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Descriptive and Computational Complexity
 COMPUTATIONAL COMPLEXITY THEORY, PROC. SYMP. APPLIED MATH
, 1989
"... Computational complexity began with the natural physical notions of time and space. Given a property, S, an important issue is the computational complexity of checking whether or not an input satisfies S. For a long time, the notion of complexity referred to the time or space used in the computatio ..."
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Computational complexity began with the natural physical notions of time and space. Given a property, S, an important issue is the computational complexity of checking whether or not an input satisfies S. For a long time, the notion of complexity referred to the time or space used in the computation. A mathematician might ask, "What is the complexity of expressing the property S?" It should not be surprising that these two questions  that of checking and that of expressing  are related. However it is startling how closely tied they are when the second question refers to expressing the property in firstorder logic. Many complexity classes originally defined in terms of time or space resources have precise definitions as classes in firstorder logic. In 1974 Fagin gave a characterization of nondeterministic polynomial time (NP) as the set of properties expressible in secondorder existential logic
Predicative Recursion and Computational Complexity
, 1992
"... The purpose of this thesis is to give a "foundational" characterization of some common complexity classes. Such a characterization is distinguished by the fact that no explicit resource bounds are used. For example, we characterize the polynomial time computable functions without making any direct r ..."
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Cited by 45 (3 self)
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The purpose of this thesis is to give a "foundational" characterization of some common complexity classes. Such a characterization is distinguished by the fact that no explicit resource bounds are used. For example, we characterize the polynomial time computable functions without making any direct reference to polynomials, time, or even computation. Complexity classes characterized in this way include polynomial time, the functional polytime hierarchy, the logspace decidable problems, and NC. After developing these "resource free" definitions, we apply them to redeveloping the feasible logical system of Cook and Urquhart, and show how this firstorder system relates to the secondorder system of Leivant. The connection is an interesting one since the systems were defined independently and have what appear to be very different rules for the principle of induction. Furthermore it is interesting to see, albeit in a very specific context, how to retract a second order statement, ("inducti...