E.G Manes and M.A. Arbib. Algebraic Approaches to Program Semantics. Text and Monographs in Computer Science. Springer Verlag, 1986. 46  Home/Search   Document Not in Database   Summary   Related Articles   Check

....identity. That is, I m ignoring some obfuscating complications that a very careful treatment would have to address, involving the foundations of mathematics and avoiding Russell s Paradox (whether the set of all sets that don t contain themselves contains itself) See [Ma71, xI] For example [MaAr86], category Mfn has sets for objects, and multivalued functions for arrows that is, an arrow f : A B maps each value a 2 A to a set of values f(a) 2 P(B) with composition de ned by g(y) while ANMfn, the category of multivalued functions with all or nothing composition has the same ....

Ernest G. Manes and Michael A. Arbib, Algebraic Approaches to Program Semantics, Springer-Verlag, 1986.

.... by now standard, turns out to be an appropriate framework for reasoning algebraically about programs and is the basis for current developments in generic programming (see e.g. 2, 19] In this section we review the relevant concepts around the categorical approach to recursive types [23, 25, 21] and its application to program calculation [24, 27, 11, 22, 5, 16] The category theoretic explanation of (recursive) types is based on the idea that types constitute objects of a category C, programs are modelled by arrows of the category, and type constructors are functors on C. In this ....

E.G. Manes and M.A. Arbib. Algebraic Approaches to Program Semantics. Texts and Monographs in Computer Science. Springer-Verlag, 1986.

....with respect to the operations needed for interpreting linear logic. The sum operation in these spaces corresponds intuitively to the superimposition of various possible results of a (non deterministic) computation: this use of the sum operation has already been considered by Arbib and Manes in [AM86], see also [Hag01] where Haghverdi uses sums (at the geometry of interaction level) for building fully complete models of multiplicative linear logic. Possible connections between our model and Haghverdi s constructions have still to be explored. For us, a K othe space is a pair X = jXj; EX ) ....

Michael A. Arbib and Ernest Manes. Algebraic approaches to program semantics. AKM Series in Theoretical Computer Science. Springer-Verlag, 1986.

....we shed some light on colimits of chains. These are of particular interest especially in theoretical computer science, because they allow the iterative construction of initial algebras of cocontinuous functors. Initial algebras can be used as a model for recursively specified data types. cf. [19]) Being able to construct initial algebras in a category of relations yields a powerful tool for the specification of non deterministic problems, for example optimization problems (cf. 2] Unfortunately, the desired colimits do not exist in general in Rel(C) However, if we impose certain ....

....133 9 Colimits of chains in Rel(C) Let A be an arbitrary category. For a given endofunctor F : A A, a pair (A; a) where a : FA A is an arrow of A, is called an F algebra. In theoretical Computer Science initial F algebras are used as models for recursively defined data types (cf. [19]) Recently, Bird and de Moor (cf. 2] have used initial algebras in allegories for the derivation of programs for formally specified optimization problems. However, they assume the existence of so called power objects to get initial algebras. At least for regular categories C, Rel(C) has power ....

Ernest G. Manes and Michael A. Arbib. Algebraic Approaches to Program Semantics. Springer Verlag, 1986.

....#) As we shall see in the following section, the latter operation is usually easier to do. The usefulness of Theorems 2 and 3 is due to this fact. Remark 4. Theorems 1, 2, and 3 compute the supremal element as the greatest fixpoint of monotone operators. This is a standard technique (see, e.g. [22], 27] which relies on the Knaster Tarski fixpoint theorem [26] 15] In this paper we are presenting a method for obtaining these monotone operators, i.e. defining a suitable S reflexive relation # and computing #( or#( We shall refer to this method as the # method. Examples given in the ....

E.G. Manes and M.A. Arbib, Algebraic Approaches to Program Semantics, Springer-Verlag, New York, 1986.

....to be continuous. Since the shu e product is distributive over arbitrary sums, x is an continuous mapping and so is the k fold shu e product. Moreover, for s 1 2 Ahh ii and s 2 2 (Ahh ii)hh ii, the scalar product s 1 s 2 2 (Ahh ii)hh ii is continuous. According to [8], Theorem 12 of Section 8.3, in nite sums of continuous functions are again continuous functions. One application of this theorem yields that t is an continuous mapping. One more application of this theorem shows that r, r 2 (Ahh ii)hh( Y ) ii, is an continuous mapping. 2 A ....

G. E. Manes and M. A. Arbib. Algebraic Approaches to Program Semantics. Springer, 1986.

....it is the colimit of the chain 0 , F0 , FF0 , Delta Delta Delta The category Fun has everything needed to make this incantation work: Fun is co complete (in fact, bi complete) and all regular functors F on Fun are cocontinuous. The proof for polynomial functors can be found in [14], and the extension to type functors is in [13] Moreover, the category Nat (Fun) inherits co completeness from the base category Fun (see [11, 7] We believe that all regular higher order functors are co continuous, though we have not yet found a proof of this in the literature, so the ....

E.G. Manes and M.A. Arbib. Algebraic Approaches to Program Semantics. Texts and Monographs in Computing Science. Springer-Verlag, 1986.

....mention the work by Arbib and Manes, notably [AM74] in which the authors give a categorical account of the duality between reachability (a simple instance of controllability) and observability. In [AM74] coalgebras are not mentioned but appear implicitly. In some of their later work (notably in [MA86], final coalgebras (called terminal there) do occur explicitly, and are used as models for the behaviour of (total) Moore automata. A crucial di#erence with recent uses of coalgebras, and with the present paper, is the absence in their work of a logical proof principle for reasoning about final ....

E.G. Manes and M.A. Arbib. Algebraic approaches to program semantics. Texts and monographs in computer science. Springer-Verlag, 1986.

....be expected to do some dicult proofs on the problem sets. There are some very good books on semantics available, such as Foundations for Programming Languages by John C. Mitchell [5] Other good books on programming language 1 theory are available by Tennent, Gordon, Manes and Arbib and Stoy[8, 1, 7, 4]. Stoy s book is the book most of my generation learned the subject from but it is quite out of date. The most wonderful reference for the matehmatical theory of domains is Plotkin s unpublished lecture notes [6] the so called Pisa Notes which contain many hard problems and even several open ....

E. Manes and M. Arbib. Algebraic Approaches to Program Semantics. Springer-Verlag, 1986.

....category with initial object 0. The initial algebra of a functor F : C C exists if F is a cocontinuous functor. The initial algebra is then given by the colimit of the chain 0 F0 F (F0) 2 The cocontinuity properties of the constructors of regular datatypes were described in [7]. It was shown there that the constant and identity functors are cocontinuous in any category C, as is the composition of any pair of cocontinuous functors. If C has coproducts, then the coproduct of two cocontinuous functors is cocontinuous. In some categories, such as Set, the product of two ....

....notation = comes from [6] and we will use the convention that composition ( binds more tightly than = We will write Colim(cn : Cn ) for the colimit of (c n : Cn ) For a category representing a preorder, colimits coincide with least upper bounds. Example 4. 1 (colimits in Set) It is shown in [7] that every right chain (c : C) in Set has a colimit (A; which is de ned as follows. Let U = f(n; x) j x 2 Cn ; n 0g and de ne an equivalence relation on U by (n; x) m; y) if 9k m;n c nk (x) c mk (y) in C k where c ij is the composition de ned in (6) Then A is the set of ....

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E.G. Manes and M.A. Arbib. Algebraic Approaches to Program Semantics. Texts and Monographs in Computer Science. Springer-Verlag, 1986.

....may, without loss of generality) that the tokens of information systems all be drawn from a common pool 7 of tokens, we nd typically that the collection of information systems under E is a cpo, even an algebraic cpo. How does this compare with more elaborate category theory approaches, as in [SP82, MA86] The answer seems clear: the categorical approach gives us a valuable characterization of the intended solution as (usually) an initial algebra; the simple order theoretic method provides no such information. This gives the clue to what we have been aiming at in our work: to nd out what needs to ....

....object, if it exists, of the category of F algebras; and we then say that F has an initial algebra, or a least xed point, in C. An initial F algebra (A; f) as above gives a canonical solution of the domain equation X = F (X) hence its importance in computing science. see, for example, [MA86] for details) We now recall some de nitions from [ES91a] A morphism f : A B of an I category K is strict if in( A) f = in( B) We can immediately see that is an initial object for the subcategory of strict morphisms K s . A functor between I categories is standard if it preserves ....

E. Manes and M. A. Arbib. Algebraic Approaches to Program Semantics. Springer-Verlag, 1986. 45

....and M a model for T in C. If T (H M = N : t) then 40 M j= H M = N : t. 2 3 Bibliographical remarks The first section is just a brief exposure to cartesian closed categories. Our presentation is based on [2] 9] and [10] Especially, examples 1.2.4 and 1.2. 5 are borrowed from [9] and [3], respectively. A more complete account on the subject can be found in [13] Most parts of the material presented on this report are based mainly on [10] chapters 2 and 3, and [2] chapter 8. An extremely succint presentation on the categorical model of the calculus can be found in [17] and ....

ARBIB, Michael A; MANES, Ernest G. Algebraic Approaches to Program Semantics. New York: Springer-Verlag, 1986. 351p.

....is based on and fixes some notation. We describe the essentials of the category theoretic explanation of inductive and coinductive datatypes, the definition of structural recursive functions in that setting and some of their algebraic laws. Further details on these topics can be found in e.g. [22, 11, 5, 1, 15]. 2.1 Preliminaries In the categorical approach to recursive types, types are modeled by objects of a category C, and functions (operations, programs) are modelled by morphisms of this category. In this setting, type constructors correspond to endofunctors on C (i.e. functors from C to C) We ....

E.G. Manes and M.A. Arbib. Algebraic Approaches to Program Semantics. Texts and Monographs in Computer Science. Springer-Verlag, 1986.

....functors on C. In this setting, a datatype T is understood as a solution to a type equation X = FX, for an appropriate endofunctor F : C C that captures the shape (or signature) of the type. This section reviews the relevant concepts concerning the categorical approach to recursive datatypes [19, 21, 15] and its application to program calculation [20, 23, 7, 16, 5] We show then how recursive operators and their calculational properties are derived from elementary categorical constructions. An interesting feature of those constructions is that they are parameterized by the signature of the ....

....C) A Theta B) Theta C. The sum bifunctor : C Theta C C is defined as separated sum: A B = f0g Theta A [ f1g Theta B) The sum inclusions inl : A A B and inr : B A B are defined by inl(a) 0; a) and inr(b) 1; b) The separated sum fails to be a coproduct in Cpo (see [21]) actually Cpo does not have coproducts. This means that, given two continuous functions f : A C and g : B C, case analysis, defined as a strict function [f; g] A B C, is not the unique mediating morphism between A B and C that satisfies the equations [f; g] ffi inl = f and [f; g] ffi ....

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E.G. Manes and M.A. Arbib. Algebraic Approaches to Program Semantics. Texts and Monographs in Computer Science. Springer-Verlag, 1986.

....was a function into the carrier set A . Again the example is typical because A 1 is a final coalgebra, which generalizes the notion of greatest fixed point. Coalgebras had previously been found to be suitable for the description of the dynamics of systems such as deterministic automata (cf. [AM80, MA86]) Traditionally these are represented as tuples hQ; A; ffi : Q Theta A Q; fi : Q Bi; consisting of a set of states Q, an input alphabet A, a next state function ffi, and an output function fi (in addition an initial state is often specified as well) Alternatively, such an automaton can ....

....of a set Q of states and a state transition function ffi that for every state q and input symbol a in A determines the next state ffihq; ai. Often an initial state and a set of final states is specified as well, but they can be dealt with separately. As observed in the introduction, in [AM82, MA86], such automata are precisely the deterministic systems with input mentioned above, because of the following bijection: ff : Q Theta A Qg = ff : Q Q A g: A homomorphism between (S; ff S ) and (T ; ff T ) is any function f : S T satisfying for all s in S, a in A, s a Gamma s 0 ) ....

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E.G. Manes and M.A. Arbib. Algebraic approaches to program semantics. Texts and monographs in computer science. Springer-Verlag, 1986.

....using which efficient functional programs can be derived from specifications by using equational reasoning. Squiggol was subsequently generalised from lists to polynomial (sumof product) datatypes [20] by using the categorical approach of modelling recursive datatypes as fixed points of functors [21, 14]. This approach allows foldr, unfold and other recursion functionals to be uniformly generalised from lists to polynomial datatypes. The generalised functionals are given special names (such as catamorphism and anamorphism) and are written symbolically using special brackets (such as banana ....

E.G. Manes and M.A. Arbib. Algebraic Approaches to Program Semantics. Texts and Monographs in Computer Science. Springer-Verlag, 1986.

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E.G Manes and M.A. Arbib. Algebraic Approaches to Program Semantics. Text and Monographs in Computer Science. Springer Verlag, 1986. 46

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E.G. Manes and M.A. Arbib. Algebraic Approaches to Program Semantics. Texts and Monographs in Computer Science. Springer-Verlag, 1986.

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Manes, E.G. and Arbib, M.A. (1986). Algebraic approaches to program semantics. Springer-Verlag.

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E.G. Manes and M.A. Arbib. Algebraic approaches to program semantics. Texts and monographs in computer science. SpringerVerlag, 1986.

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E. Manes and A. Arbib. Algebraic Approaches to Program Semantics. Texts and Monographs in Computer Science. Springer Verlag, 1986.

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E. G. Manes and M. A. Arbib. Algebraic Approaches to Program Semantics. Texts and Monographs in Computer Science. Springer-Verlag, 1986. D. Gries, series editor.

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E. G. Manes and M. A. Arbib. Algebraic Approaches to Program Semantics. Springer-Verlag, New York, 1986.

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Ernest G. Manes and Michael A. Arbib. 1986. Algebraic Approaches to Program Semantics. Springer, New York.

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Arbib, Michael A; Manes, Ernest G. Algebraic Approaches to Program Semantics. New York: Springer-Verlag, 1986. 351p.

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