lect3-4

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Lecture 3 Graph Representation

for Regular Expressions

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digraph (directed graph)

• A digraph is a pair of sets (V, E) such that

each element of E is an ordered pair of

elements in V.

• A path is an alternative sequence of

vertices and edges such that all edges are

in the same direction.

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string-labeled digraph

• A string-labeled digraph is a digraph in

which each edge is labeled by a string.

• In a string-labeled digraph, every path is

associated with a string which is obtained

by concatenating all strings on the path.

• This string is called the label of the path.

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G(r)

• For each regular expression r , we can

construct a digraph G(r) with edges

labeled by symbols and ε as follows.

• If r=Φ, then

• If r≠Φ, then

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Φ*

ε ε

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Theorem 1

• G(r) has a property that a string x belongs

to r if and only if x is the label of a path

from the initial vertex to the final vertex.

• Proof is done by induction on r .

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Graph Representation

• A graph representation of a regular

expression r is a string-labeled graph with

an initial vertex s and a final vertex f such

that a string x belongs to r if and only if x isassociated with a path from s to f .

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Corollary 2

• For any regular expression r , there exists

a string-labeled digraph with two special

vertices, a initial vertex s and a final vertex

f , such that a string x belongs to r if andonly if x is associated with a path from s to

f .

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Lecture 4 Deterministic Finite

Automata (DFA)

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• The head scans at a cell on the tape and

can read a symbol on the cell. In each

move, the head can move to the right cell.

a

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• The finite control has finitely many states

which form a set Q. For each move, thestate is changed according to the

evaluation of a transition function

δ : Q x Σ → Q .

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• δ (q, a) = p means that if the head reads

symbol a and the finite control is in the

state q, then the next state should be p,and the head moves one cell to the right.

pq

a a

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• There are some special states: an initial

state s and a set F of final states.

• Initially, the DFA is in the initial state s and

the head scans the leftmost cell. The tapeholds an input string.

s

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• When the head gets off the tape, the DFA

stops. An input string x is accepted by the

DFA if the DFA stops at a final state.

• Otherwise, the input string is rejected.

h

x

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• The DFA can be represented by

M = (Q, Σ , δ , s, F)

where Σ is the alphabet of input symbols.

• The set of all strings accepted by a DFA M

is denoted by L(M). We also say that the

language L(M) is accepted by M .

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• The transition diagram of a DFA is analternative way to represent the DFA.

• For M = (Q, Σ , δ , s, F), the transition diagram of

M is a symbol-labeled digraph G=(V, E) satisfying the following:

V = Q (s = , f = for f \in F )

E = { q p | δ (q, a) = p}.a

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L(M) = (0+1)*00(0+1)*.

δ 0 1

s p s

p q s

q q q

s p q

1

0

1

0

0, 1

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The transition diagram of the DFA M has the

following properties:

• For every vertex q and every symbol a,

there exists an edge with label a from q.

• For each string x, there exists exactly one

path starting from the initial state s

associated with x.

• A string x is accepted by M if and only if

this path ends at a final state.

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