thumb|300px| The automaton described by this [[state diagram starts in state S<sub>1</sub>, and changes states following the arrows marked 0 or 1 according to the input symbols as they arrive. The double circle marks S<sub>1</sub> as an accepting state. Since all paths from S<sub>1</sub> to itself contain an even number of arrows marked 0, this automaton accepts strings containing even numbers of 0s.]]

Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science with close connections to cognitive science and mathematical logic. The word automata comes from the Greek word αὐτόματος, which means "self-acting, self-willed, self-moving". An automaton (automata in plural) is an abstract self-propelled computing device which follows a predetermined sequence of operations automatically. An automaton with a finite number of states is called a finite automaton (FA) or finite-state machine (FSM). The figure on the right illustrates a finite-state machine, which is a well-known type of automaton. This automaton consists of states (represented in the figure by circles) and transitions (represented by arrows). As the automaton sees a symbol of input, it makes a transition (or jump) to another state, according to its transition function, which takes the previous state and current input symbol as its arguments.

Automata theory is closely related to formal language theory. In this context, automata are used as finite representations of formal languages that may be infinite. Automata are often classified by the class of formal languages they can recognize, as in the Chomsky hierarchy, which describes a nesting relationship between major classes of automata. Automata play a major role in the theory of computation, compiler construction, artificial intelligence, parsing and formal verification.

History

The theory of abstract automata was developed in the mid-20th century in connection with finite automata. Automata theory was initially considered a branch of mathematical systems theory, studying the behavior of discrete-parameter systems. Early work in automata theory differed from previous work on systems by using abstract algebra to describe information systems rather than differential calculus to describe material systems. The theory of the finite-state transducer was developed under different names by different research communities. The earlier concept of Turing machine was also included in the discipline along with new forms of infinite-state automata, such as pushdown automata.

1956 saw the publication of Automata Studies, which collected work by scientists including Claude Shannon, W. Ross Ashby, John von Neumann, Marvin Minsky, Edward F. Moore, and Stephen Cole Kleene. With the publication of this volume, "automata theory emerged as a relatively autonomous discipline".

In the same year, Noam Chomsky described the Chomsky hierarchy, a correspondence between automata and formal grammars, and Ross Ashby published An Introduction to Cybernetics, an accessible textbook explaining automata and information using basic set theory.

The study of linear bounded automata led to the Myhill–Nerode theorem, which gives a necessary and sufficient condition for a formal language to be regular, and an exact count of the number of states in a minimal machine for the language. The pumping lemma for regular languages, also useful in regularity proofs, was proven in this period by Michael O. Rabin and Dana Scott, along with the computational equivalence of deterministic and nondeterministic finite automata.

In the 1960s, a body of algebraic results known as "structure theory" or "algebraic decomposition theory" emerged, which dealt with the realization of sequential machines from smaller machines by interconnection. While any finite automaton can be simulated using a universal gate set, this requires that the simulating circuit contain loops of arbitrary complexity. Structure theory deals with the "loop-free" realizability of machines. By the end of the decade, automata theory came to be seen as "the pure mathematics of computer science".

Automata

What follows is a general definition of an automaton, which restricts a broader definition of a system to one viewed as acting in discrete time-steps, with its state behavior and outputs defined at each step by unchanging functions of only its state and input.

;Recognizable languages

:The recognizable languages are the set of languages that are recognized by some automaton. For finite automata the recognizable languages are regular languages. For different types of automata, the recognizable languages are different.

Variant definitions of automata

Automata are defined to study useful machines under mathematical formalism. So the definition of an automaton is open to variations according to the "real world machine" that we want to model using the automaton. People have studied many variations of automata. The following are some popular variations in the definition of different components of automata.

;Input

  • Finite input: An automaton that accepts only finite sequences of symbols. The above introductory definition only encompasses finite words.
  • Infinite input: An automaton that accepts infinite words (ω-words). Such automata are called ω-automata.
  • Tree input: The input may be a tree of symbols instead of sequence of symbols. In this case after reading each symbol, the automaton reads all the successor symbols in the input tree. It is said that the automaton makes one copy of itself for each successor and each such copy starts running on one of the successor symbols from the state according to the transition relation of the automaton. Such an automaton is called a tree automaton.
  • Infinite tree input : The two extensions above can be combined, so the automaton reads a tree structure with (in)finite branches. Such an automaton is called an infinite tree automaton.

;States

  • Single state: An automaton with one state, also called a combinational circuit, performs a transformation which may implement combinational logic.

{| class="wikitable" style="text-align:center"

|-

! Automaton

|-

| Deterministic Finite Automaton (DFA) -- Lowest Power <br/>

(same power) &nbsp;&nbsp; <math>||</math> &nbsp;&nbsp;(same power) <br/>

Nondeterministic Finite Automaton (NFA)<br/>

(above is weaker) &nbsp;&nbsp; <math>\cap</math> &nbsp;&nbsp; (below is stronger) <br/>

Deterministic Push Down Automaton (DPDA-I) <br/>

with 1 push-down store <br/>

<math>\cap</math> <br/>

Nondeterministic Push Down Automaton (NPDA-I) <br/>

with 1 push-down store <br/>

<math>\cap</math> <br/>

Linear Bounded Automaton (LBA) <br/>

<math>\cap</math> <br/>

Deterministic Push Down Automaton (DPDA-II) <br/>

with 2 push-down stores <br/>

<math>||</math> <br/>

Nondeterministic Push Down Automaton (NPDA-II) <br/>

with 2 push-down stores <br/>

<math>||</math> <br/>

Deterministic Turing Machine (DTM) <br/>

<math>||</math> <br/>

Nondeterministic Turing Machine (NTM) <br/>

<math>||</math> <br/>

Probabilistic Turing Machine (PTM) <br/>

<math>||</math> <br/>

Multitape Turing Machine (MTM) <br/>

<math>||</math> <br/>

Multidimensional Turing Machine

|}

Applications

Each model in automata theory plays important roles in several applied areas. Finite automata are used in text processing, compilers, and hardware design. Context-free grammar (CFGs) are used in programming languages and artificial intelligence. Originally, CFGs were used in the study of human languages. Cellular automata are used in the field of artificial life, the most famous example being John Conway's Game of Life. Some other examples which could be explained using automata theory in biology include mollusk and pine cone growth and pigmentation patterns. Going further, a theory suggesting that the whole universe is computed by some sort of a discrete automaton, is advocated by some scientists. The idea originated in the work of Konrad Zuse, and was popularized in America by Edward Fredkin. Automata also appear in the theory of finite fields: the set of irreducible polynomials that can be written as composition of degree two polynomials is in fact a regular language.

Another problem for which automata can be used is the induction of regular languages.

Automata simulators

Automata simulators are pedagogical tools used to teach, learn and research automata theory. An automata simulator takes as input the description of an automaton and then simulates its working for an arbitrary input string. The description of the automaton can be entered in several ways. An automaton can be defined in a symbolic language or its specification may be entered in a predesigned form or its transition diagram may be drawn by clicking and dragging the mouse. Well known automata simulators include Turing's World, JFLAP, VAS, TAGS and SimStudio.

Category-theoretic models

One can define several distinct categories of automata following the automata classification into different types described in the previous section. The mathematical category of deterministic automata, sequential machines or sequential automata, and Turing machines with automata homomorphisms defining the arrows between automata is a Cartesian closed category, it has both categorical limits and colimits. An automata homomorphism maps a quintuple of an automaton A<sub>i</sub> onto the quintuple of another automaton

A<sub>j</sub>. Automata homomorphisms can also be considered as automata transformations or as semigroup homomorphisms, when the state space, S, of the automaton is defined as a semigroup S<sub>g</sub>. Monoids are also considered as a suitable setting for automata in monoidal categories.

;Categories of variable automata

One could also define a variable automaton, in the sense of Norbert Wiener in his book on The Human Use of Human Beings via the endomorphisms <math>A_{i}\to A_{i}</math>. Then one can show that such variable automata homomorphisms form a mathematical group. In the case of non-deterministic, or other complex kinds of automata, the latter set of endomorphisms may become, however, a variable automaton groupoid. Therefore, in the most general case, categories of variable automata of any kind are categories of groupoids or groupoid categories. Moreover, the category of reversible automata is then a

2-category, and also a subcategory of the 2-category of groupoids, or the groupoid category.

See also

  • Boolean differential calculus
  • Petri net

References

Further reading

  • Part One: Automata and Languages, chapters 1–2, pp.&nbsp;29–122. Section 4.1: Decidable Languages, pp.&nbsp;152–159. Section 5.1: Undecidable Problems from Language Theory, pp.&nbsp;172–183.
  • John M. Howie (1991) Automata and Languages, Clarendon Press
  • dk.brics.automaton
  • libfa