We present a specialised (polynomial-based) rule for the propositional logic called the Independence Rule, which is useful to compute the conservative retractions of propositional logic theories. In this paper we show the soundness and completeness of the logical calculus based on this rule, as well as other applications. The rule is defined by means of a new kind of operator on propositional f...

We consider the problem of approximating the size of a minimum non-extendible independent set of a graph, also known as the minimum dominating independence number. We strengthen a result of Irving [2] to show that there is no constant > 0 for which this problem can be approximated within a factor of n 10 in polynomial time, unless P = NP. This is the strongest lower bound we are aware of for po...

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Let Γ be a simple graph and I ( x ) its multivariate independence polynomial. The main result of this paper is the characterization chordal graphs as only for which power series expansion − 1 Horn hypergeometric.

A stable set in a graph G is a set of pairwise non-adjacent vertices and α(G) is the size of a maximum stable set in the graph G. The polynomial I(G; x) = s0 + s1x + s2x + ... + sαx, α = α(G), is called the independence polynomial of G (Gutman and Harary, 1983), where sk is the number of stable sets of cardinality k in G. I(G; x) is partial unimodal if the sequence of its coefficients (sk) is p...

The independence number of a graph and its chromatic number are known to be hard to approximate. Due to recent complexity results, unless coRP = NP , there is no polynomial time algorithm which approximates any of these quantities within a factor of n1−2 for graphs on n vertices. We show that the situation is significantly better for the average case. For every edge probability p = p(n) in the ...

This work analyzes the statistical properties of the SHA-3 candidate cryptographic hash algorithms CubeHash and Skein to try to find nonrandom behavior. Cube tests were used to probe each algorithm’s internal polynomial structure for a large number of choices of the polynomial input variables. The cube test data were calculated on a 40-core hybrid SMP cluster parallel computer. The cube test da...

We show that Csanky’s fast parallel algorithm for computing the characteristic polynomial of a matrix can be formalized in the logical theory LAP, and can be proved correct in LAP from the principle of linear independence. LAP is a natural theory for reasoning about linear algebra introduced in [8]. Further, we show that several principles of matrix algebra, such as linear independence or the C...

Matroids were introduced in the early 20th century as a way of uniting disparate notions of “independence” from across mathematics. Among these notions were linear independence of vectors and graphic independence – defined by acyclicity on the subgraph corresponding to a set of edges. Algebraic independence over a field k, defined by the non-existence of polynomial relations with coefficients i...