## On finite alphabets and infinite bases III: Simulation (2006)

Venue: | Proc. CONCUR’06, LNCS 4137 |

Citations: | 5 - 1 self |

### BibTeX

@INPROCEEDINGS{Chen06onfinite,

author = {Taolue Chen and Wan Fokkink},

title = {On finite alphabets and infinite bases III: Simulation},

booktitle = {Proc. CONCUR’06, LNCS 4137},

year = {2006},

pages = {421--434},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. This paper studies the (in)equational theory of simulation preorder and equivalence over the process algebra BCCSP. We prove that in the presence of a finite alphabet with at least two actions, the (in)equational theory of BCCSP modulo simulation preorder or equivalence does not have a finite basis. In contrast, in the presence of an alphabet that is infinite or a singleton, the equational theory for simulation equivalence does have a finite basis. 1

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Citation Context ...sed terms p and q. Finally, E is ω-complete if for any open terms t, u with E ⊢ ρ(t) � ρ(u) (or E ⊢ ρ(t) ≈ ρ(u)) for all closed substitutions ρ, we have E ⊢ t � u (or E ⊢ t ≈ u). The core axioms A1-4 =-=[22]-=- for BCCSP(A) below are ω-complete, and sound and ground-complete modulo bisimulation equivalence, which is the finest semantics in the linear time - branching time spectrum (see Figure 1). A1 x + y ≈... |

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Citation Context ...tion semantics in this spectrum. A relation R between processes is a simulation if s0 R s1 and s0 → s ′ 0 implies s1 → s ′ 1 with s ′ 0 R s ′ 1. It was introduced by Milner in his seminal work on CCS =-=[21]-=-, and the first branching-time semantics to be used studied in the setting of process algebra (before the formulation of bisimulation by Park [27] appeared). The notion of simulation is well studied i... |

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Citation Context ...t was introduced by Milner in his seminal work on CCS [21], and the first branching-time semantics to be used studied in the setting of process algebra (before the formulation of bisimulation by Park =-=[27]-=- appeared). The notion of simulation is well studied in the literatures, both from the theoretical and from the practical point of view, see e.g. [14, 17]. ⋆ Partially supported by the Dutch Bsik proj... |

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Citation Context ...a (before the formulation of bisimulation by Park [27] appeared). The notion of simulation is well studied in the literatures, both from the theoretical and from the practical point of view, see e.g. =-=[14, 17]-=-. ⋆ Partially supported by the Dutch Bsik project BRICKS (Basic Research in Informatics for Creating the Knowledge Society), 973 Program of China (No. 2002CB312002), NNSFC (No. 60233010, No. 60273034,... |

94 | The linear time – branching time spectrum I; the semantics of concrete, sequential processes
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Citation Context ...nsensus on what constitutes an appropriate notion of observable behavior for reactive systems has led to a large number of proposals for behavioral equivalences for concurrent processes. Van Glabbeek =-=[9]-=- presented the linear time - branching time spectrum of behavioral preorders and equivalences for finitely branching, concrete, sequential processes. In this paper we focus on the simulation semantics... |

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Citation Context ...d in the initial model but which cannot be derived, are often added to the latter equational theory. For such extended equational theories, ω-completeness results were presented in the setting of CCS =-=[23, 3]-=- and ACP [6]. A number of positive and negative results regarding finite ω-complete axiomatizations for BCCSP occur in the literature. For a comprehensive survey and discussion of open problems, the i... |

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Citation Context ...uestion whether a finite algebra is finitely based is undecidable. Notable examples of ω-incomplete axiomatizations in the literature are the λKβη-calculus (see [28]) and the equational theory of CCS =-=[24]-=-. Therefore laws such as commutativity of parallelism, which are valid in the initial model but which cannot be derived, are often added to the latter equational theory. For such extended equational t... |

28 | Finite equational bases in process algebra: Results and open questions
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(Show Context)
Citation Context ...ve and negative results regarding finite ω-complete axiomatizations for BCCSP occur in the literature. For a comprehensive survey and discussion of open problems, the interested reader is referred to =-=[2]-=-. – Infinite alphabets: 1 Moller [23] proved that the ground-complete axiomatization for BCCSP modulo bisimulation equivalence is ω-complete. Groote [10] presented ω-completeness proofs for completed ... |

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Citation Context ...in [2, p355] that if A is infinite, then the axiomatization A1-4 + S is ω-complete. In both articles it was claimed that this could be proved using the technique of inverted substitutions from Groote =-=[10]-=-, but the proof itself was never given. For the sake of completeness, here we present a proof that A1-4 + S is ωcomplete, using inverted substitutions. This technique works as follows. Consider an axi... |

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Citation Context ...] showed that the algebra of naturals with addition and multiplication is finitely based, while Gurevi˘c [11] showed that after adding exponentiation the algebra is no longer finitely based. McKenzie =-=[19]-=- settled Tarski’s Finite Basis Problem in the negative, by showing that the general question whether a finite algebra is finitely based is undecidable. Notable examples of ω-incomplete axiomatizations... |

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Citation Context ...r theorem proving applications, it is convenient if an axiomatization is ω-complete, because it means that proofs by (structural) induction can be avoided in favor of purely equational reasoning; see =-=[16]-=-. In [12] it was argued that ω-completeness is desirable for the partial evaluation of programs. The existence of a finite basis for an equational theory is a classic topic of study in universal algeb... |

17 | On the Axiomatizability of Ready Traces, Ready Simulation and Failure Traces
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Citation Context ... alphabet, occurrences of action names in axioms should be interpreted as variables, as otherwise most of the axiomatizations mentioned in this paragraph would be infinite. 3s4 Blom, Fokkink and Nain =-=[4]-=- proved that BCCSP modulo ready trace equivalence does not have a finite sound and ground-complete axiomatization if |A| = ∞. Aceto, Fokkink, van Glabbeek and Ingolfsdottir [1] proved such a negative ... |

17 | PAM: A Process Algebra Manipulator
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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 =-=[15]-=-. In particular, for theorem proving applications, it is convenient if an axiomatization is ω-complete, because it means that proofs by (structural) induction can be avoided in favor of purely equatio... |

15 | Nested semantics over finite trees are equationally hard
- Aceto, Fokkink, et al.
(Show Context)
Citation Context ... Blom, Fokkink and Nain [4] proved that BCCSP modulo ready trace equivalence does not have a finite sound and ground-complete axiomatization if |A| = ∞. Aceto, Fokkink, van Glabbeek and Ingolfsdottir =-=[1]-=- proved such a negative result for 2-nested simulation and possible futures equivalence, independent of the cardinality of A. – Finite alphabets: Fokkink and Nain [8] obtained an ω-complete axiomatiza... |

15 |
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Citation Context ...is desirable for the partial evaluation of programs. The existence of a finite basis for an equational theory is a classic topic of study in universal algebra (see, e.g., [20]), dating back to Lyndon =-=[18]-=-. Murskiĭ [26] proved that “almost all” finite algebras (namely all quasi-primal ones) are finitely based, while in [25] he presented an example of a three-element algebra that has no finite basis. He... |

13 |
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Citation Context ... model but which cannot be derived, are often added to the latter equational theory. For such extended equational theories, ω-completeness results were presented in the setting of CCS [23, 3] and ACP =-=[6]-=-. A number of positive and negative results regarding finite ω-complete axiomatizations for BCCSP occur in the literature. For a comprehensive survey and discussion of open problems, the interested re... |

12 |
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Citation Context ...ic topic of study in universal algebra (see, e.g., [20]), dating back to Lyndon [18]. Murskiĭ [26] proved that “almost all” finite algebras (namely all quasi-primal ones) are finitely based, while in =-=[25]-=- he presented an example of a three-element algebra that has no finite basis. Henkin [13] showed that the algebra of naturals with addition and multiplication is finitely based, while Gurevi˘c [11] sh... |

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Citation Context ...skiĭ [26] proved that “almost all” finite algebras (namely all quasi-primal ones) are finitely based, while in [25] he presented an example of a three-element algebra that has no finite basis. Henkin =-=[13]-=- showed that the algebra of naturals with addition and multiplication is finitely based, while Gurevi˘c [11] showed that after adding exponentiation the algebra is no longer finitely based. McKenzie [... |

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Citation Context ...d in the initial model but which cannot be derived, are often added to the latter equational theory. For such extended equational theories, ω-completeness results were presented in the setting of CCS =-=[23, 3]-=- and ACP [6]. A number of positive and negative results regarding finite ω-complete axiomatizations for BCCSP occur in the literature. For a comprehensive survey and discussion of open problems, the i... |

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Citation Context ... proving applications, it is convenient if an axiomatization is ω-complete, because it means that proofs by (structural) induction can be avoided in favor of purely equational reasoning; see [16]. In =-=[12]-=- it was argued that ω-completeness is desirable for the partial evaluation of programs. The existence of a finite basis for an equational theory is a classic topic of study in universal algebra (see, ... |

10 | Simulation and bisimulation over one-counter processes
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Citation Context ...a (before the formulation of bisimulation by Park [27] appeared). The notion of simulation is well studied in the literatures, both from the theoretical and from the practical point of view, see e.g. =-=[14, 17]-=-. ⋆ Partially supported by the Dutch Bsik project BRICKS (Basic Research in Informatics for Creating the Knowledge Society), 973 Program of China (No. 2002CB312002), NNSFC (No. 60233010, No. 60273034,... |

10 |
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Citation Context ...negative, by showing that the general question whether a finite algebra is finitely based is undecidable. Notable examples of ω-incomplete axiomatizations in the literature are the λKβη-calculus (see =-=[28]-=-) and the equational theory of CCS [24]. Therefore laws such as commutativity of parallelism, which are valid in the initial model but which cannot be derived, are often added to the latter equational... |

9 | On finite alphabets and infinite bases: From ready pairs to possible worlds
- Fokkink, Nain
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Citation Context ...ity of A. – Finite alphabets: Fokkink and Nain [8] obtained an ω-complete axiomatization for BCCSP modulo failures equivalence if |A| < ∞, by adding one extra axiom that uses the cardinality of A. In =-=[7]-=- they proved that if 1 < |A| < ∞, BCCSP modulo any semantics in between readiness and possible worlds equivalence does not have a finite basis. In [5], Chen, Fokkink and Nain proved that BCCSP modulo ... |

9 | A finite basis for failure semantics
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Citation Context ...k, van Glabbeek and Ingolfsdottir [1] proved such a negative result for 2-nested simulation and possible futures equivalence, independent of the cardinality of A. – Finite alphabets: Fokkink and Nain =-=[8]-=- obtained an ω-complete axiomatization for BCCSP modulo failures equivalence if |A| < ∞, by adding one extra axiom that uses the cardinality of A. In [7] they proved that if 1 < |A| < ∞, BCCSP modulo ... |

8 |
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Citation Context ...or the partial evaluation of programs. The existence of a finite basis for an equational theory is a classic topic of study in universal algebra (see, e.g., [20]), dating back to Lyndon [18]. Murskiĭ =-=[26]-=- proved that “almost all” finite algebras (namely all quasi-primal ones) are finitely based, while in [25] he presented an example of a three-element algebra that has no finite basis. Henkin [13] show... |

6 |
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Citation Context ...in [25] he presented an example of a three-element algebra that has no finite basis. Henkin [13] showed that the algebra of naturals with addition and multiplication is finitely based, while Gurevi˘c =-=[11]-=- showed that after adding exponentiation the algebra is no longer finitely based. McKenzie [19] settled Tarski’s Finite Basis Problem in the negative, by showing that the general question whether a fi... |

3 | On finite alphabets and infinite bases II: Completed and ready simulation
- Chen, Fokkink, et al.
- 2006
(Show Context)
Citation Context ...ne extra axiom that uses the cardinality of A. In [7] they proved that if 1 < |A| < ∞, BCCSP modulo any semantics in between readiness and possible worlds equivalence does not have a finite basis. In =-=[5]-=-, Chen, Fokkink and Nain proved that BCCSP modulo completed simulation equivalence does not have a finite basis if |A| > 1, and that BCCSP modulo ready simulation equivalence does not have a finite ba... |