Absence of chaos in Digital Memcomputing Machines with solutions

TL;DR

This paper proves that digital memcomputing machines with solutions do not exhibit chaos, as their dynamical properties prevent strange attractors and chaotic behavior, ensuring predictable convergence to solutions.

Contribution

It demonstrates that DMMs with solutions cannot support chaos due to their topological and dynamical properties, confirming a previous conjecture.

Findings

No strange attractors coexist with solutions in DMMs.

DMMs with solutions lack topological transitivity and chaos.

Without solutions, DMMs only have invariant tori or periodic orbits.

Abstract

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, 023107 (2017)] 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).

Equations

$$\dot{\mathbf{x}}(t)=F(\mathbf{x}(t)),$$

$$T(t)\mathbf{x}(0)=\mathbf{x}(0)+{\int }_0^{t}F(\mathbf{x}(t^{\prime }))d{t}^{\prime }$$

Full text

Memcomputing is a recently suggested computing paradigm that employs memory to both store and process information on the same physical location Di Ventra and Pershin (2013); Traversa and Di Ventra (2015); Traversa et al. (2015); Traversa and Di Ventra (2017). It is based on the mathematical notion of Universal Memcomputing Machines introduced by us in Ref Traversa and Di Ventra, 2015, which shows that these machines have the same computational power of non-deterministic Turing machines. As such, they can, in principle, solve Non-deterministic polynomial (NP)-complete problems with polynomial resources.

In a subsequent work Traversa and Di Ventra (2017), we have shown that a physical and scalable realization of such machines –we call this subclass Digital Memcomputing Machines (DMMs)– can be implemented using self-organizing logic gates (SOLGs). Unlike standard Boolean gates, SOLGs satisfy their logic proposition by also working “in reverse”, in the sense that they are able to adapt to “input” signals from any terminal, without the distinction between input and output terminals as in conventional logic.

From a physical point of view, these gates can be made as combination of standard circuit elements (including transistors) and/or memory elements such as memristors, memcapacitors and meminductors Di Ventra et al. (2009). Irrespective of their practical realization (which incidentally is not even unique given the Boolean problem to solve) SOLGs and their combination into networks –self-organizing logic circuits– are nothing other than non-linear dynamical systems, namely they can be mathematically described by an equation of motion of the type:

[TABLE]

where $\mathbf{x}\in X$ is the set of state variables (such as voltages, currents and internal memory variables) belonging to the phase space $X$, and $F$ is the flow vector field representing the laws of temporal evolution of $\mathbf{x}$. This way, the solutions of the problem at hand can be found by driving the corresponding dynamical system to its equilibrium points.

However, DMMs possess an additional, important feature that is not common to all dynamical systems described by Eq. (1): DMMs are point dissipative Hale (2010). This mathematical property implies that there is always a bounded and compact set $J\subset X$ that attracts every point of $X$ under $F$. Hale (2010) This is indeed a critical feature for DMMs to succeed in solving complex problems efficiently since it implies that, irrespective of initial conditions, all system’s orbits will eventually enter the attractor $J$ and remain there.

However, such attractor could be the union of equilibrium points (the solutions of the Boolean problem to be solved), invariant tori (that include periodic and quasi-periodic orbits), and strange attractors. Formally, if $B_{i}$’s are the subsets of $J$ that attract the different solutions (stable fixed points, invariant tori and strange attractors), we can write $J=\bigcup_{B_{i}}\omega(B_{i})$, with the $\omega$-limit sets $\omega(B_{i})$ of $B_{i}$ the sets $\omega(B_{i})={\textstyle\bigcap_{s\geq 0}}\mathrm{Cl}{\textstyle\bigcup_{t\geq s}}T(t)B_{i}$ with $T:\mathbb{R}\times X\rightarrow X$,

[TABLE]

a $C^{r}$ semigroup with $r\geq 1$, and Lipschitz continuous. The fact that $J=\bigcup_{B_{i}}\omega(B_{i})$ is a consequence of $J$ being uniformly asymptotically stable (namely it is stable and attracts a neighborhood of J) Hale (2010). In addition, the Lipschitz continuity implies that $\omega(B_{i})\cap\omega(B_{j})=\emptyset$, $\forall B_{i},B_{j}\subset X$. Note also that the phase space, $X$, of DMMs is just a metric compact space, not a Banach (vector) space. This means that Eq. (1) does not always have a fixed point Hale (2010).

In Ref. Traversa and Di Ventra, 2017, we have provided physical and mathematical arguments as to why, in the presence of equilibrium points, neither invariant tori nor strange attractors can coexist with equilibria. This conjecture is also supported by numerical simulations of DMMs. Our argument was that since the equilibrium points of DMMs belong to an open ball with finite radius in which the convergence rate to equilibrium is exponential, then if there are any orbits that evolve into a strange attractor or a limit cycle, they would eventually enter that ball and converge exponentially fast to one equilibrium point. However, we could not prove these statements rigorously. In this paper, we prove one of the above two statements, the one concerning chaos.

Despite considerable research on chaotic dynamics, there is yet no agreed-upon definition of what constitutes chaos, so in order to prove its absence, one would be forced to prove the absence of most of its features Ovchinnikov and Di Ventra (2017). However, there is a fairly general understanding, in both the mathematics and science communities, that deterministic chaotic dynamics is equivalent to these three properties Devaney (1992); Aulbach and Kieninger (2001) that need to be satisfied by an invariant subset $A$ of $X$ ($T(A)=A$): (i) $T$ is topologically transitive in $A$, (ii) the set of periodic orbits of $T$ is dense in $A$, and (iii) $T|_{A}$ depends sensitively on initial conditions. Topological transitivity means that, given any two points in the phase space, we can find an orbit that comes arbitrarily close to both.

Mathematically, we say that the flow is topologically transitive Gottschalk and Hedlund (1955) if for any pair of non-vacuous open sets $U$, $V$ $\subset A$, there exists a positive time $t^{\prime}$ such that $T|_{A}(t^{\prime},U)\cap V\neq\emptyset$, where $T|_{A}(t^{\prime},U)=\{t^{\prime}\in\mathbb{R}^{+};\mathbf{x}(0)\in U|\;T|_{A}(t^{\prime})\mathbf{x}(0):U\to A\}$ 111For maps this is replaced by the requirement that there exists a positive $n$ such that $T|_{A}^{n}(U)\cap V\neq\emptyset$, where $T_{A}^{n}$ is the $n-$th iterate of the map restricted on $A$.. Property (iii) is redundant since one can prove Banks et al. (1992) that if the dynamical system satisfies (i) and (ii), then (iii) follows 222Since DMMs are not maps on intervals of $\mathbb{R}$, transitivity does not imply a dense set of periodic orbits Vellekoop and Berglund (1994).. Incidentally, this means that only topological (global) features are enough to characterize a flow as chaotic, without the need of introducing a metric (local property).

A stronger condition than transitivity is topological mixing which states that any open set of $A\subset X$ overlaps with any other open set if the former is propagated long enough. Mathematically, this means that for any pair of non-vacuous open sets $U$, $V$ $\subset A$, there exists a positive time $t^{\prime}$ such that $T|_{A}(t\geq t^{\prime},U)\cap V\neq\emptyset$, where $T|_{A}(t\geq t^{\prime},U)=\{t^{\prime}\leq t\in\mathbb{R}^{+};\mathbf{x}(t^{\prime})\in U|\;T|_{A}(t)\mathbf{x}(t^{\prime}):U\to A\}$ 333For maps this is replaced by the requirement that there exists a positive $n$ such that $T|_{A}^{m}(U)\cap V\neq\emptyset$, $\forall m\geq n$.. Topological mixing implies transitivity, but not the other way around Hasselblatt and Katok (2003). Note also that if $A=X$ then chaos spans the whole phase space.

If we take at face value the above properties as definitions of chaos, it is then sufficient to prove the absence of either (i) or (ii) to prove the original statement for our DMMs. We will focus on property (i), namely topological transitivity. We can also go a step further and prove absence of topological mixing in DMMs with equilibrium points. We will then discuss which invariant dynamical properties DMMs show if there are no solutions to the Boolean problems they solve.

Absence of chaos with equilibrium points – We consider a DMM described by the dynamical system $(X,T)$, with $X$ a metric compact space of dimension $D=\dim X$ and $T$ given in Eq. (2). We do not need to fix the topology of this space, but without loss of generality we may choose the Hausdorff topology. Our aim is then to prove the following:

If we take these features (in particular transitivity) as indications of the absence of strange attractors (chaos), we have indeed proved the conjecture advanced in Ref. Traversa and Di Ventra, 2017 that in DMMs with solutions chaos cannot emerge during dynamics.

Absence of chaos without equilibrium points – Let us now consider the case in which DMMs do not support equilibrium points: the Boolean problems represented by DMMs have no solutions. Then the compact metric nature of the support of the corresponding dynamical system $(X,T)$ implies the existence of at least one recurrent point Devaney (1992). Recurrent points, other than fixed points, are (high-dimensional) invariant tori Devaney (1992). This means that DMMs with no solution support at least one invariant torus. If this is the case, we can then prove the following theorem:

In other words, if the DMMs do not support solutions, they cannot manifest strange attractors (chaotic behavior), only invariant tori. However, the type and structure of these cycles is not obvious and they may indeed have periods that vary (increase) with the dimension of the phase space.

In fact, as the size of the phase space increases we cannot exclude the existence of quasi-attractors, namely stable periodic trajectories with extremely large periods and very small attraction basins Afraimovich and Shilnikov (1991). If these quasi-attractors exist, then their features may be practically indistinguishable from true chaotic behavior. The reason is that the system may, due to some small fluctuations, jump from one of these long-period cycles to another, thus rendering the dynamics practically chaotic Mikhailov and Loskutov (1996).

Conclusions – In summary, we have proved a conjecture advanced by us in Ref. Traversa and Di Ventra, 2017 that in the presence of solutions of a given Boolean problem solved by DMMs, chaotic behavior cannot coexist with such dynamics. This is an important consequence of the point dissipative property of the dynamical systems representing these machines. Point dissipative means that there is always a bounded compact subset of the whole phase space that attracts any orbit of the system. This global attractor then fully characterizes the dynamics at sufficiently long times.

We have also demonstrated that, in view of the (additional) compact metric nature of the dynamical systems describing DMMs, in the absence of equilibrium points, (high-dimensional) invariant tori are the only possible attractors of the system. However, we cannot exclude that such invariant tori are quasi-attractors. If that is the case, then their period and number could make them indistinguishable from chaotic behavior in numerical or experimental situations.

In order to complete the theoretical program initiated in Ref. Traversa and Di Ventra, 2017 we still need to prove the other conjecture: that in the presence of equilibrium points DMMs cannot support periodic orbits. We leave this (difficult) open problem for future work.

Acknowledgments – M.D. and F.L.T. acknowledge partial support from the Center for Memory Recording Research at UCSD and LoGate Computing, Inc.