We present in a uniform manner simple two-sorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus.

We demonstrate that the class of rst order functional programs over lists which terminate by multiset path ordering and admit a polynomial quasi-interpretation, is exactly the class of function computable in polynomial time. The interest of this result lies (i) on the simplicity of the conditions on programs to certify their complexity, (ii) on the fact that an important class of natural programs is captured, (iii) and on potential applications on program optimizations. 1 Introduction This paper is part of a general investigation on the implicit complexity of a specication. To illustrate what we mean, we write below the recursive rules that computes the longest common subsequences of two words. More precisely, given two strings u = u1 um and v = v1 vn of f0; 1g , a common subsequence of length k is dened by two sequences of indices i 1 < < i k and j1 < < jk satisfying u i q = v j q . lcs(; y) ! 0 lcs(x; ) ! 0 lcs(i(x); i(y)) ! lcs(x; y) + 1 lcs(i(...

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 &quot; techniques of Leivant and Bellantoni&Cook. Key words. Circuit complexity � subrecursion. Subject classi cations. 68Q15, 03D20, 94C99. 1.

This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cut-free proof length and in terms of the lengths of resolution refutations. We then define the RazborovRudich notion of natural proofs of P NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various "feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e#ectively computable in the sense of being recursively computable.

We study the effect of polynomial interpretation termination proofs of deterministic (resp. non-deterministic) algorithms defined by confluent (resp. non-confluent) rewrite systems over data structures which include strings, lists and trees, and we classify them according to the interpretations of the constructors. This leads to the definition of six classes which turn out to be exactly the deterministic (resp. non-deterministic) poly-time, linear exponential-time and doubly linear exponential time computable functions when the class is based on conuent (resp. nonconfluent) rewrite systems. We also obtain a characterisation of the linear space computable functions. Finally, we demonstrate that functions with exponential interpretation termination proofs are super-elementary.

We design functional algebras that characterize various complexity classes of global functions. For this purpose, classical schemata from recursion theory are tailored for capturing complexity. In particular we present a functional analogue of first-order logic and describe algebras of the functions computable in nondeterministic logarithmic space, deterministic and nondeterministic polynomial time, and for the functions computable by AC¹-circuits.

this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quite young eld

Copyright Stephen Austin Bloch, 1992 All rights reserved. The dissertation of Stephen Bloch is approved, and it is acceptable in quality and form for publication on micro-

We de ne the notion of adding \small amounts &quot; of nondeterminism to a deterministic function class, and give a machine model � the result is a functional AC 0 closure of the deterministic class. We characterize, by the \safe parameters &quot; technique, the classes of functions computable in linear and in quasilinear time on a multi-tape Turing machine. We thencombine these two results by extending the \safe parameters &quot; characterizations to the functions computable in (quasi)linear time with small amounts of nondeterminism, and discuss implications for both sequential and parallel complexity.

Abstract not found