We give a recursion-theoretic characterization of FP which describes polynomial time computation independently of any externally imposed resource bounds. In particular, this syntactic characterization avoids the explicit size bounds on recursion (and the initial function 2 |x||y| ) of Cobham.

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 first-order system relates to the second-order 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...

Abstract. The main results of this paper are recursion-theoretic characterizations of two parallel complexity classes: the functions computable by uniform bounded fan-in circuit families of log and polylog depth (or equivalently, the functions bitwise computable by alternating Turing machines in log and polylog time). The present characterizations avoid the complex base functions, function constructors, and a priori size or depth bounds typical of previous work on these classes. This simplicity is achieved by extending the \tiered recursion " techniques of Leivant and Bellantoni&Cook. Key words. Circuit complexity � subrecursion. Subject classi cations. 68Q15, 03D20, 94C99. 1.

We define the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P , using quasilinear-time computation and quantification over values of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy has several alternative characterizations. We define both SBH (QL) and its corresponding hierarchy of function classes, FSBH(QL),and present a variety of problems in these classes, including ql m -complete problems for each class in SBH (QL). We discuss the structure of the hierarchy, and show that certain simple structural conditions on it would imply P 6= PSPACE. We present characterizations of SBH (QL) relations based on alternating Turing machines and on first-order definability, as well as recursion-theoretic characterizations of function classes corresponding to SBH (QL).

We de ne the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P, using quasilinear-time computation and quanti cation over strings of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy hasseveral alternative characterizations. We de ne both SBH (QL) and its corresponding hierarchy of function classes, ql and present a variety of problems in these classes, including m-complete problems for each class in SBH (QL). We discuss the structure of the hierarchy, and show that determining its precise relationship to deterministic time classes can imply P 6 = PSPACE. We present characterizations of SBH (QL) relations based on alternating Turing machines and on rst-order de nability, aswell as recursion-theoretic characterizations of function classes corresponding to SBH (QL).

. In this note we characterize iterated log depth circuit classes LD i and ND i by Cobham-like bounded recursion schemata. We also give alternative characterizations which utilizes the safe recursion method developed by Bellantoni and Cook. 1. Introduction The search for recursion theoretic characterizations of various complexity classes was began by A. Cobham [Cob], who characterized the class of polynomial time computable functions by a scheme now called bounded recursion on notation. (See also [Ro] for the proof.) The essence of this recursion scheme is two fold: firstly, on input x the recursive call is made for jxj O(1) times where jxj is the length of x, and . secondly, the growth rate is bounded by a previously defined polynomial time function. The second condition is crucial for the characterization of resource bounded computations since the computation on each recursive call takes the value of the function as an argument, so the number of steps that each recursive ...

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