Tatsuya Hagino. A typed lambda calculus with categorical type constructors. In D. H. Pitt, A. Poigne, and D. E. Rydeheard, editors, Category Theory and Computer Science, volume 283 of Lecture Notes in Computer Science, pages 140--  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Coit : 8F : G F G G F Elimination. out : 8F : F F ( F ) Reduction. out (Coit m s f t) m (Coit m s) f (s t) Syntactic in the sense of being a criterion on the shape of the constructor. This criterion is used in most the articles on inductive types [Hag87,Men87,Lei90,Geu92]. 5 Again, we require jf j = j j. Dually to the case of inductive constructors, de ne (m) Coit m out : monF mon ( F ) Hence, also monotonicity of F follows uniformly from monotonicity of F and has the desired computation rule: out (M (m) f t) m M (m) f (out ....

....types, these are just the usual de nitions, with arbitrary monotonicity witnesses (sometimes also called strength) instead of canonical ones for positive type transformers F . For the positive (covariant) case, their justi cation from the point of view of category theory has rst been given in [Hag87], a very good presentation of the ideas is to be found in [Geu92] 2.2 (Co)Inductive Functors If we specialize to kind 1, we get heterogeneous (non regular) and so called (non linear) nested datatypes. Prominent examples are powerlists [Hin00] and a monadic representation of de Bruijn terms ....

Tatsuya Hagino. A typed lambda calculus with categorical type constructors. In D. H. Pitt, A. Poigne, and D. E. Rydeheard, editors, Category Theory and Computer Science, volume 283 of Lecture Notes in Computer Science, pages 140-157. Springer Verlag, 1987.

....: a] a are in one to one correspondence with monoid structures on a. Algorithmically this property, also called First Homomorphism Theorem [8] for lists, states that any Eilenberg Moore algebra can be written in a fold of map structure. This result has been generalized to categorical data types [9,10], and is also known as the factorization of catamorphisms [11] For example, for one direction of the above correspondence, a monoid (a; 0) gives rise to an algebra [a] a satisfying (A I) and (A II) by setting malg = fold 0 : We refer to [12] for a detailed discussion of the recursion ....

....on di erent parts of a data structure. The usefulness of basic categorical properties in order to generalize results from Bird s theory for lists [8] has been shown by Spivey [9] Motivated by this observation language designs based on the use of categorical data types have been developed [10]. The use of categorical data types for specifying data parallel programs is discussed by Skillicorn [11] Concrete examples of using these properties to improve parallel performance via semantics preserving program transformation are given for example in [23,11] In a similar spirit, shapely ....

T. Hagino. A Typed Lambda Calculus with Categorical Type Constructors. In Category Theory and Computer Science, LNCS 283, pages 140-157. Springer, 1987.

....for algebraic (or inductive) types. 1 Introduction Category theory is a very convenient formalism for describing datatypes. In particular, the dual notions of initial algebra and final coalgebra provide an interesting class of datatypes. This possibility was first exploited by Hagino in [Hag87a][Hag87b], and the categorical datatypes introduced there have since been used as the basis of the functional programming language Charity [CS92] Initial algebras or term algebras provide algebraic or inductive datatypes, such as natural numbers, lists and trees. Final coalgebras provide coalgebraic ....

Tatsuya Hagino. A typed lambda calculus with categorical type constructors. In D.H. Pitt, A Poign'e, and D.E. Rydeheard, editors, Category and Computer Science, pages 140--157. Springer, September 1987.

....case study the Davis Putman procedure for testing validity of propositional formulae. Some work in progress in reported on section 5. The categorical view of datatypes, which underlies this research area, dates back to the ADJ group [5] and more recently, to the contributions of T. Hagino [7] and G. Malcolm [12] The relevance of universal properties to program derivation was first recognized by Backhouse in [3] References [14,13] and [16] introduce the recursion functionals discussed here. Reference [4] provides a tutorial introduction. 2 Data Modeling in Camila Camila has been ....

T. Hagino. A typed lambda calculus with categorical type constructors. In D. H. Pitt, A. Poign'e, and D. E. Rydeheard, editors, Category Theory and Computer Science, pages 140--157. Springer Lect. Notes Comp. Sci. (283), 1987.

.... this tradition seeks to show existence of nal transition systems, which give rise to an abstract notion of bisimulation and can be used to give a semantics for process algebras [2,5] Another thread views coalgebra as a variation on universal algebra [90] and applies it to functional programming [66,74,63], automaton theory [91,90] and the object paradigm [87,71 73,13] An interesting recent development combines algebra and coalgebra to describe denotational and operational semantics [96] Reichel [87] was the rst to apply coalgebra explicitly to the object paradigm, and his basic construction ....

Tatsuya Hagino. A typed lambda calculus with categorical type constructors. In D.H. Pitt, A. Poigne and D.E. Rydeheard, editors, Category Theory and Computer Science, pages 140-157. Lecture Notes in Computer Science, Volume 283. Springer, 1988.

....document that certain implications are not based on a well stated formal theory but require a certain amount of hand waving. 2 Background 2. 1 Categorical Datatypes Hagino introduces the concept of categorical datatypes in [Hag87] and uses them to propose an extension of the typed calculus in [Hag88]. In the following text we are using initial T algebras and terminal T coalgebras. Let us remember the definition of T algebras : Definition 1 (T algebras) Let C be a category and T : C C an endofunctor, the category of T algebras has 1. as objects : pairs hA : jCj; f : F (A) Ai; 2. as ....

....approach systematically in [BB85] in an algebraic context and prove a completeness result, which says that the representations are 6 really the initial algebras. This does not imply initiality for all T algebras, because these are more general. 12 In [Wra89] a more categorical view based on [Hag88] is taken. Wraith shows how certain categorical concepts can be translated into the 2nd order calculus 13 . So it can be shown that for every closed term F : a term called functorial strength : hF i : PiS; T : S T ) FS) FT ) can be derived by induction over the structure of F ....

Tatsuya Hagino. A typed lambda calculus with categorical type constructors. LFCS Report ECS-LFCS-88-44, Laboratory for Foundations of Computer Science, January 1988. 32

....that satisfy the following equations. acc 0 [ g acc 0 (x : xs) x 8 xs 3. Re express acc as acc xs = g 1 ; 111;g n ] xs ac The correctness of the above transformation can be easily proved by induction on the construction of the type T based on the theory of Malcolm [11] and Hagino [8]. The difficulty for such transformation is to find those g i s that meet the requirement. In BMF, this can be done by calculation. Let us see some examples. Example 3.1 Considering a function computing the initial sum of a list, we can define it naively as follows. isum [ isum (x : xs) ....

T. Hagino, A typed lambda calculus with categorical typeconstructors, Category Theory and Computer Science (D.H. Pitt, A. Poign'e, and D.E. Rydeheard, eds.), Springer Lecture Notes in Computer Science, 283, pp. 140--157.

....not confluent. We should point out that the nonconfluency does not have any effect on the results in section 2.4. 2 Inductive Types in Polymorphism There are two ways to have inductive types in polymorphism. One is to postulate rules for these types in F 2 . This external approach is adopted in [13, 10, 11, 14]. The other approach is to internally code up the inductive types in F 2 . The internal method is weaker but by no means less interesting. 2.1 Weakly Initial I algebras in Polymorphism In [10, 11] the author studies a categorical language based on the following construction (and its dual) that ....

....One is to postulate rules for these types in F 2 . This external approach is adopted in [13, 10, 11, 14] The other approach is to internally code up the inductive types in F 2 . The internal method is weaker but by no means less interesting. 2. 1 Weakly Initial I algebras in Polymorphism In [10, 11] the author studies a categorical language based on the following construction (and its dual) that generates new data types. Let P (X) be an n tuple [P 1 (X) Delta Delta Delta ; P n (X) 3 where X occurs positively in P 1 (X) Delta Delta Delta ; P n (X) We may declare a new type as ....

T. Hagino. A typed lambda calculus with categorical type constructions. In D. Pitt, A. Poign'e, and D. Rydeheard, editors, Category Theory and Computer Science, LNCS 283, pages 140--157. Springer-Verlag, 1987.

....1 Introduction Folds are appreciated by functional programmers. The benefits of encapsulating common patterns of computation as higher order operators instead of using recursion directly are well known and well understood [14] The dual notion to folds, unfolds, have been explored by Hagino [10] and Malcolm [16] and popularized at this conference by Meijer et al. [18] Unfolds are certainly not new, but they are not nearly as well appreciated as folds. For example, they merit just half a page in [4] and have disappeared altogether in [3] Co inductive types warrant a few pages in [23] ....

....promotion: if h = foldr op e . map f where op is associative with identity e, then h (xs ys) h xs op h ys fold concat promotion: if h = foldr op e . map f where op is associative with identity e, then h . concat = foldr op e . map h 2. 2 Unfolds over lists We also use unfolds [10, 16, 18], a dual to folds. The standard construction of unfolds [16] gives the characterization unfold : b Either ( a,b) b [a] unfold pfg x = case pfg x of Left ( Right (a,y) a : unfold pfg y but we will find it more convenient to use the equivalent characterization unfold : ....

[Article contains additional citation context not shown here]

Tatsuya Hagino. A typed lambda calculus with categorical type constructors. In D. H. Pitt, A. Poign'e, and D. E. Rydeheard, editors, LNCS 283: Category Theory and Computer Science, pages 140--157. Springer-Verlag, September 1987.

....could be substantially greater if we could lift the level of abstraction one level higher and view all the so called regular datatypes as instances of just one construction. A framework for doing so is evident in Lambek s work on subequalizers [20] which later got the name dialgebra [9], the name we shall use in this paper. Here we begin an initial exploration of dialgebras as the basic building block of generic programs. Now at Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands. We are concerned with the development of a relational as ....

T. Hagino. A typed lambda calculus with categorical type constructors. In D.H. Pitt, A. Poigne, and D.E. Rydeheard, editors, Category Theory and Computer Science, pages 140--57. Springer-Verlag Lecture Notes in Computer Science 283, 1988.

....illustrate the importance and usefulness of coalgebraic data types more generally, and that it will have an influence on future languages for programming and reasoning. When these languages are equipped with definition mechanisms both for algebraic and for coalgebraic data types (as suggested in [Hag87] and realised in the experimental programming language charity [CF92, CS95] then one can use (and reason about) coalgebraic data structures like trees with infinitely many branches, and possibly infinite depth with the same ease as algebraic structures like finitely branching trees. What is ....

....are not familiar with coalgebraic datatypes. In order to put our approach in a wider context, we indicate in this section how it forms part of a more general theory of coalgebraic datatypes. We will sketch a possible syntax for such coalgebraic datatypes based on the approach suggested first in [Hag87] and implemented in charity [CF92, CS95] but put in a form adapted to pvs. Additionally, we sketch how to formulate associated proof principles with invariants and bisimulations for such general coalgebraic datatypes, following [HJ95, HJ98, HJ97] What we will say will be formulated for logical ....

T. Hagino. A typed lambda calculus with categorical type constructors. In D.H. Pitt, A Poign'e, and D.E. Rydeheard, editors, Category and Computer Science, number 283 in Lect. Notes Comp. Sci., pages 140--157. Springer, Berlin, 1987.

....data types [7] The calculi surrounding these types have their foundations in category theory. A useful BMF theory can be created for any categorical data type (CDT) that can be shown to adhere to a certain set of constraints. These constraints are not difficult to satisfy in practice (see [17] for details) CDTs for which BMF theories exist include, lists, sets, bags, trees and multi dimensional arrays [18, 23] The advantage of deriving a theory for a CDT is that a number of very useful tools are created merely from the salient properties of that type. Firstly, a small number of ....

Tatsuya Hagino. A typed lambda calculus with categorical type constructors. In Category Theory and Computer Science: Edinburgh U.K., pages 140--157. Springer-Verlag, September 1987. Appears in, Lecture Notes in Computer Science Vol 283.

....co induction, traversal, breadth first, level order. 1 Introduction Folds are appreciated by functional programmers; the benefits of encapsulating common patterns of computation as higher order operators are well known and well understood. Their dual, unfolds, have been explored by Hagino [8] and Malcolm [12] and popularized at this conference by Meijer et al. [14] Unfolds are now nearly as well known as folds, but they are not nearly as well appreciated. For example, they merit just half a page in [4] and have disappeared altogether in [3] Co inductive types warrant a few pages ....

....f fold join promotion: if h = foldr op e . map f where op is associative with identity e, then h (xs ys) h xs op h ys fold concat promotion: if h = foldr op e . map f where op is associative with identity e, then h . concat = foldr op e . map h 2. 2 Unfolds over lists We also use unfolds [8, 12, 14], a dual to folds. The standard construction of unfolds [12] gives the characterization unfold : b Either ( a,b) b [a] unfold pfg x = case pfg x of Left ( Right (a,y) a : unfold pfg y but we will find it more convenient to use the equivalent characterization unfold : ....

[Article contains additional citation context not shown here]

Tatsuya Hagino. A typed lambda calculus with categorical type constructors. In D. H. Pitt, A. Poign'e, and D. E. Rydeheard, editors, LNCS 283: Category Theory and Computer Science, pages 140--157. Springer-Verlag, September 1987.

....Polynomial functors will be useful for the construction of datatypes, as detailed in the next section. An example of a non polynomial functor is P. 2.3. 2 Initial Datatypes and Catamorphisms The idea of using initiality to represent datatypes has been known for many decades, although Hagino [35, 36] and Malcolm [65] brought the idea into more prominence. More details about the ideas briefly mentioned here may be found in Fokkinga [32] for example. If we have a functor F : C C, then any arrow fi in the category C which is of type fi : B FB (for some object B) is an F algebra. A category ....

T. Hagino. A typed lambda calculus with categorical type constructors. In D.H.Pitt, A. Poigne, and D.E.Rydeheard, editors, Category Theory and Computer Science, number 283 in LNCS, pages 140--157. Springer-Verlag, 1988.

....systems, but rules for nondeterminacy and for stream types are also introduced. Among them, a kind of structural induction on stream types called (MPST ) is the heart of our extended system: With (MPST ) stream transformers can be de ned as Burge s mapstream functions [Bur75] T. Hagino [Hag87] gave a clear categorical formalization of stream types (in nite list types or lazy types) whose canonical elements are given by a schema of mapstream functions, but relation between his formulation and logic is not investigated. N. Mendler and others [PL86] introduced lazy types and the type ....

....cation over I oe . 3.1 Two Level Stream Types A stream can be viewed at least in three ways: an in nite list, an in nite process, and an output sequence of an in nite process, namely, a total function on natural numbers. The formal theories of lazy functional programming such as [PL86] and [Hag87] can be regarded as the theories of concurrent functional programming based on the rst two points of view on streams. Our system uses a lazy typed lambda calculus as the underlying programming language and has lazy types as computational stream types. Computational stream types are only used as ....

T. Hagino. A Typed Lambda Calculus with Categorical Type Constructors. In Category Theory and Computer Science, LNCS 283, 1987.

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Tatsuya Hagino. A typed lambda calculus with categorical type constructors. In D. H. Pitt, A. Poigne, and D. E. Rydeheard, editors, Category Theory and Computer Science, volume 283 of Lecture Notes in Computer Science, pages 140--

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T. Hagino, A typed lambda calculus with categorical type constructors, in: D. H. Pitt, A. Poigne, D. E. Rydeheard (Eds.), Category Theory and Computer Science, Vol. 283 of Lecture Notes in Computer Science, Springer-Verlag, 1987, pp. 140--157.

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T. Hagino. A typed lambda calculus with categorical type constructors. In D.H. Pitt and D.E. Rydeheard, editors, Category Theory and Computer Science, number 283 in Lect. Notes in Comp. Sc., pages 140--157. Springer Verlag, 1988.

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Tatsuya Hagino. A Typed Lambda Calculus with Categorical Type Constructors. In Category Theory and Computer Science, Lecture Notes in Computer Science 283, eds. D. H. Pitt, A. Poing'e, and D. E. Rydeheard, pages 140--157. September 1987.

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Tatsuya Hagino. A typed lambda calculus with categorical type constructors. In D. H. Pitt, A. Poigne, and D. E. Rydeheard, editors, Category Theory and Computer Science, volume 283 of Lecture Notes in Computer Science, pages 140--

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Tatsuya Hagino. A typed lambda calculus with categorical type constructors. In D. H. Pitt, A. Poigne, and D. E. Rydeheard, editors, LNCS 283: Category Theory and Computer Science, pages 140--157. Springer-Verlag, September 1987.

No context found.

T. Hagino. A typed lambda calculus with categorical type constructors. In D. H. Pitt, A. Poigne, and D. E. Rydeheard, editors, Category Theory and Computer Science, pages 140-157. Springer Lect. Notes Comp. Sci. (283), 1987.

No context found.

T. Hagino. A typed lambda calculus with categorical type constructors. In D. H. Pitt, A. Poigne, and D. E. Rydeheard, editors, Category Theory and Computer Science, volume 283 of Lecture Notes in Computer Science, pages 140-157. Springer, 1987.

No context found.

Tatsuya Hagino. A typed lambda calculus with categorical type constructors. In D. H. Pitt, A. Poigne, and D. E. Rydeheard, editors, Category Theory and Computer Science, volume 283 of Lecture Notes in Computer Science, pages 140-157. Springer Verlag, 1987.

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Hagino, T. (1987b). A typed lambda calculus with categorical type constructors. Pages 140-157 of: Pitt, D. H., Poigne, A, & Rydeheard, D. E. (eds), Category and Computer Science. Lecture Notes in Computer Science, vol. 283. Springer.

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