## The Temporal Logic of Coalgebras via Galois Algebras (1999)

Citations: | 33 - 7 self |

### BibTeX

@TECHREPORT{Jacobs99thetemporal,

author = {Bart Jacobs},

title = {The Temporal Logic of Coalgebras via Galois Algebras},

institution = {},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are dened for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computation Tree Logic (CTL). The mapping from coalgebras to Galois algebras turns out to be functorial, yielding indexed categorical structures. This gives many examples, for coalgebras of polynomial functors on sets. Additionally, it will be shown how \fuzzy" predicates on metric spaces, and predicates on presheaves, yield indexed Galois algebras, in basically the same coalgebraic manner. Keywords: Temporal logic, coalgebra, Galois connection, fuzzy predicate, presheaf Classication: 68Q60, 03G05, 03G25, 03G30 (AMS'91); D.2.4, F.3.1, F.4.1 (CR'98). 1 Introduction This paper combines the areas of coalgebra and of temporal logic. Coalgebras are simple mathematical structures (similar, but dual, to...

### Citations

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- Emerson
- 1990
(Show Context)
Citation Context ...ning P . The latter predicate contains all states which are reachable from P . By construction we have a new Galois connection: (= P 1 P 2 , P 1 =) P 2 . Alternative notation for ! P is XP [Eme90], or OP [Gol92], or []P [KT90] (in the style of dynamic logic). Similarly, one may write =) P as GP or as P . And : =) :P is also written as FP , or }P . Before presenting examples, we cons... |

794 | Design and Synthesis of Synchronization Skeletons Using Branching-Time Temporal Logic - CLARKE, EMERSON - 1982 |

228 | A tutorial on (co)algebras and (co)induction
- Jacobs, Rutten
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(Show Context)
Citation Context ...se for an arbitrary predicate P X , ` f (Q T ) P , Q (f (P )) T by the adjunctions , Q T (f) (P T ) by (i) , ( ` T (f) (Q)) T P: Next we turn to coalgebras of polynomial functors, see [JR97] for more information. 3.3. Denition. Let T : Sets ! Sets be a polynomial functor, and : X ! T (X) be a T -coalgebra, with a predicate P X on its state space (or carrier) X . (i) Dene a new predi... |

200 |
Logics of Time and Computation
- Goldblatt
- 1987
(Show Context)
Citation Context ...ter predicate contains all states which are reachable from P . By construction we have a new Galois connection: (= P 1 P 2 , P 1 =) P 2 . Alternative notation for ! P is XP [Eme90], or OP [Gol92], or []P [KT90] (in the style of dynamic logic). Similarly, one may write =) P as GP or as P . And : =) :P is also written as FP , or }P . Before presenting examples, we consider the interac... |

187 |
Boolean algebras with operators
- Jónsson, Tarski
- 1951
(Show Context)
Citation Context ...ed as giving examples of \Galois algebras", introduced in [Kar98]. These Galois algebras are simple structures consisting of a complete Boolean algebra carrying a Galois connection (in the spirit=-= of [JT51]-=-). The latter is interpreted as the connection between lasttime and nexttime operators. It is shown in [Kar98] that all axioms and rules of Computation Tree Logic (CTL) [MP92, Eme90, Gol92] are valid ... |

183 | Topos Theory - Johnstone - 1977 |

112 |
Categorical Logic and Type Theory
- Jacobs
- 1999
(Show Context)
Citation Context ...or polynomial functors, it does not cover examples like in Subsection 6.1 where fuzzy predicates are used, which are not subobjects. There, the more general notion of indexed category orsbration (see =-=[Jac99]-=-) is needed. This goes beyond the scope of the present paper. We have concentrated on polynomial functors because they include many important examples, and because their lifting (with left adjoint) ca... |

68 | Objects and Classes, Coalgebraically - Jacobs - 1996 |

67 |
Semantics of weakening and contraction
- Jacobs
- 1994
(Show Context)
Citation Context ...s non-emptysnite supremema. This amounts to (P 1 [ P 2 ) T = P T 1 [ P T 2 for each pair of predicates P 1 ; P 2 X on a set X . (Strict and ane functions between complete lattices are considered in [=-=Jac94]-=- as one of the running examples giving categories having tensors with diagonals or with projections, and with exponential operators ! introducing only weakening or only contraction. The issue, like he... |

35 | Introduction to Lattices and - Davey, Priestley - 2002 |

34 | Reasoning about classes in object-oriented languages: Logical models and tools
- Hensel, Huisman, et al.
- 1998
(Show Context)
Citation Context ... and for some future state y of x the size is zero. It can be proved via a slightly stronger invariant Q 0 given as Q 0 (x) , Q(x) ^ 8n 2 N: nssize(x) ) iterate(pop; n) 6= : The tool described in [H=-=HJT9-=-8] translates class specications as above into logical theories for a back-end proof tool (like PVS [ORSvH95] or Isabelle [Pau94]). It extracts the interface functor from a class specication, generate... |

16 |
Functors for coalgebras, Algebra Universalis
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Citation Context ... pullbacks of arbitrary set-indexed collections of morphisms with a common codomain. This is an essentially stronger requirement than preservation of weak pullbacks of just two morphisms, as shown in =-=[Gum-=-99]. 12 This operation preserves all meets (because T preserves arbitrary weak pullbacks), and thus has a left adjoint, say ( ) T . In this way one gets nexttime (P T ) and lasttime ( ` (P )) T ope... |

15 | Temporal algebra
- Karger
- 1998
(Show Context)
Citation Context ...so the single-step, future and past operators ! and occur there. The full impact of these operators becomes apparent when they are identied as giving examples of \Galois algebras", introduced i=-=n [Kar98]-=-. These Galois algebras are simple structures consisting of a complete Boolean algebra carrying a Galois connection (in the spirit of [JT51]). The latter is interpreted as the connection between lastt... |

15 | The metric closure powerspace construction - Kent - 1987 |

13 |
Modal and Tense Predicate Logic: Models in Presheaves and Categorical Conceptualization
- Ghilardi, Meloni
- 1987
(Show Context)
Citation Context ...as of the functor T , and where GA is the category of Galois algebras (see Theorem 6.1 below). Further, it will be shown how familiar models of temporal logic given by fuzzy predicates and presheaves =-=[GM88]-=- exhibit the same underlying structure of coalgebra-indexed Galois algebras. Probably, the contribution of this paper lies not so much in the results that are obtained, but more in the integration ofs... |

13 | Spaces. Number 3 - Stone - 1982 |

12 | Structural induction and coinduction in a setting - Hermida, Jacobs - 1998 |

7 | Definition and Proof Principles for Data and Processes - Hensel - 1999 |

6 |
bisimulations and the correctness of coalgebraic re
- Invariants
- 1997
(Show Context)
Citation Context ...er state, P holds". There is the following fundamental Galois connection: (= P 1 P 2 , P 1 =) P 2 . The denitions of =) P and (= P occur in [Rut99] (as [P ] and hP i respectively) and in [J=-=ac9-=-7] (as P and P ). In [Jac97, before Proposition 3.8] the connections with temporal logic are mentioned, but not elaborated. Also the single-step, future and past operators ! and occur there. The ful... |

3 |
Semantics of lambda-I and of other substructure lambda calculi
- Jacobs
- 1994
(Show Context)
Citation Context ...sors with diagonals or with projections, and with exponential operators ! introducing only weakening or only contraction. The issue, like here, is the distinction between at least/most once. See also =-=[Jac93]-=- for examples of models of untyped lambda calculi with variables occuring at least/most once, constructed from strict/ane functions.) 18 5.3. Lemma. If T is a polynomial functor with a strict/ane lift... |