The Boolean formula value problem is in alternating log time and, more generally, parenthesis context-free languages are in alternating log time. The evaluation of reverse Polish notation Boolean formulas is also in alternating log time. These results are optimal since the Boolean formula value problem is complete for alternating log time under deterministic log time reductions. Consequently, it is also complete for alternating log time under AC reductions.

We show that polynomial time truth-table reducibility via Boolean circuits to SAT is the same as logspace truth-table reducibility via Boolean formulas to SAT and the same as logspace Turing reducibility to SAT . In addition, we prove that a constant number of rounds of parallel queries to SAT is equivalent to one round of parallel queries.

We present a new complexity theoretic approach to incremental computation. We define complexity classes that capture the intuitive notion of incremental efficiency and study their relation to existing complexity classes. We show that problems that have small sequential space complexity also have small incremental time complexity. We show that all common LOGSPACE-complete problems for P are also incr-POLYLOGTIME-complete for P. We introduce a restricted notion of completeness called NRP-completeness and show that problems which are NRP-complete for P are also incr-POLYLOGTIME-complete for P. We also give incrementally complete problems for NLOGSPACE, LOGSPACE, and non-uniform NC¹. We show that under certain restrictions problems which have efficient dynamic solutions also have efficient parallel solutions. We also consider a non-uniform model of incremental computation and show that in this model most problems have almost linear complexity. In addition, we present some techniques f...

The class \Theta p 2 of languages polynomial-time truth-table reducible to sets in NP has a wide range of different characterizations. We consider several functional versions of \Theta p 2 based on these characterizations. We show that in this way the three function classes FL NP log , FP NP log , and FP NP k are obtained. In contrast to the language case the function classes seem to all be different. We give evidence in support of this fact by showing that FL NP log coincides with any of the other classes then L = P, and that the equality of the classes FP NP log and FP NP k would imply that the number of nondeterministic bits needed for the computation of any problem in NP can be reduced by a polylogarithmic factor, and that the problem can be computed deterministically with a sub-exponential time bound of order 2 n O(1= log log n) . 1 Introduction The study of nondeterministic computation is a central topic in structural complexity theory. The acceptance mechanism of...

A new parallel algorithm is given to evaluate a straight line program. The algorithm evaluates a program over a commutative semi-ring R of degree d and size n in time O(log n(log nd)) using M(n) processors, where M(n) is the number of processors required for multiplying n \Theta n matrices over the semi-ring R in O(log n) time. Appears in SIAM J. Comput., 17/4, pp. 687--695 (1988). Preliminary version of this paper appeared in [6]. y Research supported in part by National Science Foundation Grant MCS-800756 A01. z Research supported by NSF under ECS-8404866, the Semiconductor Research Corporation under RSCH 84-06-049-6, and by an IBM Faculty Development Award. x Research Supported in part by NSF Grant DCR-8504391 and by an IBM Faculty Development Award. 1 INTRODUCTION 1 1 Introduction In this paper we consider the problem of dynamic evaluation of a straight line program in parallel. This is a generalization of the result of Valiant et al [10]. They consider the problem of ta...

We analyze the computational complexity of type inference for untyped -terms in the second-order polymorphic typed -calculus (F 2 ) invented by Girard and Reynolds, as well as higher-order extensions F 3 ; F 4 ; : : : ; F ! proposed by Girard. We prove that recognizing the F 2 - typable terms requires exponential time, and for F ! the problem is nonelementary. We show as well a sequence of lower bounds on recognizing the F k -typable terms, where the bound for F k+1 is exponentially larger than that for F k . The lower bounds are based on generic simulation of Turing Machines, where computation is simulated at the expression and type level simultaneously. Non-accepting computations are mapped to non-normalizing reduction sequences, and hence non-typable terms. The accepting computations are mapped to typable terms, where higher-order types encode reduction sequences, and first-order types encode the entire computation as a circuit, based on a unification simulation of Boolean logic. ...

It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatorial tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. Surprisingly, we conclude that no particularly good or convincing examples are known. The examples of combinatorial tautologies that we consider seem to give at most a quasipolynomial speed-up of extended Frege proofs over Frege proofs, with the sole possible exception of tautologies based on a theorem of Frankl.

Recently a 1978 conjecture by Hartmanis [Har78] was resolved [CS95], following progress made by [Ogi95]. It was shown that there is no sparse set that is hard for P under logspace many-one reductions, unless P = LOGSPACE. We extend the results to the case of sparse sets that are hard under more general reducibilities. Our main results are as follows. (1) If there exists a sparse set that is hard for P under bounded truth-table reductions, then P = NC 2 . (2) If there exists a sparse set that is hard for P under randomized logspace reductions with one-sided error, then P = Randomized LOGSPACE. (3) If there exists an NP-hard sparse set under randomized polynomial-time reductions with one-sided error, then NP = RP. (4) If there exists a 2 (log n) O(1) -sparse hard set for P under truth-table reductions, then P ` DSPACE[(logn) O(1) ]. As a by-product of (4), we obtain a uniform O(log 2 n log log n) time parallel algorithm for computing the rank of a 2 log 2 n \Theta n matrix o...

foremost recognition of technical contributions to the computing community. The citation of Cook's achievements noted that "Dr. Cook has advanced our understanding of the complexity of computation in a significant and profound way. His seminal paper, The Complexity of Theorem Proving Procedures, presented at the 1971 ACM SIGACT Symposium on the Theory of Computing, laid the foundations for the theory of NP-completeness. The ensuing exploration of the boundaries and nature of the NP-complete class of problems has been one of the most active and important research activities in computer science for the last decade. Cook is well known for his influential results in fundamental areas of computer science. He has made significant contributions to complexity theory, to time-space tradeoffs in computation, and to logics for programming languages. His work is characterized by elegance and insights and has illuminated the very nature of computation." During 1970-1979, Cook did extensive work under grants from the

In 1985, Goldwasser, Micali and Rackoff formulated interactive proof systems as a tool for developing cryptographic protocols. Indeed, many exciting cryptographic results followed from studying interactive proof systems and the related concept of zero-knowledge. Interactive proof systems also have an important part in complexity theory merging the well established concepts of probabilistic and nondeterministic computation. This thesis will study the complexity of various models of interactive proof systems. A perfect zero-knowledge interactive protocol convinces a verifier that a string is in a language without revealing any additional knowledge in an information theoretic sense. This thesis will show that for any language that has a perfect zero-knowledge proof system, its complement has a short interactive protocol. This result implies that there are not any perfect zero-knowledge protocols for NP-complete languages unless the polynomial-time hierarchy collapses. Thus knowledge comp...