R. Crole and A. Pitts. New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic. Information and Computation, 98:171--210, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to be made, in particular in connection with the research in [2] Let WC denote the category of (well )complete objects of Eff . In WC, the object F carries both the initial algebra and final coalgebra structures for L, and they are each other s inverse (F is a fixed point object in the sense of [1]) Now, in Eff , F is the internal limit of 1 L(1) L (1) whereas I 0 is the internal colimit of 0 L(0) L (0) This shows, that F , although a fixed point object, is not inductive in the sense of [2] 34 Moreover, for abstract reasons (see the final section of ....

R.L. Crole and A.M. Pitts. New Foundations for Fixpoint Computations: FIX-Hyperdoctrines and the FIX-Logic. Information and Computation, 98:171--210, 1992.

....Brauner has already presented an extension of Proposition 4. 1 in a linear model with parameterised fixpoints of all endomaps in the Kleisli category [8] Bearing the case where C is predomains in mind, we instead use stronger version, due to Freyd, of Crole and Pitts s notion of fixpoint object [9] so that we only require certain fixpoints in C. Translation [ m a (extract) 1 C (TX) Theta u: Y ) C u ( Theta [t] TX) C GF t : GFX ( Theta let x s in u: TY ) C G(let Fx = G u ) GFY Translation ] l a (extract) ....

R. L. Crole and A. M. Pitts. New foundations for fixpoint computations: FIX-hyperdoctines and the FIXlogic. Information and Computation, 98(2), June 1992.

....the compactness axiom, he established a mixed induction coinduction property of abstract relations on recursive domains in Cppo (the category of pointed cpos and strict continuous functions) Abstract category theoretic accounts of these issues can be found in [Fio93, HJ95] In type theory. In [CP92] Crole and Pitts introduced a higher order typed predicate logic for fixed point computations. This was done by exploiting Moggi s treatment of computations using monads [Mog91] and by introducing the key notion of fixpoint object . Fixpoint objects were partly inspired by Martin Lof s ....

R.L. Crole and A.M. Pitts. New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic. Information and Computation, 98:171--210, 1992.

....of an algebra oe : L , and Axiom 2 equips with a global element 1 invariant under iteration. A possible definition of 1 from a universal property for , taken from [Fre91] is given in Section 5. The essential difference between our fixed point object and the one previously considered in [CP92] is its constructive nature, in that we require it to arise as the colimit for the standard chain generated by iterating L. An intensional notion of approximation, the path relation (v) is introduced in Section 6 and characterised for the examples of Section 1. In the context of synthetic ....

R.L. Crole and A.M. Pitts. New foundations for fixpoint computations: FIXhyperdoctrines and the FIX-logic. Information and Computation, 98:171--210, 1992.

.... term of The exact connection to the structure of a monad can be found in Moggi s original work [48, page 61] Moggi originally gave a categorical interpretation for his computational meta language [48] Crole and Pitts extended Moggi s work with a fix point operator and an associated logic [13]. Later, Gordon developed an elegant operational theory for the meta language with a fixpoint operator and inductive and coinductive types [31] ml is basically a sublanguage of Gordon s except that we include directly type nat and associated operations whereas Gordon constructs them via ....

Roy L. Crole and Andrew M. Pitts. New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic. Information and

....now given by well complete focal S spaces or the full subcategory of replete focal S spaces. 8 Fixpoints For any focal L algebra ff : LA Gamma A over a complete A any map f : A Gamma A admits a canonical fixpoint which can be constructed a la Kleene analogous to classical domain theory c.f. [1, 16]. Since completeness encompasses wellcompleteness and S repleteness this can be applied to all notions of domains considered in this paper. Lemma 8.1 Let ff : LA Gamma A be a focal L algebra. For any f : A Gamma A there is a canonical map kl ff;f : Gamma A called Kleene chain of f (w.r.t. ....

....kl ff;f : Gamma A called Kleene chain of f (w.r.t. ff) such that kl ff;f (step(n) f n ( ff ) for all n 2 N. Accordingly, if A is complete for the unique map kl ff;f : Gamma A with kl ff;f = kl ff;f ffi it holds that kl ff;f (step(n) f n ( ff ) for all n 2 N. Theorem 8. 2 ([16, 1]) Let A be a complete set and ff : LA Gamma A be a focal L algebra structure on A. Then for any endomap f : A Gamma A the element fix ff (f) kl ff;f (1) 2 A is a fixpoint of f , i.e. f(fix ff (f) fix ff (f) 9 Domain Theoretic Induction Principles For the purposes of program ....

R.L. Crole and A.M. Pitts. New foundations for fixpoint computations: Fixhyperdoctrines and the fix-logic. Information and Computation, 98:171--210, 1992.

....: AB where V = fix v A (#f A .F 1 #f, fix v B (#g B .F 2 #f, g#)#) A. For example, from Bekic property and uniformity, we can show equations like fix v F = fix v (F # (#f.#x. F f x) Fixpoint objects Another promising direction is the approach based on fixpoint objects [2], as a uniform T fixpoint operator is canonically derived from a fixpoint object whose universal property implies strong proof principles. For instance, in Example 3, a uniform iterator is unique because the monad R R ( has a fixpoint object. For the setting with first class continuations, ....

Crole, R.L. and Pitts, A.M. (1992) New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic. Inform. and Comput. 98(2), 171--210.

....that would yield the above diagram with A Theta B instead of C and with D instead of C. In this section we compare this categorical generalization with three forms of logical relations that have already appeared in the literature, Kripke logical relations [Plo80, MM91] cpo logical relations [MS76, Rey74, CP92], and the relational setting over PER models discussed in Section 4 of [BFS90] Kripke logical relations over ordinary Henkin models were first used in [Plo80] in a characterization of lambda definability. Kripke logical relations were then adapted to Kripke lambda models in [MM91] Inductive ....

R.L. Crole and A.M. Pitts. New foundations for fixpoint computations: FIXhyperdoctrines and the FIX-logic. Information and Computation, 98:171--210, 1992.

....theories [1, 9] One may further consider the call by value version of the Bekic property (another equivalent axiomatization of these properties [9] along this line, which could be used for reasoning about mutual recursion. Another promising direction is the approach based on fixpoint objects [2], as a uniform T fixpoint operator is canonically derived from a fixpoint object whose universal property implies strong proof principles. For instance, in Example 1, a uniform iterator is unique because the monad R R ( has a fixpoint object. For the setting with first class controls, it ....

Crole, R.L. and Pitts, A.M. (1992) New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic. Inform. and Comput. 98(2), 171--210.

.... fi] f y) x) corresponds to that on the fixpoint operator fix v (f:F f f) fix v (f:fix v (g:F f g) These can be seen axiomatizing the call by value counterpart of the Conway theories [1, 7] Another promising direction is the approach based on fixpoint objects of Crole and Pitts [2], as a uniform T fixpoint operator is canonically derived from a fixpoint object whose universal properties imply stronger proof principles. For the setting with first class controls, it might be fruitful to study the implications of the existence of a fixpoint object for the monad T = R R ....

Crole, R.L. and Pitts, A.M. (1992) New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic. Inform. and Comput. 98, 171--210.

....In Section 5, we give a quick overview of initial algebras, final coalgebras and bifree algebras, including a couple of minor new propositions. Then, in Section 6, we show how bifree algebras in S can induce properties of fixed point operators in D. This programme was begun by Freyd and others [13, 5, 24, 28]. A further step was taken by Moggi, who, in unpublished work, gave a direct verification of the Bekic equality. Here, we give the complete story, showing how the presence of sufficiently many bifree algebras determines a unique parametrically uniform Conway operator (hence iteration operator) ....

....f : TA TA, f ffi f = f . 2. For any f : TA TA, g : TA TA and h : TA TB, if h ffi = ffi Th and g ffi h = h ffi f then g = h ffi f . One familiar setting in which a (unique) uniform T fixedpoint operator exists is when C has a fixpoint object in the sense of Crole and Pitts [5], see [24, 28] In this paper, we take the weaker notion of uniform T fixed point operator as primitive. However, we shall see circumstances below in which the two notions are equivalent. In this section, our aim is to show how T fixed point operators give rise to fixed point operators as ....

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R. Crole and A. Pitts. New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic. Information and Computation, 98:171--210, 1992.

....notion of monad see [28] for the notion of commutative monad see [24, 29] for the notion of lifting monad see Appendix A. Free algebras and fixed point objects. An algebra for an endofunctor is said to be free [17] if it is initial and if its inverse is a final coalgebra. A fixed point object [7] for a monad L = L; j; on a category with a terminal object 1 is an initial L algebra Lw w equipped with a global element 1 w invariant under the successor loop succ def = w jw Lw w. Theorem 1.1 For a commutative monad (L; j; t) on a cartesian closed category, the following are ....

....when either condition is satisfied, the diagram 1 1 succ id is an equaliser. The fixed point object for a monad provides a unique uniform fixed point operator for endomorphisms on objects with bottom (viz. objects equipped with an Eilenberg Moore algebra structure) see [7, 17]. Proposition 1.2 Let L be a commutative monad on a cartesian closed category C. Assume that L has a free algebra. Then, 1. the terminal object 1 in C becomes a zero object in the category of Eilenberg Moore algebras C L , and 2. if C has an initial object 0 then L0 = 1 and so 0 becomes a ....

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R.L. Crole and A.M. Pitts. New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic. Information and Computation, 98:171--210, 1992.

....satisfying these conditions. The other difficulty is determining the correct axiomatization of a fixed point operator. It appears that the axiom (Y V ) V (Y V ) fix) does not suffice. In models of c , fixed point operator can be defined using the so called fixpoint object. Crole and Pitts [CP92] define such an object in models of c , and discuss a logical system for reasoning about fixpoint computations, which may hold the answer to above questions. Another class of extensions is motivated by the application of the Retraction Theorem developed by Riecke and Viswanathan [RV95] where ....

Roy L. Crole and Andrew M. Pitts. New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic. Information and Computation, 98(2):171--210, jun 1992.

....Department of Mathematics and Computer Science, University of Leicester, University Road, LEICESTER, LE1 7RH, United Kingdom. tel: 44) 0)116 252 3404 email: Roy.Crole mcs.le. ac.uk February 27, 2002 Abstract Resum e: Nous montrons que la th eorie du type FIX introduite par Crole et Pitts [3] peut etre cod ee dans une variante du calcul d objets de Abadi et Cardelli. Plus pr ecisement, nous montrons que la th eorie du type FIX pr esent ee avec les jugements d egalit e et de r eduction op erationnelle, peut etre traduite dans le calcul d objets, cette translation pr eservant la ....

....donnent une description de certains objets r ecursifs int eressants en termes d expressions plus simples de la th eorie du type FIX. Ces traductions serviront egalement l automatisation des s emantiques op erationnelles. Abstract: We show that the FIX type theory introduced by Crole and Pitts [3] can be encoded in variants of Abadi and Cardelli s object calculi. More precisely, we show that the FIX type theory presented with judgements of both equality and operational reduction can be translated into object calculi, and the translation proved sound. The translations we give can be ....

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R. L. Crole and A. M. Pitts. New Foundations for Fixpoint Computations: FIX Hyperdoctrines and the FIX Logic. Information and Computation, 98:171-210, 1992. LICS '90 Special Edition of Information and Computation.

....in particular: the logic arising from the terms and equations associated with the types just listed provides a good foundation on which to build. 2.3 Fixpoint Objects We define the notion of a fixpoint object in a suitably structured category. This concept is due to Pitts; see also [CP90] and [CP92]. Definition 2.3.1 Given a category C with finite products, the C indexed category is specified by: ffl The objects of C are those of C, A; B) C(C Theta A; B) where composition of g and h in C is hh; gi in C, A and C = g(k Theta id) where k: C C in C and A ....

R. L. Crole and A. M. Pitts. New Foundations for Fixpoint Computations: FIX Hyperdoctrines and the FIX Logic. Information and Computation, 98:171--210, 1992. LICS '90 Special Edition of Information and Computation.

....in two stages. First, we give a denotational semantics to a metalogic M in the category CPPO of cppos and Scott continuous functions. Second, we give a formal translation of the types and expressions of O into those of M. M is based on the equational fragment of Crole and Pitts FIX logic [5], but contains a single parameterised recursive datatype which is used to model computations engaged in I O, and does not (explicitly) contain a fixpoint type. Following Plotkin s use of a metalogic to study object languages [24] we equip the programs (closed expressions) of M with an operational ....

....of a program. Any two communicators are contextually equivalent whether or not they are bisimilar. 3 The metalogic M We outline a Martin Lof style type theory which will be used as a metalogic, M, into which O may be translated and reasoned about it is based on ideas from the FIX Logic [5, 6], though M does not explicitly contain a fixpoint type. The (simple) types of M are given by oe : X 0 j Unit j Bool j Int j oe Theta oe j oe oe j oe j U(oe) together with a single top level recursive datatype declaration datatype U(X 0 ) c 1 of oe 1 j Delta Delta Delta j c n of oe ....

Roy. L. Crole and A. M. Pitts. New foundations for fixpoint computations: FIX hyperdoctrines and the FIX-logic. Information and Computation, 98:171--210, 1992.

....FIX in Object Calculi Roy L. Crole email: R.Crole mcs.le. ac.uk January 4, 1999 Abstract We show that the FIX type theory introduced by Crole and Pitts [3] can be encoded in variants of Abadi and Cardelli s object calculi. More precisely, we show that the FIX type theory presented with judgements of both equality and operational reduction can be translated into object calculi, and the translation proved sound. In the case of operational reductions, ....

....Brno, Czech Republic, and as Leicester MCS Technical Report 1998 10. The full paper, reproduced here, has been submitted to RAIRO Journal of Theoretical Informatics and Applications. 1 Motivation The results presented here are founded on the FIX type theory introduced by Crole and Pitts in [3]; and the object type theories (calculi) introduced by Abadi and Cardelli in [1] We shall assume that readers are familiar with these theories, and more generally with type theories possessing judgements of both equality, and operational reduction. A good general reference for (dependent) type ....

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R. L. Crole and A. M. Pitts. New Foundations for Fixpoint Computations: FIX Hyperdoctrines and the FIX Logic. Information and Computation, 98:171--210, 1992. LICS '90 Special Edition of Information and Computation.

....in two stages. First, we give a denotational semantics to a metalogic M in the category CPPO of cppos and Scott continuous functions. Second, we give a formal translation of the types and expressions of O into those of M. M is based on the equational fragment of Crole and Pitts FIX logic [5], but contains a single parameterised recursive datatype which is used to model computations engaged in I O, and does not (explicitly) contain a fixpoint type. Following Plotkin s use of a metalogic to study object languages [24] we equip the programs (closed expressions) of M with an operational ....

....of a program. Any two communicators are contextually equivalent whether or not they are bisimilar. 3 The metalogic M We outline a Martin Lof style type theory which will be used as a metalogic, M, into which O may be translated and reasoned about it is based on ideas from the FIX Logic [5, 6], though M does not explicitly contain a fixpoint type. The (simple) types of M are given by oe : X 0 j Unit j Bool j Int j oe Theta oe j oe oe j oe j U(oe) together with a single top level recursive datatype declaration datatype U(X 0 ) c 1 of oe 1 j Delta Delta Delta j c n of oe ....

Roy. L. Crole and A. M. Pitts. New foundations for fixpoint computations: FIX hyperdoctrines and the FIX-logic. Information and Computation, 98:171--210, 1992.

....April 21, 1997 Abstract This article 1 has two parts: In the first part, we present some general results about fixpoint objects. The minimal categorical structure required to model soundly the equational type theory which combines higher order recursion and computation types (introduced by [4]) is shown to be precisely a let category possessing a fixpoint object. Functional completeness for such categories is developed. We also prove that categories with fixpoint operators do not necessarily have a fixpoint object. In the second part, we extend Freyd s gluing construction for cartesian ....

....community; this paper concentrates on the correspondence between appropriate equational theories, and let categories with fixpoint objects. Moggi introduced equational theories for let categories in [14] The equational theories corresponding to categories with fixpoint objects were introduced in [4]. We shall give a brief summary of the definitions of these categorical structures, and the corresponding equational type theories, in Section 2. The aims of this paper are twofold. The first goal is to present some miscellaneous results about fixpoint objects. In particular, we are aiming to give ....

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Crole R. L. and Pitts A. M.:1992, New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic, Information and Computation,Vol. no. 98 pp. 171--210.

....is specified in two stages. First, we give a denotational semantics to a metalanguage M in the category CPPO of cppos and Scott continuous functions. Second, we give a formal translation of the types and expressions of O into those of M. M is based on the equational fragment of the FIX logic of [CP92] but contains a single parameterised recursive datatype which is used to model computations engaged in I O, and does not (explicitly) contain a fixpoint type. Following Plotkin s use of a metalanguage to study object languages [Plo85] we equip the programs (closed expressions) of M with an ....

....using the properties of minimal invariant objects. Finally, we prove Theorem 2 using the formal approximation relations. We outline a Martin Lof style type theory which will be used as a metalanguage, M, into which O may be translated and reasoned about it is based on ideas from the FIX Logic [CP92, Cro92] though M does not explicitly contain a fixpoint type. For a general account of similar type theories and their semantics, see for example [Cro93] First we describe the types of M. The (open and simple) types are given by the grammar oe : X 0 j Unit j Bool j Int j oe Theta oe j oe ....

R.L. Crole and A.M. Pitts. New foundations for fixpoint computations: FIX hyperdoctrines and the FIX logic. Information and Control, 98:171-- 210, 1992.

.... in which the defined type only occurs positively, then the induction property coincides with an initial algebra induction principle in the sense of Lehmann and Smyth [13] In particular it includes the induction principle for the fixed point object of the lifting monad, studied by Crole and Pitts [3]. Scott s principle of induction for admissible properties of least fixed points of continuous functions is a consequence of this case. We show that the induction property can yield an interesting proof principle even for problematic recursively defined domains, such as those that model untyped ....

....y 2 D) Pretty though it is, the author has yet to find useful applications of this principle. Example 3. 4 (Fixpoint induction) When Phi(ff) ff , recff: Phi(ff) is the ordinal 1, which is a fixpoint object for the lifting monad on the category of cpo s, in the sense of Crole and Pitts [3]. In this case Theorem 3.2 yields the induction principle which motivated the one for fixpoint objects of strong monads in general, studied in loc. cit. Scott s principle of induction for proving admissible properties of least fixed points of continuous functions is a formal consequence of this ....

[Article contains additional citation context not shown here]

R. L. Crole and A. M. Pitts, New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic, Information and Computation 98(1992) 171--210.

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R. Crole and A. Pitts. New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic. Information and Computation, 98:171--210, 1992.

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R. Crole and A. M. Pitts. New foundations for fixpoint computations: FIXhyperdoctrines and FIX-logic. Information and Computation, 98:171-- 210, 1992.

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R. L. Crole and A. M. Pitts. New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic. Information and Computation, 98:171--210, 1992.

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R.L. Crole and A.M. Pitts. New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic. Information and Computation, 98:171--210, 1992.

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