Automata theory

Automata theory, body of physical and logical principles underlying the operation of any electromechanical device (an automaton) that converts information from one form into another according to a definite procedure. Real or hypothetical automata of varying complexity have become indispensable tools for the investigation and implementation of systems that have structures amenable to mathematical analysis.

An example of a typical automaton is a pendulum clock. In such a mechanism the gears can assume only one of a finite number of positions, or states, with each swing of the pendulum. Each state, through the operation of the escapement, determines the next succeeding state, as well as a discrete output, which is displayed as the discrete positions of the hands of the clock. As long as such a clock is wound and its operation is not interfered with, it will continue to operate unaffected by outside influences except the effect of gravity on the pendulum.

More general automata are designed to respond to changes in external conditions or to other inputs. For example, thermostats, automatic pilots of aircraft, missile guidance systems, telephone networks, and controls of certain kinds of automatic elevators are all forms of automata.

The internal states of such devices are not determined solely by their initial state, as is the case of the pendulum clock, but may be determined by an input from a human operator, from another automaton, or by an event or series of events in the environment. A thermostat, for instance, has an “on” or “off” state that depends on the temperature. The best known general automaton is the modern electronic computer, the internal states of which are determined by the data input and which operates to produce a certain output.

Nature and origin of modern automata

The components of automata consist of specific materials and devices, such as wires, transistors, levers, relays, gears, and so forth, and their operation is based on the mechanics and electronics of these parts. The principles of their operation as a sequence of discrete states can, however, be understood independently of the nature or arrangement of their components. In this way, an automaton may be considered, abstractly, as a set of physically unspecified states, inputs, outputs, and rules of operation, and the study of automata as the investigation of what can be accomplished with these. This mode of abstraction yields mathematical systems that in certain respects resemble logical systems. Thus, an automaton can be described as a logically defined entity that can be embodied in the form of a machine, with the term automaton designating both the physical and the logical constructions.

In 1936 an English mathematician, Alan Mathison Turing, in a paper published in the Proceedings of the London Mathematical Society (“On Computable Numbers with an Application to the Entscheidungsproblem”), conceived a logical machine the output of which could be used to define a computable number. For the machine, time was considered to be discrete and its internal structure, at a given moment, was described simply as one of a finite set of states. It performed its functions by scanning an unbounded tape divided into squares, each of which either contained specific information in the form of one of a finite number of symbols or was blank. It could scan only one square at a time, and, if in any internal state except one called “passive,” it was capable of moving the tape forward or backward one square at a time, erasing a symbol, printing a new symbol if the square was blank, and altering its own internal state. The number it computed was determined by symbols (the “program”) provided on a finite portion of the tape and the rules of operation, which included stopping when the passive state was reached. The output number was then interpreted from the symbols remaining on the tape after the machine stopped.

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Automata theory since the middle of the 20th century has been extensively refined and has often found practical application in civilian and military machines. The memory banks of modern computers can store large (though finite) amounts of information. (For further information on computers and their applications, see information processing.) The original Turing machine had no limit to the memory bank because each square on the unbounded tape could hold information. The Turing machine continues to be a standard reference point in basic discussions of automata theory, and many mathematical theorems concerning computability have been proved within the framework of Turing’s original proposal.

Neural nets and automata

The finite automata of McCulloch and Pitts

Part of automata theory lying within the area of pure mathematical study is often based on a model of a portion of the nervous system in a living creature and on how that system with its complex of neurons, nerve endings, and synapses (separating gap between neurons) can generate, codify, store, and use information. The “all or none” nature of the threshold of neurons is often referred to in formulating purely logical schemata or in constructing the practical electronic gates of computers. Any physical neuron can be sufficiently excited by an oncoming impulse to fire another impulse into the network of which it forms a part, or else the threshold will not be reached because the stimulus is absent or inadequate. In the latter case, the neuron fails to fire and remains quiescent. When several neurons are connected together, an impulse travelling in a particular part of the network may have several effects. It can inhibit another neuron’s ability to release an impulse; it can combine with several other incoming impulses each of which is incapable of exciting a neuron to fire but that, in combination, may provide the threshold stimulus; or the impulse might be confined within a section of the nerve net and travel in a closed loop, in what is called “feedback.” Mathematical reasoning about how nerve nets work has been applied to the problem of how feedback in a computing machine can result in an essential ingredient in the calculational process.

Original work on this aspect of automata theory was done by Warren S. McCulloch and Walter Pitts at the Research Laboratory of Electronics at the Massachusetts Institute of Technology starting in the 1940s.

The definitions of various automata as used here are based on the work of two mathematicians, John von Neumann and Stephen Cole Kleene, and the earlier neurophysiological researches of McCulloch and Pitts, which offer a mathematical description of some essential features of a living organism. The neurological model is suggested from studies of the sensory receptor organs, internal neural structure, and effector organs of animals. Certain responses of an animal to stimuli are known by controlled observation, and, since the pioneering work of a Spanish histologist, Santiago Ramón y Cajal, in the latter part of the 19th and early part of the 20th century, many neural structures have been well known. For the purposes of this article, the mathematical description of neural structure, following the neurophysiological description, will be called a “neural net.” The net alone and its response to input data are describable in purely mathematical terms.

A neural net may be conveniently described in terms of the kind of geometric configuration that suggests the physical structure of a portion of the brain. The component parts in the geometric form of a neural net are named (after the physically observed structures) neurons. Diagrammatically they could be represented by a circle and a line (together representing the body, or soma, of a physiological neuron) leading to an arrowhead or a solid dot (suggesting an endbulb of a neuron). A neuron may be assumed to have either an excitatory or an inhibitory effect on a succeeding one; and it may possess a threshold, or minimum number of unit messages, so to speak, that must be received from other neurons before it can be activated to fire an impulse. The process of transmission of excitation mimics that which is observed to occur in the nervous system of an animal. Messages of unit excitation are transmitted from one neuron to the next, and excitation is passed along the neural net in quantized form, a neuron either becoming excited or remaining non-excited, depending on the states (excitatory or quiescent) of neurons whose endbulbs impinge upon it. Specifically, neuron N, with threshold h, will be excited at time t, if and only if h or more neurons whose excitatory endbulbs impinge upon it are excited at time t - 1 and no neuron whose inhibitory endbulb impinges upon it is excited at time t - 1. A consistent picture can be made of these conditions only if time and excitation are quantized (or pulsed). It is assumed conventionally that a unit of time is required for the transmission of a message by any neuron.

Certain neurons in the configuration mathematically represent the physiological receptors that are excited or left quiescent by the exterior environment. These are called input neurons. Other neurons called output neurons record the logical value, excited or quiescent, of the whole configuration after time delay t and transmit an effect to an exterior environment. All the rest stimulate inner neurons.

Any geometric or logical description of the neural structure of an organism formulated as the basis of physical construction must be sufficiently simple to permit mechanical, electric, or electronic simulation of the neurons and their interconnections.

The basic logical organs

The types of events that can excite the automaton and the kinds of responses that it can make must next be considered. By stripping the description down to the most simple cases, the basic organs from which more complicated robots can be constructed may be discovered. Three basic organs (or elementary automata) are necessary, each corresponding to one of the three logical operations of language: the binary operations of disjunction and conjunction, leading to such propositions as AB (read “A or B”), AB (read “A and B”), and the unary operation of negation or complementation, leading to such propositions as Ac (read “not A” or “complement of A”). First to be considered are the stimulus-response pattern of these elementary automata.

Assuming that a neuron can be in only one of two possible states—i.e., excited or quiescent—an input neuron at a given instant of time t - 1 must be either excited or nonexcited by its environment. An environmental message transmitted to two input neurons N1 and N2 at time t - 1 can then be represented numerically in any one of the four following ways, in which binary digit 1 represents excitation and binary digit 0 represents quiescence: (0, 0), (0, 1), (1, 0), (1, 1). The disjunction automaton must be such that a single output neuron M correspondingly registers at time t the response: 0, 1, 1, 1. The conjunction automaton must be such that a single output neuron M correspondingly registers at time t the response: 0, 0, 0, 1. The negation automaton considered as having two input neurons N1 and N2, of which N1 is always excited, must respond to the environmental messages (1, 0) and (1, 1) with 1, 0, respectively, at the output neuron M.

The generalized automaton and Turing’s machine

The construction of more complicated robots from these basic building blocks constitutes a large part of the theory of automata. The first step in the direction of generalization is to define the neural nets that correspond to formal expressions in n variables of the propositional calculus—that is, the formal system that concerns “or,” “and,” “not,” and “implies.” A single output automaton (of which the above three are simple examples) is a neural net with n input neurons, one output neuron, and with interconnections between neurons that conform to the rule that no neuron stimulated at time t can impinge upon a neuron that could have experienced its first stimulation at the same or an earlier time. The latter rule is the requirement of no feedback. Given this concept of a single output automaton, it is possible to examine the output response at time t + s, considered as a function of the configuration of stimuli at the n input neurons at time t. This response can be compared with the truth value of a logical statement (polynomial) from the propositional calculus. A logical statement is formed from n component propositions, each of which can assume the truth value either true or false. The comparison between automaton and logical statement is accomplished by matching response at the output neuron at time t + s with truth value of the statement for every one of the 2n cases in which the configuration of stimuli conforms to the configuration of truth values of the component propositions. If, in this sense of comparison, the functional response of the automaton is identical to the functional value of the logical statement (polynomial), the automaton is then said to compute the statement (polynomial) or the statement is said to be computable. A wider class of computable statements is introduced with the general automaton, yet to be defined, as with the more general Turing machine.

The important distinction between the logical statement and the automaton that computes it is that the first is free of any time ingredient while the second is defined only with reference to a time delay of length s.

A basic theorem states that for any polynomial P of the propositional calculus, there exists a time delay s and a single output automaton A, such that A computes P with time delay s. The proof of the theorem rests on the fact from the propositional calculus that all statements are composed from component propositions with the operations of disjunction, conjunction, and negation and the fact from the automata theory that all single output automata can be composed by interconnecting elementary automata of the disjunctive, conjunctive, and negative types.

A second step of generalization in the construction of robots proceeds from the single output automata to the neural net that possesses more than one output neuron and in which the internal connections may include feedback. Such a construction is called a “general automaton.” The class of general automata includes all-purpose, electronic digital computers the memory-storage units of which are of fixed, though possibly of very considerable, size. It is within the context of the general automaton that the purely automated decision-making, computing, controlling, and other sophisticated neural functions so suggestive of the mental ability of human beings may appropriately be discussed.

The Turing machine can be defined not only as it was in the introduction (roughly following Turing’s approach) but as a general automaton to which an unbounded memory unit (such as an unbounded tape) is added. Thus, the general automaton and the Turing machine differ in logical design only with respect to the extent of memory storage.

The distinction is critical, however, for Turing proposed that the class of numbers computable on his machine (a wider class than can be obtained by general automata) coincide with those that are effectively computable in the sense of constructive logics. A simple convention also makes it possible to interpret the output of a Turing machine as the computation of a function. The class of functions so computed, called “Turing computable” or “computable,” are of basic importance at the foundations of mathematics and elsewhere. It can also be stated that a useful class of functions that are definable without reference to machines, namely, the so-called partial recursive functions, has the same membership as the class of computable functions. For the present purposes, then, no effort need be made to define the partial recursive functions.

Turing’s approach admitted mathematical formalization to the extent that a finite list of symbols q1, q2, q3, · · · , qn could be used to denote internal states and a finite list of symbols a, b, c, · · · , λ could designate abstractly what is called “the alphabet”—that is, the list from which individual members could be chosen and printed into the squares of the machine’s tape. If the symbols R and L, respectively, designate a move of the tape one square to the right and one square to the left, it remains only to list in some orderly fashion the alternative possible steps in the machine’s operation in order to define it completely. Turing himself chose to list alternate steps, or instructions, in the form of quintuples of the above symbols. It is also possible to use quadruples to define a machine. Such a list, then, of, say, quadruples of instructions is equivalent to a Turing machine, and it is significant that the list is finite.

The finiteness of the list of quadruples of instructions leads to the idea that all Turing machines can be listed—that is, they are at most countable in number. This being the case, it can be proved that there is what Turing called a “universal” machine capable of operating like any given Turing machine. For a given partial recursive function of a single argument, there is a corresponding integer, called the Gödel number, that identifies the Turing machine capable of computing the given function. The Gödel number and the argument value of the function to be computed can be given as input data on the tape of the universal machine. From the Gödel number, the list of instructions, defined in the form of quadruples, that are necessary for the computation of the given recursive function at the specific argument value can be encoded by the universal machine on its own tape, and, from that point on, the universal machine will duplicate the required Turing machine.

Input: events that affect an automaton

Once having reached the definition of the general automaton and the more general universal Turing machine, a general definition of the events in the environment that stimulate it may be introduced. The automaton, which computes logical statements, is not defined without reference to time, a characteristic that distinguishes the machine itself from the logic. In the same way, stimuli are not definable, in general, without reference to time. These facts are indicative of the simulation features that the computing machine bears with respect to man.

For an automaton with n input neurons, N1, N2, · · · , Nn, an individual history of stimulation, starting with the present moment, t = 0, and continuing to the remote past, can be recorded as a sequence of n-tuples, (β1, β2, · · · , βn), in which each binary digit, βk, is either a 0 or a 1. Thus, the beginning of one such individual history for an automaton of four neurons might be recorded in tabular form as an unending list of quadruples of the type (1, 0, 1, 1) (see Box, display 1).

An event is a collection of individual histories. This is a generalization of the idea already used to characterize an environmental message transmitted to the two input neurons of an elementary automaton at time t - 1. As an example, the stimulus (0, 1) is the same as the collection of all individual histories in which neuron N2 was stimulated at time t - 1 and neuron N1 was not. As another example, the event that neuron N2 (of a two-neuron automaton) is presently stimulated and has always been stimulated on alternate second can be represented as the collection of two individual histories (see 2). While some events require an infinite tabulation, others that specify the states of each neuron over a finite past (allowing that anything might have occurred before) permit a finite tabulation. Events of the second kind are called definite events, or stimuli.

The construction (either actual or theoretical) of a general automaton with the help of the logical components and interconnections of a neural net results in an entity that responds in reproducible ways to stimuli. A response becomes recorded as a configuration of binary digits, corresponding to the states of the finite number of output neurons at a specified time t in the future, while a stimulus is a collection of individual histories extending over the past and including the present. The logical construction implies a behaviour in the guise of a listing of responses to all possible stimuli. Reciprocally, for a given behaviour of the type defined, the possible structure of a machine that could produce such behaviour can be investigated.

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