notion of equivalence also arises naturally as the consecutive approximations of bisimulation equivale=-=nce [24, 25]-=-. For finitely branching transition graphs (and thus for BPP processes) the limit of the n-bounded-tr-bisimulations for n ! ! coincides with bisimulation equivalence: Theorem 16. [25] For any finitely... |

2 | Behavioural Equivalences and Infinite Transition Graphs - Dedidability - 1991 |

2 | Refusal testing - Philips - 1987 |