Tarmo Uustalu, Varmo Vene Primitive (co)recursion and course-of-value (co)iteration, categorically. 1999. http://www.cs.ut.ee/~varmo/papers/inf.ps.gz (16.05.2005) 45  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the unique function satisfying [f; g] in 1 = f and [f; g] in 2 = g, where in 1 and in 2 are the coproduct injections. Theorem 4.4. For all f : C T (C Z) there exists a unique g : C Z such that g = T [g; id] f . The proof can be found in [6] or (for the dual case) in Uustalu and Vene [23]. In our example TX = R X, the uniquely de ned function g satis es hd g = f 1 and tl g = id; g] f 2 (1) if f = hf 1 ; f 2 i : C R (C Z) 12 Since coproducts in set are disjoint unions, we can rewrite the second equation to tl g(s) f 2 (s) if f 2 (s) 2 Z g f 2 (s) if f 2 (s) 2 ....

T. Uustalu and V. Vene. Primitive (Co)Recursion and Course of Value (Co)Iteration, Categorically. INFORMATICA (IMI, Lithuania), 10:5 26, 1999.

....schemata range from primitive recursion and course of value iteration to well founded recursion. However, the situation with coalgebras seems less advanced. Dualisations of de nition by primitive recursion and course of value iteration have been stated on an abstract level (see e.g. Vene Uustalu [UV99]) but for many frequently encountered situations solutions are known for special cases only. One example is the use of operators in the speci cation of dynamical systems like sequential or parallel composition, as in process algebra. Similarly, one uses operators like addition and multiplication ....

....from above it is a sucient condition for all related states to be bisimilar. A small example demonstrates that the technique enables simpler proofs involving less complex relations. We show that primitive corecursion and the dual of course of value iteration as presented by Vene and Uustalu [UV99] can be obtained from the coiteration schema for suitable instantiations of T and . Moreover, we brie y explain how it can be used to justify the validity of speci cations involving operators of a certain type. The same operators were considered by Turi and Plotkin [TP97] and shown to be ....

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Tarmo Uustalu and Varmo Vene. Primitive (co)recursion and course-of-value (co)iteration, categorically. INFORMATICA (IMI, Lithuania), 10(1):5-26, 1999.

....Mendler s formulations are inspired by categorical considerations. In the context of simple type theory, there has been a great deal of work, and some quite interesting formulations have been proposed. In particular, one may cite recent work by Uustalu and Vene, for example the papers [107] and [108] . One problem has been to deal properly with the analogues of the side arguments in recursion schemes, that distinguish recursion from mere iteration. Uustalu and Vene have managed (in [106] to systematise a variety of recursive schemes into a cube , analogous to Barendregt s cube of pure ....

T. Uustalu and V. Vene. Primitive (co)recursion and course-of-values (co)iteration, categorically. Informatica, 10(1):5--26, March 1999.

No context found.

Uustalu, T., Vene, V.: Primitive (co)recursion and course-of-value (co)iteration, categorically. Informatica 10(1) (1999) 5--26

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T. Uustalu and V. Vene, Primitive (co)recursion and course-of-value (co)iteration, categorically, Informatica 10(1) (1999) 5--26.

....are briefly introduced. For a good introduction to initial algebras and final coalgebras from the programming perspective and to the categorical approach to functional programming in general, we refer to [Fok92, BdM97] The recursion and corecursion schemes used in the paper are described in [UV99, UVP01] The classic category theory texts treating (co)monads are [Man76, BW84] Throughout the paper, we work in one base category C about which we do not make any specific assumptions other than the existence of the particular coproducts, initial algebras etc. that we name. The category Set ....

T. Uustalu and V. Vene. Primitive (Co)recursion and Course-of-value (Co)iteration, Categorically. Informatica, 10(1):5--26, 1999.

....iteration and primitive recursion. Some other schemes that are more general in terms of direct expression power, e.g. the course of value strengthenings of iteration and primitive recursion, are equally useful in programming and do not lead to considerably higher sophistication in the formulation [1]. Too particular generalizations, however, get too esoteric and sophisticated. To make a library primitive out of every powerful imaginable recursion scheme is therefore not a good idea. It might be the case that a better approach is to add another dimension of genericity. In this paper, we ....

....specific recursion schemes. The product comonad, for example, gives us the asymmetric form of mutual iteration (zygomorphisms, 2] including the most important special case, primitive recursion (paramorphisms, 3] The generalized stream comonad gives course of value iteration (histomorphisms, [1]) and an interesting generalization permitting, for instance, recursive calls of the definiendum on rearrangements of recursive components of the given argument value. Another type of comonad giving rise to meaningful recursion schemes is that of the state in context comonad. In the paper, we ....

T Uustalu and V Vene. Primitive (co)recursion and course-of-value (co)iteration, categorically. INFORMATICA, 10(1):5--26, 1999.

....; s 2 ) hd 1 (s 1 ; s 2 ) hd(s 1 ) and tl merge(s 1 ; s 2 ) merge h 2 ; tl 1 i(s 1 ; s 2 ) merge(s 2 ; tl(s 1 ) as desired. 2 Primitive (Co)Recursion The idea behind the categorical formulation of primitive (Co)recursion is (to my knowledge) due to [2] and has been treated in [5, 4] and, in a much more abstract setting, in [1] Theorem 2.1 (Primitive Recursion) For all f : T ( X:TX) A) A there exists a unique g : X:TX A such that T X:TX in ## T hid;gi ## T ( X:TX A) f ## X:TX g ## A (1) commutes. Proof. Consider the diagram X:TX in ## Th ## T ....

....describes what happens when one phrases course of value recursion in terms of initial algebras and then reverses the arrows. I was not able to nd the material presented in this section in the literature. In particular, the course of value (co)recursion scheme is di erent from the one discussed in [1, 4], the main di erence being that our scheme subsumes primitive (co)recursion. We denote the initial algebra for the functor TA (X) T (A X) by ( X:T (A X) in A ) Similarly, X:T (C X) out C ) is the nal coalgebra for TC , given by TC (X) T (C X) Corollary 3.1. For all g : X:TX A ....

T. Uustalu and V. Vene. Primitive (Co)Recursion and Course of Value (Co)Iteration, Categorically. INFORMATICA (IMI, Lithuania), 10:526, 1999.

....This looks promising for the reason that in Set like categories, where exponents exist but coexponents do not, there are more monads than comonads. A third direction would be to nd out if and how the present work may be restated for the so called Mendler style of formulating recursion schemes [Uustalu and Vene 1999b, 2000] As Mendler style formulations tend to be simpler than formulations in the conventional style, there is hope that the constructions of the present paper, especially those related to distributivity, may admit a smoother restatement in the Mendler style. And nally, since the recursion scheme ....

Uustalu, T. and Vene, V. 1999a. Primitive (Co)recursion and Course-of-Value (Co)iteration, Categorically. INFORMATICA 10, 1, 5-26.

....This looks promising for the reason that in Set like categories, where exponents exist but coexponents do not, there more monads than comonads. A third direction would be to nd out if and how the present work may be restated for the so called Mendler style of formulating recursion schemes [Uustalu and Vene 1999b, 2000] As Mendler style formulations tend to be simpler than formulations in the conventional style, there is hope that the constructions of the present paper, especially those related to distributivity, may admit a smoother restatement in the Mendler style. And nally, since the recursion scheme ....

Uustalu, T. and Vene, V. 1999a. Primitive (Co)recursion and Course-of-Value (Co)iteration, Categorically. INFORMATICA 10, 1, 5-26.

....iteration and primitive recursion. Some other schemes that are more general in terms of direct expression power, e.g. the course of value strengthenings of iteration and primitive recursion, are equally useful in programming and do not lead to considerably higher sophistication in the formulation [1]. Too particular generalizations, however, get too esoteric and sophisticated. To make a library primitive out of every powerful imaginable recursion scheme is therefore not a good idea. It might be the case that a better approach is to add another dimension of genericity. In this paper, we ....

....specific recursion schemes. The product comonad, for example, gives us the asymmetric form of mutual iteration (zygomorphisms, 2] including the most important special case, primitive recursion (paramorphisms, 3] The generalized stream comonad gives course of value iteration (histomorphisms, [1]) and an interesting generalization permitting, for instance, recursive calls of the definiendum on rearrangements of recursive components of the given argument value. Another type of comonad giving rise to meaningful recursion schemes is that of the state in context comonad. In the paper, we ....

T Uustalu and V Vene. Primitive (co)recursion and course-of-value (co)iteration, categorically. INFORMATICA, 10(1):5--26, 1999.

....from speci cations; this concerns both speci cation methodology and computer assistance in synthesis. We have also started to study the relating categorical deduction systems (typed combinatory logics a la Curien) their utility in program calculation and the relevant categorical theory [38,36,39,40]. We also intend to nd out the details of the apparent 24 close relationship of enhanced course of value Mendler style (co)recursion to Gim enez new formulation of guarded (co)recursion [9] for systems with suband supertyping and quanti cation with upper and lower bounds; radically di erent ....

T. Uustalu and V. Vene, Primitive (co)recursion and course-of-value (co)iteration, categorically, INFORMATICA 10 (1999) 5-26.

....concentrate on recursion combinators de nable via the cata combinator, i.e. the conventional style iterator that the value constructor for an inductive type comes together with. The discussions of conventional style combinators for course of value iteration and simultaneous iteration appeared in [UV99a,Uus99] those of Mendlerstyle combinators for these recursion schemes are a novel contribution of this paper. A typetheoretic presentation of the Mendler style course of value recursion combinator can be found in [UV97,UV00,Uus98] The rest of the paper is organized as follows. In Section 2, ....

....coding in the Mendler style is course of value iteration: if no references are allowed in an equation to inputs of the function de ned, then the equation has to refer to candidate courses of its outputs. In the conventional style, course of value iteration admits the following coding [UV97,UV00,UV99a] Let F again be a functor. De ne (C n F )A = C F A for any A and let out nF stand for a nal C n F coalgebra, C n F ) for its carrier, and [ nF for the corresponding ana combinator. C nF ) is the coinductive type of F streams for an element type C, out nF the value destructor, ....

T Uustalu and V Vene. Primitive (co)recursion and course-of-value (co)iteration, categorically. INFORMATICA, 10(1):5-26, 1999.

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Tarmo Uustalu, Varmo Vene Primitive (co)recursion and course-of-value (co)iteration, categorically. 1999. http://www.cs.ut.ee/~varmo/papers/inf.ps.gz (16.05.2005) 45

No context found.

Tarmo Uustalu and Varmo Vene. Primitive (co)recursion and course-of-value (co)iteration, categorically. INFORMATICA (IMI, Lithuania), 10(1):5--26, 1999.

No context found.

T. Uustalu and V. Vene. Primitive (co)recursion and course-of-values (co)iteration, categorically. Informatica, 10(1):5--26, March 1999.

No context found.

Tarmo Uustalu and Varmo Vene. Primitive (co)recursion and course-of-value (co)iteration, categorically. INFORMATICA, 10(1):5--26, 1999.

No context found.

Tarmo Uustalu and Varmo Vene. Primitive (co)recursion and course-of-value (co)iteration, categorically. Informatica (IMI, Lithuania), 10(1):5--26, 1999.

No context found.

T.Uustalu, V.Vene. Primitive (co)recursion and course-of-value (co)iteration, categorically, Informatica (IMI, Lithuania) 10(1), 1999.

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