![]() Indeed, consider the category of finite state deterministic automata that accept a fixed given language L, equipped with the standard notion of automata morphism, then this category fails to have an initial and a final object. Emptiness of such automata is decidable.īut, even for studying finite state automata, it makes sense to talk about infinite automata. Other examples include register automata (these automata have a finite set of registers, and work over an infinite alphabet: these can compare letters with registers and store letters in registers), and the more modern form of nominal automata (that have the same expressiveness, but have better foundations and properties). These can be minimized and tested for equality. These are known as weighted automata over a base field (due to Schützenberger in 61). For instance one considers the class of automata that have as state space a (finite dimension) vector space, and as transition functions linear maps (plus some initial an final things). In the same vein, there ar other examples of infinite automata for which the state space is infinite, but with a lot of structure. ![]() Pushdown automata, for instance, are automata that have an infinite set of configurations (these have a finite number of states, but the reality is that these should be thought as 'infinite automata'). In fact, in automata theory (which departs a lots from the origins of Kleene, Rabin and Scott), there are many forms of automata that are not finite. In short, non-finite automata are theoretically well-defined, and occasionally even encountered. The state of this system is the combination of a discrete Markov state and a tuple of real numbers ("mixed state Markov model"), hence it can be in an infinite number of different states. However, this article ( full text) develops a variation where the transition probabilities also depend on the real-number vector of prior outputs. The filters compute the next real-valued output from a vector of the previous outputs ("autoregressive" model), but the underlying Markov model is finite since its transition probabilities only depend on the current Markov state. Hidden Markov Models can be used to model a number sequence as the output of a probabilistic system consisting of one output filter for each state of a (discrete, finite) Markov model with unobservable states. This might not ever happen in theoretical computer science, but it's not too exotic in the domain of modeling real number sequences. This means that if the (relevant) state of the automaton involves a real-valued variable, there is an infinite number of potential states (putting aside the finiteness of floating point representations), and the automaton is not finite. The crucial part is that the state of the automaton can be fully characterized by an element of some finite set of discrete states. ![]() The full name is "finite state automata".
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