Fabio L. Traversa

Universal Memcomputing Machines

We introduce the notion of universal memcomputing machines (UMMs): a class of brain-inspired general-purpose computing machines based on systems with memory, whereby processing and storing of information occur on the same physical location. We analytically prove that the memory properties of UMMs endow them with universal computing power (they are Turing-complete), intrinsic parallelism, functional polymorphism, and information overhead, namely, their collective states can support exponential data compression directly in memory. We also demonstrate that a UMM has the same computational power as a nondeterministic Turing machine, namely, it can solve nondeterministic polynomial (NP)-complete problems in polynomial time. However, by virtue of its information overhead, a UMM needs only an amount of memory cells (memprocessors) that grows polynomially with the problem size. As an example, we provide the polynomial-time solution of the subset-sum problem and a simple hardware implementation of the same. Even though these results do not prove the statement NP = P within the Turing paradigm, the practical realization of these UMMs would represent a paradigm shift from the present von Neumann architectures, bringing us closer to brain-like neural computation.

Oscillator Noise: A Nonlinear Perturbative Theory Including Orbital Fluctuations and Phase-Orbital Correlation

We derive a full statistical characterization of the noise spectrum of a free running oscillator perturbed by white Gaussian noise sources, including the effect of orbital fluctuations and of their correlation with phase noise, thus extending the previous theory based on the Floquet decomposition of the linearized oscillator equations . This allows to derive explicit relationships for the relevant phase, amplitude and correlation spectra. The examples provide a validation of the theoretical results, and allow to assess the importance of the Floquet exponents and eigenvectors on the magnitude of the orbital noise contribution.

Dynamic computing random access memory

The present von Neumann computing paradigm involves a significant amount of information transfer between a central processing unit and memory, with concomitant limitations in the actual execution speed. However, it has been recently argued that a different form of computation, dubbed memcomputing (Di Ventra and Pershin 2013 Nat. Phys. 9 200-2) and inspired by the operation of our brain, can resolve the intrinsic limitations of present day architectures by allowing for computing and storing of information on the same physical platform. Here we show a simple and practical realization of memcomputing that utilizes easy-to-build memcapacitive systems. We name this architecture dynamic computing random access memory (DCRAM). We show that DCRAM provides massively-parallel and polymorphic digital logic, namely it allows for different logic operations with the same architecture, by varying only the control signals. In addition, by taking into account realistic parameters, its energy expenditures can be as low as a few fJ per operation. DCRAM is fully compatible with CMOS technology, can be realized with current fabrication facilities, and therefore can really serve as an alternative to the present computing technology.

Time-Dependent Many-Particle Simulation for Resonant Tunneling Diodes: Interpretation of an Analytical Small-Signal Equivalent Circuit

A full many-particle (beyond the mean-field approximation) electron quantum-transport simulator, which is named BITLLES, is used to analyze the transient current response of resonant tunneling diodes (RTDs). The simulations have been used to test an analytical (free-fitting parameters) small-signal equivalent circuit for RTDs under stable direct-current-biased conditions. The comparison provides an excellent agreement and furnishes a way to physically interpret each circuit element. In addition, a nonlinear novel RTD behavior in the negative differential conductance region has been established, i.e. asymmetric time constants in the RTD current response when high-low or low-high voltage steps are considered.

Memory Models of Adaptive Behavior

Adaptive response to varying environment is a common feature of biological organisms. Reproducing such features in electronic systems and circuits is of great importance for a variety of applications. We consider memory models inspired by an intriguing ability of slime molds to both memorize the period of temperature and humidity variations and anticipate the next variations to come, when appropriately trained. Effective circuit models of such behavior are designed using: 1) a set of LC contours with memristive damping and 2) a single memcapacitive system-based adaptive contour with memristive damping. We consider these two approaches in detail by comparing their results and predictions. Finally, possible biological experiments that would discriminate between the models are discussed. In this paper, we also introduce an effective description of certain memory circuit elements.

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.

Frequency-domain evaluation of the adjoint Floquet eigenvectors for oscillator noise characterisation

The calculation of orbital fluctuations and of the phase-orbital correlation within Floquet-based noise analysis of autonomous systems requires the availability of all the direct and adjoint Floquet eigenvectors associated with the noiseless limit cycle. Here the authors introduce a novel numerical technique for their frequency domain determination. The algorithm is entirely based on the Jacobian matrices already available from the harmonic balance-based calculation of the limit cycle, thus avoiding any time-domain integration. The Floquet eigenvalues and adjoint eigenvectors are calculated from a generalised eigenvalue problem, thus making the approach readily implementable into CAD tools provided that the Jacobian matrices are made available.

Generalized Floquet theory: application to dynamical systems with memory and Bloch's theorem for nonlocal potentials.

Floquet theory is a powerful tool in the analysis of many physical phenomena, and extended to spatial coordinates provides the basis for Blochs theorem. However, in its original formulation it is limited to linear systems with periodic coefficients. Here, we extend the theory by proving a theorem for the general class of systems including linear operators commuting with the period-shift operator. The present theorem greatly expands the range of applicability of Floquet theory to a multitude of phenomena that were previously inaccessible with this type of analysis, such as dynamical systems with memory. As an important extension, we also prove Blochs theorem for nonlocal potentials.

Robust weak-measurement protocol for Bohmian velocities

We present a protocol for measuring Bohmian - or the mathematically equivalent hydrodynamic - velocities based on an ensemble of two position measurements, defined from a Positive Operator Valued Measure, separated by a finite time interval. The protocol is very accurate and robust as long as the first measurement uncertainty divided by the finite time interval between measurements is much larger than the Bohmian velocity, and the system evolves under flat potential between measurements. The difference between the Bohmian velocity of the unperturbed state and the measured one is predicted to be much smaller than 1% in a large range of parameters. Counter-intuitively, the measured velocity is that at the final time and not a time-averaged value between measurements.

Memcomputing with membrane memcapacitive systems

We show theoretically that networks of membrane memcapacitive systems-capacitors with memory made out of membrane materials-can be used to perform a complete set of logic gates in a massively parallel way by simply changing the external input amplitudes, but not the topology of the network. This polymorphism is an important characteristic of memcomputing (computing with memories) that closely reproduces one of the main features of the brain. A practical realization of these membrane memcapacitive systems, using, e.g., graphene or other 2D materials, would be a step forward towards a solid-state realization of memcomputing with passive devices.