Example of a Deterministic Finite Automata, DFA Machine Definition M = (Q, Sigma, delta, q0, F) Q = { q0, q1, q2, q3, q4 } the set of states (finite) Sigma = { 0, 1 } the input string alphabet (finite) delta the state transition table - below q0 = q0 the starting state F = { q2, q4 } the set of final states (accepting when in this state and no more input) inputs delta | 0 | 1 | ---------+---------+---------+ q0 | q3 | q1 | q1 | q1 | q2 | states q2 | q2 | q2 | q3 | q4 | q3 | q4 | q4 | q4 | ^ ^ ^ | | | | +---------+-- every transition must have a state +-- every state must be listed

An exactly equivalent diagram description for the machine M. Each circle is a unique state. The machine is in exactly one state and stays in that state until an input arrives. Connection lines with arrows represent a state transition from the present state to the next state for the input symbol(s) by the line. L(M) is the notation for a Formal Language defined by a machine M. Some of the shortest strings in L(M) = { 00, 11, 000, 001, 010, 101, 110, 111, 0000, 0001, 0010, 0011, 0100, 0101, 0110, 1001, ... } In words, L is the set of strings over { 0, 1} that contain at least two 0's starting with 0, or that contain at least two 1's starting with 1. Every input sequence goes through a sequence of states, for example 00 q0 q3 q4 11 q0 q1 q2 000 q0 q3 q4 q4 001 q0 q3 q4 q4 010 q0 q3 q3 q4 011 q0 q3 q3 q3 0110 q0 q3 q3 q3 q4 Rather abstract, there are tens of millions of DFA being used today.

Definition of a Regular Expression ------------------ A regular expression may be the null string, r = epsilon A regular expression may be an element of the input alphabet, sigma, r = a A regular expression may be the union of two regular expressions, r = r1 + r2 A regular expression may be the concatenation (no symbol) of two regular expressions, r = r1 r2 A regular expression may be the Kleene closure (star) of a regular expression r = r1* (the asterisk should be a superscript, but this is plain text) A regular expression may be a regular expression in parenthesis r = (r1) Nothing is a regular expression unless it is constructed with only the rules given above. The language represented or generated by a regular expression is a Regular Language, denoted L(r). The regular expression for the machine M above is r = (1(0*)1(0+1)*)+(0(1*)0(0+1)*) Later we will give an algorithm for generating a regular expression from a machine definition. For simple DFA's, start with each accepting state and work back to the start state writing the regular expression. The union of these regular expressions is the regular expression for the machine. For every DFA there is a regular language and for every regular language there is a regular expression. Thus a DFA can be converted to a regular expression and a regular expression can be converted to a DFA.

Given a DFA and one or more strings, determine if the string(s) are accepted by the DFA. This may be error prone and time consuming to do by hand. Fortunately, there is a program available to do this for you. On linux.gl.umbc.edu do the following: ln -s /afs/umbc.edu/users/s/q/squire/pub/dfa dfa cp /afs/umbc.edu/users/s/q/squire/pub/download/ab_b.dfa . dfa < ab_b.dfa # or dfa < ab_b.dfa > ab_a.out

### Practical definition of a DFA

## Here, we start with the state diagram: From this we see the states Q = { Off, Starting forward, Running forward, Starting reverse, Running reverse } The alphabet (inputs) Sigma = { off request, forward request, reverse request, time-out } The initial state would be Off. The final state would be Off. The state transition table, Delta,is obvious yet very wordy. Note: that upon entering a state, additional actions may be specified. Inputs could be from buttons, yet could come from a computer.