14,962 research outputs found
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
Ćukasiewicz-Moisil Many-Valued Logic Algebra of Highly-Complex Systems
A novel approach to self-organizing, highly-complex systems (HCS), such as living organisms and artificial intelligent systems (AIs), is presented which is relevant to Cognition, Medical Bioinformatics and Computational Neuroscience. Quantum Automata (QAs) were defined in our previous work as generalized, probabilistic automata with quantum state spaces (Baianu, 1971). Their next-state functions operate through transitions between quantum states defined by the quantum equations of motion in the Schroedinger representation, with both initial and boundary conditions in space-time. Such quantum automata operate with a quantum logic, or Q-logic, significantly different from either Boolean or Ćukasiewicz many-valued logic. A new theorem is proposed which states that the category of quantum automata and automata--homomorphisms has both limits and colimits. Therefore, both categories of quantum automata and classical automata (sequential machines) are bicomplete. A second new theorem establishes that the standard automata category is a subcategory of the quantum automata category. The quantum automata category has a faithful representation in the category of Generalized (M,R)--Systems which are open, dynamic biosystem networks with defined biological relations that represent physiological functions of primordial organisms, single cells and higher organisms
Fredkin Gates for Finite-valued Reversible and Conservative Logics
The basic principles and results of Conservative Logic introduced by Fredkin
and Toffoli on the basis of a seminal paper of Landauer are extended to
d-valued logics, with a special attention to three-valued logics. Different
approaches to d-valued logics are examined in order to determine some possible
universal sets of logic primitives. In particular, we consider the typical
connectives of Lukasiewicz and Godel logics, as well as Chang's MV-algebras. As
a result, some possible three-valued and d-valued universal gates are described
which realize a functionally complete set of fundamental connectives.Comment: 57 pages, 10 figures, 16 tables, 2 diagram
Counting inequivalent monotone Boolean functions
Monotone Boolean functions (MBFs) are Boolean functions satisfying the monotonicity condition for any . The number of MBFs in n variables is
known as the th Dedekind number. It is a longstanding computational
challenge to determine these numbers exactly - these values are only known for
at most 8. Two monotone Boolean functions are inequivalent if one can be
obtained from the other by renaming the variables. The number of inequivalent
MBFs in variables was known only for up to . In this paper we
propose a strategy to count inequivalent MBF's by breaking the calculation into
parts based on the profiles of these functions. As a result we are able to
compute the number of inequivalent MBFs in 7 variables. The number obtained is
490013148
Do Hard SAT-Related Reasoning Tasks Become Easier in the Krom Fragment?
Many reasoning problems are based on the problem of satisfiability (SAT).
While SAT itself becomes easy when restricting the structure of the formulas in
a certain way, the situation is more opaque for more involved decision
problems. We consider here the CardMinSat problem which asks, given a
propositional formula and an atom , whether is true in some
cardinality-minimal model of . This problem is easy for the Horn
fragment, but, as we will show in this paper, remains -complete (and
thus -hard) for the Krom fragment (which is given by formulas in
CNF where clauses have at most two literals). We will make use of this fact to
study the complexity of reasoning tasks in belief revision and logic-based
abduction and show that, while in some cases the restriction to Krom formulas
leads to a decrease of complexity, in others it does not. We thus also consider
the CardMinSat problem with respect to additional restrictions to Krom formulas
towards a better understanding of the tractability frontier of such problems
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