6 research outputs found
Set systems: order types, continuous nondeterministic deformations, and quasi-orders
By reformulating a learning process of a set system L as a game between
Teacher and Learner, we define the order type of L to be the order type of the
game tree, if the tree is well-founded. The features of the order type of L
(dim L in symbol) are (1) We can represent any well-quasi-order (wqo for short)
by the set system L of the upper-closed sets of the wqo such that the maximal
order type of the wqo is equal to dim L. (2) dim L is an upper bound of the
mind-change complexity of L. dim L is defined iff L has a finite elasticity (fe
for short), where, according to computational learning theory, if an indexed
family of recursive languages has fe then it is learnable by an algorithm from
positive data. Regarding set systems as subspaces of Cantor spaces, we prove
that fe of set systems is preserved by any continuous function which is
monotone with respect to the set-inclusion. By it, we prove that finite
elasticity is preserved by various (nondeterministic) language operators
(Kleene-closure, shuffle-closure, union, product, intersection,. . ..) The
monotone continuous functions represent nondeterministic computations. If a
monotone continuous function has a computation tree with each node followed by
at most n immediate successors and the order type of a set system L is
{\alpha}, then the direct image of L is a set system of order type at most
n-adic diagonal Ramsey number of {\alpha}. Furthermore, we provide an
order-type-preserving contravariant embedding from the category of quasi-orders
and finitely branching simulations between them, into the complete category of
subspaces of Cantor spaces and monotone continuous functions having Girard's
linearity between them. Keyword: finite elasticity, shuffle-closur
Complete Randomized Cutting Plane Algorithms for Propositional Satisfiability
The propositional satisfiability problem (SAT) is a fundamental problem in computer science and combinatorial optimization. A considerable number of prior researchers have investigated SAT, and much is already known concerning limitations of known algorithms for SAT. In particular, some necessary conditions are known, such that any algorithm not meeting those conditions cannot be efficient. This paper reports a research to develop and test a new algorithm that meets the currently known necessary conditions.
In chapter three, we give a new characterization of the convex integer hull of SAT, and two new algorithms for finding strong cutting planes. We also show the importance of choosing which vertex to cut, and present heuristics to find a vertex that allows a strong cutting plane. In chapter four, we describe an experiment to implement a SAT solving algorithm using the new algorithms and heuristics, and to examine their effectiveness on a set of problems. In chapter five, we describe the implementation of the algorithms, and present computational results. For an input SAT problem, the output of the implemented program provides either a witness to the satisfiability or a complete cutting plane proof of satisfiability. The description, implementation, and testing of these algorithms yields both empirical data to characterize the performance of the new algorithms, and additional insight to further advance the theory. We conclude from the computational study that cutting plane algorithms are efficient for the solution of a large class of SAT problems