## Temporal Structures (1990)

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Citations: | 29 - 20 self |

### BibTeX

@MISC{Casley90temporalstructures,

author = {Ross Casley and Roger F. Crew and José Meseguer and Vaughan Pratt},

title = {Temporal Structures},

year = {1990}

}

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### Abstract

We combine the principles of the Floyd-Warshall-Kleene algorithm, enriched categories, and Birkhoff arithmetic, to yield a useful class of algebras of transitive vertex-labeled spaces. The motivating application is a uniform theory of abstract or parametrized time in which to any given notion of time there corresponds an algebra of concurrent behaviors and their operations, always the same operations but interpreted automatically and appropriately for that notion of time. An interesting side application is a language for succinctly naming a wide range of datatypes. 1 Introduction Posets, metric spaces, "closed" automata, and categories have in common the notion of a space of points with distances between points. These distances are respectively truth values, reals, languages, and sets. Distances have two facets, logical and metrical. The logical facet is expressed respectively via implications p ! q between truth values, comparisons x y between reals, inclusions L ` M between langua...

### Citations

2573 |
The design and analysis of computer algorithms
- Aho, Hopcroft, et al.
- 1974
(Show Context)
Citation Context ...automaton M having n states, viewed as an n × n matrix of (finitely presented) regular languages, from which one may then easily read off a regular expression denoting the language L(M) accepted by M =-=[AHU74]-=-. Even Gaussian elimination was found to be so describable in the (nonidempotent) ring of reals made a field via 1/(1 − x) = 1 + x + x 2 + . . .. Independently and working in a categorical setting, Ei... |

1005 |
Categories for the Working Mathematician
- Lane
- 1971
(Show Context)
Citation Context ...e on these topics that we will assume of the reader, and to coordinate this perspective with the rest of the paper. Considerably more information on these topics can be found in the books of Mac Lane =-=[Mac71]-=- and Kelly [Kel82]. 3.1 Monoidal Categories Informally, a monoidal category amounts to a structure that is both a monoid and a category. Formally a strict monoidal category D = (D, ⊗, I) is a category... |

398 | Basic Concepts of Enriched Category Theory, volume 64
- Kelly
(Show Context)
Citation Context ...ched categories in their full generality, their motivating applications were confined to V ’s forming classes rather than sets, an emphasis continued in Kelly’s subsequent book on enriched categories =-=[Kel82, p22]-=-. Yet in 1974 F.W. Lawvere [Law73] in an excellent advertisement for the utility of enriched categories emphasized V ’s that were semirings, hence sets rather than classes, and with category structure... |

394 |
Algorithm 97: Shortest Path
- Floyd
- 1962
(Show Context)
Citation Context ...that of S. Warshall [War62] for computing the transitive closure of a binary relation on n elements presented as an n × n Boolean (0- and 1-valued) matrix. The unification was hinted at by R.W. Floyd =-=[Flo62]-=- who observed that Warshall’s procedure would compute shortest paths instead of transitive closure if Booleans were replaced by nonnegative reals, disjunction by min, and conjunction by addition. The ... |

297 |
Regular Algebra and Finite Machines
- Conway
- 1971
(Show Context)
Citation Context ...ction by addition. The uniform expression of this common algorithm in terms of semirings was first described by Robert and Ferland [RF68]. Synonyms for this notion of semiring include regular algebra =-=[Con71]-=-, Kleene algebra [Koz80, Koz81], and quantale [Vic89]. Other instantiations of this abstract algorithm were subsequently found. Replacing Booleans in Warshall’s algorithm by languages, conjunction by ... |

242 | Modeling Concurrency with Partial Orders
- Pratt
- 1986
(Show Context)
Citation Context ...l define only operations on individual behaviors, hence including concatenation but excluding union and Kleene star. The operations we treat include all behavior operations of the process language of =-=[Pra86]-=-, namely concurrence, orthocurrence, concatenation, and local concatenation, 1 We prefer D to V as connoting distance or delay. 4sas well as new operations synchronized concurrence, exponentiation, pr... |

229 |
Closed categories
- Eilenberg, Kelly
- 1965
(Show Context)
Citation Context ...imination was found to be so describable in the (nonidempotent) ring of reals made a field via 1/(1 − x) = 1 + x + x 2 + . . .. Independently and working in a categorical setting, Eilenberg and Kelly =-=[EK65]-=- developed a more categorically sophisticated version of the same generalization of metric space under the name enriched category or V -category. In place of a semiring they used a monoidal category V... |

224 | A theorem on Boolean matrices
- Warshall
- 1962
(Show Context)
Citation Context ...s all sups and whose monoidal operation preserves (i.e. distributes over) those sups. The basis for this unification is the O(n 3 ) time procedure of Roy [Roy59] and independently that of S. Warshall =-=[War62]-=- for computing the transitive closure of a binary relation on n elements presented as an n × n Boolean (0- and 1-valued) matrix. The unification was hinted at by R.W. Floyd [Flo62] who observed that W... |

131 | Modeling concurrency with geometry, in
- Pratt
- 1991
(Show Context)
Citation Context ...ave made considerable progress towards understanding the roles of duality and linear logic in this model. From one point of view there is a duality between order in schedules and geometry in automata =-=[Pra91]-=-, whereby true concurrency in schedules has a natural representation in automata in terms of surfaces and volumes. From a slightly different viewpoint a more symmetric duality ideally suited to modeli... |

117 |
Metric spaces, generalized logic and closed categories
- Lawvere
(Show Context)
Citation Context ...y, their motivating applications were confined to V ’s forming classes rather than sets, an emphasis continued in Kelly’s subsequent book on enriched categories [Kel82, p22]. Yet in 1974 F.W. Lawvere =-=[Law73]-=- in an excellent advertisement for the utility of enriched categories emphasized V ’s that were semirings, hence sets rather than classes, and with category structure merely that of a poset. This brou... |

116 |
Topology via Logic
- Vickers
- 1989
(Show Context)
Citation Context ...mon algorithm in terms of semirings was first described by Robert and Ferland [RF68]. Synonyms for this notion of semiring include regular algebra [Con71], Kleene algebra [Koz80, Koz81], and quantale =-=[Vic89]-=-. Other instantiations of this abstract algorithm were subsequently found. Replacing Booleans in Warshall’s algorithm by languages, conjunction by concatenation, and disjunction by union, yields Kleen... |

50 |
Bits and pieces of the theory of institutions
- Tarlecki
- 1986
(Show Context)
Citation Context ...It is routine to extend this mapping to a functor (U ↓ V ) J → (U J ↓ V J ) and show that it is an isomorphism of categories. From this lemma we may obtain the following very useful theorem. Tarlecki =-=[Tar85]-=- proves the weaker version of this theorem in which J is not a parameter of the theorem, in that A and B are required to have all limits rather than just J-limits. The lemma in Mac Lane’s treatment of... |

36 |
Variation through enrichment
- Betti, Carboni, et al.
- 1983
(Show Context)
Citation Context ...has a right adjoint E: Set → D! taking the set X to the corresponding chaotic (codiscrete) D-category (δDS(u, v) = 1 for all u, v). Meanwhile, for the case of general colimits we refer to a result of =-=[BCSW83]-=-. Theorem 7 If D is cocomplete and D is closed then D! is cocomplete. There is also a similar but much easier result about limits. PSL needs limits for local product and hence local concatenation. The... |

31 | A logic of concrete time intervals - Lewis - 1990 |

27 |
The pomset model of parallel processes: Unifying the temporal and the spatial
- Pratt
- 1984
(Show Context)
Citation Context ... notion of a partial order as a behavior of a “truly concurrent” process, and uniformly extends it to other classes of spaces via the above correspondences. This extension was first proposed by Pratt =-=[Pra84]-=- with just the semiring view in mind; here we extend that proposal to take advantage of the enriched category perspective as well as additional basic operations whose utility were not at all apparent ... |

22 | Event Spaces and Their Linear Logic
- Pratt
- 1991
(Show Context)
Citation Context ...d to modeling linear logic as envisaged above may be obtained by the trick of moving some of the automaton structure over to the schedule side to make the two sides of the duality perfectly symmetric =-=[Pra92]-=-. This symmetry then permits both sides of the duality to be regarded as elements of a single model, by analogy with the negative and positive propositions of propositional logic, which are mixed in t... |

20 | On induction vs. *-continuity - Kozen |

19 |
V.R.: Partial order models of concurrency and the computation of functions
- Gaifman, Pratt
- 1987
(Show Context)
Citation Context ...aint. In the noncartesian case, call it 3 ′ , 1 represents nonstrict temporal precedence and 2 strict, with 1 ⊗ 2 = 2 (u ≤ v < w → u < w). This structure is hidden in the two-relation “prosset” model =-=[GP87]-=- used in the proof of Kahn’s principle relative to a pomset-based semantics of nets [Pra86]. For the cartesian closed 3, 1 denotes “temporal” or accidental order and 2 causal [Gai89]. Here 1 ⊗ 2 = 1, ... |

16 |
Transitivité et Connexité
- Roy
- 1959
(Show Context)
Citation Context ... monoid whose underlying partial order contains all sups and whose monoidal operation preserves (i.e. distributes over) those sups. The basis for this unification is the O(n 3 ) time procedure of Roy =-=[Roy59]-=- and independently that of S. Warshall [War62] for computing the transitive closure of a binary relation on n elements presented as an n × n Boolean (0- and 1-valued) matrix. The unification was hinte... |

13 |
Functors: The Categorical Imperative
- Arrows
- 1975
(Show Context)
Citation Context ...ore accessible via definitions that are not only close at hand but presented from the same perspective as the applications. Enriched categories or V -categories are treated briefly by Arbib and Manes =-=[AM75]-=- and exhaustively and precisely by Kelly [Kel82]. Section I-8 of Mac Lane also touches on them, without calling them such. As far as we are aware ours is the first proposed engineering application of ... |

13 | Generalized arithmetic - BIRKHOFF - 1942 |

12 | A representation theorem for models of *-free PDL - Kozen - 1980 |

10 |
Modeling Concurrency by Partial Orders and Nonlinear Transition Systems," Linear Time, Branching Time
- Gaifman
- 1988
(Show Context)
Citation Context ...on “prosset” model [GP87] used in the proof of Kahn’s principle relative to a pomset-based semantics of nets [Pra86]. For the cartesian closed 3, 1 denotes “temporal” or accidental order and 2 causal =-=[Gai89]-=-. Here 1 ⊗ 2 = 1, that is, if u accidentally precedes v whereas v causes (necessarily precedes) w, then we may only infer that u accidentally precedes w. Thus one may identify the logic of causal and ... |

4 | Enriched categories and the Floyd–Warshall connection
- Pratt
- 1989
(Show Context)
Citation Context ...ories into close proximity to the parallel computer science development. Nevertheless this connection between enriched categories and shortest-path algorithms remained undetected for another 15 years =-=[Pra89]-=-. The present paper starts from the notion of a partial order as a behavior of a “truly concurrent” process, and uniformly extends it to other classes of spaces via the above correspondences. This ext... |

4 |
Généralisation de l’algorithme de Warshall
- Robert, Ferland
- 1968
(Show Context)
Citation Context ...ns were replaced by nonnegative reals, disjunction by min, and conjunction by addition. The uniform expression of this common algorithm in terms of semirings was first described by Robert and Ferland =-=[RF68]-=-. Synonyms for this notion of semiring include regular algebra [Con71], Kleene algebra [Koz80, Koz81], and quantale [Vic89]. Other instantiations of this abstract algorithm were subsequently found. Re... |