L. H. Harper

A class of boolean functions with linear combinational complexity

Abstract In this paper we investigate the combinational complexity of Boolean functions satisfying a certain property, P nk,m. A function of n variables has the P nk,m property if there are at least m functions obtainable from each way of restricting it to a subset of n - - k variables. We show that the complexity of a P n3,5 function is no less than 7n−4 6 , and this bound cannot be much improved. Further, we find that for each k, there are P nk,2k functions with complexity linear in n.

On the bandwidth of a Hamming graph

The bandwidth of the Hamming graph (the product, (Kn)d, of complete graphs) has been an open question for many years. Recently Berger-Wolf and Rheingold 1] pointed out that the bandwidth of a numbering of the Hamming graph may be interpreted as a measure of the effects of noise in the multi-channel transmission of data with that numbering. They also gave lower and upper bounds for it. In this paper we present better lower and upper bounds, showing that the bandwidth of (Kn)d is asymptotic to (2/?d)nd as d?∞.

Lower Bounds on Synchronous Combinational Complexity

Synchronous combinational complexity, a measure of the size of logic circuits without races, is investigated in this paper. The first author has presented a method for obtaining an

Some remarks on normalized matching

O(n\log n)

Linear programming, the global approach

lower bound to synchronous combinational complexity and has shown that this bound applies to “almost all” Boolean functions in n variables. However, he could not constructively exhibit functions to which the lower bound applied (although Wolfgang Paul did produce an example). In this paper we weaken and extend the hypothesis of the lower bound so that a larger class of functions satisfies it and apply it to the determinant and marriage functions of

Morphisms for the Maximum Weight Ideal Problem

GF(2)

A Sperner theorem for the interval order of a projective geometry

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A solution of the Sperner-Erdös problem

Abstract A new class of LYM orders is obtained, and several results about general LYM orders are proved. (1) Let A 1 ⊂ A 2 ⊂ … ⊂ A r be a chain of subsets of [ n ] = { l ,…, n }. Let 〈 a i 〉 and 〈 b i 〉 be two nondecreasing sequences with a i ⩽ b i for l ⩽ i ⩽ r . Then { X ⊂ [ n ]: a i ⩽ | ∩ A i |⩽ b i }, ordered by inclusion, is a poset having the LYM property. (2) The smallest regular covering of an LYM order has M ( P ) chains, where M ( P ) is the least common multiple of the rank sizes. (3) Every LYM order has a smallest regular covering with at most || − h ( P ) classes of distinct chains, where h ( P ) is the height of P . To obtain (3), we discuss “minimal sets” of covering relations between two adjacent levels of an LYM-order.

Accidental combinatorist, an autobiography

Many optimization problems require the maximizing or minimizing of a linear function subject to linear constraints. Such problems, called linear programs (l.p.‘s), were intensively investigated in the years after WWII and a beautiful theory developed. The basic results of this theory, such as 1.~. duality and the simplex algorithm, have become standard fare for combinatorial mathematicians. (Cf., Hall’s book 141, or the more recent one by Lawler [ 71.) In this paper we shall investigate l.p.‘s from the point of view of category theory. We first ask whether l.p.‘s and related problems have morphisms, i.e., transformations which preserve the defining characteristics of the problem. If the answer is yes, we then ask what structure the resulting category has and what that implies about the problem. For the definitions and results of category theory which we use here, the reader is referred to MacLane’s book [S].

Optimal numberings and isoperimetric problems on graphs

A notion of morphism for the Maximum Weight Ideal Problem is defined and applied to solve the Edge?Isoperimetric Problem on the graphs of regular solids and their products.