Mathematical Logic and Theoretical Computer Science

2020 ◽  
Author(s):  
David Kueker
Keyword(s):  
Mathematical Logic ◽  
Computer Science ◽  
Theoretical Computer Science ◽  
Theoretical Computer

Preface to Special Issue devoted to the memory of Sauro Tulipani

2008 ◽  
Vol 18 (1) ◽  
pp. 1-4 ◽  
Author(s):  
FLAVIO CORRADINI ◽  
CARLO TOFFALORI
Keyword(s):  
Mathematical Logic ◽  
Computer Science ◽  
Crucial Role ◽  
Recursion Theory ◽  
Theoretical Computer Science ◽  
Special Issue ◽  
Nineteen Thirties ◽  
Theoretical Computer ◽  
The Subject ◽  
The Church

2006 was a special year for both mathematical logic and computer science, as it celebrated Gödel's centenary. Although Gödel's work was mainly concerned with mathematics and metamathematics, the crucial role it had in the foundation of modern theoretical computer science is undeniable: for instance, one only has to remember Gödel's contributions to the birth of recursion theory as well as his part in the debate in the nineteen thirties on the subject of the Church Thesis.

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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