By Julius T. Tou (Eds.)
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Note that the word 011 is obtained in two ways: as 0 concatenated with 11, and as 01 concatenated with 1. This example is enough to show that concatenation is not a Cartesian product. Before explaining the star operator, something should be said about the null word. First, the null word λ and the empty set 0 must be distinguished: for the null word is the word of zero length, whereas the empty set is the set that has no wTords at all as members; these behave quite differently in regular expressions.
2. FINITE AUTOMATA A finite automaton is a kind of dynamic system which, at the discrete moments of time under consideration, satisfies the following conditions: (a) At each of these moments of time, the system subject to an input can be in just one of a finite number of possible states. (b) At these moments of time, the inputs of the system can be chosen from a finite number of possible states. (c) At any of these moments of time, the state of the system is uniquely defined by the state of the input and the state of system at the previous moment of time.
Before explaining the star operator, something should be said about the null word. First, the null word λ and the empty set 0 must be distinguished: for the null word is the word of zero length, whereas the empty set is the set that has no wTords at all as members; these behave quite differently in regular expressions. If the null word is mysterious, just think of a word of length 3. If you throw away one symbol, you are left with a word of length 2. Then if you throw away another symbol, you are left with a word of length 1, a single symbol.
Applied Automata Theory by Julius T. Tou (Eds.)