By P. Braffort and D. Hirschberg (Eds.)

ISBN-10: 0444533915

ISBN-13: 9780444533913

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Extra resources for Computer Programming and Formal Systems

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From a certain point of view, having a convenient representation of one’s behavior available for modification is what is meant by consciousness. 4. To represent computers as well as computations in a formalism that permits a treatment of the relation between a computation and the computer that carries out the computation. 5. To give a quantitative theory of computation. There might be a quantitative measure of the size of a computation analogous to Shannon’s measure of information. The present paper contains no information about this.

Namely, the expression gets quite complicated and we must admit functionals capable of taking themselves as arguments. These escape our orderly hierarchy of functionals. 4. Non-Computable Functions and Functionals. It might be supposed that in a mathematical theory of computation one need only consider computable functions. However, mathematical physics is carried out in terms of real valued functions which are not computable but only approximable by computable functions. We shall consider several successive extensions of the class C { 9 } .

We will not regard the sets A x ( B x C ) and ( A x B) x C as the same, but there is a canonical 1-1 mapping between them, gA,B,C:(A X B) X c -+ A X ( B X c) to express the fact that these sets are canonically isomorphic. Other canonical isomorphisms are 1. fA,B:A X B + B X A defined by t(u) = ~B,A(eA,B(u),~A,B(~)) 2. d ~ : A X ( B O C ) - + A X B O A x C 3. u ~ : ( A O B ) O C +- A O ( B Q C ) 1. &:AC x BC + ( A X B)c 5. d3:ABX Ac +- AB@C 6. s~:(AB)C+- ABxC We shall denote the null set (containing no elements) by 0 and the set A BASIS FOR A MATHEMATICAL THEORY OF COMPUTATION 51 consisting of the integers from 1 to n by n.

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Computer Programming and Formal Systems by P. Braffort and D. Hirschberg (Eds.)

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