Lecture 3 Graph Representation for Regular Expressions

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# Lecture 3 Graph Representation for Regular Expressions - PowerPoint PPT Presentation

Lecture 3 Graph Representation for Regular Expressions. 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 .

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### Lecture 3 Graph Representation for Regular Expressions

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.
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.
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

Φ*

ε

ε

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.
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 is associated with a path from s to f.
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 and only if x is associated with a path from s to f.
Puzzle: If a regular expression r contains u

+''s, v ·''s, and w *''s, how many

ε-edges does G(r) contain?

Question: How to reduce the number of

ε-edges?

Theorem 3
• An ε-edge (u,v) in G(r) which is a unique out-edge from a nonfinal vertex u or a unique in-edge to a noninitial vertex v can be shrunk to a single vertex. (If one of u and v is the initial vertex or the final vertex, so is the resulting vertex.)
• Remark: Shrinking should be done one by one.

### Lecture 4 Deterministic Finite Automata (DFA)

tape

Finite Control

DFA

e

p

h

b

t

a

l

a

The tape is divided into finitely many cells. Each cell contains a symbol in an alphabet Σ.

a

• 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.
The finite control has finitely many states which form a set Q. For each move, the state is changed according to the evaluation of a transition function

δ : Q x Σ → Q .

a

a

• 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.

q

p

There are some special states: an initial states and a set F of final states.
• Initially, the DFA is in the initial state s and the head scans the leftmost cell. The tape holds an input string.

s

x

• 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

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.
The transition diagram of a DFA is an alternative 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

δ 0 1

s p s

p q s

q q q

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

1

0, 1

0

0

s

p

q

1

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.