R. L. Crole. Categories for Types. Cambridge University Press, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and hyperdoctrines respectively. The presentation is aligned to the game models of the next chapter, which will be instances of the framework presented here. The results build upon research into categorical semantics of Intuitionistic Linear Logic [19, 86, 20, 76] and second order polymorphism [112, 106, 65, 68, 31]. 5.1 Autonomous categories We begin with IMLL whose proofs can be interpreted as morphisms in symmetric monoidal closed categories (also called autonomous or closed) 87] In fact, IMLL cut free proofs (up to congruence) represent morphisms in free such categories [86, 76] De nition 5.1. A ....

....and functors preserving (all elements of) their structure in a strict way (not only up to isomorphism) 5. 7 Type variables and universal quanti cation The interpretation of second order variables necessitates the availability of still more complicated categorical objects as outlined, e.g. in [31] for System F. We sketch the desired structure adapting it to our case as appropriate. First we show how to interpret types possibly containing (free) type variables. The basic idea is that each second order type with n distinct free variables is a description of an n ary operation on some ....

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R. L. Crole. Categories for Types. Cambridge University Press, 1993.

.... axiomatics of the situation, see [AHS02] 6 The Model We shall use the hyper doctrine formulation of model of System F, as originally proposed by Seeley [See87] based on Lawvere s notion of hyperdoctrines [Law70] and simpli ed by Pitts [Pit88] a good textbook presentation can be found in [Cro93]. We begin with a key de nition: G U (k) Sub(U) G(k) where U is the universe of System F types constructed in Section 6. 6.1 The Base Category We rstly de ne a base category B . The objects are natural numbers. A morphism n m is an m tuple hA 1 ; Am i; A i 2 G U (n) 1 i ....

R. Crole, Categories for Types, Cambridge University Press, 1993.

....with the p property has a terminal semantics iff the functor Spec: BEHA THEOP has a right adjoint. proof: This is a consequence of a general result of category theory a functor F:C D has a right adjoint iff for every object B of D there is a terminal object in the category (FSB) See [Crole 93] for a proof. 1 Corollary 4.5: A O institution with the p property has a terminal semantics iff Beha is a right adjoint of Spec. 1 These results on adjointness and exactness (3.9) which generalise [Fiadeiro and Costa 93] are one of the motivating forces behind the construction of ....

R.Crole, Categories for Types, Cambridge University Press 1993

....features in the same way that PCF has been studied as a prototypical functional language. 2 Models for Idealized Algol What should a model for Idealized Algol look like Since the language is an applied simply typed calculus, we should expect to model it in a cartesianclosed category C [9]. To accommodate the recursion in the language, we can ask for C to be cpo enriched [5] or more minimally, to be rational [2] This says that C is enriched over pointed posets, and that the least upper bounds of chains ff k ffi j k 2 g of iterated endomorphisms f : A A exist in a suitably ....

R. Crole. Categories for Types. Cambridge University Press, 1994.

....the monad ( Delta; JffiP K = I ffiI Gamma ffi I ffiJP K Gamma ffi A J DeltaP K = I DeltaI Gamma DeltaI DeltaJP K Gamma DeltaA i Here t A is the morphism A I defined by t A = f( a; t A ) j a 2 Sigma A g. 4. 5 Guarded Recursion By standard techniques [16], we can interpret any SCCS term P (X) containing a process variable X as a morphism JP K : A Gamma A: An interpretation of the recursive term rec X: P is provided by a morphism p : I Gamma A such that ( A ( Ap I Omega (JP K) der I ; p) unit If X is guarded in P , ....

R. L. Crole. Categories for Types. Cambridge University Press, 1994.

....functional computation, for both theoretical and practical reasons. On the foundational side there are elegant connections between the typed calculus, intuitionistic logic and cartesian closed categories, leading to the Propositions as Types paradigm [14] and the development of categorical logic [9,17]. From a practical point of view, compile time type reconstruction is a boon to the programmer in languages such as Standard ML and Haskell. Turning to concurrency, the situation is much less satisfactory. There is no generally accepted foundation for typed concurrent programming, and even a ....

.... GammaK = JQ GammaK; as = and coincide in SProc. 6 Categorical Logic A significant aspect of the theory of the typed calculus is the close connection between syntax and semantics as formalised by the construction of syntactic categories and the proof of various correspondence theorems [9, 17]. Some progress has been made towards a similar connection for interaction categories. The idea is to present a process theory as a process signature together with a collection of axioms, which are expressions of the form P = Q Gamma with P Gamma and Q Gamma proved processes. There is ....

R. L. Crole. Categories for Types. Cambridge University Press, 1994.

....[12] a system for the specification and formal development of software. It has a rigorous mathematical foundation, based on logic and category theory. It provides an ordersorted higher order logic representation language called Slang, whose semantics are founded on categorical type theory [3, 10]. It includes a rich set of primitives for composing specifications by reusing and parameterizing one or more copies of other specifications. The user specifies what and how various component specifications are to be included in the whole, and the colimit operation from category theory is used to ....

R.L. Crole. Categories for Types. Cambridge University Press, 1993.

....proving equalities between definable functions. It is well known that Lawvere s elegant definition of a natural numbers object, which works very well in cartesian closed categories, is not powerful enough in categories with weaker structure. Instead, a modified parameterized definition is needed [21, 5]. In a category with finite products, the notion of parameterized natural numbers object supports the definition of functions by primitive recursion. Moreover, in a cartesian closed category, any ordinary (Lawvere) natural numbers objects is automatically parameterized. Much the same situation ....

R.L. Crole. Categories for Types. Cambridge University Press, Cambridge, 1993.

....object in Set. 4 Parameterized interval objects It is well known that Lawvere s elegant definition of a natural numbers object, which works very well in cartesian closed categories, is not powerful enough in categories with weaker structure. Instead, a modified parameterized definition is needed [21, 7]. In a category with finite products, the notion of parameterized natural numbers object supports the definition of functions by primitive recursion. Moreover, in a cartesian closed category, any ordinary natural numbers objects is automatically parameterized. Much the same situation arises for ....

R.L. Crole. Categories for Types. Cambridge University Press, Cambridge, 1993.

....functional computation, for both theoretical and practical reasons. On the foundational side there are elegant connections between the typed calculus, intuitionistic logic and cartesian closed categories, leading to the Propositions as Types paradigm [14] and the development of categorical logic [9,17]. From a practical point of view, compile time type reconstruction is a boon to the programmer in languages such as Standard ML and Haskell. Turning to concurrency, the situation is much less satisfactory. There is no generally accepted foundation for typed concurrent programming, and even a ....

.... GammaK = JQ GammaK; as = and coincide in SProc. 6 Categorical Logic A significant aspect of the theory of the typed calculus is the close connection between syntax and semantics as formalised by the construction of syntactic categories and the proof of various correspondence theorems [9, 17]. Some progress has been made towards a similar connection for interaction categories. The idea is to present a process theory as a process signature together with a collection of axioms, which are expressions of the form P = Q Gamma with P Gamma and Q Gamma proved processes. There is ....

R. L. Crole. Categories for Types. Cambridge University Press, 1994.

....specification, synthesis, and maintenance. In addition, we give an industrial perspective on what is needed to make this technology have broader appeal to industry. We begin with some formal preliminaries and a brief discussion of Specware TM. 2. Formal Preliminaries The field of category theory [4,10,12] provides a foundational theory. This theory was applied to systems theory and systems engineering [3,6] This theory was embodied in the software development tool Specware TM [14,16,17] 2.1. Category of Signatures A signature consists of the following: 1. A set S of sort symbols 2. A triple ....

Crole, Roy, Categories for Types, Cambridge University Press, 1993.

.... T : N Gamma B that sends the arrow ( X; Phi 1 ] X ; Phi k ] in N(n; k) to the arrow Omega t n [X ; Phi 1 ] t n [X ; Phi k ] ff in B (n; k) Each ccc representation t n : Pn Gamma Fn is defined by induction on the structure of terms using the ccc structure of Fn (see [Cro93] for details) 3. Sconing with iml categories The construction of sconing with iml categories is facilitated by introducing certain functors Delta n k that lie conceptually close to the notion of implicit polymorphism: Each type f = X ; Phi] in Ob(Pn ) induces functors Delta n k f : Ob(P ....

R. Crole, Categories for Types, Cambridge University Press, 1993.

....with the p property has a terminal semantics iff the functor Spec: BEHATHEO op has a right adjoint. proof: This is a consequence of a general result of category theory a functor F:CD has a right adjoint iff for every object B of D there is a terminal object in the category (FB) See [Crole 93] for a proof. z Corollary 4.5: A b institution with the p property has a terminal semantics iff Beha is a right adjoint of Spec. z These results on adjointness and exactness (3.9) which generalise [Fiadeiro and Costa 93] are one of the motivating forces behind the construction of ....

R.Crole, Categories for Types, Cambridge University Press 1993

....combinators Y T . The base type nat is interpreted by f N , the constants n by n, n 2 N, and Omega by f N . The constant case k is interpreted by der f N ; f i j i 2 N] where f i = k ( i ) 0 i k ; i k: This interpretation is extended to all terms in the standard way [Cro94]. To accommodate recursion, we need another definition. Let K be a cartesian closed category. A fixpoint operator on K is a family of maps ( r A : K(A;A) Gamma K(1; A) satisfying f ffi f r = f r : Given such an operator, we can interpret the fixpoint combinator YA : 1 (A ) A) A ....

R. Crole. Categories for Types. Cambridge University Press, 1994.

....faithful functor C oe C S . The above example of structure on C is illustrative. Exactly similar definitions can be given for a range of structures, including: ffl models of classical (or intuitionistic) linear logic including the additives and exponentials [13] ffl cartesian closed categories [20] ffl models of polymorphism [20] 5 2.1 Examples of Specification Structures In each case we specify the category C , the assignment of properties P S to objects and the Hoare triple relation. 1) C = Set , P S X = X, affgb def , f(a) b. In this case, C S is the category of pointed ....

....above example of structure on C is illustrative. Exactly similar definitions can be given for a range of structures, including: ffl models of classical (or intuitionistic) linear logic including the additives and exponentials [13] ffl cartesian closed categories [20] ffl models of polymorphism [20]. 5 2.1 Examples of Specification Structures In each case we specify the category C , the assignment of properties P S to objects and the Hoare triple relation. 1) C = Set , P S X = X, affgb def , f(a) b. In this case, C S is the category of pointed sets. 2) C = Rel , P S X = fg, fRg ....

R. L. Crole. Categories for Types. Cambridge University Press, 1994.

.... Realizability Samson Abramsky 1 Introduction Realizability has proved to be a fruitful approach to the semantics of computation, see e.g. [AL91, Cro93, Lon95, AC98]. The scope of realizability methods has been limited to Intuitionistic Logic (with some extensions to Classical Logic) on the logical side, and to functional computation on the computational side. Our aim in the present work is to explore the possibilities for broadening the scope of ....

....and process algebra, on the one hand, and type theory, categorical models and realizability on the other. Background in process algebra may be found in standard texts such as [Hoa85, Hen88, Mil89, Ros97] while background in realizability, categorical models etc. is provided by texts such as [GLT89, AL91, Cro93, AC98, BW99]. A modest background in either or both of these fields should suffice to understand the main ideas. Most of the detailed verification of properties of the formal definitions we will present is left as a series of exercises. The diligent reader who attempts a number of these should get some ....

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R. L. Crole. Categories for Types. Cambridge University Press, 1993.

....of isomorphism. 4.5 The Categorical Type Theory Correspondence Our aim here is to sketch a correspondence between autonomous type theories and autonomous categories of a kind that is well known to exist between simply typed calculi and cartesian closed categories. For a careful account, see [8] and [16] Let T = h Sg; Ax i be a autonomous theory, C a autonomous category, and M and N models of T in C . A homomorphism h : M Gamma N of models of T in C is given by a collection h a : a ] M Gamma [ a ] N of isomorphisms in C for every base type a of Sg, such that the ....

R. L. Crole. Categories for Types. Cambridge University Press, 1993.

.... Theta ThetaB . Soundness and completeness A axiom system is specified by a pair h Sig; Ax i where Sig is a set of signatures and Ax a set of equations in context regarded as axioms. The precise sense of soundness and completeness in the context of categorical type theory can be found in e.g. [7]. Theorem 3.3 (Soundness) For any category 0 B B B E # B 1 C C C A , and for any interpretation of any axiom system h Sig; Ax i in the category, the same interpretation satisfies every equation in context that is a theorem of the axiom system. Theorem 3.4 (Completeness) The ....

R. L. Crole. Categories for Types. Cambridge University Press, 1993.

....operations and constructing their fixpoints. The actual definition of the guarded operations and the construction of their fixpoints are in section 4. 2 Inductive operations 2.1 Polynomial categories Let S be fixed as a cartesian closed category. 2 The well known polynomial construction [8, 12], yields the free cartesian closed category S[ generated by S and a formal arrow : A Gamma B, where A and B are arbitrary objects of S. It comes with a structure preserving functor I : S Gamma S[ and a distinguished arrow 2 S[ IA; IB) They are universal for all such pairs, ....

R. Crole, Cateories for Types. (Cambridge University Press 1994)

....where P Gamma and Q Gamma. We then require the equational theory over proved process terms and the set of equations in context Ax so that the free category will generate a autonomous category with an endofunctor fl. category. Since the way these rules are generated they will be omitted see [3, 4]. Finally the set of transitions in conext, Tran a set of transitions of the form: P : ff 1 ; ff n 1 ; n Gamma P 0 : fi 1 : fi n Such that each P ff 1 ; ff n and P fi 1 ; fi n are processes symbols and the prefix judgments Prefix 1 : ff fi , ....

R.L. Crole. Categories for types. Cambridge University Press, 1993.

....the variables occurring free in the term) and returns a denotation of the term as an element of the appropriate domain. Following the style of denotational semantics, the valuation function should be defined compositionally. Models of the simply typed calculus 4 are cartesian closed categories [13, 4]. The interpretation is standard: given a cartesian closed category, types are interpreted as objects, and terms as maps. An important feature of PCF is recursion in the form of a fixed point operator at every type. Here we appeal to Knaster Tarski s Fixed point Theorem [30] to construct the fixed ....

R. L. Crole. Categories for Types. Cambridge University Press, 1993.

....models will not make the distinction between context concatenation and the tensor connective that we are aiming at with L categories. We proceed in the same sequence as the definition of the categorical model. This is fairly traditional in categorical type theory, for a textbook example see Crole [Cro94]. We only define the structure involved and (mostly) omit the (lengthy, but routine) verification that the structure has the required properties. We start by defining the functor L: B op Set T . The objects of B are contexts Gammaj Delta and the morphisms of B are given by substitution ....

Roy L. Crole. Categories for Types. Cambridge University Press, 1994.

....1 ) ff M 1 (k) where k FV(M ) 3 Domain theoretic semantics of simply typed calculi In this section we shall discuss the domain theoretic semantics of simply typed lambda calculi in general. Although the constructions below are standard (see e.g. the books of Lambek Scott [10] or Crole [6]) we discuss them in some detail in order to make the paper accessible also for readers not familiar with this subject. Most constructions make sense in an arbitrary cartesian closed category (ccc) However we will confine ourselves to the domain semantics and will only occasionally comment on ....

Roy L. Crole. Categories for Types. Cambridge University Press, 1993.

....features in the same way that PCF has been studied as a prototypical functional language. 2 Models for Idealized Algol What should a model for Idealized Algol look like Since the language is an applied simply typed calculus, we should expect to model it in a cartesianclosed category C [9]. To accommodate the recursion in the language, we can ask for C to be cpo enriched [5] or more minimally, to be rational [2] This says that C is enriched over pointed posets, and that the least upper bounds of chains ff k ffi j k 2 g of iterated endomorphisms f : A A exist in a suitably ....

R. Crole. Categories for Types. Cambridge University Press, 1994.

.... 2 ] I[ 1 ] I[ 2 ] 3 Relating the type theories In this section we relate the various type theories that interest us in the paper. We do this by relating their associated categorical free constructions using a generalisation of a glueing method due to Lafont [20, Annexe C] see also [9, 31]. x : x : hi : 1 t i : i (i = 1; 2) ht 1 ; t 2 i : 1 2 i (t) i ; x : 1 t : x : 1 : t : 1 t : 1 t 1 : 1 t(t 1 ) t) t : i i (t) 1 2 t : 1 2 ; x i : i t i : i = 1; 2) ....

R. Crole. Categories for Types. Cambridge University Press, 1994.

....faithful functor C oe C S . The above example of structure on C is illustrative. Exactly similar definitions can be given for a range of structures, including: ffl models of Classical (or Intuitionistic) Linear Logic including the additives and exponentials [10] ffl cartesian closed categories [15] ffl models of polymorphism [15] 2.1 Examples of Specification Structures In each case we specify the category C , the assignment of properties P to objects and the Hoare triple relation. 1. C = Set; PX = X; affgb j f(a) b: In this case, C S is the category of pointed sets. 2. C = Rel ; ....

....above example of structure on C is illustrative. Exactly similar definitions can be given for a range of structures, including: ffl models of Classical (or Intuitionistic) Linear Logic including the additives and exponentials [10] ffl cartesian closed categories [15] ffl models of polymorphism [15]. 2.1 Examples of Specification Structures In each case we specify the category C , the assignment of properties P to objects and the Hoare triple relation. 1. C = Set; PX = X; affgb j f(a) b: In this case, C S is the category of pointed sets. 2. C = Rel ; PX = X; SfRgT j 8x 2 S:fy j ....

R. L. Crole. Categories for Types. Cambridge University Press, 1994.

....satisfactory denotational models fully complete for the whole class of ML types. This paper consists of three parts. In the first part, we provide an axiomatization of models fully complete for ML types. This axiomatization is given on the models of system F which are called hyperdoctrines ([Cro93]) As in [Abr97] our axiomatization also works in the context of adjoint models. It consists of two crucial steps. First, we axiomatize the fact that every morphism f : 1 # [ T ] where T is an ML type generates, under decomposition, a possibly infinite typed Bohm tree. Then, we introduce an ....

....computations in the simply typed # calculus can be handled just by operations strategies which do nothing more than copying information, without producing any new result. 7 2 Models of System F We focus on hyperdoctrine models of system F. First, we recall the notion of 2# hyperdoctrine (see [Cro93]) This essentially corresponds to the notion of external model (see [AL91] Then, we give the formal definition of full (and faithful) complete hyperdoctrine model. Finally, we carry out a linear analysis of the notion of 2# hyperdoctrine. This will allow us to express conditions which guarantee ....

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R.Crole, Categories for Types, Cambridge University Press, 1993.

....satisfactory denotational models fully complete for the whole class of ML types. This paper consists of three parts. In the first part, we provide an axiomatization of models fully complete for ML types. This axiomatization is given on the models of system F which are called hyperdoctrines ([Cro93]) As in [Abr97] our axiomatization also works in the context of adjoint models. It consists of two crucial steps. First, we axiomatize the fact that every morphism f : 1 [ T ] where T is an ML type generates, under decomposition, a possibly infinite typed Bohm tree. Then, we introduce an ....

....in the simply typed calculus can be handled just by operations strategies which do nothing more than copying information, without producing any new result. 7 2 Models of System F We focus on hyperdoctrine models of system F. First, we recall the notion of 2 Theta hyperdoctrine (see [Cro93]) This essentially corresponds to the notion of external model (see [AL91] Then, we give the formal definition of full (and faithful) complete hyperdoctrine model. Finally, we carry out a linear analysis of the notion of 2 Theta hyperdoctrine. This will allow us to express conditions which ....

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R.Crole, Categories for Types, Cambridge University Press, 1993.

....of the model w.r.t. ML types. In particular, we introduce a categorical notion of adjoint hyperdoctrine. Adjoint hyperdoctrines arise as co Kleisli indexed categories of linear indexed categories. In what follows, we assume that all indexed categories which we consider are strict (see e.g. [AL91,Cro93] for more details on indexed categories) Definition 2 (2 Theta hyperdoctrine, Law70,Pit88] A 2 Theta hyperdoctrine is a triple (C; G;8) where: C is the base category, it has finite products, and it consists of a distinguished object U which generates all other objects using the product ....

.... ; Xm T ] U m U : Well typed terms, i.e. X 1 ; Xm ; x 1 : T 1 ; xn : Tn M : T , are interpreted by morphisms in the category G(U m ) X 1 ; Xm ; x 1 : T 1 ; xn : Tn M : T ] X T 1 ] Theta: Theta[ X Tn ] X T ] See e.g. [Cro93] for more details. Definition 3 (Full and Faithful Completeness) Let M = C; G;8; be a 2 Theta hyperdoctrine. M is fully and faithfully complete w.r.t. the class of closed types T if, for all T 2 T , 8f 2 Hom G(1) 1; T ] 9( fij normal form M: M : T f = M : T ] Before ....

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R.Crole, Categories for Types, Cambridge University Press, 1993.

....information between input and output ports. The proof of full completeness consists in showing that this model satisfies the axioms in the axiomatization of fully complete models for ML types given in [AL99a] This axiomatization is given on the models of system F which are called hyperdoctrines ([Cro93]) In particular, it works in the context of adjoint models. It consists of two main steps. The first is an axiomatization of the fact that every morphism f : 1 [ T ] where T is an ML type generates, under decomposition, a possibly infinite typed Bohm tree. Then, an axiom which rules out ....

....apart normal forms from unsolvable terms. Corollary 2. In any theory satisfying Typical Ambiguity on 1 , a term in whose normal form appears cannot be equated to a term, in whose normal form does not appear. 2 Models of System F We recall first the notion of 2 Theta hyperdoctrine (see [Cro93]) This essentially corresponds to the notion of external model (see [AL91] Then, we give the formal definition of fully (and faithfully) complete hyperdoctrine model. Finally, we define the categorical notion of adjoint hyperdoctrine, on which the axiomatization of full completeness at ML types ....

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R.Crole, Categories for Types, Cambridge University Press, 1993.

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R. L. Crole. Categories for Types. Cambridge University Press, 1993.

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R. Crole. Categories for Types. Cambridge University Press, 1993.

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Crole, Roy, Categories for Types, Cambridge University Press, 1993.

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, Module categories of analytic groups, Cambridge University Press, 1982.

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