The Clifford algebra of $R^{n,n}$ and the Boolean Satisfiability Problem

TL;DR

This paper explores a novel algebraic approach to the Boolean Satisfiability Problem using Clifford algebra, establishing a continuous formulation within the orthogonal group O(n) that offers new theoretical insights.

Contribution

It introduces a Boolean algebra within Clifford algebra Cl(R^{n,n}) and formulates the SAT problem continuously in the orthogonal group O(n), bridging algebraic and geometric perspectives.

Findings

Boolean algebra formulated in Clifford algebra Cl(R^{n,n})

Continuous formulation of SAT in the orthogonal group O(n)

New perspectives on SAT problem through algebraic-geometric connection

Abstract

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.