Albertus is recorded as having made a mechanical automaton in the form of a brass head that would answer questions put to it.

The languages described by these grammars are exactly all languages that can be recognized by a linear bounded automaton ( a nondeterministic Turing machine whose tape is bounded by a constant times the length of the input.

Computationally, a context-sensitive language is equivalent with a linear bounded nondeterministic Turing machine, also called a linear bounded automaton.

is generated by the grammar, and is accepted by the pushdown automaton where is defined as follows:

Modern methods include the use of lossless data compression for incremental parsing, prediction suffix tree and string searching by factor oracle algorithm ( basically a factor oracle is a finite state automaton constructed in linear time and space in an incremental fashion ).

In the Percy Jackson & the Olympians book The Battle of the Labyrinth, Daedalus is the enigmatic Quintus ( Latin which means 5 or fifth ), and has preserved himself since antiquity by placing his animus, his life force, into an automaton, an idea pioneered by his nephew Perdix.

A finite-state machine ( FSM ) or finite-state automaton ( plural: automata ), or simply a state machine, is a mathematical model of computation used to design both computer programs and sequential logic circuits.

In 1967, Zuse also suggested that the universe itself is running on a cellular automaton or similar computational structure ( digital physics ); in 1969, he published the book Rechnender Raum ( translated into English as Calculating Space ).

In computer science, a pushdown automaton ( PDA ) is

The automaton can alternatively ignore the stack, and leave it as it is.

Thus we have a model which is technically known as a " nondeterministic pushdown automaton " ( NDPDA or NPDA ).

If in every situation only one transition is available as continuation of the computation, then the result is a deterministic pushdown automaton ( DPDA ), a strictly weaker device.

A linear bounded automaton is a device which is more powerful than a pushdown automaton but less so than a Turing machine.

In principle it is enough to create in every such case new automaton instances that will handle the extra choices.

In order to formalize the semantics of the pushdown automaton a description of the current situation is introduced.

For each single pushdown automaton these two languages need to have no relation: they may be equal but usually this is not the case.

As a result we obtain a single state pushdown automaton, the state here is, accepting the context-free language by empty stack.

In The Sign of the Four, Watson quotes Holmes as being " an automaton, a calculating machine ", and Holmes is quoted as saying, " It is of the first importance not to allow your judgement to be biased by personal qualities.

This automaton consists of states ( represented in the figure by circles ), and transitions ( represented by arrows ).

Minds, Machines and Gödel is J. R. Lucas's 1959 philosophical paper in which he argues that a human mathematician cannot be accurately represented by an algorithmic automaton.

This type of processing can be represented by an embedded pushdown automaton.

These languages are exactly all languages that can be recognized by a non-deterministic pushdown automaton.

These languages are exactly all languages that can be decided by a finite state automaton.

As a serious proposal, it was first suggested by mathematician John Von Neumann in the late 1940s when he proposed a kinematic self-reproducing automaton model as a thought experiment.

Context-sensitive grammars are more general than context-free grammars but still orderly enough to be parsed by a linear bounded automaton.

A special subclass of context-free languages are the deterministic context-free languages which are defined as the set of languages accepted by a deterministic pushdown automaton and are identical to the set of languages accepted by a LR ( k ) parser.

* Conway's Game of Life, a cellular automaton devised by mathematician John Horton Conway

A stack automaton, by contrast, does allow access to and operations on deeper elements.

Situations such as these can be identified in the design phase of the automaton by examining the grammar the automaton uses.

The pushdown automaton either accepts by final state, which means after reading its input the automaton reaches an accepting state ( in ), or it accepts by empty stack (), which means after reading its input the automaton empties its stack.

A deterministic finite automaton M is a 5-tuple,

In automata theory, a nondeterministic finite automaton with ε-moves ( ε-NFA ) is defined as a 5-tuple, ( Q, Σ, δ, q < sub > 0 </ sub >, F ), consisting of

Such an automaton may be defined as a 5-tuple, in which is the set of states, is the set of input symbols, is the transition function ( mapping a state and an input symbol to a set of states ), is the initial state, and is the set of accepting states.

:* q < sub > 0 </ sub > is the start state, that is, the state of the automaton before any input has been processed, where q < sub > 0 </ sub >∈ Q.

: A sequence of states q < sub > 0 </ sub >, q < sub > 1 </ sub >, q < sub > 2 </ sub >,...., q < sub > n </ sub >, where q < sub > i </ sub > ∈ Q such that q < sub > 0 </ sub > is the start state and q < sub > i </ sub > = δ ( q < sub > i-1 </ sub >, a < sub > i </ sub >) for 0 < i ≤ n, is a run of the automaton on an input word w

A classic form of state diagram for a finite state machine or finite automaton ( FA ) is a directed graph with the following elements ( Q, Σ, Z, δ, q < sub > 0 </ sub >, F ):

Formally, a deterministic Büchi automaton is a tuple A = ( Q, Σ, δ, q < sub > 0 </ sub >, F ) that consists of the following components:

In a non-deterministic Büchi automaton, the transition function δ is replaced with a transition relation Δ that returns a set of states and initial state is q < sub > 0 </ sub > is replaced by a set of initial states Q < sub > 0 </ sub >.

Let A =( Q < sub > A </ sub >, Σ, Δ < sub > A </ sub >, I < sub > A </ sub >, F < sub > A </ sub >) and B =( Q < sub > B </ sub >, Σ, Δ < sub > B </ sub >, I < sub > B </ sub >, F < sub > B </ sub >) be Büchi automata and C =( Q < sub > C </ sub >, Σ, Δ < sub > C </ sub >, I < sub > C </ sub >, F < sub > C </ sub >) be a finite automaton.

: Proof: If we assume, w. l. o. g., Q < sub > A </ sub >∩ Q < sub > B </ sub > is empty then L ( A )∪ L ( B ) is recognized by the Büchi automaton ( Q < sub > A </ sub >∪ Q < sub > B </ sub >, Σ, Δ < sub > A </ sub >∪ Δ < sub > B </ sub >, I < sub > A </ sub >∪ I < sub > B </ sub >, F < sub > A </ sub >∪ F < sub > B </ sub >).

: Proof: The Büchi automaton A '=( Q ', Σ, Δ ', I ', F ') recognizes L ( A )∩ L ( B ), where

In the automata formulation, the Krohn – Rhodes theorem for finite automata states that given a finite automaton A with states Q and input set I, output alphabet U, then one can expand the states to Q ' such that the new automaton A ' embeds into a cascade of " simple ", irreducible automata: In particular, A is emulated by a feed-forward cascade of ( 1 ) automata whose transitions semigroups are finite simple groups and ( 2 ) automata which are banks of flip-flops running in parallel.