Title:
Optimized Blist Form (OBF)
Optimized Blist Form (OBF)
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Author(s)
Rossignac, Jarek
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Abstract
Any Boolean expressions may be converted into positive-form, which has only union and intersection operators.
Let E be a positive-form expression with n literals. Assume that the truth-values of the literals are read one at a
time. The numbers s(n) of steps (operations) and b(n) of working memory bits (footprint) needed to evaluate E
depend on E and on the evaluation technique. A recursive evaluation performs s(n)=n–1 steps but requires
b(n)=log(n)+1 bits. Evaluating the disjunctive form of E uses only b(n)=2 bits, but may lead to an exponential
growth of s(n). We propose a new Optimized Blist Form (OBF) that requires only s(n)=n steps and b(n)=⌈log2j⌉
bits, where j=⌈log2(2n/3+2)⌉. We provide a simple and linear cost algorithm for converting positive-form
expressions to their OBF. We discuss three applications: (1) Direct CSG rendering, where a candidate surfel
stored at a pixel is classified against an arbitrarily complex Boolean expression using a footprint of only 6
stencil bits; (2) the new Logic Matrix (LM), which evaluates any positive form logical expression of n literals in
a single cycle and uses a matrix of at most n×j wire/line connections; and (3) the new Logic Pipe (LP), which
uses n gates that are connected by a pipe of ⌈log2j⌉ lines and when receiving a staggered stream of input vectors
produces a value of a logical expression at each cycle.
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Date Issued
2007-05-23
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Text
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Technical Report