Simplified theory of Boolean functions
Date
2018-06-22
Authors
Lui, Patrick Kam
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Abstract
A new, intuitive approach to the study of a Boolean function using its set of parities of subfunctions called the parity spectrum is presented. This approach simplifies the classical theory of Boolean difference, and serves to unify and extend a number of previous results on the modulo-2 logic design and fault detection of digital logic networks. Fundamental properties of the parity spectrum are established. They are instrumental in developing the principal results.
New algebraic and geometric representations for fixed polarity and fixed basis modulo-2 canonical expansions (FPEs and FBEs) are obtained by identifying coefficients in these expansions to subfunction parities in the parity spectrum. These representations offer new insights into the underlying structure of modulo-2 canonical expansions as well as algorithms that manipulate them.
Boolean matrix transforms among the parity spectrum, the FPEs, and the FBEs are described in a unified manner using Kronecker products, and efficient recursive algorithms derived for these and other transforms are applied to two different approaches to the minimization of FPEs and FBEs.
By verifying subfunction parities from the parity spectrum of the function implemented by a digital logic network, the generalized constrained parity testing technique is developed. It is considered for detecting multiple stuck-at faults in single-output combinational networks.
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Keywords
Computer science, Mathematics, Algebra, Boolean