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In order to study the axiomatization of the construct over possibly non-halting programs and tests, this paper introduces the notion of -sets by considering the tests from an abstract -algebra. When the -algebra is an ada, the axiomatization is shown to be complete by obtaining a subdirect representation of -sets. Further, this paper considers the equality test with the construct and gives a complete axiomatization through the notion of agreeable -sets. [ABSTRACT FROM AUTHOR]

Intuitionistic fuzzy (IF) rough sets are the generalization of traditional rough sets obtained by combining the IF set theory and the rough set theory. The existing research on IF rough sets mainly concentrates on the establishment of lower and upper approximation operators using constructive and axiomatic approaches. Less effort has been put on the attribute reduction of databases based on IF rough sets. This paper systematically studies attribute reduction based on IF rough sets. Firstly, attribute reduction with traditional rough sets and some concepts of IF rough sets are reviewed. Then, we introduce some concepts and theorems of attribute reduction with IF rough sets, and completely investigate the structure of attribute reduction. Employing the discernibility matrix approach, an algorithm to find all attribute reductions is also presented. Finally, an example is proposed to illustrate our idea and method. Altogether, these findings lay a solid theoretical foundation for attribute reduction based on IF rough sets. [ABSTRACT FROM AUTHOR]

It is here shown how to use pieces of macroscopic thermodynamics to generate microscopic probability distributions for generalized ensembles, thereby directly connecting macro-state-axiomatics with microscopic results. [ABSTRACT FROM AUTHOR]

In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2ω = ω2 and that $L({\mathscr{P}}(\omega_1))$ satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that □(κ) fails for all regular κ > ω1. [ABSTRACT FROM AUTHOR]