We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is deenable in Monadic Second Order Logic. We show that for bounded tree-width these problems are solvable in polynomial time. The same holds for bounded clique width in the cases, where the decomposition , which establishes the bound on the clique{width, can be computed in polynomia...

In this paper, we study the arithmetics of skew polynomial rings over finite fields, mostly from an algorithmic point of view. We give various algorithms for fast multiplication, division and extended Euclidean division. We give a precise description of quotients of skew polynomial rings by a left principal ideal, using results relating skew polynomial rings to Azumaya algebras. We use this des...

An algorithm and its first implementation in C# are presented for assembling arbitrary quantum circuits on the base of Hadamard and Toffoli gates and for constructing multivariate polynomial systems over the finite field Z2 arising when applying the Feynman’s sum-over-paths approach to quantum circuits. The matrix elements determined by a circuit can be computed by counting the number of common...

Let S be an abelian semigroup, and A a finite subset of S. The sumset hA consists of all sums of h elements of A, with repetitions allowed. Let |hA| denote the cardinality of hA. Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial p(t) such that |hA| = p(h) for all sufficiently large h. Lattice point ...

The parallel complexity class NC has many equivalent models such as polynomial size formulae and bounded width branching programs. Caussinus et al. [CMTV98] considered arithmetizations of two of these classes, #NC and #BWBP. We further this study to include arithmetization of other classes. In particular, we show that counting paths in branching programs over visibly pushdown automata is in FLo...

We investigate the complexity of 1) computing the characteristic polynomial, the minimal polynomial, and all the invariant factors of an integer matrix, and of 2) verifying them, when the coefficients are given as input. It is known that each coefficient of the characteristic polynomial of a matrix A is computable in GapL, and the constant term, the determinant of A, is complete for GapL. We sh...

We provide some applications of a polynomial criterion for difference sets. These include counting the sets with specified parameters in terms Hilbert functions, particular count bent functions. also consider question about bentness certain Boolean functions introduced by Carlet when $\mathcal{C}$-condition him doesn't hold.

We design deterministic fully polynomial-time approximation scheme (FPTAS) for computing the partition function for a class of multi-spin systems, extending the known approximable regime by an exponential scale. As a consequence, we have an FPTAS for the Potts models with inverse temperature β up to a critical threshold |β| = O( 1 ∆ ) where ∆ is the maximum degree, confirming a conjecture in [1...

First-order model counting emerged recently as a novel reasoning task, at the core of efficient algorithms for probabilistic logics such as MLNs. For certain subsets of first-order logic, lifted model counters were shown to run in time polynomial in the number of objects in the domain of discourse, where propositional model counters require exponential time. However, these guarantees apply only...

An algorithm is presented for exactly solving (in fact, counting) the number of maximum weight satisfying assignments of a 2-Cnf formula. The worst case running time of O(1.246) for formulas with n variables improves on the previous bound of O(1.256) by Dahllöf, Jonsson, and Wahlström. The algorithm uses only polynomial space. As a consequence we get an O(1.246) time algorithm for counting maxi...