In this paper we describe a new method for establishing lower bounds for the number of multiplications and divisions required to compute rational functions. We shall start by reminding the reader of some standard notations.

This paper presents new algorithms for the parallel evaluation of certain polynomial expression. In particular, for the parallel evaluation of xⁿ,we introduce an algorithm which takes two steps of parallel division and [log₂n] steps of parallel addition,...

One of the most important open questions in the field of computational complexity is the question of whether there is a polynomial time decision procedure for the classical propositional calculus.

The purpose of the present paper is to study a question ...

A striking example of practical tradeoffs between storage space and execution time is provided by the IBM 1401 Fortran compiler.

On another level, there is an interesting relation between the time and storage required to recognize context free ...

It is well known from the work of Motzkin [55], Belaga [58] and Pan [66], that “most” n^th degree polynomials p ε R[x] require about n/2 ×, ÷ ops and n ± ops and that these bounds can always be achieved within the framework of preconditioned evaluation ^(...

Limiting consideration to algorithms satisfying various numerical stability requirements may change lower bounds for computational complexity and/or make lower bounds easier to prove. We will show that, under a sufficiently strong restriction upon ...

The parallel derivative of a set of strings is introduced. Given a serial, repetition-free parallel program schema, its closure is constructed by taking parallel derivatives of its set of computations. The construction resembles the construction of a ...

The complexity of decision procedures for the Weak Monadic Second-Order Theories of the Natural Numbers are considered. If only successor is allowed as a primitive, then every alternation of second-order quantifiers causes an exponential increase in the ...

Random access machines (RAMs) are usually defined to have registers that hold integers. While this captures in part the structure of a commercial computer, it overlooks an implementation-dependent feature of most binary oriented machines, namely their ...

We define polynomial time computable operator. Our definition generalizes Cook's definition to arbitrary function inputs. Polynomial classes are defined in terms of these operators; the properties of these classes are investigated. Honest polynomial ...

An open problem, posed by A. Rosenberg [R], motivates the consideration of representations of graphs and the effect of these representations on the efficiency of algorithms which determine properties of unlabelled graphs. In this paper we investigate ...

We are interested in combinational circuits synthesized from and-gates and or-gates. We first show that n³ distinct and-gate inputs are needed to form the product of two Boolean matrices, and hence O(n³) two-input and-gates are needed to compute the ...

The results of Cook and Karp ([K], [C]) aroused considerable interest for at least two reasons. First, the answer to a long-standing open question which had seemed peculiar to automata theory—whether deterministic and nondeterministic polynomial-time-...