An Abstract Complexity Theory for Boolean Functions
Koob, Gary Michael
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https://hdl.handle.net/2142/69353
Description
Title
An Abstract Complexity Theory for Boolean Functions
Author(s)
Koob, Gary Michael
Issue Date
1987
Department of Study
Electrical Engineering
Discipline
Electrical Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Computer Science
Abstract
Traditional theories measure the complexity of a function as a lower bound on the cost of its implementation. Since the results of the theory are biased by the chosen implementation paradigm and its associated cost measure, little insight is gained into those structural properties of a problem that affect all implementations. It is our assertion that--at least in the limited domain of Boolean functions--these properties can be revealed through rigorous, mathematical analysis and summarized in a single number characterizing the problem's inherent complexity. The proposed complexity measure is based on a fusion of dependence concepts from switching theory and structural analysis from information theory and, along with its extension to incompletely-specified functions, serves as the foundation of a theory of Boolean functions. This theory comprises three branches, each of which describes a different fundamental mode of interaction between functions. Joint Computation Theory (JCT) describes the "parallel" interaction of two functions through a measure called correlation. Direct Computation Theory (DCT) defines a function-containment relation ostensibly based on the Boolean operation of projection. Finally, Relative Computation Theory (RCT) describes the "sequential" interaction between two functions through a measure called effective work. Direct Computation Theory permits the rigorous definition of a family of Boolean functions, whose study, in turn, reveals interesting relationships among the three theories. The theory of families is used to characterize the computations of finite state machines.
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