arXiv subjects
Publications and source records attributed to Marco Budinich.
We refine the formulation of the Boolean satisfiability problem with $n$ Boolean variables in Clifford algebra ${\cal C}\ell(\mathbb{R}^{n,n})$ [3] and exploit this continuous setting to outline a new unsatisfiability test. This algorithm is not combinatorial and can prove unsatisfiability in polynomial time.
We explore the relations between the Boolean Satisfiability Problem with $n$ Boolean variables and the orthogonal group $\mbox{O}(n)$. We show that all $2^n$ possible solutions induce involutions of $\mathbb{R}^n$ that lie in the compact, disconnected real manifold of dimension $n (n-1)/2$ of $\mbox{O}(n)$. This result in turn gives a new unsatisfiability test within group $\mbox{O}(n)$.
We formulate a Boolean algebra in the set of idempotents of Clifford algebra Cl($R^{n,n}$) and within this frame we examine different formulations of the Boolean Satisfiability Problem in Clifford algebra. Exploiting the isomorphism between null subspaces of $R^{n,n}$ associated to simple spinors and the orthogonal group O(n) we ultimately give a continuous formulation of the Boolean Satisfiability Problem within this group that opens unexplored perspectives.
We explore the three separate isomorphisms that link together simple spinors, null vectors and the orthogonal group O(n) and exploit them to look back at these arguments from a unified viewpoint.
We present a formulation of the Boolean Satisfiability Problem in spinor language that allows to give a necessary and sufficient condition for unsatisfiability. With this result we outline an algorithm to test for unsatisfiability with possibly interesting theoretical properties.
We show that complex representations of Clifford algebra can always be reduced either to a real or to a quaternionic algebra depending on signature of complex space thus showing that complex spinors are unavoidably either real Majorana spinors or quaternionic spinors. We use this result to support (1,3) signature for Minkowski space.
We present a necessary and sufficient condition for a spinor $ω$ to be of nullity zero, i.e. such that for any null vector $v$, $v ω\ne 0$. This dives deeply in the subtle relations between a spinor $ω$ and $ω_c$, the (complex) conjugate of $ω$ belonging to the same spinor space.
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 $φ$, transform as $v \to x v x^{-1}$ and $φ\to x φx^{-1}$ for some algebra element $x$. We show that while under combined $PT$ spinor $φ\to x φx^{-1}$ remain in its spinor space, under $P$ or $T$ separately $φ$ 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.
We show that the binary representation of the integers has a role to play in many aspects of Clifford algebras.
We investigate the relations between spinors and null vectors in Clifford algebra with particular emphasis on the conditions that a spinor must satisfy to be simple (also: pure). In particular we prove: i) a new property for null vectors: each of them bisects spinor space into two parts of equal size; ii) that simple spinors form one-dimensional subspaces of spinor space; iii) a necessary and sufficient condition for a spinor to be simple that generalizes a theorem of Cartan and Chevalley that appears now as a corollary of this result. We also show how to write down easily the most general spinor with a given associated totally null plane.
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.
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 Γ. 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).
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.
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.