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Movie Title Year Distributor Notes Rev Formats Body Talk 1982 VCX Facial DRO Chubby Chicks Need Cock Too 2013 Western Visuals DO Classic Tits Galore 2013 Western Visuals DRO Famous Ta Ta's 1986 Essex Video / Electric Hollywood Facial Kay Parker Collection 2 2005 Alpha Blue Archives LezOnly DRO Kitten Natividad Collection 2006 Alpha Blue Archives DRO Sex Game 1986 Essex Video / Electric Hollywood DO Stalag 69 1982 Video Home Library Titillation 1982 TVX 1 DRO Way Up Front Unknown 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. The word automata (the plural of automaton) comes from the Greek word a?t?µata, which means "self-making". The figure at right illustrates a finite-state machine, which belongs to 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 current state and the recent symbol as its inputs.
Automata theory is closely related to formal language theory. An automaton is a finite representation of a formal language that may be an infinite set. Automata are often classified by the class of formal languages they can recognize, typically illustrated by the Chomsky hierarchy, which describes the relations between various languages and kinds of formalized logics. Automata play a major role in theory of computation, compiler construction, artificial intelligence, parsing and formal verification. Contents 1 Automata 1.1 Very informal description 1.2 Informal description 1.3 Formal definition 1.3.1 definition of finite state automata 2 Variant definitions of automata 3 Classes of automata 3.1 Discrete, continuous, and hybrid automata 4 Hierarchy in terms of powers 5 Applications 6 Automata simulators 7 Connection to category theory 8 History 9 See also 10 References 11 Further reading 12 External links Automata Following is an introductory definition of one type of automaton, which attempts to help one grasp the essential concepts involved in automata theory/theories.

Very informal description An automaton is a construct made of states designed to determine if the input should be accepted or rejected. It looks a lot like a basic board game where each space on the board represents a state. Each state has information about what to do when an input is received by the machine (again, rather like what to do when you land on the Go To Jail spot in a popular board game). As the machine receives a new input, it looks at the state and picks a new spot based on the information on what to do when it receives that input at that state. When there are no more inputs, the automaton stops and the space it is on when it completes determines whether the automaton accepts or rejects that particular set of inputs. Informal description An automaton runs when it is given some sequence of inputs in discrete (individual) time steps or steps. An automaton processes one input picked from a set of symbols or letters, which is called an alphabet. The symbols received by the automaton as input at any step are a finite sequence of symbols called words. An automaton has a finite set of states. At each moment during a run of the automaton, the automaton is in one of its states. When the automaton receives new input it moves to another state (or transitions) based on a function that takes the current state and symbol as parameters. This function is called the transition function. The automaton reads the symbols of the input word one after another and transitions from state to state according to the transition function until the word is read completely. Once the input word has been read, the automaton is said to have stopped. The state at which the automaton stops is called the final state. Depending on the final state, it's said that the automaton either accepts or rejects an input word. There is a subset of states of the automaton, which is defined as the set of accepting states. If the final state is an accepting state, then the automaton accepts the word. Otherwise, the word is rejected. The set of all the words accepted by an automaton is called the language recognized by the automaton. In short, an automaton is a mathematical object that takes a word as input and decides whether to accept it or reject it. Since all computational problems are reducible into the accept/reject question on inputs, (all problem instances can be represented in a finite length of symbols)[citation needed], automata theory plays a crucial role in computational theory. Formal definition Automaton definition of finite state automata A deterministic finite automaton is represented formally by a 5-tuple , where: Q is a finite set of states. S is a finite set of symbols, called the alphabet of the automaton. d is the transition function, that is, d: Q × S ? Q. q0 is the start state, that is, the state of the automaton before any input has been processed, where q0? Q. F is a set of states of Q (i.e. F?Q) called accept states. Input word An automaton reads a finite string of symbols a1,a2,...., an , where ai ? S, which is called an input word. The set of all words is denoted by S*. Run A sequence of states q0,q1,q2,...., qn, where qi ? Q such that q0 is the start state and qi = d(qi-1,ai) for 0 < i = n, is a run of the automaton on an input word w = a1,a2,...., an ? S*. In other words, at first the automaton is at the start state q0, and then the automaton reads symbols of the input word in sequence. When the automaton reads symbol ai it jumps to state qi = d(qi-1,ai). qn is said to be the final state of the run. Accepting word A word w ? S* is accepted by the automaton if qn ? F. Recognized language An automaton can recognize a formal language. The language L ? S* recognized by an automaton is the set of all the words that are accepted by the automaton. Recognizable languages The recognizable languages are the set of languages that are recognized by some automaton. For the above definition of automata the recognizable languages are regular languages. For different definitions 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", which we want to model using the automaton. People have studied many variations of automata. The most standard variant, which is described above, is called a deterministic finite automaton. The following are some popular variations in the definition of different components of automata. Input Finite input: An automaton that accepts only finite sequence of symbols. The above introductory definition only encompasses finite words. Infinite input: An automaton that accepts infinite words (?-words). Such automata are called ?-automata. Tree word 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 Finite states: An automaton that contains only a finite number of states. The above introductory definition describes automata with finite numbers of states. Infinite states: An automaton that may not have a finite number of states, or even a countable number of states. For example, the quantum finite automaton or topological automaton has uncountable infinity of states. Stack memory: An automaton may also contain some extra memory in the form of a stack in which symbols can be pushed and popped. This kind of automaton is called a pushdown automaton Transition function Deterministic: For a given current state and an input symbol, if an automaton can only jump to one and only one state then it is a deterministic automaton. Nondeterministic: An automaton that, after reading an input symbol, may jump into any of a number of states, as licensed by its transition relation. Notice that the term transition function is replaced by transition relation: The automaton non-deterministically decides to jump into one of the allowed choices. Such automata are called nondeterministic automata. Alternation: This idea is quite similar to tree automaton, but orthogonal. The automaton may run its multiple copies on the same next read symbol. Such automata are called alternating automata. Acceptance condition must satisfy all runs of such copies to accept the input. Acceptance condition Acceptance of finite words: Same as described in the informal definition above. Acceptance of infinite words: an omega automaton cannot have final states, as infinite words never terminate. Rather, acceptance of the word is decided by looking at the infinite sequence of visited states during the run. Probabilistic acceptance: An automaton need not strictly accept or reject an input. It may accept the input with some probability between zero and one. For example, quantum finite automaton, geometric automaton and metric automaton have probabilistic acceptance. Different combinations of the above variations produce many classes of automaton. Automata theory is a subject matter that studies properties of various types of automata. For example, the following questions are studied about a given type of automata. Which class of formal languages is recognizable by some type of automata? (Recognizable languages) Are certain automata closed under union, intersection, or complementation of formal languages? (Closure properties) How expressive is a type of automata in terms of recognizing a class of formal languages? And, their relative expressive power? (Language hierarchy) Automata theory also studies the existence or nonexistence of any effective algorithms to solve problems similar to the following list: Does an automaton accept any input word? (Emptiness checking) Is it possible to transform a given non-deterministic automaton into deterministic automaton without changing the recognizable language? (Determinization) For a given formal language, what is the smallest automaton that recognizes it? (Minimization) Classes of automata

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