Claudio Hermida. Fibrations, Logical Predicates and Indeterminates. PhD thesis, University of Edinburgh, 1993. Techn. rep. LFCS-93-277. Also available as Aarhus Univ. DAIMI Techn. rep. PB-462.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....bisimulation from a to [36, Theorem 2.5] a morphism is essentially a functional bisimulation. When Dom a C Dom , a is a subcoalgebra of iff the identity relation on Dom a is a bisimulation from a to . The above categorical definition of bisimulation appeared in [1] It has a chaxacterisation [18, 19] in terms of liftings of relations R C A x B to relations R T C TAx TB. This in turn was transformed in [10] to another characterisation of bisimulations that uses the idea of paths between functors, an idea introduced in [24, Section 6] A path is a finite list of symbols of the kinds rj, ....

Claudio Hermida. Fibrations, Logical Predicates and Indeterminates. PhD thesis, University of Edinburgh, 1993. Techn. rep. LFCS-93-277. Also available as Aarhus Univ. DAIMI Techn. rep. PB-462.

....2.5] a morphism is essentially a functional bisimulation. When Dom c C Dom, c is a subcoalgebra of iff the identity relation on Dom c is a bisimulation from to. The above categorical definition of bisimulation appeared in [1] It has a characterisation in terms of liftings of relations [14, 15]. For R C A x B, define a relation R T C TAx TB by induction on the formation of the polynomial functor T: R D = idD R Id = R R rXr2 = x,y) xRry and 2xr22y = x, y) xy U (2x, xry Rr = f,g) Vd D, f(d)r#(a) These liftings preserve many basic properties of relations. Thus if ....

Claudio Hermida. Fibrations, Logical Predicates and Indeterminates. PhD thesis, University of Edinburgh, 1993. Techn. rep. LFCS-93-277. Also available as Aarhus Univ. DAIMI Techn. rep. PB-462.

....2.5] a morphism is essentially a functional bisimulation. When Dom Dom , is a subcoalgebra of i the identity relation on Dom is a bisimulation from to . The above categorical de nition of bisimulation appeared in [1] It has a characterisation in terms of liftings of relations [14, 15]. For R A B, de ne a relation R TA TB by induction on the formation of the polynomial functor T : D = id D Id T1 T2 = f(x; y) 1 xR 1 y and 2 xR 2 yg T1 T2 = f( 1 x; 1 y) xR f( 2 x; 2 y) xR = f(f; g) 8d 2 D; f(d)R g(d)g: These ....

Claudio Hermida. Fibrations, Logical Predicates and Indeterminates. PhD thesis, University of Edinburgh, 1993. Techn. rep. LFCS-93-277. Also available as Aarhus Univ. DAIMI Techn. rep. PB-462.

....bisimulation from to [17, Theorem 2.5] a morphism is essentially a functional bisimulation. When Dom Dom , is a subcoalgebra of i the identity relation on Dom is a bisimulation from to . 7 The above categorical de nition of bisimulation appeared in [1] It has a characterisation [8, 9] in terms of liftings of relations R A B to relations R T TA TB. This in turn was transformed in [6] to another characterisation of bisimulations that uses the idea of paths between functors, an idea introduced in [12, Section 6] A path is a nite list of symbols of the kinds j , ....

Claudio Hermida. Fibrations, Logical Predicates and Indeterminates. PhD thesis, University of Edinburgh, 1993. Techn. rep. LFCS-93-277. Also available as Aarhus Univ. DAIMI Techn. rep. PB-462.

....Theorem 2.5] a morphism is essential a functional bisimulation. When Dom Dom , is a subcoalgebra of i the identity relation on Dom is a bisimulation from to . The above categorial de nition of bisimulation appeared in [1] It has a characterisation in terms of liftings of relations [5,6]. For R A B, de ne a relation R T TA TB by induction on the formation of the polynomial functor T : R D = id D R Id = R R T1 T2 = f(x; y) 1 xR T1 1 y and 2 xR T2 2 yg R T1 T2 = f( 1 x; 1 y) xR T1 yg S f( 2 x; 2 y) xR T2 yg R T D = f(f; g) 8d 2 ....

Hermida, C., \Fibrations, Logical Predicates and Indeterminates," Ph.D. thesis, University of Edinburgh (1993), techn. rep. LFCS-93-277. Also available as Aarhus Univ. DAIMI Techn. rep. PB-462.

....category over I 2 Sets is the poset hP(I ThetaI ) i of relations on I with reverse inclusion. The category Rel (op) is bicartesian closed, and the functor Rel (op) Sets strictly preserves this BiCCC structure. The BiCCC structure may be obtained from general fibred principles (see e.g. [4, 5]) or from the isomorphism Rel = Rel (op) given by negation: R I Theta I) 7 ( R I Theta I) But it may also be described concretely: ffl The terminal object in Rel (op) is the empty relation ; 1 2 on the (oneelement) terminal set 1 2 Sets. ffl The cartesian product R Theta S ....

C. Hermida. Fibrations, Logical Predicates and Indeterminates. PhD thesis, Univ. Edinburgh, 1993. Techn. rep. LFCS-93-277. Also available as Aarhus Univ. DAIMI Techn. rep. PB-462.

....the categories Pred and Rel of predicates and relations on sets, and mention their basic properties. Then we describe how polynomial functors T : Sets Sets can be lifted to polynomial functors Pred(T ) Pred Pred and Rel(T ) Rel Rel by induction on the structure of T . This is as in [7, 9, 8]. Algebras and coalgebras of these lifted functors Pred(T ) and Rel(T ) give important (logical) information about T . Definition 2. i) The category Pred has as objects pairs (I; P ) where P I is a subset of I or a predicate on I. Morphisms (P I) Q J) are functions u: I J for which ....

.... logical formulas; e.g. R I Theta I) Theta (S J Theta J) Theta ) R) 0 Theta 0 ) S) f(hi; ji; hi 0 ; j 0 i) j R(i; i 0 ) S(j; j 0 )g (P I) Q J) P ) 0 (Q) fz 2 I J j (9i 2 I : z = i P (i) 9j 2 J: z = 0 j Q(j)g: See [7, 8] for a more abstract description. 2 A polynomial functor T : Sets Sets acting on sets can be lifted to two new polynomial functors Pred(T ) Pred Pred acting on predicates and to Rel(T ) Rel Rel acting on relations, in commuting squares Pred F NaN F NaN Pred(T ) F NaN F NaN fflffl ....

C. Hermida. Fibrations, Logical Predicates and Indeterminates. PhD thesis, Univ. Edinburgh, 1993. Techn. rep. LFCS-93-277. Also available as Aarhus Univ. DAIMI Techn. rep. PB-462.

.... RP P, obtained by reindexing against the counit of the adjunction at the base, P ) h i(P ) which we interpret logically as S : Y ) j 9y 2 Y: y S (y) We obtain thus a monad : P P bred over : B B which is the logical predicate for the powerobject in the framework of [Her93]. We carry out a similar analysis for the partial map classi er, viz. the right adjoint to B : B Ptl M (B) whose value at an object X we denote X . The counit of the adjunction is the partial map X : X X, that is X (x) x if x #, using the is de ned predicate #. With the ....

.... the following formula: y : Y j y # (y) where we have used the convention that y is the (unique) value in Y of y 2 Y when y is in the image of Y : Y Y (which is the meaning of #) We have thus deduced the logical predicate for the lifting monad according to our framework [Her93]. Part II Satisfaction of modal formulae vs. Opsimulation 4 Modal formulae over a transition system In this section we recall the basic background about transition systems, simulations and modal formulae. A transition system S over a set of labels (or actions ) L consists of: a set S, ....

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C. Hermida. Fibrations, logical predicates and related topics. PhD thesis, University of Edinburgh, 1993. Tech. Report ECS-LFCS-93-277. Also available as Aarhus Univ. DAIMI Tech. Report PB-462.

....properties of Fib was to give a category theoretic account of certain phenomena arising in logic and computer science, taking the point of view that a bration is the proper abstract counterpart of a (constructive) predicate logic, over the simple theory corresponding to its base category. See [Her93], which also contains fuller details of the main constructions occurring in this paper. Such categorical logic applications have been elaborated in [HJ95a, HJ95b, HJ95c] Dept. of Mathematics and Statistics, McGill University, 805 Sherbrooke St. W. Montreal, QC, Canada H3A 2K6. ....

....B A , change ofbase along F yields a cartesian bred adjunction F (E) ## F (q) ## q (F ) E q ## G A ## F B ## G Proof. We will only spell out the data of the resulting adjunction, leaving the veri cation of the adjunction laws to the reader. Details may be found in [Her93]. To simplify the presentation, we assume the bration is split, which allows us to ignore the coherent isomorphisms arising from the pseudo functorial nature of a cleavage. We use the following abbreviations q 0 = F (q) F 0 = q (F ) G 00 = q 0 ) G) Consider G (F ....

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C. Hermida. Fibrations, logical predicates and related topics. PhD thesis, University of Edinburgh, 1993. Tech. Report ECS-LFCS-93277. Also available as Aarhus Univ. DAIMI Tech. Report PB-462.

....also the right technical framework. In particular, the relationship between inductive predicates and logical predicates is best presented in this setting, as logical predicates for type constructors given by adjoints arise uniformly from an intrinsic property of adjunctions between fibrations, cf. [Her93]. ii) The categorical framework which we work in takes explicit account of proofs of entailments between predicates. Thus this work can be seen as a generalisation of induction principles from the usual proof irrelevant setting to the type theoretic (or constructive) one. See Remark 2.5 below. ....

....2.7. Definition. A bicartesian fibration P #p B is a fibration over a bicartesian category B , such that P is bicartesian and p strictly preserves such structure. 2.8. Remark. A bicartesian fibration is a bicartesian object in Fib, the 2 category of fibrations described in 2.6 above. See [Her93] and the references there for details on such matters. 2.9. Examples. i) Classical logic. The fibration corresponding to classical first order logic is the subobject fibration Sub(Set) #cod Set . The category Sub(Set) is the category of subobjects: its objects are pairs (S; X) where S X, ....

[Article contains additional citation context not shown here]

C. Hermida. Fibrations, logical predicates and indeterminates. PhD thesis, University of Edinburgh, 1993. Tech. Report ECS-LFCS-93-277. Also available as Aarhus Univ. DAIMI Tech. Report PB-462.

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Claudio Hermida. Fibrations, Logical Predicates and Indeterminates. PhD thesis, University of Edinburgh, 1993. Techn. rep. LFCS-93-277. Also available as Aarhus Univ. DAIMI Techn. rep. PB-462.

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C. Hermida. Fibrations, Logical Predicates and Indeterminates. PhD thesis, Univ. Edinburgh, 1993. Techn. rep. LFCS-93-277. Also available as Aarhus Univ. DAIMI Techn. rep. PB-462.

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