1 / 26

Lecture 3 Graph Representation for Regular Expressions

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 .

kemal
Download Presentation

Lecture 3 Graph Representation for Regular Expressions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

  5. Φ* ε ε

  6. 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.

  7. 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.

  8. 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.

  9. 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?

  10. 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.

  11. Lecture 4 Deterministic Finite Automata (DFA)

  12. tape head Finite Control DFA

  13. 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.

  14. 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.

  15. 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 .

  16. 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

  17. 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

  18. 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

  19. 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.

  20. 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

  21. 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

  22. 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.

More Related