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Discrete, continuous, and hybrid automata Normally automata theory describes the states of abstract machines but there are analog automata or continuous automata or hybrid discrete-continuous automata, which use analog data, continuous time, or both. Hierarchy in terms of powers The following is an incomplete hierarchy in terms of powers of different types of virtual machines. The hierarchy reflects the nested categories of languages the machines are able to accept.[1] Automaton Deterministic Finite Automaton (DFA) -- Lowest Power (same power) {\displaystyle ||}|| (same power) Nondeterministic Finite Automaton (NFA) (above is weaker) {\displaystyle \cap }\cap (below is stronger) Deterministic Push Down Automaton (DPDA-I) with 1 push-down store {\displaystyle \cap }\cap Nondeterministic Push Down Automaton (NPDA-I) with 1 push-down store {\displaystyle \cap }\cap Linear Bounded Automaton (LBA) {\displaystyle \cap }\cap Deterministic Push Down Automaton (DPDA-II) with 2 push-down stores {\displaystyle ||}|| Nondeterministic Push Down Automaton (NPDA-II) with 2 push-down stores {\displaystyle ||}|| Deterministic Turing Machine (DTM) {\displaystyle ||}|| Nondeterministic Turing Machine (NTM) {\displaystyle ||}||

Probabilistic Turing Machine (PTM) {\displaystyle ||}|| Multitape Turing Machine (MTM) {\displaystyle ||}|| 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 the human languages. Cellular automata are used in the field of biology, the most common example being John Conway's Game of Life. Some other examples which could be explained using automata theory in biology include mollusk and pine cones 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 which can be written as composition of degree two polynomials is in fact a regular language.[2] 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.[3] Connection to category theory One can define several distinct categories of automata[4] 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,[5][6] it has both categorical limits and colimits. An automata homomorphism maps a quintuple of an automaton Ai onto the quintuple of another automaton Aj.[7] 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 Sg. Monoids are also considered as a suitable setting for automata in monoidal categories.[8][9][10] 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 {\displaystyle A_{i}\to A_{i}}A_{i}\to A_{i}. 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

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