## Towards a Mathematical Operational Semantics (1997)

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Venue: | In Proc. 12 th LICS Conf |

Citations: | 139 - 9 self |

### BibTeX

@INPROCEEDINGS{Turi97towardsa,

author = {Daniele Turi and Gordon Plotkin},

title = {Towards a Mathematical Operational Semantics},

booktitle = {In Proc. 12 th LICS Conf},

year = {1997},

pages = {280--291},

publisher = {IEEE, Computer Society Press}

}

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### Abstract

We present a categorical theory of `well-behaved' operational semantics which aims at complementing the established theory of domains and denotational semantics to form a coherent whole. It is shown that, if the operational rules of a programming language can be modelled as a natural transformation of a suitable general form, depending on functorial notions of syntax and behaviour, then one gets both an operational model and a canonical, internally fully abstract denotational model for free; moreover, both models satisfy the operational rules. The theory is based on distributive laws and bialgebras; it specialises to the known classes of well-behaved rules for structural operational semantics, such as GSOS.

### Citations

1310 | A structural approach to operational semantics
- Plotkin
- 1981
(Show Context)
Citation Context ...language. A convenient way of specifying such a transition relation is by induction on the structure of the programs, starting from suitable operational rules for the basic constructs of the language =-=[21]-=-. Research supported by EuroFOCS. y Research supported by an EPSRC Senior Fellowship. Traditionally, operational semantics is contrasted with the mathematical interpretation of programs called denotat... |

743 | Notions of computation and monads
- Moggi
- 1991
(Show Context)
Citation Context ...t yet clear whether this fits our purposes. ) We would also like to obtain adequacy results when working in categories of partial maps. There is an obvious question about Moggi's computational monads =-=[19]-=- and our behaviour functors which remains to be investigated. In a different direction, we would like to understand the relationship between the transitional approach considered here and others, such ... |

275 |
An initial algebra approach to the specification, correctness and implementation of abstract data types
- Goguen, Thatcher, et al.
- 1978
(Show Context)
Citation Context ...asic constructs. The programs themselves form the initial such \Sigma-algebra and the corresponding unique homomorphism from the programs to the denotational model is called initial algebra semanticss=-=[12]-=-. The semantic domain, ie the carrier of the denotational model, can often be regarded as the final solution of a domain equation X = B(X), for a suitable `behaviour' functor B. In other words, the se... |

264 | Models for concurrency - Winskel, Nielsen - 1995 |

207 |
Process Algebra
- Baeten, Weijland
- 1991
(Show Context)
Citation Context ...semantics [23]. Adequacy proofs can be quite demanding, hence general criteria ensuring adequacy are of interest. For process algebras, as used for specifying nondeterministic and concurrent programs =-=[17, 5]-=-, there exist syntactic restrictions on the format of the operational rules which ensure that bisimulation [17] is a congruence. Among the rules in these formats, GSOS rules [8] are the best known and... |

197 |
Outline of a mathematical theory of computation
- Scott
- 1970
(Show Context)
Citation Context ...or reasoning about them in terms of abstract, mathematical entities. It is therefore fundamental that a denotational semantics be adequate, ie that it determines the operational behaviour of programs =-=[24]-=-. For languages without variable binding, but possibly multi-sorted, a denotational model can be seen as a \Sigma-algebra, where \Sigma is the signature of the language corresponding to the basic cons... |

196 |
Bisimulation canâ€™t be traced
- Bloom, Istrail, et al.
- 1995
(Show Context)
Citation Context ...oncurrent programs [17, 5], there exist syntactic restrictions on the format of the operational rules which ensure that bisimulation [17] is a congruence. Among the rules in these formats, GSOS rules =-=[8]-=- are the best known and (negative) tree rules [11] are the most general. In [22], the `processes as terms' method, based on such a congruence result, is presented which allows for the systematic deriv... |

191 |
F.W.: Structured operational semantics and bisimulation as a congruence
- Groote, Vaandrager
- 1992
(Show Context)
Citation Context ...he most general. In [22], the `processes as terms' method, based on such a congruence result, is presented which allows for the systematic derivation of adequate denotational models from `tyft rules' =-=[13]-=-, a class of rules equivalent to tree rules. We present here a categorical reformulation and generalisation of the above adequacy meta-results. First, we show that certain sets R of GSOS rules can be ... |

163 |
A final coalgebra theorem
- Aczel, Mendler
- 1989
(Show Context)
Citation Context ...t every ae-model is adequate with respect to the intended operational model in the sense that the behaviour of the programs can be determined from any ae-model up to a generalised, coalgebraic notion =-=[4, 16]-=- of bisimulation. The other adjunction implies that there exists a final ae-model -- the canonical denotational model D ae (1) over the final coalgebra of abstract, global behaviours. It is necessaril... |

131 |
Terminal coalgebras in well-founded set theory
- Barr
- 1993
(Show Context)
Citation Context ...has a final object 1, and the forgetful functor UB has a right adjoint GB : C ! B-Coalg, then GB 1 is the final B-coalgebra. For the endofunctor BX = (P fi X) A on Set, such a right adjoint GB exists =-=[6]-=-. It follows [29, x13] that the final coalgebra GB 1 is the set of rooted, image finite trees, with branches labelled by a 2 A, quotiented by (ordinary) bisimulation. This is the set of `abstract glob... |

128 |
The formal theory of monads
- Street
- 1972
(Show Context)
Citation Context ... (Cf [15].) When both T and e T are monads, e T lifts the monad T to the B-coalgebras if the forgetful functor UB : B-Coalg ! C (together with the identity natural transformation) is a monad morphism =-=[27]-=- from e T to T . 2 6 Remark 5.1 A monad e T lifts a monad T = hT ; j; i to the B-coalgebras if and only if UB e T = TUB and, for every B-coalgebra k : X ! BX , the diagram TX X T 2 X BTX BT 2 X BX e T... |

120 |
A domain equation for bisimulation
- Abramsky
- 1991
(Show Context)
Citation Context ...l behaviours', ie the (abstract) nondeterministic processes. Semantically, the above strong extensionality result specialises then to the fact that such a final coalgebra is internally fully-abstract =-=[1]-=- with respect to bisimulation, ie its largest bisimulation is the equality, hence bisimilar elements are indistinguishable. 5 Operational Monads Definition 5.1 Let T and B be endofunctors on the same ... |

91 | Full abstraction for a simple parallel programming language - Hennessy, Plotkin - 1979 |

89 | Categorical logic
- Pitts
- 1996
(Show Context)
Citation Context ...ntiation, etc, which need parameters and `accumulators'. (By structural recursion we mean definition by structural induction.) Here we shall need the following `folklore' structural recursion theorem =-=[20]-=- with accumulators, ie with terms as parameters of the recursive definition. Theorem 5.1 (Structural Recursion) Let T be a monad freely generated by an endofunctor \Sigma on a cartesian category C and... |

83 |
A Calculus for Communicating Systems, volume 92 of LNCS
- Milner
- 1980
(Show Context)
Citation Context ...semantics [23]. Adequacy proofs can be quite demanding, hence general criteria ensuring adequacy are of interest. For process algebras, as used for specifying nondeterministic and concurrent programs =-=[17, 5]-=-, there exist syntactic restrictions on the format of the operational rules which ensure that bisimulation [17] is a congruence. Among the rules in these formats, GSOS rules [8] are the best known and... |

65 |
Distributive laws
- Beck
- 1969
(Show Context)
Citation Context ...ad up to these two isomorphisms of categories \Sigma-Alg = T -Alg B-Coalg = D-Coalg 7.1 Distributive Laws Given a monad T = hT ; j; i and a comonad D = hD; "; ffii on a category C, a distributive=-= law [7] of the mo-=-nad T over the comonad D is a natural transformations: TD ) DT satisfying the lawssffi jD = Djsffi D = D ffisT ffi T and their dual T " = " T ffisD ffi D ffi T ffi = ffi T ffisThe following ... |

55 | Initial Algebra and Final Coalgebra Semantics for Concurrency, Report CS-R9409
- Rutten, Turi
- 1994
(Show Context)
Citation Context ...ded operational model of a language. The corresponding unique coalgebra homomorphism, given by finality, from the intended operational model to the semantic domain is called final coalgebra semantics =-=[2, 23]-=-; under suitable assumptions on B, it is fully abstract with respect to behavioural equivalence. When initial algebra and final coalgebra semantics coincide, 1 one has an adequate denotational semanti... |

54 | Ntyft/ntyxt rules reduce to ntree rules
- Fokkink, Glabbeek
- 1996
(Show Context)
Citation Context ... restrictions on the format of the operational rules which ensure that bisimulation [17] is a congruence. Among the rules in these formats, GSOS rules [8] are the best known and (negative) tree rules =-=[11]-=- are the most general. In [22], the `processes as terms' method, based on such a congruence result, is presented which allows for the systematic derivation of adequate denotational models from `tyft r... |

53 | Functorial Operational Semantics and its Denotational Dual - Turi - 1996 |

46 |
Non-well-founded sets. Number 14
- Aczel
- 1988
(Show Context)
Citation Context ...ded operational model of a language. The corresponding unique coalgebra homomorphism, given by finality, from the intended operational model to the semantic domain is called final coalgebra semantics =-=[2, 23]-=-; under suitable assumptions on B, it is fully abstract with respect to behavioural equivalence. When initial algebra and final coalgebra semantics coincide, 1 one has an adequate denotational semanti... |

36 | Bisimulation and open maps
- Joyal, Nielsen, et al.
- 1993
(Show Context)
Citation Context ...t every ae-model is adequate with respect to the intended operational model in the sense that the behaviour of the programs can be determined from any ae-model up to a generalised, coalgebraic notion =-=[4, 16]-=- of bisimulation. The other adjunction implies that there exists a final ae-model -- the canonical denotational model D ae (1) over the final coalgebra of abstract, global behaviours. It is necessaril... |

23 | A category-theoretic account of program modules
- Moggi
- 1989
(Show Context)
Citation Context ... the present theory could lead to applications in modular compiler development technology. Perhaps there will be a useful theory of the combination of operational semantics of different languages (cf =-=[18]-=-). Again, perhaps one can relate the operational semantics of a language with that of its translation into another target language (cf [26]). Acknowledgements. Thanks to Marcelo Fiore and Alex Simpson... |

22 |
Adjoint Lifting Theorems for Categories of Algebras
- Johnstone
- 1975
(Show Context)
Citation Context ...nceforth they are called ae-models ; they specialise to the GSOS models of [25] and to models of tree rules (with an appropriate definition). The primary fact about ae-models is that, from results in =-=[15]-=-, it easily follows that the forgetful functors to each of the categories of denotational and operational models have adjoints. One adjunction implies that there exists an initial ae-model -- the inte... |

21 |
Processes as terms: Non-well-founded models for bisimulation
- Rutten
- 1992
(Show Context)
Citation Context ... the operational rules which ensure that bisimulation [17] is a congruence. Among the rules in these formats, GSOS rules [8] are the best known and (negative) tree rules [11] are the most general. In =-=[22]-=-, the `processes as terms' method, based on such a congruence result, is presented which allows for the systematic derivation of adequate denotational models from `tyft rules' [13], a class of rules e... |

19 | Final universes of processes
- Aczel
- 1994
(Show Context)
Citation Context ...ith weak bisimulation in this setting. As shown, eg in [13], weak bisimulation for a given set of rules can be reduced to strong bisimulation by adding three special rules for thes- action. (See also =-=[3]-=-.) These rules are in the tyft/tyxt format, but they can be compiled into safe tree rules, hence the present theory can be applied. This way of dealing with weak bisimulation is quite indirect, but th... |

16 | The tyft/tyxt format reduces to tree rules
- Fokkink
- 1994
(Show Context)
Citation Context ... edges, and v j 6 b j \Gamma! as `negative', targetless edges. A rule of type (13) is well-founded if all backwards chains of edges in its dependency graph are finite [13]. Definition 6.1 (Tree rules =-=[10, 11]-=-) A (simple negative) tree rule is a well-founded rule of type (13) such that the x k and the y i are all distinct variables and are the only variables occurring in the rule (ie the z i and v j are al... |

16 | Compositionality via cut-elimination: Hennessy-Milner logic for an arbitrary GSOS
- Simpson
- 1995
(Show Context)
Citation Context ...onal rules ae, the -bialgebras can be seen as combinations of operational and denotational models which satisfy the rules. Henceforth they are called ae-models ; they specialise to the GSOS models of =-=[25]-=- and to models of tree rules (with an appropriate definition). The primary fact about ae-models is that, from results in [15], it easily follows that the forgetful functors to each of the categories o... |

12 |
A type-theoretic account of Standard ML
- Harper, Stone
- 1996
(Show Context)
Citation Context ...bination of operational semantics of different languages (cf [18]). Again, perhaps one can relate the operational semantics of a language with that of its translation into another target language (cf =-=[26]-=-). Acknowledgements. Thanks to Marcelo Fiore and Alex Simpson for discussions. Part of this study is based on the first author's thesis; he wishes to thank Jaco de Bakker and Bart Jacobs for their gui... |

8 | Categorical modeling of structural operational rules: case studies
- Turi
- 1997
(Show Context)
Citation Context ...eventually, led to the discovery of the safe tree rules. A third advantage is that by varying \Sigma and B a wide variety of notions of program constructs and behaviour can be accommodated. (See also =-=[30]-=-.) Further, one can study abstract notions of operational rules ae, such as `abstract GSOS' and `abstract tree rules', applicable to languages other than process algebras and whose properties can be s... |

4 | Categories for synchrony and asynchrony
- Cockett, Spooner
- 1995
(Show Context)
Citation Context ...of dealing with weak bisimulation is quite indirect, but that just reflects the absence of an established denotational model for it. A more direct treatment of weak bisimulation might arise following =-=[9]-=-. 1 The Motivating Example: GSOS Consider the language with signature \Sigma consisting of a constant symbol `nil', a set of unary action prefixing operators indexed by a finite set A of actions range... |