Critical branching processes in digital memcomputing machines

Memcomputing is a novel computing paradigm that employs time non-locality (memory) to solve combinatorial optimization problems. It can be realized in practice by means of non-linear dynamical systems whose point attractors represent the solutions of the original problem. It has been previously shown that during the solution search digital memcomputing machines go through a transient phase of avalanches (instantons) that promote dynamical long-range order. By employing mean-field arguments we predict that the distribution of the avalanche sizes follows a Borel distribution typical of critical branching processes with exponent $\tau= 3/2$. We corroborate this analysis by solving various random 3-SAT instances of the Boolean satisfiability problem. The numerical results indicate a power-law distribution with exponent $\tau = 1.51 \pm 0.02$, in very good agreement with the mean-field analysis. This indicates that memcomputing machines self-tune to a critical state in which avalanches are characterized by a branching process, and that this state persists across the majority of their evolution.

[1]  Massimiliano Di Ventra,et al.  The parallel approach , 2013 .

[2]  D. Freedman,et al.  On the histogram as a density estimator:L2 theory , 1981 .

[3]  Robert A. Nawrocki,et al.  A Mini Review of Neuromorphic Architectures and Implementations , 2016, IEEE Transactions on Electron Devices.

[4]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[5]  Fabio L. Traversa,et al.  Topological Field Theory and Computing with Instantons , 2016, ArXiv.

[6]  Isaac L. Chuang,et al.  Quantum Computation and Quantum Information (10th Anniversary edition) , 2011 .

[7]  Fabio L. Traversa,et al.  Instantons in self-organizing logic gates , 2017, ArXiv.

[8]  Fabio L. Traversa,et al.  MemComputing Integer Linear Programming , 2018, ArXiv.

[9]  M. A. Muñoz,et al.  Simple unified view of branching process statistics: Random walks in balanced logarithmic potentials. , 2016, Physical review. E.

[10]  Fabio L. Traversa,et al.  Memcomputing: Leveraging memory and physics to compute efficiently , 2018, ArXiv.

[11]  Fabio L. Traversa,et al.  Evidence of an exponential speed-up in the solution of hard optimization problems , 2017, Complex..

[12]  E. Lu,et al.  Avalanches and the Distribution of Solar Flares , 1991 .

[13]  Zohar Manna,et al.  The calculus of computation - decision procedures with applications to verification , 2007 .

[14]  J. Tanner,et al.  A derivation of the Borel distribution , 1961 .

[15]  Massimiliano Di Ventra,et al.  Polynomial-time solution of prime factorization and NP-hard problems with digital memcomputing machines , 2015, Chaos.

[16]  Fabio L. Traversa,et al.  Accelerating Deep Learning with Memcomputing , 2018, Neural Networks.