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 ε as follows.
If r=Φ, then
If r≠Φ, then

6. Φ* ε ε

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
ε-edges does G(r) contain?
Question: How to reduce the number of
ε-edges?

12. 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.

15. Lecture 4 Deterministic Finite Automata (DFA)

16. DFA
Finite Control
tape
head

17. p h b e t a l a
The tape is divided into finitely many cells. Each cell contains a symbol in an alphabet Σ.

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
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 .

20. a a q p
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.

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. Otherwise, the input string is rejected.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, 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.

24. 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

25. δ
s p s
p q s
q q q 1
0, 1 s p q 1
L(M) = (0+1)*00(0+1)*.

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.