The Boolean SATisfiability Problem and the orthogonal group O(n)O(n)

We explore the relations between the Boolean Satisfiability Problem with $n$ literals and the orthogonal group $O(n)$ and show that all solutions lie in the compact and disconnected real manifold of dimension $n (n-1)/2$ of this group.Comment: 11 pages, no figures, 6 reference

A Spinorial Formulation of the Maximum Clique Problem of a Graph

We present a new formulation of the maximum clique problem of a graph in complex space. We start observing that the adjacency matrix A of a graph can always be written in the form A = B B where B is a complex, symmetric matrix formed by vectors of zero length (null vectors) and the maximum clique problem can be transformed in a geometrical problem for these vectors. This problem, in turn, is translated in spinorial language and we show that each graph uniquely identifies a set of pure spinors, that is vectors of the endomorphism space of Clifford algebras, and the maximum clique problem is formalized in this setting so that, this much studied problem, may take advantage from recent progresses of pure spinor geometry

The Extended Fock Basis of Clifford Algebra

We investigate the properties of the Extended Fock Basis (EFB) of Clifford algebras introduced in [1]. We show that a Clifford algebra can be seen as a direct sum of multiple spinor subspaces that are characterized as being left eigenvectors of \Gamma. We also show that a simple spinor, expressed in Fock basis, can have a maximum number of non zero coordinates that equals the size of the maximal totally null plane (with the notable exception of vectorial spaces with 6 dimensions).Comment: Minimal corrections to the published versio

On Spinors of Zero Nullity

We present a necessary and sufficient condition for a spinor \u3c9 to be of nullity zero, i.e. such that for any null vector v, v\u3c9 \u338= 0. This dives deeply in the subtle relations between a spinor \u3c9 and \u3c9c, the (complex) conjugate of \u3c9 belonging to the same spinor space

On Spinors Transformations

We begin showing that for even dimensional vector spaces $V$ all automorphisms of their Clifford algebras are inner. So all orthogonal transformations of $V$ are restrictions to $V$ of inner automorphisms of the algebra. Thus under orthogonal transformations $P$ and $T$ - space and time reversal - all algebra elements, including vectors $v$ and spinors $\varphi$, transform as $v \to x v x^{-1}$ and $\varphi \to x \varphi x^{-1}$ for some algebra element $x$. We show that while under combined $PT$ spinor $\varphi \to x \varphi x^{-1}$ remain in its spinor space, under $P$ or $T$ separately $\varphi$ goes to a 'different' spinor space and may have opposite chirality. We conclude with a preliminary characterization of inner automorphisms with respect to their property to change, or not, spinor spaces.Comment: Minor changes to propositions 1 and

On Computational Complexity of Clifford Algebra

After a brief discussion of the computational complexity of Clifford algebras, we present a new basis for even Clifford algebra Cl(2m) that simplifies greatly the actual calculations and, without resorting to the conventional matrix isomorphism formulation, obtains the same complexity. In the last part we apply these results to the Clifford algebra formulation of the NP-complete problem of the maximum clique of a graph introduced in a previous paper.Comment: 13 page

Neural Relax

We present an algorithm for data preprocessing of an associative memory inspired to an electrostatic problem that turns out to have intimate relations with information maximization