The Computational Beauty of Nature
Computer Explorations of Fractals, Chaos,
Complex Systems, and Adaptation

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Glossary - S

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Saddle     A type of surface that is neither a peak nor a valley but still has a 0 gradient. Saddle points are situated such that moving in one direction takes one uphill, while moving in another direction would be downhill. Hence, saddles look like the things that cowboys ride on.

Scalar     A single number, as opposed to a multidimensional vector or matrix.

Schema/Schemata     A similarity template used to analyze genetic algorithms. By using wild-card characters, a schema defines an entire class of strings that may be found in a population.

Search/Search Method     A method for finding a region of interest in a search space. Usually, the interesting regions correspond to solutions to a specific problem. Hill-climbing, steepest descent (ascent), simulated annealing, and genetic algorithms are all search methods.

Search Space     A characterization of every possible solution to a problem instance. For a neural network the search space is defined as all possible assignments to the network weights; for a genetic algorithm, it is every conceivable value assignment to the strings in the population.

Selection     See natural selection.

Self-Organization     A spontaneously formed higher-level pattern of structure or function that is emergent through the interactions of lower-level objects.

Self-Organized Criticality (SOC)     A mathematical theory that describes how systems composed of many interacting parts can tune themselves toward dynamical behavior that is critical in the sense that it is neither stable nor unstable but at a region near a phase transition. SOC systems display events in a power-law distribution and are never quite at equilibrium. See also edge of chaos and self-organization.

Self-Referential     Referring directly back to oneself through information flow, influence, or cause and effect. See Self-Referential.

Self-Similar     An object that is structurally recursive in that a part will look like the whole. See also fractal.

Sensitivity     The tendency of a system (sometimes chaotic) to change dramatically with only small perturbations.

Set     A collection of things, usually numbers. Sets may be infinite in size.

Shadowing Lemma     Implies that a numerical simulation of chaos may ``shadow'' a real trajectory of a real chaotic system.

Sigmoidal     An ``S'' shaped function that is often used as an activation function in a neural network.

Simulate/Simulation     Experimentation in the space of theories, or a combination of experimentation and theorization. Some numerical simulations are programs that represent a model for how nature works. Usually, the outcome of a simulation is as much a surprise as the outcome of a natural event, due to the richness and uncertainty of computation.

Simulated Annealing     A partially random method of search and optimization usually used for combinatorial optimization problems. The technique is modeled on how the molecular structure of metals is disordered at high temperatures but very ordered and crystalline at low temperatures. In simulated annealing, a problem instance is reformulated so that it loosely resembles disordered material. Gradually, the temperature is lowered such that the ordered states correspond to good solutions to a problem.

Space Complexity     A function that describes the amount of memory required for a program to run on a computer to perform a particular task. The function is parameterized by the length of the program's input. See also time complexity.

Space-Filling     Refers to a curve that manages to twist and turn in such a way that it actually fills a space or volume. All space-filling curves are fractal.

Special Function     In Lisp or Stutter, a built-in function that may or may not fully evaluate its arguments, such as the if primitive.

Stable     Having a basin of attraction that is non-zero in size; an attractor that can withstand some form of perturbation.

Standard Deviation     A measure of the spread of a set of data. For a Gaussian distribution, the standard deviation hints at the width of the tails of the distribution function.

Statement     In a formal system, a string of characters that is formed according to well-defined rules such that it is legal for the language that is the formal system. For example, in the formal system of arithmetic, the expression ``5 + 3 * (2 - 4)'' is a valid and well-formed statement, but ``5 + )3 * * (2 (- 4)'' is not.

State Space     In this book, another name for the phase space of a dynamical system. Roughly speaking, if the dynamics of a dynamical system can be described by n values, then the state space is the n-dimensional volume that the system moves through. Systems that are continuous in time will form a smooth trajectory through this volume, while discrete systems may jump to different locations on subsequent time steps. In either case, if a system ever returns to a previously visited location in the state space, then the system is in either a fixed point or a limit cycle. For chaotic systems, or for programs that never halt, the system will always be at a previously unvisited portion of the state space.

Steepest Descent (Ascent)     A search method that uses the gradient information of a search space and moves in the opposite direction from the gradient until no further downhill (or uphill) progress can be made. See also hill-climbing.

Stochastic     Something that is random.

Strange Attractor     An attractor of a dynamical system that is usually fractal in dimension and is indicative of chaos.

Strategy     In game theory, a policy for playing a game. A strategy is a complete recipe for how a player should act in a game under all circumstances. Some policies may employ randomness, in which case they are referred to as mixed strategies.

Strength     For a classifier system, a classifier's relative ability to win a bidding match for the right to post its message on the message list.

String     Any sequence of letters, numbers, digits, bits, or symbols.

Stutter     A silly programming language used in this book that is based on Lisp and is capable of universal computation.

Symmetric Matrix     A matrix with the lower-left half equal to the mirror image of the upper-right half.

Synapse     The junction between two neurons in which neural activity is propagated from one neuron to another. See also excitatory, inhibitory, and weight.

Synchronous     Acting in a lockstep fashion, with each event occurring in a precise order, or in such a way as to eliminate the notion of order entirely.

System     Something that can be studied as a whole. Systems may consist of subsystems that are interesting in their own right. Or they may exist in an environment that consists of other similar systems. Systems are generally understood to have an internal state, inputs from an environment, and methods for manipulating the environment or themselves. Since cause and effect can flow in both directions of a system and environment, interesting systems often posses feedback, which is self-referential in the strongest case.

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Copyright © Gary William Flake, 1998-2002. All Rights Reserved. Last modified: 30 Nov 2002