A Family of Polymodal Systems and its Application to Generalized Possibility Measures and Multi-Rough Sets

2006 ◽  
Vol 10 (5) ◽  
pp. 625-632 ◽  
Author(s):  
Sadaaki Miyamoto ◽  
◽  
Tetsuya Murai ◽  
Yasuo Kudo ◽  
◽  
...  
Keyword(s):  
Information System ◽  
Computer Science ◽  
Relational Database ◽  
Rough Sets ◽  
Theoretical Computer Science ◽  
Theoretical Frameworks ◽  
Theoretical Computer ◽  
The One ◽  
Information Tables ◽  
Polymodal Logic

Polymodal systems generally have large areas of applications to theoretical computer science including the theory of programming, while other applications are not yet fully explored. In this paper we consider a family of polymodal systems with the structure of lattices on the polymodal indices. After investigating theory of the polymodal systems such as the completeness, we study two applications. One is generalized possibility measures in which lattice-valued measures are proposed and relations with the ordinary possibility and necessity measures are uncovered. Second application is consideration of an information system as a table such as the one in the relational database. It is known that rough sets are used to discover regularities from such information tables. Applying polymodal logic concept, we generalize rough sets which are called multi-rough sets here. Our consideration is mainly to establish theoretical frameworks in these two application areas and hence no real examples are shown here.

scholarly journals Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials

2021 ◽  
Author(s):  
Mareike Dressler ◽  
Adam Kurpisz ◽  
Timo de Wolff
Keyword(s):  
Computer Science ◽  
Affine Transformation ◽  
Optimization Problems ◽  
Polynomial Optimization ◽  
Theoretical Computer Science ◽  
Sums Of Squares ◽  
Theoretical Computer ◽  
Complexity Bounds ◽  
Polynomial Optimization Problems ◽  
Transformation Of Variables

AbstractVarious key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems is based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS-based certificates remain valid: First, for polynomials, which are nonnegative over the n-variate boolean hypercube with constraints of degree d there exists a SONC certificate of degree at most $$n+d$$ n + d . Second, if there exists a degree d SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree d SONC certificate that includes at most $$n^{O(d)}$$ n O ( d ) nonnegative circuit polynomials. Moreover, we prove that, in opposite to SOS, the SONC cone is not closed under taking affine transformation of variables and that for SONC there does not exist an equivalent to Putinar’s Positivstellensatz for SOS. We discuss these results from both the algebraic and the optimization perspective.

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