Tatsuya Hagino. A Categorical Programming Language. Ph.D. thesis. University of Edinburgh. September 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....may be able to reason about certain term rewriting problems caused by the non linear use of variables. In typed calculi, the logical rules describe how terms of a compound type may be formed and how they may used to construct other terms. The semantic counterpart to this was described by Hagino [36] (and see [65] using the concept of a dialgebra. Definition 2.4.4 Let F; G : C D be functors. The category of F; G dialgebras has as objects pairs (X; h) where FX Those type constructors we call final are characterised in Hagino s calculus as being final dialgebras of certain functors, ....

T. Hagino. A Categorical Programming Language. PhD thesis, University of Edinburgh, Department of Computer Science, 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 ....

Tatsuya Hagino. A categorical programming language. PhD thesis, University of Edinburgh, 1987.

....a propositional expression Phi(P ) s.t. P appears strictly positive, it is possible to construct the greatest fixpoint of Phi, s.t. co intro) Phi( Q Phi(Q) co elim) Q 1. 1 Related work Lambda calculi with inductive types have been considered by a number of authors, e.g. see [Hag87,Men88,Dyb91,CM89,Geu92,Loa97,Alt98]. Loader notes that strong normalisation can be shown by using the techniques from System F. This is carried out for monotone inductive types with primitive recursion by Ralph Matthes [Mat98] using an impredicative meta theory. Benl presents a predicative strong normalisation proof for ....

Tatsuya Hagino. A Categorical Programming Language. PhD thesis, University of Edinburgh, September 1987.

....fixed point of a contracting function Phi : S P ) S P ) using Banach s fixed point theorem. There the domain is the same, but its finality is not recognized. 2 A final remark. There is a notion which generalizes and combines both algebras and coalgebras of functors: An F; G dialgebra [Hag87] of two functors F and G from a category D to a category C is still a pair (A; ff) but with ff an arrow in C from F (A) to G(A) It is a notion useful in type theory. 9 3 F Bisimulation The final semantics example in the previous section has the property that it maps two states into the same ....

T. Hagino. A Categorical Programming Language. PhD thesis, University of Edinburgh, September 1987.

....programs into ones which use resources efficiently. The calculational style of programmingwas later extended by Malcolm to accommodate datatypes other than lists ( Mal89] Malcolm introduced a generic promotion theorem, which has its origins in a categorical description of programming ([Hag87]) and which describes conditions under which catamorphisms over regular datatypes may be fused. The significance of the theorem derives primarily from the facts that most of the types used in functional programming are regular, and that many functions of interest are expressible as catamorphisms ....

T. Hagino. A Categorical Programming Language. Dissertation, University of Edinburgh, 1987.

....programs into ones which use resources efficiently. The calculational style of programmingwas later extended by Malcolm to accommodate datatypes other than lists ( Mal89] Malcolm introduced a generic promotion theorem, which has its origins in a categorical description of programming ([Hag87]) and which describes conditions under which catamorphisms over regular datatypes may be fused. The significance of the theorem derives primarily from the facts that most of the types used in functional programming are regular, and that many functions of interest are expressible as catamorphisms ....

T. Hagino. A Categorical Programming Language. Dissertation, University of Edinburgh, 1987.

....Recursion to Catamorphism Finally, the recursive de nition of map## is turned into a catamorphism by means of fold promotion. Fold promotion is based on a generic promotion theorem introduced by Malcolm [18] The promotion theorem, which has its origins in a categorical description of programming [12], describes conditions under which the composition of an arbitrary (strict) function and a catamorphism over a regular data type may be fused to arrive at a new catamorphism equivalent to the original composition. For map## the promotion theorem takes the form 16 map## Nil = h1, map##(Cons(y1, ....

T. Hagino. A Categorical Programming Language. PhD thesis, University of Edinburgh, 1987.

....de nition of map## is turned into a catamorphism by means of fold promotion. Fold promotion is based on a 14 Johann Visser Warm Fusion in Stratego generic promotion theorem introduced by Malcolm [18] The promotion theorem, which has its origins in a categorical description of programming [11], describes conditions under which the composition of an arbitrary (strict) function and a catamorphism over a regular data type may be fused to arrive at a new catamorphism equivalent to the original composition. For map## the promotion theorem takes the form map## Nil = h1, map##(Cons(y1, y2) ....

T. Hagino. A Categorical Programming Language. PhD thesis, University of Edinburgh, 1987.

.... function Such uniqueness questions make sense in many contexts: arithmetic theories [29] first order term rewriting theories [9] primitive recursive arithmetics [14] simply and higher order typed lambda calculi and related functional languages[13, 20, 29] categorical programming languages [5, 15] and more generally wherever we define a procedure iteratively on an inductive data type[15, 16] If mathematical induction is provided explicitly within the formal theory, the problem of uniqueness often becomes trivial (e.g. 1 Research supported by Aids in Research of the Ministry of Education, ....

.... term rewriting theories [9] primitive recursive arithmetics [14] simply and higher order typed lambda calculi and related functional languages[13, 20, 29] categorical programming languages [5, 15] and more generally wherever we define a procedure iteratively on an inductive data type[15, 16]. If mathematical induction is provided explicitly within the formal theory, the problem of uniqueness often becomes trivial (e.g. 1 Research supported by Aids in Research of the Ministry of Education, Science and Culture of Japan, and the Oogata Kenkyu Josei grant of Keio University 2 Research ....

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T. Hagino. A categorical programming language, Phd Thesis, Univ. of Edinburgh (1987).

....Type Theory for programming. 2 should be clear by the nature of this 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 ....

....define datatype junk = c of junk unit; fun d(c f) f; val sa = c(fn (x) d x) x) the expression (d sa) sa; does not terminate (this corresponds to Omega = x:xx) x:xx) in the untyped calculus) ffl There is no equivalent of a terminal T coalgebra. Hagino therefore proposes (see [Hag87] pp 168) a codatatype construct , which enables us to define streams by codatatype a stream = head is a tail is a stream This point might be the most important in practice, because codatatypes represent certain applications of continuations. So this is possibly a way of taming the power of ....

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Tatsuya Hagino. A Categorical Programming Language. PhD thesis, University of Edinburgh, September 1987.

....The Coq proof development system developed at INRIA Rocqencourt and ENS Lyon is an implementation of this last system. In category theory, basic conventional style) inductive and coinductive types are modelled by initial algebras and terminal coalgebras for covariant functors. Hagino [11] designed a typed functional language CPL based on distributive categories with initial algebras and terminal coalgebras for strong covariant functors. The implemented Charity language by Cockett et al. 3] is a similar programming language. The program calculation community is rooted in the ....

T. Hagino, A categorical programming language, Ph.D. Thesis CST-47-87, Lab. for Foundations of Computer Science, Dept. of Computer Science, Univ. of Edinburgh (1987).

....given a propositional expression (P ) s.t. P appears strictly positive, it is possible to construct the greatest xpoint of , s.t. co intro) Q (Q) co elim) Q 1. 1 Related work Lambda calculi with inductive types have been considered by a number of authors, e.g. see [Hag87,Men88,Dyb91,CM89,Geu92,Loa97,Alt98]. Loader notes that strong normalisation can be shown by using the techniques from System F. This is carried out for monotone inductive types with primitive recursion by Ralph Matthes [Mat98] using an impredicative meta theory. Benl presents a predicative strong normalisation proof for ....

Tatsuya Hagino. A Categorical Programming Language. PhD thesis, University of Edinburgh, September 1987.

....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 Categorical Programming Language. PhD thesis, LFCS, University of Edinburgh, 1987.

....that given a propositional expression (P ) s.t. P appears strictly positive, it is possible to construct the greatest xpoint of , s.t. co intro ( Q (Q) co elim Q 1. 1 Related work Lambda calculi with inductive types have been considered by a number of authors, i.e. see [Hag87,Men88,Dyb91,CM89,Geu92,Loa97,Alt98]. Loader notes that strong normalisation can be shown by using the techniques from System F. This is carried out for monotone inductive types with primitive recursion by Ralph Matthes [Mat98] using an impredicative meta theory. Benl presents a predicative strong normalisation proof for ....

Tatsuya Hagino. A Categorical Programming Language. PhD thesis, University of Edinburgh, September 1987.

....behavior classes on the other hand has been suggested. Till now terminal and initial constraints have been used alternatively within algebraic specifications. But in the algebraically more general setting of the simultaneous use of algebras and co algebras, which may be subsumed by Di algebras [452], initial constraints and terminal constraints supplementary and allow in a uniform manner the algebraic specification of data types, of behavior classes and of dynamic systems. The application of the combined use of initial and terminal constraints on Di algebras is subject of future research in ....

T. Hagino. A Categorical Programming Language. Technical Report ECS-LFCS-87-38, University of Edinburgh, Department od Computer Science, Sept. 1987.

....construction presented here to extend the D set semantics to general T algebras, where T is internal and preserves monomorphisms. However, in the spirit of the model construction in chapter 2.1. 2 we will concentrate here on the concrete model construction and the direct interpretation 1 See [Hag87] or [Alt90] for some examples. 2 According to definition 5.3.8, p. 71, Fu92] a decidable functors is one which preserves sets with enumerable domain. 3 The proof of proposition 5.3.9, p.71, ibid, seems incorrect: It is in general not possible to construct a finite Turing machine from any ....

Tatsuya Hagino. A Categorical Programming Language. PhD thesis, University of Edinburgh, September 1987.

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Tatsuya Hagino. A Categorical Programming Language. Ph.D. thesis. University of Edinburgh. September 1987.

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Tatsuya Hagino. A categorical programming language. PhD thesis, University of Edinburgh, 1987.

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T. Hagino. A categorical programming language. PhD thesis, Edinburgh University, 1987.

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Tatsuya Hagino. A Categorical Programming Language. PhD thesis, Edinburgh University, 1987.

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Tatsuya Hagino. A categorical programming language. PhD thesis, University of Edinburgh, 1987.

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T. Hagino. A categorical programming language. PhD thesis, Edinburgh University, 1987.

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Hagino,T.: A categorical programming language. Ph.D. thesisreport CST-47-87, Edinburgh University, 1987.

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Tatsuya Hagino. A categorical programming language. PhD thesis, University of Edinburgh, 1987.

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Hagino, T. (1987a). A categorical programming language. Ph.D. thesis, University of Edinburgh.

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