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Polynomial-time solution of prime factorization and NP-complete problems with digital memcomputing machines.

Fabio L Traversa1, Massimiliano Di Ventra1

  • 1Department of Physics, University of California, San Diego, La Jolla, California 92093-0319, USA.

Chaos (Woodbury, N.Y.)
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Digital Memcomputing Machines (DMMs) can solve complex problems, including NP-hard ones, using polynomial resources. This is achieved through self-organizing logic circuits with memory, offering a potential solution to the P versus NP problem.

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Area of Science:

  • Computer Science
  • Computational Complexity Theory
  • Materials Science

Background:

  • Non-deterministic Polynomial (NP) problems are computationally intensive.
  • Existing computational models face limitations in solving NP problems efficiently.
  • The P versus NP problem remains a fundamental question in computer science.

Purpose of the Study:

  • Introduce Digital Memcomputing Machines (DMMs) capable of solving NP problems with polynomial resources.
  • Establish the mathematical and physical constraints for poly-resource resolvability.
  • Propose a practical implementation using self-organizing logic circuits (SOLCs).

Main Methods:

  • Connection to dynamical systems theory to derive physical constraints.
  • Development of self-organizing logic gates and circuits (SOLCs) with memory elements.
  • Application of functional analysis to prove mathematical constraints for poly-resource resolvability.

Main Results:

  • SOLCs exhibit a global attractor with equilibrium points corresponding to problem solutions.
  • Exponential convergence to solutions with a polynomial scaling of convergence rate.
  • Demonstrated solutions for prime factorization and subset-sum problems.
  • DMMs show robustness against noise and scalability.

Conclusions:

  • DMMs, implemented via SOLCs, offer a novel approach to solving NP-hard problems efficiently.
  • The proposed architecture satisfies mathematical and physical constraints for poly-resource resolvability.
  • This work provides insights into the NP = P question and the feasibility of poly-resource resolvability.