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...

Let k be a positive integer. There is a longest finite sequence x 1 ,...,x n in k letters in which no consecutive block x i ,...,x 2i is a subsequence of any other consecutive block x j ,...,x 2j . Let n(k) be this longest length. We prove that n(1) = 3, n(2) = 11, and n(3) is incomprehensibly large. We give a lower bound for n(3) in terms of the familiar Ackerman hierarchy. We also give asymptotic upper and lower bounds for n(k). We view n(3) as a particularly elemental description of an incomprehensibly large integer. Related problems involving binary sequences (two letters) are also addressed. We also report on some recent computer explorations of R. Dougherty which we use to raise the lower bound for n(3). TABLE OF CONTENTS 1. Finiteness, and n(1),n(2). 2. Sequences of fixed length sequences. 3. The Main Lemma. 4. Lower bound for n(3). 5. The function n(k). 6. Related problems and computer explorations. 1. FINITENESS, AND n(1),n(2) We use Z for the set of all integers, Z + f...

This paper is a contribution to the formal study and analysis of vernacular forms of program derivation. Specifically, in this paper, our vernacular derivations are elementary program transformations over the natural numbers. We provide an intensional semantics for these transformations within the derivations of the Elementary theory of Operations and Numbers, EON, [Bee85]. This semantics is intensional in the sense that the computational content of a derivation associated with a transformation is equal, up to the intensional equality underlying the theory EON, to the computational content of the transformation itself. The interpretation enables us to underwrite the correctness of the program transformations and, further, provides an analysis of correctness by classifying, via schema, the operations available by these transformations. 2 Introduction What is the relationship between vernacular and formalised arguments? We are interested in developing methods and results which can be...

It is well-known by now that large parts of (non-constructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the Bolzano-Weierstra principle, the existence of a limit superior for bounded sequences etc.) which are known to be equivalent to arithmetical comprehension (relative to T ) and therefore go far beyond the strength of PRA (when added to T ). In this paper

this paper is to investigate the impact on the design of a programming language of tight integration of a language processor with a theorem prover (intelligent proof assistant). What improvements in syntax, semantics, and computation do we gain by this? We assume as self-evident the obvious gain which is quite substantial. The language obtains a sound semantics by its interpretation into a formal logical theory. By proving theorems about our programs we can prove them correct. From the many language systems with theorem provers we mention just two: PVS and Isabelle/HOL [6, 5]. Both of them are more general theorem provers than programming languages. On the other hand, our system CL (Clausal Language) , has been designed as a programming language from the start. It is not a language which can be used for industrial applications (yet, we hope), but neither is it a toy language. It has been used in our undergraduate teaching for seven years now. About four hundred students yearly actively use it in the four courses based on CL [2]. While, the formal basis for PVS and Isabelle is high order logic with typed functionals, CL is based on the simplest non-trivial formal theory: Peano Arithmetic (PA)

We give two simple uniform characterizations of the complexity classes L NL P NP by register machine programs. Both characterizations are intrinsic because they do not refer to any time and space bounds. The rst characterization, which permits programs to peek into their computation history, captures also Pspace and it is completely syntactical in that the restrictions on programs are decided from their syntax. The programs computing predicates of larger classes are permitted to access larger parts of their history. The second characterization restricts the access of programs to parts of their input which are then projected away by existential quantication. 1

We present some new decision and comparison problems of unusually high computational complexity. Most of the problems are strictly combinatorial in nature; others involve basic logical notions. Their complexities range from iterated exponential time completeness to # 0 time completeness to #(# # ,0) time completeness to #(# # ,,0) time completeness. These three ordinals are well known ordinals from proof theory, and their associated complexity classes represent new levels of computational complexity for natural decision problems. Proofs will appear in an extended version of this manuscript to be published elsewhere. 1. Iterated exponential time - universal relational sentences Let F be a function from A* into B*, where A,B are finite alphabets. We say that F is iterated exponential time computable if and only if there is a multitape Turing machine TM (which processes inputs from A* and outputs from B*) and an integer constant c > 0 such that TM computes F(x) with run time at most...

. 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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