COMS W3261
Computer Science Theory
Lecture 3: September 16, 2009
Deterministic Finite Automata

1. Outline

• Review
• Deterministic finite automata
• Example of a DFA
• Practice problems

2. Review

• Inductive proofs
• Regular expressions
• Examples of regular expressions

3. Deterministic Finite Automata

• A deterministic finite automaton (DFA) has five components:
1. A finite set of states Q.
2. An input alphabet consisting of a finite set of symbols Σ.
3. A transition function δ that maps Q × &Sigma to Q.
This transition function can be represented by a transition diagram in which the nodes are labeled by states and arcs by symbols.
4. A start state which is one of the states in Q.
5. A set of final (or accepting) states F.
• A DFA accepts an input string x if there is a path of transitions from the initial state to a final state that spells out x.
• L(A), the language defined by a DFA A is the set of strings accepted by A.
• A language accepted by a DFA is called a regular language.

4. Example of a DFA

• Consider the following DFA D with:
1. The finite set of states Q = {A, B, C}
2. The input alphabet Σ = {0, 1}
3. The transition function δ as represented by the following transition table:



State	Input Symbol
	0	1
A	A	B
B	B	C
C	C	C



4. Start state A
5. The set of final states F = {B}


• Given the input string 0010, D makes the following transitions started off in its initial start state A:
   0   0   1   0
 A   A   A   B   B
This sequence of transitions takes D from its start state A to the final state B. Thus, the input 0010 is accepted by D.


• Given the input string 101, D makes the following transitions started off in its initial start state A:
   1   0   1
 A   B   B   C
This sequence of transitions takes D from its start state A to the nonfinal state B. Therefore, the input 101 is not accepted by D.


• The language defined by D is the set of strings of 0's and 1's having exactly one 1.
• The regular expression 0*10* defines the same language.

5. Practice Problems

1. Construct a DFA for a light switch.
2. Let L be the language { w | w is any string of a's and b's containing at least one a and at least one b }.
1. Construct a DFA for L.
2. Show the behavior of your DFA processing the input string aabaa.
3. Construct a regular expression for L.
3. Let L be the language { abxba | x is any string of a's, b's, and c's not containing ba }. This language models comments in the programming language C.
1. Construct a DFA for L.
2. Show the behavior of your DFA processing the input string abcbaba.
3. Construct a regular expression for L.
4. Show how your regular expression generates ababcba.
4. Construct a DFA for the language L((a+b)*abba(a+b)*).
5. Let L be the language consisting of all strings of a's and b's having an even number of a's and an even number of b's.
   1. Construct a DFA for L.
   2. Show the behavior of your DFA processing the input string abbaabab.
   3. Construct a regular expression for L.
   4. Show how your regular expression generates abbaabab.

6. Reading Assignment

• HMU: Sects. 2.1-2.2