Digital Memcomputing Research Articles

Transfer-Free Analog and Digital Flexible Memristors Based on Boron Nitride Films.

The traditional von Neumann architecture of computers, constrained by the inherent separation of processing and memory units, faces challenges, for instance, memory wall issue. Neuromorphic computing and in-memory computing offer promising paradigms to overcome the limitations of additional data movement and to enhance computational efficiency. In this work, transfer-free flexible memristors based on hexagonal boron nitride films were proposed for analog neuromorphic and digital memcomputing. Analog memristors were prepared; they exhibited synaptic behaviors, including paired-pulse facilitation and long-term potentiation/depression. The resistive switching mechanism of the analog memristors were investigated through transmission electron microscopy. Digital memristors were prepared by altering the electrode materials, and they exhibited reliable device performance, including a large on/off ratio (up to 106), reproducible switching endurance (>100 cycles), non-volatile characteristic (>60 min), and effective operating under bending conditions (>100 times).

Self-averaging of digital memcomputing machines.

Digital memcomputing machines (DMMs) are a new class of computing machines that employ nonquantum dynamical systems with memory to solve combinatorial optimization problems. Here, we show that the time to solution (TTS) of DMMs follows an inverse Gaussian distribution, with the TTS self-averaging with increasing problem size, irrespective of the problem they solve. We provide both an analytical understanding of this phenomenon and numerical evidence by solving instances of the 3-SAT (satisfiability) problem. The self-averaging property of DMMs with problem size implies that they are increasingly insensitive to the detailed features of the instances they solve. This is in sharp contrast to traditional algorithms applied to the same problems, illustrating another advantage of this physics-based approach to computation.

Directed percolation and numerical stability of simulations of digital memcomputing machines.

Digital memcomputing machines (DMMs) are a novel, non-Turing class of machines designed to solve combinatorial optimization problems. They can be physically realized with continuous-time, non-quantum dynamical systems with memory (time non-locality), whose ordinary differential equations (ODEs) can be numerically integrated on modern computers. Solutions of many hard problems have been reported by numerically integrating the ODEs of DMMs, showing substantial advantages over state-of-the-art solvers. To investigate the reasons behind the robustness and effectiveness of this method, we employ three explicit integration schemes (forward Euler, trapezoid, and Runge-Kutta fourth order) with a constant time step to solve 3-SAT instances with planted solutions. We show that (i) even if most of the trajectories in the phase space are destroyed by numerical noise, the solution can still be achieved; (ii) the forward Euler method, although having the largest numerical error, solves the instances in the least amount of function evaluations; and (iii) when increasing the integration time step, the system undergoes a "solvable-unsolvable transition" at a critical threshold, which needs to decay at most as a power law with the problem size, to control the numerical errors. To explain these results, we model the dynamical behavior of DMMs as directed percolation of the state trajectory in the phase space in the presence of noise. This viewpoint clarifies the reasons behind their numerical robustness and provides an analytical understanding of the solvable-unsolvable transition. These results land further support to the usefulness of DMMs in the solution of hard combinatorial optimization problems.

Flexible boron nitride-based memristor forin situdigital and analogue neuromorphic computing applications

The data processing efficiency of traditional computers is suffering from the intrinsic limitation of physically separated processing and memory units. Logic-in-memory and brain-inspired neuromorphic computing are promising in-memory computing paradigms for improving the computing efficiency and avoiding high power consumption caused by extra data movement. However, memristors that can conduct digital memcomputing and neuromorphic computing simultaneously are limited by the difference in the information form between digital data and analogue data. In order to solve this problem, this paper proposes a flexible low-dimensional memristor based on boron nitride (BN), which has ultralow-power non-volatile memory characteristic, reliable digital memcomputing capabilities, and integrated ultrafast neuromorphic computing capabilities in a single in situ computing system. The logic-in-memory basis, including FALSE, material implication (IMP), and NAND, are implemented successfully. The power consumption of the proposed memristor per synaptic event (198 fJ) can be as low as biology (fJ level) and the response time (1 μs) of the neuromorphic computing is four orders of magnitude shorter than that of the human brain (10 ms), paving the way for wearable ultrahigh efficient next-generation in-memory computing architectures.

Efficient solution of Boolean satisfiability problems with digital memcomputing

Boolean satisfiability is a propositional logic problem of interest in multiple fields, e.g., physics, mathematics, and computer science. Beyond a field of research, instances of the SAT problem, as it is known, require efficient solution methods in a variety of applications. It is the decision problem of determining whether a Boolean formula has a satisfying assignment, believed to require exponentially growing time for an algorithm to solve for the worst-case instances. Yet, the efficient solution of many classes of Boolean formulae eludes even the most successful algorithms, not only for the worst-case scenarios, but also for typical-case instances. Here, we introduce a memory-assisted physical system (a digital memcomputing machine) that, when its non-linear ordinary differential equations are integrated numerically, shows evidence for polynomially-bounded scalability while solving “hard” planted-solution instances of SAT, known to require exponential time to solve in the typical case for both complete and incomplete algorithms. Furthermore, we analytically demonstrate that the physical system can efficiently solve the SAT problem in continuous time, without the need to introduce chaos or an exponentially growing energy. The efficiency of the simulations is related to the collective dynamical properties of the original physical system that persist in the numerical integration to robustly guide the solution search even in the presence of numerical errors. We anticipate our results to broaden research directions in physics-inspired computing paradigms ranging from theory to application, from simulation to hardware implementation.

Taming a nonconvex landscape with dynamical long-range order: Memcomputing Ising benchmarks.

Recent work on quantum annealing has emphasized the role of collective behavior in solving optimization problems. By enabling transitions of clusters of variables, such solvers are able to navigate their state space and locate solutions more efficiently despite having only local connections between elements. However, collective behavior is not exclusive to quantum annealers, and classical solvers that display collective dynamics should also possess an advantage in navigating a nonconvex landscape. Here we give evidence that a benchmark derived from quantum annealing studies is solvable in polynomial time using digital memcomputing machines, which utilize a collection of dynamical components with memory to represent the structure of the underlying optimization problem. To illustrate the role of memory and clarify the structure of these solvers we propose a simple model of these machines that demonstrates the emergence of long-range order. This model, when applied to finding the ground state of the Ising frustrated-loop benchmarks, undergoes a transient phase of avalanches which can span the entire lattice and demonstrates a connection between long-range behavior and their probability of success. These results establish the advantages of computational approaches based on collective dynamics of continuous dynamical systems.

Digital memcomputing: From logic to dynamics to topology

Digital memcomputing machines (DMMs) are a class of computational machines designed to solve combinatorial optimization problems. A practical realization of DMMs can be accomplished via electrical circuits of highly non-linear, point-dissipative dynamical systems engineered so that periodic orbits and chaos can be avoided. A given logic problem is first mapped into this type of dynamical system whose point attractors represent the solutions of the original problem. A DMM then finds the solution via a succession of elementary instantons whose role is to eliminate solitonic configurations of logical inconsistency (“logical defects”) from the circuit. By employing a supersymmetric theory of dynamics, a DMM can be described by a cohomological field theory that allows for computation of certain topological matrix elements on instantons that have the mathematical meaning of intersection numbers on instantons. We discuss the “dynamical” meaning of these matrix elements, and argue that the number of elementary instantons needed to reach the solution cannot exceed the number of state variables of DMMs, which in turn can only grow at most polynomially with the size of the problem. These results shed further light on the relation between logic, dynamics and topology in digital memcomputing.

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 . 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 , 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.

Accelerating deep learning with memcomputing

Restricted Boltzmann machines (RBMs) and their extensions, often called “deep-belief networks”, are powerful neural networks that have found applications in the fields of machine learning and artificial intelligence. The standard way to train these models resorts to an iterative unsupervised procedure based on Gibbs sampling, called “contrastive divergence”, and additional supervised tuning via back-propagation. However, this procedure has been shown not to follow any gradient and can lead to suboptimal solutions. In this paper, we show an efficient alternative to contrastive divergence by means of simulations of digital memcomputing machines (DMMs) that compute the gradient of the log-likelihood involved in unsupervised training. We test our approach on pattern recognition using a modified version of the MNIST data set of hand-written numbers. DMMs sample effectively the vast phase space defined by the probability distribution of RBMs, and provide a good approximation close to the optimum. This efficient search significantly reduces the number of generative pretraining iterations necessary to achieve a given level of accuracy in the MNIST data set, as well as a total performance gain over the traditional approaches. In fact, the acceleration of the pretraining achieved by simulating DMMs is comparable to, in number of iterations, the recently reported hardware application of the quantum annealing method on the same network and data set. Notably, however, DMMs perform far better than the reported quantum annealing results in terms of quality of the training. Finally, we also compare our method to recent advances in supervised training, like batch-normalization and rectifiers, that seem to reduce the advantage of pretraining. We find that the memcomputing method still maintains a quality advantage (>1% in accuracy, corresponding to a 20% reduction in error rate) over these approaches, despite the network pretrained with memcomputing defines a more non-convex landscape using sigmoidal activation functions without batch-normalization. Our approach is agnostic about the connectivity of the network. Therefore, it can be extended to train full Boltzmann machines, and even deep networks at once.

Stress-Testing Memcomputing on Hard Combinatorial Optimization Problems.

Memcomputing is a novel computing paradigm that employs time non-local dynamical systems to compute with and in memory. The digital version of these machines [digital memcomputing machines or (DMMs)] is scalable, and is particularly suited to solve combinatorial optimization problems. One of its possible realizations is by means of standard electronic circuits, with and without memory. Since these elements are non-quantum, they can be described by ordinary differential equations. Therefore, the circuit representation of DMMs can also be simulated efficiently on our traditional computers. We have indeed previously shown that these simulations only require time and memory resources that scale linearly with the problem size when applied to finding a good approximation to the optimum of hard instances of the maximum-satisfiability problem. The state-of-the-art algorithms, instead, require exponential resources for the same instances. However, in that work, we did not push the simulations to the limit of the processor used. Since linear scalability at smaller problem sizes cannot guarantee linear scalability at much larger sizes, we have extended these results in a stress-test up to 64×106 variables (corresponding to about 1 billion literals), namely the largest case that we could fit on a single core of an Intel Xeon E5-2860 with 128 GB of dynamic random-access memory (DRAM). For this test, we have employed a commercial simulator, Falcon of MemComputing, Inc. We find that the simulations of DMMs still scale linearly in both time and memory up to these very large problem sizes versus the exponential requirements of the state-of-the-art solvers. These results further reinforce the advantages of the physics-based memcomputing approach compared with traditional ones.

Perspective: Memcomputing: Leveraging memory and physics to compute efficiently

It is well known that physical phenomena may be of great help in computing some difficult problems efficiently. A typical example is prime factorization that may be solved in polynomial time by exploiting quantum entanglement on a quantum computer. There are, however, other types of (non-quantum) physical properties that one may leverage to compute efficiently a wide range of hard problems. In this perspective, we discuss how to employ one such property, memory (time non-locality), in a novel physics-based approach to computation: Memcomputing. In particular, we focus on digital memcomputing machines (DMMs) that are scalable. DMMs can be realized with non-linear dynamical systems with memory. The latter property allows the realization of a new type of Boolean logic, one that is self-organizing. Self-organizing logic gates are “terminal-agnostic,” namely, they do not distinguish between the input and output terminals. When appropriately assembled to represent a given combinatorial/optimization problem, the corresponding self-organizing circuit converges to the equilibrium points that express the solutions of the problem at hand. In doing so, DMMs take advantage of the long-range order that develops during the transient dynamics. This collective dynamical behavior, reminiscent of a phase transition, or even the “edge of chaos,” is mediated by families of classical trajectories (instantons) that connect critical points of increasing stability in the system's phase space. The topological character of the solution search renders DMMs robust against noise and structural disorder. Since DMMs are non-quantum systems described by ordinary differential equations, not only can they be built in hardware with the available technology, they can also be simulated efficiently on modern classical computers. As an example, we will show the polynomial-time solution of the subset-sum problem for the worst cases, and point to other types of hard problems where simulations of DMMs' equations of motion on classical computers have already demonstrated substantial advantages over traditional approaches. We conclude this article by outlining further directions of study.

Impact of semiconducting electrodes on the electroresistance of ferroelectric tunnel junctions

Ferroelectric tunnel junctions are promising candidates for the realization of energy-efficient digital memories and analog memcomputing devices. In this work, we investigate the impact of a semiconducting layer in series to the junction on the sign of electroresistance. To this scope, we compare tunnel junctions fabricated out of Pt/BaTiO3/La1/3Sr2/3MnO3 (LSMO) and Pt/BaTiO3/Nb:SrTiO3 (Nb:STO) heterostructures, displaying an opposite sign of the electroresistance. By capacitance-voltage profiling, we observe a behavior typical of Metal-Oxide-Semiconductor tunnel devices in both cases but compatible with the opposite sign of charge carriers in the semiconducting layer. While Nb:STO displays the expected n-type semiconducting character, metallic LSMO develops an interfacial p-type semiconducting layer. The different types of carriers at the semiconducting interfaces and the modulation of the depleted region by the ferroelectric charge have a deep impact on electroresistance, possibly accounting for the different sign observed in the two systems.

Absence of periodic orbits in digital memcomputing machines with solutions.

In Traversa and Di Ventra [Chaos 27, 023107 (2017)] we argued, without proof, that if the non-linear dynamical systems with memory describing the class of digital memcomputing machines (DMMs) have equilibrium points, then no periodic orbits can emerge. In fact, the proof of such a statement is a simple corollary of a theorem already demonstrated in Traversa and Di Ventra [Chaos 27, 023107 (2017)]. Here, we point out how to derive such a conclusion. Incidentally, the same demonstration implies absence of chaos, a result we have already demonstrated in Di Ventra and Traversa [Phys. Lett. A 381, 3255 (2017)] using topology. These results, together with those in Traversa and Di Ventra [Chaos 27, 023107 (2017)], guarantee that if the Boolean problem the DMMs are designed to solve has a solution, the system will always find it, irrespective of the initial conditions.

Absence of chaos in digital memcomputing machines with solutions

Digital memcomputing machines (DMMs) are non-linear dynamical systems designed so that their equilibrium points are solutions of the Boolean problem they solve. In a previous work [Chaos 27 (2017) 023107] it was argued that when DMMs support solutions of the associated Boolean problem then strange attractors cannot coexist with such equilibria. In this work, we demonstrate such conjecture. In particular, we show that both topological transitivity, and the strongest property of topological mixing, are inconsistent with the point dissipative property of DMMs when equilibrium points are present. This is true for both the whole phase space and the global attractor. Absence of topological transitivity is enough to imply absence of chaotic behavior. In a similar vein, we prove that if DMMs do not have equilibrium points, the only attractors present are invariant tori/periodic orbits with periods that may possibly increase with system size (quasi-attractors).

Memcomputing Numerical Inversion With Self-Organizing Logic Gates.

We propose to use digital memcomputing machines (DMMs), implemented with self-organizing logic gates (SOLGs), to solve the problem of numerical inversion. Starting from fixed-point scalar inversion, we describe the generalization to solving linear systems and matrix inversion. This method, when realized in hardware, will output the result in only one computational step. As an example, we perform simulations of the scalar case using a 5-bit logic circuit made of SOLGs, and show that the circuit successfully performs the inversion. Our method can be extended efficiently to any level of precision, since we prove that producing -bit precision in the output requires extending the circuit by at most bits. This type of numerical inversion can be implemented by DMM units in hardware; it is scalable, and thus of great benefit to any real-time computing application.

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

We introduce a class of digital machines, we name Digital Memcomputing Machines, (DMMs) able to solve a wide range of problems including Non-deterministic Polynomial (NP) ones with polynomial resources (in time, space, and energy). An abstract DMM with this power must satisfy a set of compatible mathematical constraints underlying its practical realization. We prove this by making a connection with the dynamical systems theory. This leads us to a set of physical constraints for poly-resource resolvability. Once the mathematical requirements have been assessed, we propose a practical scheme to solve the above class of problems based on the novel concept of self-organizing logic gates and circuits (SOLCs). These are logic gates and circuits able to accept input signals from any terminal, without distinction between conventional input and output terminals. They can solve boolean problems by self-organizing into their solution. They can be fabricated either with circuit elements with memory (such as memristors) and/or standard MOS technology. Using tools of functional analysis, we prove mathematically the following constraints for the poly-resource resolvability: (i) SOLCs possess a global attractor; (ii) their only equilibrium points are the solutions of the problems to solve; (iii) the system converges exponentially fast to the solutions; (iv) the equilibrium convergence rate scales at most polynomially with input size. We finally provide arguments that periodic orbits and strange attractors cannot coexist with equilibria. As examples, we show how to solve the prime factorization and the search version of the NP-complete subset-sum problem. Since DMMs map integers into integers, they are robust against noise and hence scalable. We finally discuss the implications of the DMM realization through SOLCs to the NP = P question related to constraints of poly-resources resolvability.