Efficient Analog Circuits for Boolean Satisfiability

Efficient solutions to nonpolynomial (NP)-complete problems would significantly benefit both science and industry. However, such problems are intractable on digital computers based on the von Neumann architecture, thus creating the need for alternative solutions to tackle such problems. Recently, a deterministic, continuous-time dynamical system (CTDS) was proposed <xref ref-type="bibr" rid="ref1">[1]</xref> to solve a representative NP-complete problem, Boolean Satisfiability (SAT). This solver shows polynomial analog time-complexity on even the hardest benchmark <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-SAT (<inline-formula> <tex-math notation="LaTeX">$k \geq 3$ </tex-math></inline-formula>) formulas, but at an energy cost through exponentially driven auxiliary variables. This paper presents a novel analog hardware SAT solver, <monospace>AC-SAT</monospace>, implementing the CTDS via incorporating novel, analog circuit design ideas. <monospace>AC-SAT</monospace> is intended to be used as a coprocessor and is programmable for handling different problem specifications. It is especially effective for solving hard <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-SAT problem instances that are challenging for algorithms running on digital machines. Furthermore, with its modular design, <monospace>AC-SAT</monospace> can readily be extended to solve larger size problems, while the size of the circuit grows linearly with the product of the number of variables and the number of clauses. The circuit is designed and simulated based on a 32-nm CMOS technology. Simulation Program with Integrated Circuit Emphasis (SPICE) simulation results show speedup factors of ~10<sup>4</sup> on even the hardest 3-SAT problems, when compared with a state-of-the-art SAT solver on digital computers. As an example, for hard problems with <inline-formula> <tex-math notation="LaTeX">$N=50$ </tex-math></inline-formula> variables and <inline-formula> <tex-math notation="LaTeX">$M=212$ </tex-math></inline-formula> clauses, solutions are found within from a few nanoseconds to a few hundred nanoseconds.

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