Book contents
- Frontmatter
- Contents
- Preface
- List of Participants
- Relationships Between Monotone and Non-Monotone Network Complexity
- On Read-Once Boolean Functions
- Boolean Function Complexity: a Lattice-Theoretic Perspective
- Monotone Complexity
- On Submodular Complexity Measures
- Why is Boolean Complexity Theory so Difficult?
- The Multiplicative Complexity of Boolean Quadratic Forms, a Survey
- Some Problems Involving Razborov-Smolensky Polynomials
- Symmetry Functions in AC° can be Computed in Constant Depth with Very Small Size
- Boolean Complexity and Probabilistic Constructions
- Networks Computing Boolean Functions for Multiple Input Values
- Optimal Carry Save Networks
Relationships Between Monotone and Non-Monotone Network Complexity
Published online by Cambridge University Press: 23 September 2009
- Frontmatter
- Contents
- Preface
- List of Participants
- Relationships Between Monotone and Non-Monotone Network Complexity
- On Read-Once Boolean Functions
- Boolean Function Complexity: a Lattice-Theoretic Perspective
- Monotone Complexity
- On Submodular Complexity Measures
- Why is Boolean Complexity Theory so Difficult?
- The Multiplicative Complexity of Boolean Quadratic Forms, a Survey
- Some Problems Involving Razborov-Smolensky Polynomials
- Symmetry Functions in AC° can be Computed in Constant Depth with Very Small Size
- Boolean Complexity and Probabilistic Constructions
- Networks Computing Boolean Functions for Multiple Input Values
- Optimal Carry Save Networks
Summary
Abstract
Monotone networks have been the most widely studied class of restricted Boolean networks. It is now possible to prove superlinear (in fact exponential) lower bounds on the size of optimal monotone networks computing some naturally arising functions. There remains, however, the problem of obtaining similar results on the size of combinational (i.e. unrestricted) Boolean networks. One approach to solving this problem would be to look for circumstances in which large lower bounds on the complexity of monotone networks would provide corresponding bounds on the size of combinational networks.
In this paper we briefly review the current state of results on Boolean function complexity and examine the progress that has been made in relating monotone and combinational network complexity.
Introduction
One of the major problems in computational complexity theory is to develop techniques by which non-trivial lower bounds, on the amount of time needed to solve ‘explicitly defined’ decision problems, could be proved. By ‘nontrivial’ we mean bounds which are superlinear in the length of the input; and, since we may concentrate on functions with a binary input alphabet, the term ‘explicitly defined’ may be taken to mean functions for which the values on all inputs of length n can be enumerated in time 2cn for some constant c.
Classical computational complexity theory measures ‘time’ as the number of moves made by a (multi-tape) deterministic Turing machine. Thus a decision problem, f, has time complexity, T(n) if there is a Turing machine program that computes f and makes at most T(n) moves on any input of length n.
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- Boolean Function Complexity , pp. 1 - 24Publisher: Cambridge University PressPrint publication year: 1992
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