A. Simpson. Recursive types in Kleisli categories. Unpublished manuscript, available from http://www.dcs.ed.ac.uk/home/als/, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....sense) for solving recursive type equations. There he introduced algebraically compact categories and established their fundamental property: that bifunctors on them have canonical and minimal fixed points. This has been a first important step towards an axiomatic theory of recursive types (see [Sim92] and [Fio94a, Chapters 6 8] Other work on algebraic compactness can be found in [Ad a93, Bar92] Concerning fixed points of endomorphisms, it was noticed by [HP90] after studying the work of [Law64, Law69] that in the presence of cartesian closure they are inconsistent with coproducts ....

A.K. Simpson. Recursive types in Kleisli categories. Manuscript (available from http://www.dcs.ed.ac.uk/home/als/kleisli.dvi.Z), 1992.

.... calculus plus rstclass continuations) there is a bijective correspondence between stable uniform cbv xpoint operators and uniform iterators, via Filinski s construction of recursion from iteration [5] The notion of uniform T xpoint operators arose from the context of Axiomatic Domain Theory [7, 26]. By letting T be a lifting monad on a category of predomains, a uniform T xpoint operator amounts to a uniform xpoint operator on domains (the least xpoint operator in the standard order theoretic setting) In general, T can be any strong monad on a category with nite products, thus a uniform ....

....we shall consider a computational model with the base category C and a strong monad T . 6.1. Uniform T Fixpoint Operators We rst recall the notion of uniform T xpoint operator of Simpson and Plotkin [27] which arose from considerations on xpoint operators in Axiomatic Domain Theory (ADT) [7, 26]. In ADT, we typically start with a category C of predomains, for example the category of complete partial orders (possibly without bottom) and continuous functions. Then we consider the lifting monad T on C , which adds a bottom element to cpo s. Then objects of the form TX are pointed cpo s ....

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Simpson, A.K. (1992) Recursive types in Kleisli categories. Manuscript.

....is that the induced morphism X [Y ) Z] is not uniquely determined. The canonical evaluation morphism is a functional closed relation and for the induced morphism we can always choose a functional one, and as such it is unique, i.e. these morphisms come from SCS rather than SCS . In [23] such a situation is called a Kleisli exponential. There is an alternative description of the relation space by observing SCS (Y; Z) SCS Y; K(Z) Thus the normal function space [Y K(Z) with the compact open topology, which is simply the Scott topology, yields a space that is ....

A. K. Simpson. Recursive types in Kleisli categories. Manuscript (available from http://www.dcs.ed.ac.uk), 1992.

.... plus firstclass continuations) there is a bijective correspondence between stable uniform cbv fixpoint operators and uniform iterators, via Filinski s construction of recursion from iteration [5] The notion of uniform T fixpoint operators arose from the context of Axiomatic Domain Theory [7, 25]. By letting T be a lifting monad on a category of predomains, a uniform T fixpoint operator amounts to a uniform fixpoint operator on domains (the least fixpoint operator in the standard order theoretic setting) In general, T can be any strong monad on a category with finite products, thus a ....

....Operators In this section we shall consider a computational model with the base category C and a strong monad T . We first recall the notion of uniform T fixpoint operator of Simpson and Plotkin [26] which arose from considerations on fixpoint operators in Axiomatic Domain Theory (ADT) [7, 25]. In ADT, we typically start with a category C of predomains, for example the category of # complete partial orders (possibly without bottom) and continuous functions. Then we consider the lifting monad T on C , which adds a bottom element to # cpo s. Then objects of the form TX are just the ....

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Simpson, A.K. (1992) Recursive types in Kleisli categories. Manuscript, available from http://www.dcs.ed.ac.uk/home/als/.

....a computation Gamma c M : B denotes a function from [ Gamma ] to the 24 Paul Blain Levy carrier of [ B] The Scott semantics of Sect. 3.1 can also be seen as an algebra semantics, because a domain corresponds to an algebra for the lifting monad on PreDom. This notion is exploited in [22, 35]. We illustrate this approach in the case of global store, where we use the strong monad S (S Theta Gamma) A closed computation c M : B will denote an element of X , where [ B] X; Now suppose we prefix to M a command such as X : 3. This immediately gives us an element of TX . ....

Alex K. Simpson. Recursive types in Kleisli categories. Unpublished manuscript, August 1992.

....of FPC is given in table 1. The type decorations on various constructs above guarantee uniqueness of typing: if Gamma r : A and Gamma r : A 0 , then A = A 0 . We shall usually omit type decorations. 1 INTRODUCTION 4 1. 3 Models Categorical models of FPC are discussed in [Fio94] and [Sim92]. In [Fio94] chapter 8 and section 9.1) models of FPC are carefully developed in a categorical axiomatisation of partial map structures. We use the more general monadic setting of [Sim92] the details of the interpretation of FPC generalise more or less verbatim to this setting. Definition 1.1 ....

....omit type decorations. 1 INTRODUCTION 4 1.3 Models Categorical models of FPC are discussed in [Fio94] and [Sim92] In [Fio94] chapter 8 and section 9.1) models of FPC are carefully developed in a categorical axiomatisation of partial map structures. We use the more general monadic setting of [Sim92]; the details of the interpretation of FPC generalise more or less verbatim to this setting. Definition 1.1 A model of FPC is given by A1 4: A1. A category K with finite products and co products, both chosen. A2. A commutative strong monad ( on K. The Kleisli category of ( is denoted by ....

Alex K. Simpson. Recursive types in Kleisli categories. Unpublished paper, available electronically, August 1992.

....a restricted form of type expressions, which are guaranteed to have a corresponding functor. At this point we can introduce some constructions with universal properties, whose existence follows from algebraic compactness: the x type (introduce by [CP92] and a uniform x point combinator (see [Sim92]) 38 5.4.1 The x type The monad L extends to an endofunctor L 0 on the category CL of predomains and partial maps, namely L 0 f = Lf ; Y ; Y whenever f : X LY . Let L : L( be the free L 0 algebra in CL , then one can prove that it is also the free L algebra in C. In fact, ....

....L 0 algebra in CL , then one can prove that it is also the free L algebra in C. In fact, CP92] introduces the equivalent (but apparently weaker) notion of x type, which is enough for de ning a uniform x point combinator and prove the consistent algebraic compactness of CL and C L (see [Sim92]) LF signature extension for the x type types x type : Pdom operations : L I L : X : Pdom: LX X) X : we write y for I L (X; 0 for L ( and s for ; L axioms :1 X : Pdom; LX X) c : L : y ( c) L( y )c) X : Pdom; LX X) ....

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A.K. Simpson. Recursive types in kleisli categories. available via FTP from theory.doc.ic.ac.uk, 1992.

....p the two notions of replete coincide, so one gets the best of both. It avoids appeals to internal category theory, which many nd obscure In studying the replete construction, we have focused on those properties which are more interesting in relation to SDT , Axiomatic Domain Theory (see [Fre91, Sim92]) and Evaluation Logic (see [Mog94] 1.1 Summary of main de nitions and results This section summarizes the main de nitions and results, but for the sake of simplicity they are not stated in the most general form. In this section we x a category B with binary products, and 1 a cartesian ....

A.K. Simpson. Recursive types in kleisli categories. available via FTP from theory. doc.ic.ac.uk, 1992.

....a first step, we concentrate on the enrichment of models of ADT. The intention is that the enriched Yoneda Grothendieck Dedekind Cayley embedding [27] will provide the desired representation (c.f. 15, 11] Axiomatic versions of various traditional results in domain theory can be found in e.g. [39, 16, 17, 38, 13, 9, 11, 32]. For instance, in [39] the crucial role Research supported by EPSRC grant GR J84205, Frameworks for Programming Language Semantics and Logic. y Research supported by an EPSRC Senior Fellowship. of Cpo enrichment in the solution of recursive domain equations was recognised and made the central ....

A.K. Simpson. Recursive types in Kleisli categories. Manuscript (available from http://www.dcs.ed.ac.uk/home/als/), 1992.

....a T algebra, and a computation Gamma c M : B denotes a function from [ Gamma ] to the carrier of [ B] The Scott semantics of Sect. 3.1 can also be seen as an algebra semantics, because a domain corresponds to an algebra for the lifting monad on PreDom. This notion is exploited in [22, 35]. We illustrate this approach in the case of global store, where we use the strong monad S (S Theta Gamma) A closed computation c M : B will denote an element of X , where [ B] X; Now suppose we prefix to M a command such as X : 3. This immediately gives us an element of TX . ....

A. K. Simpson. Recursive types in Kleisli categories. Unpublished manuscript, 1992.

.... the role of lifting and the category Cpo by an arbitrary strong monad and its associated category of algebras thereby fitting these results into Moggi s [17] monadic approach to denotational semantics and the associated approach to program logic introduced by the author in [21] Simpson [29] studies the key properties of minimal invariance and uniformity of fixpoints in this setting, and their relationship to the notion of a fixpoint object for the monad in the sense of Crole and Pitts [3] see also Mulry [19] His results require a commutativity condition to hold of the monad. ....

A. K. Simpson, Recursive Types in Kleisli Categories, preprint, University of Edinburgh Department of Computer Science, July 1992.

....and recursion. This system did not provide a close integration of the two kinds of recursive type, and suffered from a reliance on the heavy machinery of fixpoint induction for reasoning about terms involving general recursion. Following recent work by Freyd and others on algebraic compactness [Fre90, Fre91, Fre92, Bar92, Sim92, Fio94], the more elegant solution presented in this paper was developed. In brief, Freyd showed how to reduce the problem of finding solutions for general recursive domain equations to that of building inductive types, provided the functors involved are algebraically bounded, i.e. the inductive and ....

.... do not want to assert that all corresponding least and greatest fixpoints are isomorphic; for instance, if there were any function from t: t to t: t then all terms at each type would be provably equal (given that we desire categorical finite sums as well as cartesian closure) Following Simpson [Sim92], we will identify the algebraically bounded functors by considering a faithful commutative strong monad whose functor T is algebraically bounded. Call a type pointed if it is the underlying object of an Eilenberg Moore T algebra, i.e. if there is a retraction ae : T ( for the unit j ....

Alex K. Simpson. Recursive types in kleisli categories. Available as kleisli.dvi.Z by anonymous ftp from ftp.dcs.ed.ac.uk in directory pub/als, August 1992.

....sense) for solving recursive type equations. There he introduced algebraically compact categories and established their fundamental property: that bifunctors on them have canonical and minimal fixed points. This has been a first important step towards an axiomatic theory of recursive types (see [Sim92] and [Fio94a, Chapters 6 8] Other work on algebraic compactness can be found in [Ad a93, Bar92] Concerning fixed points of endomorphisms, it was noticed by [HP90] after studying the work of [Law64, Law69] that in the presence of cartesian closure they are inconsistent with coproducts (empty ....

....f = ff. Roughly at the same time, Peter Freyd discovered the universal properties of the solutions of domain equations clearing the way for their full categorytheoretic treatment (see [Fre91, Fre92] This discovery spurred research toward an axiomatic presentation of categories of domains (see [Sim92, Fio94a, FP94] which encompassed that centered on O categories. Freyd s axiomatic presentation sets an important criterion about properties of functors in a model of SDT (see [Hyl91, Ros95] By pursuing the SDT approach in the setting of realizability toposes it is easy to model both ....

A. K. Simpson. Recursive types in Kleisli categories. Manuscript (available from http://www.dcs.ed.ac.uk/), 1992.

....F S A ffi Note that the construction of the adjunction f U a f F making the mentioned diagrams commute could have been done with an arbitrary distributive law of one comonad over another comonad. Now, with any object D in a cartesian category (D; Theta; 1) we may associate a comonad, [Sim92]; the functor part is given by D Theta ( Gamma) a component of the counit is given by D Theta B 2 B and a component of the comultiplication is given by D Theta B 1 ;Id D Theta (D Theta B) which is equal to the composite D Theta B Delta ThetaI d (D Theta D) Theta ....

A. Simpson. Recursive types in kleisli categories. Manuscript, 1992.

....formulation based on internal categories. All this is also described in the language of fibred category theory in a forthcoming book by Jacobs [34] For the axiomatic theory of domains, the already mentioned work by Freyd [19, 18, 20] on algebraically compact categories has been crucial. Simpson [73] generalized this work further in 1992 and Fiore [14] gave a comprehensive treatment of axiomatic domain theory in categories of partial maps in 1996. The work of Fiore, Plotkin and Power on complete cuboidal sets in one of the most recent (1997) works on axiomatic domain theory [16] and more ....

A.K. Simpson. Recursive types in kleisli categories. Unpublished manuscript, August 1992.

....Moggi 1991) All this works well for those type theories that type only total functions. But real programming languages allow arbitrary recursively defined functions and recursively defined datatypes, and consequently include programs that do not terminate. Traditionally such J.R. Longley and A.K. Simpson 2 features are modelled using the techniques of domain theory. In this paper we provide a uniform approach to modelling them in categories of modest sets. To do this, we identify appropriate structure for doing domain theory in such realizability models . In Sections 2 and 3 we introduce PCAs ....

....kxy x; sxyz xy(xz) sxy# : Note that, as x #, the Kleene equality of the first equation can be replaced by strict equality. Clearly a total PCA is just an ordinary combinatory algebra (Hindley and Seldin 1986) We now present some examples of both partial and total PCAs. J.R. Longley and A.K. Simpson 4 (i) The Kleene PCA) Consider the set N of natural numbers equipped with Kleene application: m Delta n fmg(n) where fmg denotes the partial recursive function coded by m under some standard enumeration. The existence of k; s with the required properties is an immediate consequence of the ....

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A.K. Simpson. Recursive types in Kleisli categories. Unpublished manuscript. Available by FTP from ftp.dcs.ed.ac.uk/pub/als/kleisli.ps.Z, 1992.

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A. Simpson. Recursive types in Kleisli categories. Unpublished manuscript, available from http://www.dcs.ed.ac.uk/home/als/, 1992.

....The research in the present paper constitutes a development of the techniques of synthetic domain theory. Nevertheless, our applications to axiomatically given classes of models demonstrate that our results should be viewed equally much as a contribution to the eld of axiomatic domain theory [9, 10, 39, 4, 3, 5, 6]. It is the author s view that embedding categories of predomains within models of intuitionistic set theory is the correct approach to obtaining an axiomatic account of domain theoretic constructions that applies uniformly across the di erent types of model. At present, it is the only known ....

....K(A; A) to be the unique strict function such that the diagram below commutes. We say that a is special F invariant if i z i = id A . The notion of special invariant object was rst introduced for cpo enriched categories in [8] A generalisation to an axiomatic setting appears in [39], from where the following result is taken. For completeness, we include a proof. Lemma 7.6 For any isomorphism FA A, the following are equivalent: A is special F invariant. A is an initial F algebra. 3. A FA is a nal F coalgebra. PROOF. Given any F algebra, c : FC C, de ....

A.K. Simpson. Recursive types in Kleisli categories. Unpublished manuscript, University of Edinburgh, 1992. 77

....algebra for F . Accordingly, let (B, l ( c ( be the specified bilimit of (F 0, x ( Define a morphism FB B by b = i (c si F l i ) Lemma 6.3 (B, b) is a bifree F algebra. The proof is by establishing that FB B is a special F invariant object in the sense of [6, 29], and that this property is characteristic of bifree F algebras, again see [6, 29] This concludes the proof of Proposition 3. We now complete the proof of Theorem 1 by establishing the result below. Proposition 4 If Axiom 1 holds then the internal category pP is suitable. The proof of this ....

....of (F 0, x ( Define a morphism FB B by b = i (c si F l i ) Lemma 6.3 (B, b) is a bifree F algebra. The proof is by establishing that FB B is a special F invariant object in the sense of [6, 29] and that this property is characteristic of bifree F algebras, again see [6, 29]. This concludes the proof of Proposition 3. We now complete the proof of Theorem 1 by establishing the result below. Proposition 4 If Axiom 1 holds then the internal category pP is suitable. The proof of this proposition is very long. In this conference version of the paper, we just state the ....

A.K. Simpson. Recursive types in Kleisli categories. Unpublished manuscript, University of Edinburgh, 1992.

....below commutes. I z ( # C(A, A) I s # z ( # C(A, A) a # F ( # a 1 # We say that a is special F invariant if # i z i = id A . The notion of special invariant object was first introduced for cpo enriched categories in [5] A generalisation to other enrichments appears in [22], from where the following result is taken. Proposition 8.6 For any isomorphism FA a # A, the following are equivalent: 1. FA a # A is special F invariant. 2. FA a # A is an initial F algebra. 20 3. A a 1 # FA is a final F coalgebra. Although our context here is mildly ....

....is taken. Proposition 8.6 For any isomorphism FA a # A, the following are equivalent: 1. FA a # A is special F invariant. 2. FA a # A is an initial F algebra. 20 3. A a 1 # FA is a final F coalgebra. Although our context here is mildly di#erent, one easily adapts the proofs in [5, 22]. Lemma 8.7 The isomorphism FB b # B is special F invariant. Proof Consider the diagram below. I c ( # l ( # C(B, B) I s # c ( # l ( # C(B, B) b # F ( # b # # By equation (b) of of Theorem 2(2) we have that # i c i # l i = id B . Thus, for b to be special F ....

A.K. Simpson. Recursive types in Kleisli categories. Unpublished manuscript, University of Edinburgh, 1992. 26

....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) ....

....on the category Cpo of, not necessarily pointed, complete partial orders. Axiomatically, we assume that C is a category with finite products, a monad embodying the equational properties of partial map classifiers (an equational lifting monad [4] partial function spaces (Kleisli exponentials [22, 28]) and a (parameterized) natural numbers object. These conditions are always satisfied by the categories of predomains that arise in axiomatic and synthetic domain theory [10, 12, 20, 11, 26, 30] Theorem 4 states that such categories support at most one uniform recursion operator (a T ....

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A. Simpson. Recursive types in Kleisli categories. Unpublished manuscript, available from http://www.dcs.ed.ac.uk/home/als/, 1992.

....in RC. This does not follow by elementary diagram chasing, but it does follow using an alternative equational characterization of free algebras as special invariant objects. These were introduced in the Cpo enriched case by Freyd [6] Our treatment, which fits into the general setting of [17], involves regarding C as an P enriched category and RC as a Q P enriched one. To emphasize the enrichment under consideration, we shall write P C for the P enriched version of C and Q P RC for the Q P enriched version of RC. The hom objects P C(A; B) and Q P RC(A;B) are the same realized ....

....So, by the uniformity of Qfix, it holds that (x ffi Gamma) ffi Qfix(ff ffi Phi( Gamma) ffi ff Gamma1 ) Qfix(fi ffi Phi( Gamma) ffi ff Gamma1 ) Whence, as ff is special Phi invariant, it follows that x = Qfix(fi ffi Phi( Gamma) ffi ff Gamma1 ) as required. This argument is from [17]. 5 j 6. Dual to 4 j 6. 1 j 2 j 3. By a similar proof to 4 j 5 j 6. Theta Part 1 of Theorem 6.3 follows easily. By earlier remarks, C has a free F algebra FA a A. So, by the proposition, PhiA R(a) A is a free Phi algebra in RC. It remains to prove part 2. Suppose that Psi : SD ....

A. K. Simpson. Recursive types in Kleisli categories. Available by FTP from ftp.dcs.ed.ac.uk/pub/als/kleisli.ps.Z, 1992.

....functions between domains, with respect to which the fixed point is characterised by the property of uniformity. Further, the monad determines a category of partial functions, which is arguably the most suitable category for program semantics. The development of this viewpoint can be found in [23, 2, 18, 29, 3]. In recent years it has become apparent that many natural categories of predomains arise as full subcategories of elementary toposes. For example, the category of complete partial orders and continuous functions is a full reflective subcategory of the Grothendieck topos, H, considered in [6, ....

A.K. Simpson. Recursive types in Kleisli categories. Unpublished manuscript. Available from ftp://ftp.dcs.ed.ac.uk/pub/als/Research/, 1992.

....axiomatic setting. Freyd already showed in [5] that a number of basic equational properties (such as dinaturality) hold. Here we provide a complete characterisation of all the valid equations. We work in a general categorical setting based on the authors previous work on axiomatic domain theory [10, 4]. We show that, under mild conditions, the induced fixed point operator endows the appropriate category with a unique well behaved parameterized fixed point operator (Theorem 3) By a general completeness result (Theorem 2) it follows that the axioms of iteration theories [1] are complete for ....

....of Theorem 3 is the requirement of the existence of many bifree algebras. We expect to prove a general theorem (Conjecture 1) implying their existence in familiar situations. Let C be a category with finite products, a natural numbers object N, and a commutative monad (T ; j; t) see e.g. [10]) satisfying the extra equation t ffi Delta = T hj; 1i : TX T (TX Theta X) true of all lifting monads) and such that C has all Kleisli exponentials (again see [10] Also assume given a family of functions ( Delta) C(TA; TA) C(1; TA) such that: for all f : TA TA, f ffi f = f ....

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A.K. Simpson. Recursive types in Kleisli categories. Department of Computer Science, University of Edinburgh, ftp.dcs.ed.ac.uk/pub/als/Research/kleisli.ps.Z, 1992.

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Sim92. A. Simpson, Recursive types in Kleisli categories. Manuscript, LFCS, University of Edinburgh, 1992.

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