Showing papers on "Computability published in 1984"

Effectively given domains and lambda-calculus models

Abstract: The syntax of a formal language is effectively given. This is not immediately so for the semantics. This paper deals with the simple but sufficiently powerful applicative language (λ-calculus) and studies effectiveness properties of its semantics. In particular it analyses the effectiveness of the interpretation of λ-terms as well as different notions of computability over models.

Best Simultaneous Approximation (Chebyshev Centers)

Abstract: The problem of approximating simultaneously a set of data in a given metric space by a single element of an approximating family arises naturally in many practical problems. A common procedure is to choose the “best” approximant by a least squares principle, which has the advantages of existence, uniqueness, stability and easy computability. However, in many cases the least deviation principle makes more sense. Geometrically, this amounts to covering the given data set by a ball of minimal radius among those centered at points of the approximating family. The theory of best simultaneous approximants in this sense, called also Chebyshev centers, was initiated by A. L. Garkavi about twenty years ago. It has drawn more attention in the last decade, but is still in a developing stage. In this short survey I try to describe the main known results and to point at some of the connections between the theory of Chebyshev centers and other problems of Approximation Theory and of Banach Space Theory.

An intensional implementation technique for functional languages

Abstract: The potential of functional programming languages has not been widely accepted yet. The reason lies in the difficulties associated with their implementation. In this dissertation we propose a new implementation technique for functional languages by compiling them into 'Intensional Logic' of R. Montague and R. Carnap. Our technique is not limited to a particular hardware or to a particular evaluation strategy; nevertheless it lends itself directly to demand-driven tagged dataflow architecture. Even though our technique can handle conventional languages as well, our main interest is exclusively with functional languages in general and with Lucid-like dataflow languages in particular. We give a brief general account of intensional logic and then introduce the concept of intensional algebras as structures (models) for intensional logic. We, formally, show the computability requirements for such algebras. The target language of our compilation is the family of languages DE (definitional equations over intensional expressions). A program in DE is a linear (not structured) set of non-ambiguous equations defining nullary variable symbols. One of these variable symbols should be the symbol result. We introduce the compilation of Iswim (a first order variant of Landin's ISWIM) as an example of compiling functions into intensional expressions. A compilation algorithm is given. Iswim(A), for any algebra of data types A, is compiled into DE(Flo(A)) where Flo(A) is a uniquely defined intensional algebra over the tree of function calls. The approach is extended to compiling Luswim and Lucid. We describe the demand-driven tagged dataflow (the eduction) approach to evaluating the intensional family of target languages DE. Furthermore, for each intensional algebra, we introduce a collection of rewrite rules. A justification of correctness is given. These rules are the basis for evaluating programs in the target DE by reduction. Finally, we discuss possible refinements and extensions to our approach.

Alternation bounded auxiliary pushdown automata

Abstract: Languages accepted by alternating auxiliary pushdown automata using simultaneously a(n) alternations and s(n) space are shown to be members of the class of languages accepted by nondeterministic Turing machines using a(n) 2es(n) space for some c > 0. This result is used to show that the hierarchy of classes of languages accepted by pushdown automata based on the number of alternations collapses at the second level of the hierarchy. The power of alternation bounded pushdown automata without auxiliary storage is also investigated.

Partial closures and semantics of while: Towards an iteration-based theory of data types

Abstract: The present paper proposes first a generalization of closure theory and revisits Moore's theory in this framework. Afterwards closures of non cyclic functions are introduced and a method is given to transform cyclic into non cyclic functions. Eventually semantics of the while construct is found to be the closure of a function. Computability on inductive and non inductive data types is then studied with iterative means.

Rational approximation of fractals

Abstract: Stationary distributions for certain Markov chains of inverse branches of rational maps are put forward as the basis of an approximation theory for fractals. Results on existence and on computability of moments are proved.

Computability of Probabilistic Parameters for Some Classes of Formal Languages

Abstract: In a previous paper [BT 83], some probabilistic notions of density and waiting time for a formal language have been studied. We prove here that these probabilistic parameters are computable with an arbitrary precision for some families of languages : the languages with an end marker ; the prefix-free regular sets, with matricial algorithms on Markov chains related to deterministic finite-state automata ; at the end, the prefix-free languages of palindrom words, for which the use of counting generating series yields new results in the equally likely case, already studied in [BT 83], and allows to give partial answers in the general case.