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. By the end of the decade, automata theory came to be seen as "the pure mathematics of computer science". The theory of computational complexity also took shape in the 1960s. Structure theory deals with the "loop-free" realizability of machines. While any finite automaton can be simulated using a universal gate set, this requires that the simulating circuit contain loops of arbitrary complexity. 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. Rabin and Dana Scott, along with the computational equivalence of deterministic and nondeterministic finite automata. The pumping lemma for regular languages, also useful in regularity proofs, was proven in this period by Michael O. 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. 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 book included Kleene's description of the set of regular events, or regular languages, and a relatively stable measure of complexity in Turing machine programs by Shannon. With the publication of this volume, "automata theory emerged as a relatively autonomous discipline". Ross Ashby, John von Neumann, Marvin Minsky, Edward F. The earlier concept of Turing machine was also included in the discipline along with new forms of infinite-state automata, such as pushdown automata.ġ956 saw the publication of Automata Studies, which collected work by scientists including Claude Shannon, W. The theory of the finite-state transducer was developed under different names by different research communities. 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. Automata theory was initially considered a branch of mathematical systems theory, studying the behavior of discrete-parameter systems. The theory of abstract automata was developed in the mid-20th century in connection with finite automata. Automata play a major role in the theory of computation, compiler construction, artificial intelligence, parsing and formal verification. 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. In this context, automata are used as finite representations of formal languages that may be infinite. 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.Īutomata theory is closely related to formal language theory. This automaton consists of states (represented in the figure by circles) and transitions (represented by arrows). The figure on the right illustrates a finite-state machine, which is a well-known type of automaton. An automaton with a finite number of states is called a Finite Automaton (FA) or Finite-State Machine (FSM). An automaton (automata in plural) is an abstract self-propelled computing device which follows a predetermined sequence of operations automatically. The word automata comes from the Greek word αὐτόματος, which means "self-acting, self-willed, self-moving".
It is a theory in theoretical computer science with close connections to mathematical logic. Since all paths from S 1 to itself contain an even number of arrows marked 0, this automaton accepts strings containing even numbers of 0s.Īutomata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. The double circle marks S 1 as an accepting state.
(Clicking on each layer gets an article on that subject) The automaton described by this state diagram starts in state S 1, and changes states following the arrows marked 0 or 1 according to the input symbols as they arrive.
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