Lecture 3 Graph Representation for Regular Expressions

1
Lecture 3 Graph Representation for Regular
Expressions
2
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.

3
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.

4
G(r)

• For each regular expression r, we can construct a
  digraph G(r) with edges labeled by symbols and e
  as follows.
• If rF, then
• If r?F, then

5
(No Transcript)
6
F
e
e
7
(No Transcript)
8
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.

9
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.

10
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.

11

• Puzzle If a regular expression r contains u
• ''s, v ''s, and w ''s, how many
• e-edges does G(r) contain?
• Question How to reduce the number of
• e-edges?

12
Theorem 3

• An e-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.

13
(No Transcript)
14
(No Transcript)
15
Lecture 4 Deterministic Finite Automata (DFA)
16
DFA
17
e
p
h
b
t
a
l
a

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

18
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.

19

• 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
• d Q x S ? Q .

20

a
a
p
q

• d(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.

21
s

• 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 tape holds
  an input string.

22
x
h

• 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.

23

• The DFA can be represented by
• M (Q, S, d, s, F)
• where S 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.

24

• The transition diagram of a DFA is an
  alternative way to represent the DFA.
• For M (Q, S, d, 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 d(q, a) p.

a
25
d 0 1
s p s
p q s
q q q
1
0, 1
0
0
s
p
q
1

• L(M) (01)00(01).

26

• 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.