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Algebraic Systems

Algebraic Systems. The idea that underlays computational Theory. Components. Set of objects, S Collection of functions relating objects in S Set of axioms that specify membership in S specify properties of functions. Example.

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Algebraic Systems

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  1. Algebraic Systems The idea that underlays computational Theory

  2. Components • Set of objects, S • Collection of functions relating objects in S • Set of axioms • that specify membership in S • specify properties of functions

  3. Example • The system of natural numbers is an algebraic system (N,S) where • N is a set • S is a function that satisfies three particular axioms

  4. Axioms • There is a zero element in N (0 є N) • There is a one-to-one successor function • w • If M 0 then M = N

  5. In English • Axioms 1 and 2 assert that the o bjects 0,0’,0’’,0’’’,… are contained in N. These are the natural numbers, for which we use the standard notation, 0,1,2,3 • Axiom 3 requires that N contain only the natural numbers. That is, there must be no smaller set M that satisfies axioms 1 and 2

  6. Implications • All operations on integers may be defined in terms of (N,S) • Can be shown that the principle of mathematical induction follows from (N,S) • if p(0) is valid and p(k) -> p(k’), then p(n) is valid for each natural number, n.

  7. Source • Denning, P., Dennis, J., Qualitz, J. (1978) Machines Languages and Computation. Englewood Cliffs, NJ: Prentice-Hall.

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