Title: Introduction to Finite Automata
1Introduction to Finite Automata
- Adapted from the slides of Stanford CS154
2Informal Explanation
- Finite automata
- finite collections of states with transition
rules that take you from one state to another. - Original application
- sequential switching circuits, where the state
was the settings of internal bits. - Today, several kinds of software can be modeled
by FA.
3Representing FA
- Simplest representation is often a graph.
- Nodes states.
- Arcs indicate state transitions.
- Labels on arcs tell what causes the transition.
4Example Recognizing Strings Ending in ing
nothing
Start
5Automata to Code
- In C/C, make a piece of code for each state.
This code - Reads the next input.
- Decides on the next state.
- Jumps to the beginning of the code for that state.
6Example Automata to Code
- 2 / i seen /
- c getNextInput()
- if (c n) goto 3
- else if (c i) goto 2
- else goto 1
- 3 / in seen /
- . . .
7Deterministic Finite Automata
- A formalism for defining languages, consisting
of - A finite set of states (Q, typically).
- An input alphabet (S, typically).
- A transition function (d, typically).
- A start state (q0, in Q, typically).
- A set of final states (F ? Q, typically).
- Final and accepting are synonyms.
8The Transition Function
- Takes two arguments
- a state and an input symbol.
- d(q, a) the state that the DFA goes to when it
is in state q and input a is received.
9Graph Representation of DFAs
- Nodes states.
- Arcs represent transition function.
- Arc from state p to state q labeled by all those
input symbols that have transitions from p to q. - Arrow labeled Start to the start state.
- Final states indicated by double circles.
10Example Graph of a DFA
Accepts all strings without two consecutive 1s.
11Alternative Representation Transition Table
0
1
A A B B A C C C C
12Extended Transition Function
- We describe the effect of a string of inputs on a
DFA - by extending d to a state and a string.
- Induction on length of string.
- Basis d(q, e) q
- Induction d(q,wa) d(d(q,w),a)
- w is a string a is an input symbol.
13Extended d Intuition
- Convention
- w, x, y, x are strings.
- a, b, c, are single symbols.
- Extended d is computed for state q and inputs
a1a2an by following a path in the transition
graph, starting at q and selecting the arcs with
labels a1, a2,,an in turn.
14Example Extended Delta
d(B,011) d(d(B,01),1) d(d(d(B,0),1),1)
d(d(A,1),1) d(B,1) C
15Language of a DFA
- Automata of all kinds define languages.
- If A is an automaton, L(A) is its language.
- For a DFA A, L(A) is the set of strings labeling
paths from the start state to a final state. - Formally L(A) the set of strings w such that
d(q0, w) is in F.
16Example String in a Language
String 101 is in the language of the DFA
below. Start at A.
17Example String in a Language
String 101 is in the language of the DFA below.
Follow arc labeled 1.
0
0,1
1
1
A
C
B
Start
0
18Example String in a Language
String 101 is in the language of the DFA below.
Then arc labeled 0 from current state B.
0
0,1
1
1
A
C
B
Start
0
19Example String in a Language
String 101 is in the language of the DFA below.
Finally arc labeled 1 from current state A.
Result is an accepting state, so 101 is in the
language.
0
0,1
1
1
A
C
B
Start
0
20Example Concluded
- The language of our example DFA is
- w w is in 0,1 and w does not have
- two consecutive 1s
21Regular Languages
- A language L is regular if it is the language
accepted by some DFA. - Note the DFA must accept only the strings in L,
no others. - Some languages are not regular.
- Intuitively, regular languages cannot count to
arbitrarily high integers.
22Example A Nonregular Language
- L1 0n1n n 1
- Note ai is conventional for i as.
- Thus, 04 0000, e.g.
- Read The set of strings consisting of n 0s
followed by n 1s, such that n is at least 1. - Thus, L1 01, 0011, 000111,
23Another Example
- L2 w w in (, ) and w is balanced
- Note alphabet consists of the parenthesis
symbols ( and ). - Balanced parens are those that can appear in an
arithmetic expression. - E.g. (), ()(), (()), (()()),
24But Many Languages are Regular
- Regular Languages can be described in many ways,
e.g., regular expressions. - They appear in many contexts and have many useful
properties. - Example the strings that represent floating
point numbers in your favorite language is a
regular language.
25Nondeterminism
- A nondeterministic finite automaton has the
ability to be in several states at once. - Transitions from a state on an input symbol can
be to any set of states.
26Nondeterminism (2)
- Start in one start state.
- Accept if any sequence of choices leads to a
final state. - Intuitively the NFA always guesses right.
27Formal NFA
- A finite set of states, typically Q.
- An input alphabet, typically S.
- A transition function, typically d.
- A start state in Q, typically q0.
- A set of final states F ? Q.
28Transition Function of an NFA
- d(q, a) is a set of states.
- Extend to strings as follows
- Basis d(q, e) q
- Induction d(q, wa) the union over all states p
in d(q, w) of d(p, a)
29Language of an NFA
- A string w is accepted by an NFA if d(q0, w)
contains at least one final state. - The language of the NFA is the set of strings it
accepts.
30Equivalence of DFAs, NFAs
- A DFA can be turned into an NFA that accepts the
same language. - If dD(q, a) p, let the NFA have dN(q, a)
p. - Then the NFA is always in a set containing
exactly one state the state the DFA is in after
reading the same input.
31Equivalence (2)
- Surprisingly, for any NFA there is a DFA that
accepts the same language. - Proof is the subset construction.
- The number of states of the DFA can be
exponential in the number of states of the NFA. - Thus, NFAs accept exactly the regular languages.
32Subset Construction
- Given an NFA with states Q, inputs S, transition
function dN, state state q0, and final states F,
construct equivalent DFA with - States 2Q (Set of subsets of Q).
- Inputs S.
- Start state q0.
- Final states all those with a member of F.
33Subset Construction (2)
- The transition function dD is defined by
- dD(q1,,qk, a) is the union over all i 1,,k
of dN(qi, a).
34Proof of Equivalence Subset Construction
- Show by induction on w that
- dN(q0, w) dD(q0, w)
- Basis w e dN(q0, e) dD(q0, e) q0.
35Induction
- Assume IH (Induction Hypothesis) holds for
strings shorter than w. - Let w xa IH holds for x.
- Let dN(q0, x) dD(q0, x) S.
- Let T the union over all states p in S of dN(p,
a). - Then dN(q0, w) dD(q0, w) T.
- For NFA the extension of dN.
- For DFA definition of dD plus extension of dD.
- That is, dD(S, a) T then extend dD to w xa.
36NFAs With e-Transitions
- We can allow state-to-state transitions on e
input. - These transitions are done spontaneously, without
looking at the input string. - A convenience at times, but still only regular
languages are accepted.
37Example e-NFA
38Subset construction in the Textbook
39Subset construction in the Textbook (cont.)
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