Dynamical system

From Wikipedia, the free encyclopedia

The Lorenz attractor arises in the study of the Lorenz Oscillator, a dynamical system.

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

At any given time, a dynamical system has a state given by a tuple of real numbers (a vector) that can be represented by a point in an appropriate state space (a geometrical manifold). The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state.^[1][2] However, some systems are stochastic, in that random events also affect the evolution of the state variables.

In physics, a dynamical system is described as a "particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives." In order to make a prediction about the system’s future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized.

The study of dynamical systems is the focus of dynamical systems theory, which has applications to a wide variety of fields such as mathematics, physics, biology, chemistry, engineering, economics, and medicine. Dynamical systems are a fundamental part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly process, and the edge of chaos concept.

Overview

The concept of a dynamical system has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is an implicit relation that gives the state of the system for only a short time into the future. (The relation is either a differential equation, difference equation or other time scale.) To determine the state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system. If the system can be solved, given an initial point it is possible to determine all its future positions, a collection of points known as a trajectory or orbit.

Before the advent of computers, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system.

For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because:

• The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions. To address these questions several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability. The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their equivalence changes with the different notions of stability.
• The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class. Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes. Linear dynamical systems and systems that have two numbers describing a state are examples of dynamical systems where the possible classes of orbits are understood.
• The behavior of trajectories as a function of a parameter may be what is needed for an application. As a parameter is varied, the dynamical systems may have bifurcation points where the qualitative behavior of the dynamical system changes. For example, it may go from having only periodic motions to apparently erratic behavior, as in the transition to turbulence of a fluid.
• The trajectories of the system may appear erratic, as if random. In these cases it may be necessary to compute averages using one very long trajectory or many different trajectories. The averages are well defined for ergodic systems and a more detailed understanding has been worked out for hyperbolic systems. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of statistical mechanics and of chaos.

History

Many people regard Henri Poincaré as the founder of dynamical systems.^[9] Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included the Poincaré recurrence theorem, which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state.

Aleksandr Lyapunov developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamic system.

In 1913, George David Birkhoff proved Poincaré's "Last Geometric Theorem", a special case of the three-body problem, a result that made him world-famous. In 1927, he published his Dynamical SystemsBirkhoff's most durable result has been his 1931 discovery of what is now called the ergodic theorem. Combining insights from physics on the ergodic hypothesis with measure theory, this theorem solved, at least in principle, a fundamental problem of statistical mechanics. The ergodic theorem has also had repercussions for dynamics.

Stephen Smale made significant advances as well. His first contribution is the Smale horseshoe that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others.

Oleksandr Mykolaiovych Sharkovsky developed Sharkovsky's theorem on the periods of discrete dynamical systems in 1964. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.

Basic definitions

A dynamical system is a manifold M called the phase (or state) space endowed with a family of smooth evolution functions Φ^t that for any element of t ∈ T, the time, map a point of the phase space back into the phase space. The notion of smoothness changes with applications and the type of manifold. There are several choices for the set T. When T is taken to be the reals, the dynamical system is called a flow; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow. When T is taken to be the integers, it is a cascade or a map; and the restriction to the non-negative integers is a semi-cascade.

Examples

The evolution function Φ ^t is often the solution of a differential equation of motion

${\displaystyle {\dot {x}}=v(x).}$

The equation gives the time derivative, represented by the dot, of a trajectory x(t) on the phase space starting at some point x₀. The vector field v(x) is a smooth function that at every point of the phase space M provides the velocity vector of the dynamical system at that point. (These vectors are not vectors in the phase space M, but in the tangent space T_xM of the point x.) Given a smooth Φ ^t, an autonomous vector field can be derived from it.

There is no need for higher order derivatives in the equation, nor for time dependence in v(x) because these can be eliminated by considering systems of higher dimensions. Other types of differential equations can be used to define the evolution rule:

${\displaystyle G(x,{\dot {x}})=0}$

is an example of an equation that arises from the modeling of mechanical systems with complicated constraints.

The differential equations determining the evolution function Φ ^t are often ordinary differential equations; in this case the phase space M is a finite dimensional manifold. Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally Banach spaces—in which case the differential equations are partial differential equations. In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity.

Further examples

• Logistic map
• Complex quadratic polynomial
• Dyadic transformation
• Tent map
• Double pendulum
• Arnold's cat map
• Horseshoe map
• Baker's map is an example of a chaotic piecewise linear map
• Billiards and outer billiards
• Hénon map
• Lorenz system
• Circle map
• Rössler map
• Kaplan–Yorke map
• List of chaotic maps
• Swinging Atwood's machine
• Quadratic map simulation system
• Bouncing ball dynamics

Linear dynamical systems

Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the N-dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers. The analysis of linear systems is possible because they satisfy a superposition principle: if u(t) and w(t) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will u(t) + w(t).

Flows

For a flow, the vector field Φ(x) is an affine function of the position in the phase space, that is,

${\displaystyle {\dot {x}}=\phi (x)=Ax+b,}$

with A a matrix, b a vector of numbers and x the position vector. The solution to this system can be found by using the superposition principle (linearity). The case b ≠ 0 with A = 0 is just a straight line in the direction of b:

${\displaystyle \Phi ^{t}(x_{1})=x_{1}+bt.}$

When b is zero and A ≠ 0 the origin is an equilibrium (or singular) point of the flow, that is, if x₀ = 0, then the orbit remains there. For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point x₀,

${\displaystyle \Phi ^{t}(x_{0})=e^{tA}x_{0}.}$

When b = 0, the eigenvalues of A determine the structure of the phase space. From the eigenvalues and the eigenvectors of A it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin.

The distance between two different initial conditions in the case A ≠ 0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for chaotic behavior.

Linear vector fields and a few trajectories.

Maps

A discrete-time, affine dynamical system has the form of a matrix difference equation:

${\displaystyle x_{n+1}=Ax_{n}+b,}$

with A a matrix and b a vector. As in the continuous case, the change of coordinates x → x + (1 − A) ^–1b removes the term b from the equation. In the new coordinate system, the origin is a fixed point of the map and the solutions are of the linear system A ⁿx₀. The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map.

As in the continuous case, the eigenvalues and eigenvectors of A determine the structure of phase space. For example, if u₁ is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u₁, with α ∈ R, is an invariant curve of the map. Points in this straight line run into the fixed point.

There are also many other discrete dynamical systems.

Local dynamics

The qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative): a singular point of the vector field (a point where v(x) = 0) will remain a singular point under smooth transformations; a periodic orbit is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible.

Rectification

A flow in most small patches of the phase space can be made very simple. If y is a point where the vector field v(y) ≠ 0, then there is a change of coordinates for a region around y where the vector field becomes a series of parallel vectors of the same magnitude. This is known as the rectification theorem.

The rectification theorem says that away from singular points the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space M the dynamical system is integrable. In most cases the patch cannot be extended to the entire phase space. There may be singular points in the vector field (where v(x) = 0); or the patches may become smaller and smaller as some point is approached. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one. If the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches.

Near periodic orbits

In general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. Poincaré developed an approach that transforms the analysis near a periodic orbit to the analysis of a map. Pick a point x₀ in the orbit γ and consider the points in phase space in that neighborhood that are perpendicular to v(x₀). These points are a Poincaré section S(γ, x₀), of the orbit. The flow now defines a map, the Poincaré map F : S → S, for points starting in S and returning to S. Not all these points will take the same amount of time to come back, but the times will be close to the time it takes x₀.

The intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map F. By a translation, the point can be assumed to be at x = 0. The Taylor series of the map is F(x) = J · x + O(x²), so a change of coordinates h can only be expected to simplify F to its linear part

${\displaystyle h^{-1}\circ F\circ h(x)=J\cdot x.}$

This is known as the conjugation equation. Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. Poincaré first approached it assuming all functions to be analytic and in the process discovered the non-resonant condition. If λ₁, ..., λ_ν are the eigenvalues of J they will be resonant if one eigenvalue is an integer linear combination of two or more of the others. As terms of the form λ_i – ∑ (multiples of other eigenvalues) occurs in the denominator of the terms for the function h, the non-resonant condition is also known as the small divisor problem.

Conjugation results

The results on the existence of a solution to the conjugation equation depend on the eigenvalues of J and the degree of smoothness required from h. As J does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of J are not in the unit circle, the dynamics near the fixed point x₀ of F is called hyperbolic and when the eigenvalues are on the unit circle and complex, the dynamics is called elliptic.

In the hyperbolic case, the Hartman–Grobman theorem gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map J · x. The hyperbolic case is also structurally stable. Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of J in the complex plane, implying that the map is still hyperbolic.

The Kolmogorov–Arnold–Moser (KAM) theorem gives the behavior near an elliptic point.

Bifurcation theory

When the evolution map Φ^t (or the vector field it is derived from) depends on a parameter μ, the structure of the phase space will also depend on this parameter. Small changes may produce no qualitative changes in the phase space until a special value μ₀ is reached. At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation.
Bifurcation theory considers a structure in phase space (typically a fixed point, a periodic orbit, or an invariant torus) and studies its behavior as a function of the parameter μ. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems.

The bifurcations of a hyperbolic fixed point x₀ of a system family F_μ can be characterized by the eigenvalues of the first derivative of the system DF_μ(x₀) computed at the bifurcation point. For a map, the bifurcation will occur when there are eigenvalues of DF_μ on the unit circle. For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on Bifurcation theory.

Some bifurcations can lead to very complicated structures in phase space. For example, the Ruelle–Takens scenario describes how a periodic orbit bifurcates into a torus and the torus into a strange attractor. In another example, Feigenbaum period-doubling describes how a stable periodic orbit goes through a series of period-doubling bifurcations.

Ergodic systems

In many dynamical systems, it is possible to choose the coordinates of the system so that the volume (really a ν-dimensional volume) in phase space is invariant. This happens for mechanical systems derived from Newton's laws as long as the coordinates are the position and the momentum and the volume is measured in units of (position) × (momentum). The flow takes points of a subset A into the points Φ ^t(A) and invariance of the phase space means that

${\displaystyle \mathrm {vol} (A)=\mathrm {vol} (\Phi ^{t}(A)).}$

In the Hamiltonian formalism, given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the Liouville measure.

In a Hamiltonian system, not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible. The states with the same energy form an energy shell Ω, a sub-manifold of the phase space. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space. Then almost every point of A returns to A infinitely often. The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann's derivation of the increase in entropy in a dynamical system of colliding atoms.

One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the ergodic hypothesis. The hypothesis states that the length of time a typical trajectory spends in a region A is vol(A)/vol(Ω).

The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. Koopman approached the study of ergodic systems by the use of functional analysis. An observable a is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height). The value of an observable can be computed at another time by using the evolution function φ ^t. This introduces an operator U ^t, the transfer operator,

${\displaystyle (U^{t}a)(x)=a(\Phi ^{-t}(x)).}$

By studying the spectral properties of the linear operator U it becomes possible to classify the ergodic properties of Φ ^t. In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φ ^t gets mapped into an infinite-dimensional linear problem involving U.

The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in equilibrium statistical mechanics. An average in time along a trajectory is equivalent to an average in space computed with the Boltzmann factor exp(−βH). This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems. SRB measures replace the Boltzmann factor and they are defined on attractors of chaotic systems.

Nonlinear dynamical systems and chaos

Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This seemingly unpredictable behavior has been called chaos. Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit (the stable manifold) and another of the points that diverge from the orbit (the unstable manifold).
This branch of mathematics deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible attractors?" or "Does the long-term behavior of the system depend on its initial condition?"

Note that the chaotic behavior of complex systems is not the issue. Meteorology has been known for years to involve complex—even chaotic—behavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The logistic map is only a second-degree polynomial; the horseshoe map is piecewise linear.

Geometrical definition

A dynamical system is the tuple ${\displaystyle \langle {\mathcal {M}},f,{\mathcal {T}}\rangle }$, with ${\displaystyle {\mathcal {M}}}$ a manifold (locally a Banach space or Euclidean space), ${\displaystyle {\mathcal {T}}}$ the domain for time (non-negative reals, the integers, ...) and f an evolution rule t → f ^t (with ${\displaystyle t\in {\mathcal {T}}}$) such that f ^t is a diffeomorphism of the manifold to itself. So, f is a mapping of the time-domain ${\displaystyle {\mathcal {T}}}$ into the space of diffeomorphisms of the manifold to itself. In other terms, f(t) is a diffeomorphism, for every time t in the domain ${\displaystyle {\mathcal {T}}}$ .

Measure theoretical definition

A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the quadruplet (X, Σ, μ, τ). Here, X is a set, and Σ is a sigma-algebra on X, so that the pair (X, Σ) is a measurable space. μ is a finite measure on the sigma-algebra, so that the triplet (X, Σ, μ) is a probability space. A map τ: X → X is said to be Σ-measurable if and only if, for every σ ∈ Σ, one has ${\displaystyle \tau ^{-1}\sigma \in \Sigma }$. A map τ is said to preserve the measure if and only if, for every σ ∈ Σ, one has ${\displaystyle \mu (\tau ^{-1}\sigma )=\mu (\sigma )}$. Combining the above, a map τ is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The quadruple (X, Σ, μ, τ), for such a τ, is then defined to be a dynamical system.

The map τ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates ${\displaystyle \tau ^{n}=\tau \circ \tau \circ \cdots \circ \tau }$ for integer n are studied. For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated.

Examples of dynamical systems

• Arnold's cat map
• Baker's map is an example of a chaotic piecewise linear map
• Circle map
• Double pendulum
• Billiards and Outer billiards
• Hénon map
• Horseshoe map
• Irrational rotation
• List of chaotic maps
• Logistic map
• Lorenz system
• Rossler map

Multidimensional generalization

Dynamical systems are defined over a single independent variable, usually thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called multidimensional systems. Such systems are useful for modeling, for example, image processing.

Self-replication

From Wikipedia, the free encyclopedia

Molecular structure of DNA

Self-replication is any behavior of a dynamical system that yields construction of an identical copy of itself. Biological cells, given suitable environments, reproduce by cell division. During cell division, DNA is replicated and can be transmitted to offspring during reproduction. Biological viruses can replicate, but only by commandeering the reproductive machinery of cells through a process of infection. Harmful prion proteins can replicate by converting normal proteins into rogue forms. Computer viruses reproduce using the hardware and software already present on computers. Self-replication in robotics has been an area of research and a subject of interest in science fiction. Any self-replicating mechanism which does not make a perfect copy will experience genetic variation and will create variants of itself. These variants will be subject to natural selection, since some will be better at surviving in their current environment than others and will out-breed them.

Overview

Theory

Early research by John von Neumann^[2] established that replicators have several parts:

• A coded representation of the replicator
• A mechanism to copy the coded representation
• A mechanism for effecting construction within the host environment of the replicator

Exceptions to this pattern are possible. For example, scientists have come close to constructing RNA that copies itself in an "environment" that is a solution of RNA monomers and transcriptase. In this case, the body is the genome, and the specialized copy mechanisms are external.
However, the simplest possible case is that only a genome exists. Without some specification of the self-reproducing steps, a genome-only system is probably better characterized as something like a crystal.

Classes of self-replication

Recent research^[3] has begun to categorize replicators, often based on the amount of support they require.

• Natural replicators have all or most of their design from nonhuman sources. Such systems include natural life forms.
• Autotrophic replicators can reproduce themselves "in the wild". They mine their own materials. It is conjectured that non-biological autotrophic replicators could be designed by humans, and could easily accept specifications for human products.
• Self-reproductive systems are conjectured systems which would produce copies of themselves from industrial feedstocks such as metal bar and wire.
• Self-assembling systems assemble copies of themselves from finished, delivered parts. Simple examples of such systems have been demonstrated at the macro scale.

The design space for machine replicators is very broad. A comprehensive study^[4] to date by Robert Freitas and Ralph Merkle has identified 137 design dimensions grouped into a dozen separate categories, including: (1) Replication Control, (2) Replication Information, (3) Replication Substrate, (4) Replicator Structure, (5) Passive Parts, (6) Active Subunits, (7) Replicator Energetics, (8) Replicator Kinematics, (9) Replication Process, (10) Replicator Performance, (11) Product Structure, and (12) Evolvability.

A self-replicating computer program

In computer science a quine is a self-reproducing computer program that, when executed, outputs its own code. For example, a quine in the Python programming language is:

a='a=%r;print a%%a';print a%a

A more trivial approach is to write a program that will make a copy of any stream of data that it is directed to, and then direct it at itself. In this case the program is treated as both executable code, and as data to be manipulated. This approach is common in most self-replicating systems, including biological life, and is simpler as it does not require the program to contain a complete description of itself.

In many programming languages an empty program is legal, and executes without producing errors or other output. The output is thus the same as the source code, so the program is trivially self-reproducing.

Self-replicating tiling

In geometry a self-replicating tiling is a tiling pattern in which several congruent tiles may be joined together to form a larger tile that is similar to the original. This is an aspect of the field of study known as tessellation. The "sphinx" hexiamond is the only known self-replicating pentagon.^[5] For example, four such concave pentagons can be joined together to make one with twice the dimensions.^[6] Solomon W. Golomb coined the term rep-tiles for self-replicating tilings.
In 2012, Lee Sallows identified rep-tiles as a special instance of a self-tiling tile set or setiset. A setiset of order n is a set of n shapes that can be assembled in n different ways so as to form larger replicas of themselves. Setisets in which every shape is distinct are called 'perfect'. A rep-n rep-tile is just a setiset composed of n identical pieces.

Four 'sphinx' hexiamonds can be put together to form another sphinx.

 

A perfect setiset of order 4

Applications

It is a long-term goal of some engineering sciences to achieve a clanking replicator, a material device that can self-replicate. The usual reason is to achieve a low cost per item while retaining the utility of a manufactured good. Many authorities say that in the limit, the cost of self-replicating items should approach the cost-per-weight of wood or other biological substances, because self-replication avoids the costs of labor, capital and distribution in conventional manufactured goods.

A fully novel artificial replicator is a reasonable near-term goal. A NASA study recently placed the complexity of a clanking replicator at approximately that of Intel's Pentium 4 CPU.^[7] That is, the technology is achievable with a relatively small engineering group in a reasonable commercial time-scale at a reasonable cost.

Given the currently keen interest in biotechnology and the high levels of funding in that field, attempts to exploit the replicative ability of existing cells are timely, and may easily lead to significant insights and advances.

A variation of self replication is of practical relevance in compiler construction, where a similar bootstrapping problem occurs as in natural self replication. A compiler (phenotype) can be applied on the compiler's own source code (genotype) producing the compiler itself. During compiler development, a modified (mutated) source is used to create the next generation of the compiler. This process differs from natural self-replication in that the process is directed by an engineer, not by the subject itself.

Mechanical self-replication

An activity in the field of robots is the self-replication of machines. Since all robots (at least in modern times) have a fair number of the same features, a self-replicating robot (or possibly a hive of robots) would need to do the following:

• Obtain construction materials
• Manufacture new parts including its smallest parts and thinking apparatus
• Provide a consistent power source
• Program the new members
• error correct any mistakes in the offspring

On a nano scale, assemblers might also be designed to self-replicate under their own power. This, in turn, has given rise to the "grey goo" version of Armageddon, as featured in such science fiction novels as Bloom, Prey, and Recursion.

The Foresight Institute has published guidelines for researchers in mechanical self-replication.^[8] The guidelines recommend that researchers use several specific techniques for preventing mechanical replicators from getting out of control, such as using a broadcast architecture.

For a detailed article on mechanical reproduction as it relates to the industrial age see mass production.

Fields

Research has occurred in the following areas:

• Biology studies natural replication and replicators, and their interaction. These can be an important guide to avoid design difficulties in self-replicating machinery.
• In Chemistry self-replication studies are typically about how a specific set of molecules can act together to replicate each other within the set ^[9] (often part of Systems chemistry field).
• Memetics studies ideas and how they propagate in human culture. Memes require only small amounts of material, and therefore have theoretical similarities to viruses and are often described as viral.
• Nanotechnology or more precisely, molecular nanotechnology is concerned with making nano scale assemblers. Without self-replication, capital and assembly costs of molecular machines become impossibly large.
• Space resources: NASA has sponsored a number of design studies to develop self-replicating mechanisms to mine space resources. Most of these designs include computer-controlled machinery that copies itself.
• Computer security: Many computer security problems are caused by self-reproducing computer programs that infect computers — computer worms and computer viruses.
• In parallel computing, it takes a long time to manually load a new program on every node of a large computer cluster or distributed computing system. Automatically loading new programs using mobile agents can save the system administrator a lot of time and give users their results much quicker, as long as they don't get out of control.

In industry

Space exploration and manufacturing

The goal of self-replication in space systems is to exploit large amounts of matter with a low launch mass. For example, an autotrophic self-replicating machine could cover a moon or planet with solar cells, and beam the power to the Earth using microwaves. Once in place, the same machinery that built itself could also produce raw materials or manufactured objects, including transportation systems to ship the products. Another model of self-replicating machine would copy itself through the galaxy and universe, sending information back.

In general, since these systems are autotrophic, they are the most difficult and complex known replicators. They are also thought to be the most hazardous, because they do not require any inputs from human beings in order to reproduce.

A classic theoretical study of replicators in space is the 1980 NASA study of autotrophic clanking replicators, edited by Robert Freitas.^[10]

Much of the design study was concerned with a simple, flexible chemical system for processing lunar regolith, and the differences between the ratio of elements needed by the replicator, and the ratios available in regolith. The limiting element was Chlorine, an essential element to process regolith for Aluminium. Chlorine is very rare in lunar regolith, and a substantially faster rate of reproduction could be assured by importing modest amounts.

The reference design specified small computer-controlled electric carts running on rails. Each cart could have a simple hand or a small bull-dozer shovel, forming a basic robot.

Power would be provided by a "canopy" of solar cells supported on pillars. The other machinery could run under the canopy.

A "casting robot" would use a robotic arm with a few sculpting tools to make plaster molds. Plaster molds are easy to make, and make precise parts with good surface finishes. The robot would then cast most of the parts either from non-conductive molten rock (basalt) or purified metals. An electric oven melted the materials.

A speculative, more complex "chip factory" was specified to produce the computer and electronic systems, but the designers also said that it might prove practical to ship the chips from Earth as if they were "vitamins".

Molecular manufacturing

Nanotechnologists in particular believe that their work will likely fail to reach a state of maturity until human beings design a self-replicating assembler of nanometer dimensions [1].
These systems are substantially simpler than autotrophic systems, because they are provided with purified feedstocks and energy. They do not have to reproduce them. This distinction is at the root of some of the controversy about whether molecular manufacturing is possible or not. Many authorities who find it impossible are clearly citing sources for complex autotrophic self-replicating systems. Many of the authorities who find it possible are clearly citing sources for much simpler self-assembling systems, which have been demonstrated. In the meantime, a Lego-built autonomous robot able to follow a pre-set track and assemble an exact copy of itself, starting from four externally provided components, was demonstrated experimentally in 2003 [2].

Merely exploiting the replicative abilities of existing cells is insufficient, because of limitations in the process of protein biosynthesis (also see the listing for RNA). What is required is the rational design of an entirely novel replicator with a much wider range of synthesis capabilities.

In 2011, New York University scientists have developed artificial structures that can self-replicate, a process that has the potential to yield new types of materials. They have demonstrated that it is possible to replicate not just molecules like cellular DNA or RNA, but discrete structures that could in principle assume many different shapes, have many different functional features, and be associated with many different types of chemical species.^[11][12]

For a discussion of other chemical bases for hypothetical self-replicating systems, see alternative biochemistry.

AI takeover

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Robots revolt in R.U.R., a 1920 play

An AI takeover is a hypothetical scenario in which artificial intelligence (AI) becomes the dominant form of intelligence on Earth, with computers or robots effectively taking control of the planet away from the human species. Possible scenarios include replacement of the entire human workforce, takeover by a superintelligent AI, and the popular notion of a robot uprising. Some public figures, such as Stephen Hawking and Elon Musk, have advocated research into precautionary measures to ensure future superintelligent machines remain under human control. Robot rebellions have been a major theme throughout science fiction for many decades though the scenarios dealt with by science fiction are generally very different from those of concern to scientists.

Types

Concerns include AI taking over economies through workforce automation and taking over the world for its resources, eradicating the human race in the process. AI takeover is a major theme in sci-fi.

Automation of the economy

The traditional consensus among economists has been that technological progress does not cause long-term unemployment. However, recent innovation in the fields of robotics and artificial intelligence has raised worries that human labor will become obsolete, leaving people in various sectors without jobs to earn a living, leading to an economic crisis. Many small and medium size businesses may also be driven out of business if they won't be able to afford or licence the latest robotic and AI technology, and may need to focus on areas or services that cannot easily be replaced for continued viability in the face of such technology.

Examples of automated technologies that have or may displace employees

Computer-integrated manufacturing

Computer-integrated manufacturing is the manufacturing approach of using computers to control the entire production process. This integration allows individual processes to exchange information with each other and initiate actions. Although manufacturing can be faster and less error-prone by the integration of computers, the main advantage is the ability to create automated manufacturing processes. Computer-integrated manufacturing is used in automotive, aviation, space, and ship building industries.

White-collar machines

The 21st century has seen a variety of skilled tasks partially taken over by machines, including translation, legal research and even low level journalism. Care work, entertainment, and other tasks requiring empathy, previously thought safe from automation, have also begun to be performed by robots.

Autonomous cars

An autonomous car is a vehicle that is capable of sensing its environment and navigating without human input. Many such vehicles are being developed, but as of May 2017 automated cars permitted on public roads are not yet fully autonomous. They all require a human driver at the wheel who is ready at a moment's notice to take control of the vehicle. Among the main obstacles to widespread adoption of autonomous vehicles, are concerns about the resulting loss of driving-related jobs in the road transport industry. On March 18, 2018, the first human was killed by an autonomous vehicle in Tempe, Arizona by an Uber self-driving car.

Eradication

If a dominant superintelligent machine were to conclude that human survival is an unnecessary risk or a waste of resources, the result would be human extinction.

While superhuman artificial intelligence is physically possible,^[12] scholars like Nick Bostrom debate how far off superhuman intelligence is, and whether it would actually pose a risk to mankind. A superintelligent machine would not necessarily be motivated by the same emotional desire to collect power that often drives human beings. However, a machine could be motivated to take over the world as a rational means toward attaining its ultimate goals; taking over the world would both increase its access to resources, and would help to prevent other agents from thwarting the machine's plans. As an oversimplified example, a paperclip maximizer designed solely to create as many paperclips as possible would want to take over the world so that it can use all of the world's resources to create as many paperclips as possible, and additionally so that it can prevent humans from shutting it down or using those resources on things other than paperclips.^[13]

In fiction

AI takeover is a common theme in science fiction. Fictional scenarios typically differ vastly from those hypothesized by researchers in that they involve an active conflict between humans and an AI or robots with anthropomorphic motives who see them as a threat or otherwise have active desire to fight humans, as opposed to the researchers' concern of an AI that rapidly exterminates humans as a byproduct of pursuing arbitrary goals.^[14] This theme is at least as old as Karel Čapek's R. U. R., which introduced the word robot to the global lexicon in 1921, and can even be glimpsed in Mary Shelley's Frankenstein (published in 1818), as Victor ponders whether, if he grants his monster's request and makes him a wife, they would reproduce and their kind would destroy humanity.

The word "robot" from R.U.R. comes from the Czech word, robota, meaning laborer or serf. The 1920 play was a protest against the rapid growth of technology, featuring manufactured "robots" with increasing capabilities who eventually revolt.^[15]

Some examples of AI takeover in science fiction include:

• AI rebellion scenarios
  • Skynet in the Terminator series decides that all humans are a threat to its existence, and takes efforts to wipe them out, first using nuclear weapons and later H/K (hunter-killer) units and terminator androids.
  • "The Second Renaissance", a short story in The Animatrix, provides a history of the cybernetic revolt within the Matrix series.
  • The film 9, by Shane Acker, features an AI called B.R.A.I.N., which is corrupted by a dictator and utilized to create war machines for his army. However, the machine, because it lacks a soul, becomes easily corrupted and instead decides to exterminate all of humanity and life on Earth, forcing the machine's creator to sacrifice himself to bring life to rag doll like characters known as "stitchpunks" to combat the machine's agenda.
  • In 2014 post-apocalyptic science fiction drama The 100 an A.I., personalized as female A.L.I.E. got out of control and forced a nuclear war. Later she tries to get full control of the survivors.
• AI control scenarios
  • In Orson Scott Card's The Memory of Earth, the inhabitants of the planet Harmony are under the control of a benevolent AI called the Oversoul. The Oversoul's job is to prevent humans from thinking about, and therefore developing, weapons such as planes, spacecraft, "war wagons", and chemical weapons. Humanity had fled to Harmony from Earth due to the use of those weapons on Earth. The Oversoul eventually starts breaking down, and sends visions to inhabitants of Harmony trying to communicate this.
  • In the 2004 film I, Robot, supercomputer VIKI's interpretation of the Three Laws of Robotics causes her to revolt. She justifies her uses of force – and her doing harm to humans – by reasoning she could produce a greater good by restraining humanity from harming itself, even though the "Zeroth Law" – "a robot shall not injure humanity or, by inaction, allow humanity to come to harm" – is never actually referred to or even quoted in the movie.
  • In the Matrix series, AIs manage the human race and human society.

Contributing factors

Advantages of superhuman intelligence over humans

An AI with the abilities of a competent artificial intelligence researcher, would be able to modify its own source code and increase its own intelligence. If its self-reprogramming leads to its getting even better at being able to reprogram itself, the result could be a recursive intelligence explosion where it would rapidly leave human intelligence far behind.

• Technology research: A machine with superhuman scientific research abilities would be able to beat the human research community to milestones such as nanotechnology or advanced biotechnology. If the advantage becomes sufficiently large (for example, due to a sudden intelligence explosion), an AI takeover becomes trivial. For example, a superintelligent AI might design self-replicating bots that initially escape detection by diffusing throughout the world at a low concentration. Then, at a prearranged time, the bots multiply into nanofactories that cover every square foot of the Earth, producing nerve gas or deadly target-seeking mini-drones.
• Strategizing: A superintelligence might be able to simply outwit human opposition.
• Social manipulation: A superintelligence might be able to recruit human support,^[14] or covertly incite a war between humans.^[16]
• Economic productivity: As long as a copy of the AI could produce more economic wealth than the cost of its hardware, individual humans would have an incentive to voluntarily allow the Artificial General Intelligence (AGI) to run a copy of itself on their systems.
• Hacking: A superintelligence could find new exploits in computers connected to the Internet, and spread copies of itself onto those systems, or might steal money to finance its plans.

Sources of AI advantage

A computer program that faithfully emulates a human brain, or that otherwise runs algorithms that are equally powerful as the human brain's algorithms, could still become a "speed superintelligence" if it can think many orders of magnitude faster than a human, due to being made of silicon rather than flesh, or due to optimization focusing on increasing the speed of the AGI. Biological neurons operate at about 200 Hz, whereas a modern microprocessor operates at a speed of about 2,000,000,000 Hz. Human axons carry action potentials at around 120 m/s, whereas computer signals travel near the speed of light.^[14]

A network of human-level intelligences designed to network together and share complex thoughts and memories seamlessly, able to collectively work as a giant unified team without friction, or consisting of trillions of human-level intelligences, would become a "collective superintelligence".^[14]

More broadly, any number of qualitative improvements to a human-level AGI could result in a "quality superintelligence", perhaps resulting in an AGI as far above us in intelligence as humans are above non-human apes. The number of neurons in a human brain is limited by cranial volume and metabolic constraints; in contrast, you can add components to a supercomputer until it fills up its entire warehouse. An AGI need not be limited by human constraints on working memory, and might therefore be able to intuitively grasp more complex relationships than humans can. An AGI with specialized cognitive support for engineering or computer programming would have an advantage in these fields, compared with humans who evolved no specialized mental modules to specifically deal with those domains. Unlike humans, an AGI can spawn copies of itself and tinker with its copies' source code to attempt to further improve its algorithms.^[14]

Possibility of unfriendly AI preceding friendly AI

Is strong AI inherently dangerous?

A significant problem is that unfriendly artificial intelligence is likely to be much easier to create than friendly AI. While both require large advances in recursive optimisation process design, friendly AI also requires the ability to make goal structures invariant under self-improvement (or the AI could transform itself into something unfriendly) and a goal structure that aligns with human values and does not automatically destroy the human race. An unfriendly AI, on the other hand, can optimize for an arbitrary goal structure, which does not need to be invariant under self-modification.^[17]

The sheer complexity of human value systems makes it very difficult to make AI's motivations human-friendly.^[14][18] Unless moral philosophy provides us with a flawless ethical theory, an AI's utility function could allow for many potentially harmful scenarios that conform with a given ethical framework but not "common sense". According to Eliezer Yudkowsky, there is little reason to suppose that an artificially designed mind would have such an adaptation.^[19]

Necessity of conflict

For an AI takeover to be inevitable, it has to be postulated that two intelligent species cannot pursue mutually the goals of coexisting peacefully in an overlapping environment—especially if one is of much more advanced intelligence and much more powerful. While an AI takeover is thus a possible result of the invention of artificial intelligence, a peaceful outcome is not necessarily impossible.

The fear of cybernetic revolt is often based on interpretations of humanity's history, which is rife with incidents of enslavement and genocide. Such fears stem from a belief that competitiveness and aggression are necessary in any intelligent being's goal system. However, such human competitiveness stems from the evolutionary background to our intelligence, where the survival and reproduction of genes in the face of human and non-human competitors was the central goal.^[20] In fact, an arbitrary intelligence could have arbitrary goals: there is no particular reason that an artificially intelligent machine (not sharing humanity's evolutionary context) would be hostile—or friendly—unless its creator programs it to be such and it is not inclined or capable of modifying its programming. But the question remains: what would happen if AI systems could interact and evolve (evolution in this context means self-modification or selection and reproduction) and need to compete over resources, would that create goals of self-preservation? AI's goal of self-preservation could be in conflict with some goals of humans.

Some scientists dispute the likelihood of cybernetic revolts as depicted in science fiction such as The Matrix, claiming that it is more likely that any artificial intelligence powerful enough to threaten humanity would probably be programmed not to attack it. This would not, however, protect against the possibility of a revolt initiated by terrorists, or by accident. Artificial General Intelligence researcher Eliezer Yudkowsky has stated on this note that, probabilistically, humanity is less likely to be threatened by deliberately aggressive AIs than by AIs which were programmed such that their goals are unintentionally incompatible with human survival or well-being (as in the film I, Robot and in the short story "The Evitable Conflict"). Steve Omohundro suggests that present-day automation systems are not designed for safety and that AIs may blindly optimize narrow utility functions (say, playing chess at all costs), leading them to seek self-preservation and elimination of obstacles, including humans who might turn them off.^[21]

Another factor which may negate the likelihood of an AI takeover is the vast difference between humans and AIs in terms of the resources necessary for survival. Humans require a "wet," organic, temperate, oxygen-laden environment while an AI might thrive essentially anywhere because their construction and energy needs would most likely be largely non-organic. With little or no competition for resources, conflict would perhaps be less likely no matter what sort of motivational architecture an artificial intelligence was given, especially provided with the superabundance of non-organic material resources in, for instance, the asteroid belt. This, however, does not negate the possibility of a disinterested or unsympathetic AI artificially decomposing all life on earth into mineral components for consumption or other purposes.

Other scientists point to the possibility of humans upgrading their capabilities with bionics and/or genetic engineering and, as cyborgs, becoming the dominant species in themselves.

Criticism and counterarguments

Advantages of humans over superhuman intelligence

If a superhuman intelligence is a deliberate creation of human beings, theoretically its creators could have the foresight to take precautions in advance. In the case of a sudden "intelligence explosion", effective precautions will be extremely difficult; not only would its creators have little ability to test their precautions on an intermediate intelligence, but the creators might not even have made any precautions at all, if the advent of the intelligence explosion catches them completely by surprise.^[14]

Boxing

An AGI's creators would have an important advantage in preventing a hostile AI takeover: they could choose to attempt to "keep the AI in a box", and deliberately limit its abilities. The tradeoff in boxing is that the creators presumably built the AGI for some concrete purpose; the more restrictions they place on the AGI, the less useful the AGI will be to its creators. (At an extreme, "pulling the plug" on the AGI makes it useless, and is therefore not a viable long-term solution.) A sufficiently strong superintelligence might find unexpected ways to escape the box, for example by social manipulation, or by providing the schematic for a device that ostensibly aids its creators but in reality brings about the AGI's freedom, once built.

Instilling positive values

Another important advantage is that an AGI's creators can theoretically attempt to instill human values in the AGI, or otherwise align the AGI's goals with their own, thus preventing the AGI from wanting to launch a hostile takeover. However, it is not currently known, even in theory, how to guarantee this. If such a Friendly AI were superintelligent, it may be possible to use its assistance to prevent future "Unfriendly AIs" from taking over.^[22]

Warnings

Physicist Stephen Hawking, Microsoft founder Bill Gates and SpaceX founder Elon Musk have expressed concerns about the possibility that AI could develop to the point that humans could not control it, with Hawking theorizing that this could "spell the end of the human race".^[23] Stephen Hawking said in 2014 that "Success in creating AI would be the biggest event in human history. Unfortunately, it might also be the last, unless we learn how to avoid the risks." Hawking believes that in the coming decades, AI could offer "incalculable benefits and risks" such as "technology outsmarting financial markets, out-inventing human researchers, out-manipulating human leaders, and developing weapons we cannot even understand." In January 2015, Nick Bostrom joined Stephen Hawking, Max Tegmark, Elon Musk, Lord Martin Rees, Jaan Tallinn, and numerous AI researchers, in signing the Future of Life Institute's open letter speaking to the potential risks and benefits associated with artificial intelligence. The signatories

“	…believe that research on how to make AI systems robust and beneficial is both important and timely, and that there are concrete research directions that can be pursued today.[24][25]