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Computational speed-up by effective operators1

Published online by Cambridge University Press:  12 March 2014

Albert R. Meyer  and
Patrick C. Fischer
Albert R. Meyer
Affiliation:
University of California, Berkeley, California 94720
Patrick C. Fischer
Affiliation:
University of California, Berkeley, California 94720
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Extract

The complexity of a computable function can be measured by considering the time or space required to compute its values. Particular notions of time and space arising from variants of Turing machines have been investigated by R. W. Ritchie [14], Hartmanis and Stearns [8], and Arbib and Blum [1], among others. General properties of such complexity measures have been characterized axiomatically by Rabin [12], Blum [2], Young [16], [17], and McCreight and Meyer [10].

In this paper the speed-up and super-speed-up theorems of Blum [2] are generalized to speed-up by arbitrary total effective operators. The significance of such theorems is that one cannot equate the complexity of a computable function with the running time of its fastest program, for the simple reason that there are computable functions which in a very strong sense have no fastest programs.

Let φi be the ith partial recursive function of one variable in a standard Gödel numbering of partial recursive functions. A family Φ0, Φ1, … of functions of one variable is called a Blum measure on computation providing

(1) domain (φi) = domain (Φi), and

(2) the predicate [Φi(x) = m] is recursive in i, x and m.

Typical interpretations of Φi(x) are the number of steps required by the ith Turing machine (in a standard enumeration of Turing machines) to converge on input x, the space or number of tape squares required by the ith Turing machine to converge on input x (with the convention that Φi(x) is undefined even if the machine fails to halt in a finite loop), and the length of the shortest derivation of the value of φi(x) from the ith set of recursive equations.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 37 , Issue 1 , March 1972 , pp. 55 - 68
Copyright
Copyright © Association for Symbolic Logic 1972

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Footnotes

1

Some of the results reported here were presented in preliminary form at the Ninth Annual Symposium on Switching and Automata Theory (1968), sponsored by the IEEE Computer Group. This research was supported in part by the following grants: ARPA (SD-146) at Carnegie-Mellon University; NSF GP-7701; NRC (Canada) A-5549; and Nonr-4102(01) at Project MAC, M.I.T.

References

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