## On the Foundations of Final Semantics: Non-Standard Sets, Metric Spaces, Partial Orders (1998)

Venue: | PROCEEDINGS OF THE REX WORKSHOP ON SEMANTICS: FOUNDATIONS AND APPLICATIONS, VOLUME 666 OF LECTURE NOTES IN COMPUTER SCIENCE |

Citations: | 47 - 10 self |

### BibTeX

@INPROCEEDINGS{Rutten98onthe,

author = {Jan J. M. M. Rutten and Daniele Turi},

title = {On the Foundations of Final Semantics: Non-Standard Sets, Metric Spaces, Partial Orders},

booktitle = {PROCEEDINGS OF THE REX WORKSHOP ON SEMANTICS: FOUNDATIONS AND APPLICATIONS, VOLUME 666 OF LECTURE NOTES IN COMPUTER SCIENCE},

year = {1998},

pages = {477--530},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of non-standard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of post-fixed point. They are also used here for giving a new comprehensive presentation of the (still) non-standard theory of non-well-founded sets (as non-standard sets are usually called). This paper is meant to provide a basis to a more general project aiming at a full exploitation of the finality of the domains in the semantics of programming languages --- concurrent ones among them. Such a final semantics enjoys uniformity and generality. For instance, semantic observational equivalences like bisimulation can be derived as instances of a single `coalgebraic' definition (introduced elsewhere), which is parametric of the functor appearing in the domain equation. Some properties of this general form of equivalence are also studied in this paper.

### Citations

1332 |
A Calculus of Communicating Systems
- Milner
- 1982
(Show Context)
Citation Context ...or concurrent programming languages. (To be precise, the distinctive feature of the domains under consideration is non-determinism rather than concurrency, the starting point being languages like CCS =-=[Mil80]-=- in which concurrency is reduced to sequentiality plus non-determinism.) It has resulted in a general semantic framework which could be called final semantics , as it is based on the observation that ... |

1286 | A structural approach to operational semantics
- Plotkin
- 1981
(Show Context)
Citation Context ...non-deterministic computations which can be said to be split at every state into a set of possible computations. To describe non-deterministic computations labelled transition systems in the style of =-=[Plo81b]-=- are often used: Example 2.4 Labelled Transition Systems A labelled transition system (LTS) is a triple L = (S; A; !), consisting of a set S of states, a set A of labels, and a transition relation !` ... |

656 |
Concurrency and automata on infinite sequences
- Park
- 1981
(Show Context)
Citation Context ...to each program its equivalence class and the domain is then simply defined as the image of that mapping. A popular example of such an observational equivalence is (strong) bisimulation as defined in =-=[Par81]-=-. Now, the functor used for the final semantics in [Acz88] can be shown to induce bisimulation equivalence in the sense that two programs are mapped (via the final semantics) into the same process if ... |

441 | Computational lambda-calculus and monads
- Moggi
- 1989
(Show Context)
Citation Context ...or F is to be part of a comonad and the arrows between F-coalgebras have to preserve also this extra comonadic structure. Semantics by means of comonads has been investigated in [BG91]. (But see also =-=[Mog89]-=- for semantics in terms of the dual notion --- monads.) It would be interesting to understand whether some connections can be established with that work. 7.3 Coinduction For F -algebras the following ... |

238 | The lazy lambda calculus
- ABRAMSKY
- 1989
(Show Context)
Citation Context ...algebra of the same functor, the (final) semantics of the language will immediately follow. (Ideally, this scheme would include not only concurrent languages, but also applicative ones --- see, e.g., =-=[Abr90]-=-). Alternatively, the observable computations of the class of programs of the language under study might be directly defined as a coalgebra of the chosen functor. Of the general methodology sketched a... |

218 |
Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics
- Lane
- 1998
(Show Context)
Citation Context ...the arrows one can easily define the category of F -algebras. 2 Notice that in category theory the name F -(co)algebra is usually reserved for the case when F is the functor of a (co)monad (see, e.g, =-=[Lan71]-=-). F -(co)algebras have then some extra structure. They form a different category which, however, can be regarded as a subcategory of the above category of F -(co)algebras by simply forgetting the ext... |

191 |
Admissible Sets and Structures
- Barwise
- 1975
(Show Context)
Citation Context ...tively, indeterminates could also be added as new symbols in the language. For instance, in [BE88] indeterminates are indeed treated as primitive elements (Urelemente) of a set theory like the one in =-=[Bar75]-=-. But in order to carry out this extension of the language formally, an extension of the axioms of the theory is also required. 4.4 Special Final Coalgebra Theorem The assumption that the universe (gr... |

169 |
The category-theoretic solution of recursive domain equations
- Smyth, Plotkin
- 1982
(Show Context)
Citation Context ...nsive mappings as arrows) have a final coalgebra. The proof is based on a theorem stating that such functors have fixed points. The latter theorem extends earlier results of [AR89] along the lines of =-=[SP82]-=-, and is proved in full detail. For partial orders an initial algebra theorem and the so-called limit-colimit coincidence are well-known (see [SP82]), but, apparently, it was never proved in detail th... |

162 |
A final coalgebra theorem
- Aczel, Mendler
- 1989
(Show Context)
Citation Context ...ing with final semantics is that there is a single `coalgebraic ' notion of (possibly observational) equivalence which is parametric of the functor: it is the definition of F-bisimulation as given in =-=[AM89]-=-. For a particular choice of the functor F , namely the one used in [Acz88] (but see also [BZ82]), F-bisimulation coincides with bisimulation in the traditional sense, as was observed above. Also othe... |

128 |
Terminal coalgebras in well-founded set theory
- Barr
- 1993
(Show Context)
Citation Context ...elta \Delta \Delta (9) 47 This has not been fully investigated so far, although a `schematological' approach to domain equations as in (9) is sketched in [Abr88]. A more abstract approach is taken in =-=[Bar91]-=- when dealing with the existence of final coalgebras in the category Set of sets (it is not immediately clear whether standard set theory or just basic set theory is assumed there). The existence of f... |

120 |
A domain equation for bisimulation
- Abramsky
- 1991
(Show Context)
Citation Context ... about the category of cpo's is the study of a new notion, called ordered F -bisimulation, which is a generalization of the definition of F-bisimulation. Both the notions of partial bisimulation from =-=[Abr91]-=- and that of simulation from [Pit92] (for the functorial case) can be seen to be examples of ordered F-bisimulations. Corresponding to the notion of ordered F-bisimulation is a generalized notion of s... |

104 |
The Liar: An Essay on Truth and Circularity
- Barwise, Etchemendy
- 1987
(Show Context)
Citation Context ...Here, the definition of the expanded universe is carried out within the language of set theory, but, alternatively, indeterminates could also be added as new symbols in the language. For instance, in =-=[BE88]-=- indeterminates are indeed treated as primitive elements (Urelemente) of a set theory like the one in [Bar75]. But in order to carry out this extension of the language formally, an extension of the ax... |

78 |
J.I.Zucker: Processes and the Denotational Semantics of Concurrency
- Bakker
- 1982
(Show Context)
Citation Context ...nal) equivalence which is parametric of the functor: it is the definition of F-bisimulation as given in [AM89]. For a particular choice of the functor F , namely the one used in [Acz88] (but see also =-=[BZ82]-=-), F-bisimulation coincides with bisimulation in the traditional sense, as was observed above. Also other equivalences, like for instance trace equivalence, can be obtained by instantiating F-bisimula... |

76 |
Co-induction in relational semantics
- Milner, Tofte
- 1991
(Show Context)
Citation Context ...0 coalgebra and let (B; fi) be any F -coalgebra. Ifs: (B; fi) ! (A; ff) is a mapping between F -algebras andsis epic (the generalization of surjective), thensis an isomorphism. (See also [Smy92].) In =-=[MT91]-=-, this principle is used in the basic case where the category under consideration is a lattice and the functor F a monotonic operation. At the same time, the fact that an F-coalgebra (A; ff) is final ... |

73 |
Basic Set Theory
- Levy
- 1979
(Show Context)
Citation Context ...tor s: 17 (The axioms above, as well as those given in the sequel, are written for convenience in natural language but note that they can also be expressed in the language of set theory -- see, e.g., =-=[Lev79]-=-.) Further useful notions can be derived from the above axioms, like, for instance, that of ordered pair : ! x; y ? j fx; fx; ygg: A formal definition of function can then be given as a collection f o... |

68 | Solving reflexive domain equations in a category of complete metric spaces - America, Rutten - 1989 |

67 | A Categorical Programming Language
- Hagino
- 1987
(Show Context)
Citation Context ...rem. (There the domain is the same, but its finality is not recognized.) 2 A final remark. There is a notion which generalizes and combines both algebras and coalgebras of functors: An F; G-dialgebra =-=[Hag87]-=- of two functors F and G from a category D to a category C is still a pair (A; ff), but with ff an arrow in C from F (A) to G(A). It is a notion useful in type theory. 9 3 F-Bisimulation The final sem... |

60 |
Post-graduate lecture notes in advanced domain theory
- Plotkin
- 1981
(Show Context)
Citation Context ...tion of injective), thensis an isomorphism. An immediate consequence is, for instance, the induction principle for natural numbers (viewed as initial algebra of a suitably chosen functor). (E.g., see =-=[Plo81a]-=- and [LS81].) The dualization of the induction principle yields what could be called a coinduction principle for final F -coalgebras: let (A; ff) be a final F - 50 coalgebra and let (B; fi) be any F -... |

54 |
Set theory with free construction principles. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 4 e série
- Forti, Honsell
- 1983
(Show Context)
Citation Context ...ets were considered to be sets. It was computer science that provided non-well-founded sets with one of the first significant applications: semantic processes are non-well-founded sets. (But see also =-=[FH83] for a -- -=-previous -- purely mathematical application.) 4.2.2 Decoration Lemma In [Acz88] the anti-foundation axiom is formulated in terms of graphs and their "decorations ". Corollary 4.8 shows that,... |

46 |
Non-well-founded sets. Number 14
- Aczel
- 1988
(Show Context)
Citation Context ...languages as well.) However, in the above as well as in other examples the recognized finality of the domain is not systematically exploited, except for the final coalgebra semantics for CCS given in =-=[Acz88]-=-. As mentioned in the introduction, in the forthcoming paper Observations as Functors other instances of final coalgebra semantics will be given, starting from the idea that observations can be formal... |

44 |
A Category-Theoretic Approach to the Semantics of Programming Languages
- Oles
- 1982
(Show Context)
Citation Context ...ndofunctors over them, initial algebras and final coalgebras coincide. Finally, it should be mentioned that an early reference to finality as a definition method for semantic mappings can be found in =-=[Ole82]-=-. Consider now the other distinctive characteristic of final coalgebra semantics mentioned above. An extra coalgebraic structure is added to programs (as a function from programs to their observable c... |

40 | A co-inducion principle for recursively defined domains
- Pitts
- 1994
(Show Context)
Citation Context ...study of a new notion, called ordered F -bisimulation, which is a generalization of the definition of F-bisimulation. Both the notions of partial bisimulation from [Abr91] and that of simulation from =-=[Pit92]-=- (for the functorial case) can be seen to be examples of ordered F-bisimulations. Corresponding to the notion of ordered F-bisimulation is a generalized notion of strong extensionality. A proof is giv... |

37 |
Algebraic Specification of Data Types: A Synthetic Approach
- Lehmann, Smyth
- 1981
(Show Context)
Citation Context ...tive), thensis an isomorphism. An immediate consequence is, for instance, the induction principle for natural numbers (viewed as initial algebra of a suitably chosen functor). (E.g., see [Plo81a] and =-=[LS81]-=-.) The dualization of the induction principle yields what could be called a coinduction principle for final F -coalgebras: let (A; ff) be a final F - 50 coalgebra and let (B; fi) be any F -coalgebra. ... |

34 | Mathematics Form and Function - Lane, Saunders - 1986 |

28 | Monads and comonads in intensional semantics
- BROOKES, STONE
- 1993
(Show Context)
Citation Context ...Usually, the endofunctor F is to be part of a comonad and the arrows between F-coalgebras have to preserve also this extra comonadic structure. Semantics by means of comonads has been investigated in =-=[BG91]-=-. (But see also [Mog89] for semantics in terms of the dual notion --- monads.) It would be interesting to understand whether some connections can be established with that work. 7.3 Coinduction For F -... |

21 | A cook’s tour of the finitary non-wellfounded sets
- Abramsky
(Show Context)
Citation Context ...!! F n (1) = F ! ! /\Gamma F (F ! ) /\Gamma \Delta \Delta \Delta (9) 47 This has not been fully investigated so far, although a `schematological' approach to domain equations as in (9) is sketched in =-=[Abr88]-=-. A more abstract approach is taken in [Bar91] when dealing with the existence of final coalgebras in the category Set of sets (it is not immediately clear whether standard set theory or just basic se... |

20 |
Processes as terms: Non-well-founded models for bisimulation
- Rutten
- 1992
(Show Context)
Citation Context ... means of a rather ad hoc construction. Instead, in the forthcoming paper Observations as functors , a systematic method for deriving semantic operators from transition system specifications given in =-=[Rut92]-=- is rephrased in terms of final coalgebra semantics. This amounts to deriving a \Sigma-algebra for the domain by means of finality properties. It can be then proved that the original final semantics i... |

17 | Metric Semantics for Concurrency - Meyer - 1988 |

13 |
Least fixed point of a functor
- Adámek, Koubek
- 1979
(Show Context)
Citation Context ...least and the greatest fixed point (w.r.t. ) of f are a ff2On f"ff and Y ff2On f#ff: The generalization of the above theorem from least fixed points to initial algebras has already been worked ou=-=t in [AK79]-=-. Lattices (as pre-orders) generalize to categories, bottom elements to initial objects, monotone functions to endofunctors, least upper bounds to colimits. One has then the following diagram: 0 ! \Ga... |

8 | Deriving denotational models for bisimulation from structured operational semantics, in - Rutten - 1990 |

5 | A general construction of hyperuniverses - Forti, Honsell - 1992 |

5 |
I-categories and duality
- Smyth
- 1992
(Show Context)
Citation Context ...nitial algebra not only is recognized there to be a final transition system, but also indicated to be a final coalgebra as a consequence of the limit-colimit duality. The latter has been used also in =-=[Smy92]-=- to prove that, for so-called information categories (general order-theoretic frameworks for solving domain equations) and suitable endofunctors over them, initial algebras and final coalgebras coinci... |

3 |
Deadlock and fairness in morphisms of transition systems
- Hesselink
- 1988
(Show Context)
Citation Context ...-) by defining ff : S ! P(A\ThetaS), for all s; s 0 2 S, a 2 A, by 7 ! a; s 0 ?2 ff(s) () s a \Gamma!s 0 : 2 The above is the coalgebra representation of transition systems from [Acz88] (but see also =-=[Hes88]-=-) mentioned in the introduction. The LTS associated to a language like CCS has programs as states and atomic actions as labels. Transitions are given by the inductive closure of a set of structural ru... |

1 |
Generalised finiteness conditions on labelled transition systems for operational semantics of programming languages
- Breugel
- 1992
(Show Context)
Citation Context ...e finite (a weaker notion than finitely branching), one could replace in the above definition the functor P comp by another powerset functor: P closed , which yields all metrically closed subsets. In =-=[Bre92], domains -=-are given suited for LTS's that satisfy even more general "branching" properties.) 5.3 Fixed Points in CMS In this subsection, it will be shown that every locally contracting functor has a f... |