## Characterizing Recursive Programs Up To Bisimilarity

### BibTeX

@MISC{Levy_characterizingrecursive,

author = {Paul Blain Levy},

title = {Characterizing Recursive Programs Up To Bisimilarity},

year = {}

}

### OpenURL

### Abstract

A recursive program is determined, up to bisimilarity, by the operation of the recursion body on arbitrary processes, of which it is a fixpoint. The traditional proof of this fact uses Howe’s method, but that does not tell us how the fixpoint is obtained. In this paper, we show that the fixpoint may be obtained by a least fixpoint procedure iterated through the hierarchy of countable ordinals, using Groote and Vaandrager’s notion of nested simulation. 1

### Citations

3463 |
Communication and Concurrency
- Milner
- 1989
(Show Context)
Citation Context ...hesis is that a recursive program in a transition system S is is a nesting fixpoint of the monotone endofunction on A(S ) given by the recursion body. To illustrate this, we consider the calculus CCS =-=[Mil89]-=-, over a fixed set Act of actions. As CCS is untyped, a context Γ is merely a list of distinct identifers. The syntax is given inductively by the rules in Fig. 1. We write Prog for the set of programs... |

275 | Models for Concurrency
- Winskel, Nielsen
- 1995
(Show Context)
Citation Context ...a ∈ Σ} ∪ {(0 ↦→ τ,τ)} • Let L ⊆ Σ be a subset. We express P\L as ‖ V {0 ↦→ P}, with V given by {({0 ↦→ a},a) | a ∈ Σ \ L} ∪ {({0 ↦→ a},a) | a ∈ Σ \ L} ∪ {(0 ↦→ τ,τ)} The “synchronization algebras” of =-=[WN95]-=- are likewise expressible. As explained in [Mil89], we could also incorporate into the language countably mutual recursion. We have not done so, but our results would go through without difficulty. Th... |

124 |
A Domain Equation For Bisimulation
- Abramsky
- 1990
(Show Context)
Citation Context ...ome monotone endofunctions do not have a nesting fixpoint. Perhaps restricting to the exploratory functions of [LW09] would be fruitful, as these are all definable in a sufficiently rich calculus. In =-=[Abr91]-=- a domain theoretic model is provided that captures bisimilarity between processes without divergences. For general processes it induces a more subtle preorder. The results of this paper may be adapte... |

114 | Proving congruence of bisimulation in functional programming languages - Howe - 1996 |

33 | Relational Reasoning about Functions and Nondeterminism
- Lassen
- 1998
(Show Context)
Citation Context ...sting and must-testing preorders, recursion calculates the least pre-fixed point. (In the case of must-testing, we must assume the calculus uses erratic rather than ambiguous nondeterminism, see e.g. =-=[Las98]-=-.) What if we work modulo bisimilarity? We provide a characterization of the fixpoint calculated by recursion as follows. First calculate the least pre-fixed point up to similarity. Within this equiva... |

11 |
The lazy λ-calculus
- Abramsky
- 1990
(Show Context)
Citation Context ...etween processes without divergences. For general processes it induces a more subtle preorder. The results of this paper may be adapted to lower (i.e. divergence-insensitive) applicative bisimulation =-=[Abr90]-=- in nondeterministic λ-calculus. However, in this instance Howe’s method is stronger because it shows applicative bisimilarity to be preserved not only by recursion (Corollary 1) but also by applicati... |

7 |
Friiso Groote and Frits Vaandrager, Structured operational semantics and bisimulation as a congruence
- Jan
(Show Context)
Citation Context ...cca(x ′ ) such that (x ′ ,y ′ ) ∈ (R;�) • a simulation up to � when for all (x,x ′ ) ∈ R and a ∈ Act, if y ∈ succa(x) then there exists y ′ ∈ succa(x ′ ) such that (x ′ ,y ′ ) ∈ (�;R;�) Definition 2. =-=[GV92]-=- Let Act be a set, and let S = (X,→) be an Act-labelled transition system. For each ordinal α, we shall define a preorder �α, known as α-nested similarity, with the property that any simulation contai... |

6 | Infinitary Howe’s method - Levy - 2006 |

2 |
Seeing beyond divergence. presented at BCS FACS meeting “25 Years of CSP
- Roscoe
- 2004
(Show Context)
Citation Context ... using Howe’s method [How96, Lev06]. 5Characterizing Recursive Programs Up To Bisimilarity Levy 4 Conclusions and Further Work The present paper was greatly inspired by the denotational semantics in =-=[Ros04]-=-, where recursion is interpreted by a “reflected” fixpoint calculated in two steps. The quotient of CCS by bisimilarity is an image-countable transition system S in which bisimilarity is discrete. (It... |

2 |
On the Foundations of Final Coalgebra Semantics
- Turi, Rutten
- 1998
(Show Context)
Citation Context ...cted” fixpoint calculated in two steps. The quotient of CCS by bisimilarity is an image-countable transition system S in which bisimilarity is discrete. (It can also be described as a final coalgebra =-=[TR98]-=-.) Therefore nesting fixpoints in A(S ) are genuine fixpoints and unique. This almost provides a denotational semantics, except that some monotone endofunctions do not have a nesting fixpoint. Perhaps... |

1 |
Levy and Kidane Yemane Weldemariam. Exploratory functions on nondeterministic strategies, up to lower bisimilarity
- Blain
(Show Context)
Citation Context ...enuine fixpoints and unique. This almost provides a denotational semantics, except that some monotone endofunctions do not have a nesting fixpoint. Perhaps restricting to the exploratory functions of =-=[LW09]-=- would be fruitful, as these are all definable in a sufficiently rich calculus. In [Abr91] a domain theoretic model is provided that captures bisimilarity between processes without divergences. For ge... |