The depth of all Boolean functions

It is shown that every Boolean function of n arguments has a circuit of depth n+1 over the basis {f|f:{0,1}^2 -> {0,1}}

Some results on circuit depth

An important problem in theoretical computer science is to develop methods for estimating the complexity of finite functions. For many familiar functions there remain important gaps between the best known lower and upper bound we investigate the inherent complexity of Boolean functional taking circuits as our model of computation and depth (or delay)to be the measure of complexity. The relevance of circuits as a model of computation for Boolean functions stems from the fact that Turing machine computations may be efficiently simulated by circuits. Important relations among various measures of circuit complexity are btained as well as bounds on the maximum depth of any function and of any monotone function. We then give a detailed account of the complexity of NAND circuits for several important functions and pursue an analysis of the important set of symmetric functions. A number of gap theorems for symmetric functions are exhibited and these are contrasted with uniform hierarchies for several large sets of functions. Finally, we describe several short formulae for threshold functions

Planar acyclic computation

This paper considers the following problem: given a specification consisting of a set of variables X, a multiset of functions F on those variables, and a cyclic ordering on X U F, determine whether or not there exists a planar acyclic circuit which realises the specification. An algorithm is given which produces such a circuit whenever one exists. In proving that our algorithm meets this requirement we provide a simple mathematical characterisation of those specifications which are realisable

Some results on circuit depth

An important problem in theoretical computer science is to develop methods for estimating the complexity of finite functions. For many familiar functions there remain important gaps between the best known lower and upper bound we investigate the inherent complexity of Boolean functional taking circuits as our model of computation and depth (or delay)to be the measure of complexity. The relevance of circuits as a model of computation for Boolean functions stems from the fact that Turing machine computations may be efficiently simulated by circuits. Important relations among various measures of circuit complexity are btained as well as bounds on the maximum depth of any function and of any monotone function. We then give a detailed account of the complexity of NAND circuits for several important functions and pursue an analysis of the important set of symmetric functions. A number of gap theorems for symmetric functions are exhibited and these are contrasted with uniform hierarchies for several large sets of functions. Finally, we describe several short formulae for threshold functions