3,348 research outputs found
Scaled Boolean Algebras
Scaled Boolean algebras are a category of mathematical objects that arose
from attempts to understand why the conventional rules of probability should
hold when probabilities are construed, not as frequencies or proportions or the
like, but rather as degrees of belief in uncertain propositions. This paper
separates the study of these objects from that not-entirely-mathematical
problem that motivated them. That motivating problem is explicated in the first
section, and the application of scaled Boolean algebras to it is explained in
the last section. The intermediate sections deal only with the mathematics. It
is hoped that this isolation of the mathematics from the motivating problem
makes the mathematics clearer.Comment: 53 pages, 8 Postscript figures, Uses ajour.sty from Academic Press,
To appear in Advances in Applied Mathematic
A Sound and Complete Axiomatization of Majority-n Logic
Manipulating logic functions via majority operators recently drew the
attention of researchers in computer science. For example, circuit optimization
based on majority operators enables superior results as compared to traditional
logic systems. Also, the Boolean satisfiability problem finds new solving
approaches when described in terms of majority decisions. To support computer
logic applications based on majority a sound and complete set of axioms is
required. Most of the recent advances in majority logic deal only with ternary
majority (MAJ- 3) operators because the axiomatization with solely MAJ-3 and
complementation operators is well understood. However, it is of interest
extending such axiomatization to n-ary majority operators (MAJ-n) from both the
theoretical and practical perspective. In this work, we address this issue by
introducing a sound and complete axiomatization of MAJ-n logic. Our
axiomatization naturally includes existing majority logic systems. Based on
this general set of axioms, computer applications can now fully exploit the
expressive power of majority logic.Comment: Accepted by the IEEE Transactions on Computer
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