1 / 12

Regular Grammars

Regular Grammars. Lecture 11 Naveen Z Quazilbash. Overview. Examples Definition Language defined by the grammar Linearity of grammar Equivalence with NFAs. Examples. Example 1: S  aAb AcA A ε The corresponding language is ac*b Example 2: S  aA A bB BaA B  ε

aelwen
Download Presentation

Regular Grammars

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. Regular Grammars Lecture 11 Naveen Z Quazilbash

  2. Overview • Examples • Definition • Language defined by the grammar • Linearity of grammar • Equivalence with NFAs

  3. Examples • Example 1: S aAb AcA Aε • The corresponding language is ac*b • Example 2: S aA AbB BaA Bε • The corresponding language is (ab)+ S=>aAb=>acAb=>accAb=>accb S=>aAb=>ab S=>aA=>abB=>ab S=>aA=>abB=>abaA=>ababB=>abab

  4. Some Definitions • The substitution rules that form a grammar are called “Productions”. • The left side of a rule is a “variable”. • The right side is a mixture of variables and symbols of the alphabet (a.k.a “Terminal symbols”).

  5. Some Definitions Contd. (2) • A grammar defines the language of all strings that can be derived from the “Start variable, S”, by successive applications of productions. • Note that the strings of this language do not contain variables.

  6. Some Definitions Contd. (3) • A sequence of substitutions that leads from S to some string of the language is called a “Derivation”. • Example: • S=>aA=>abB=>ab is a derivation; we may write it in a shorter form as S=>*ab, • where “=>*” means “several steps”.

  7. Formal Definition of Grammar • Formally, a grammar is a four tuple: G= {V, T, S, P}, where • V= set of variables, e.g. V={S, A, B} • T= set of terminal symbols, e.g. T={a, b} • S= start variable (analogous to the start state in automaton). • P= set of productions, e.g. P={SaA, AbB, BaA, Bε} • Note that V and T must be non-empty and disjoint.

  8. Language defined by the Grammar • The set of all strings derivable from S is the “language defined (or generated) by the grammar”.

  9. Linearity of Grammar • A Grammar is “Linear” if all productions are of the form Aw1Bw2 or Aw3, where A and B are variables and w1,w2 and w3 are strings (with no variables in them). • Example of a linear grammar: SaB Sε BSb • Shorter notation: Sab | ε • Language: L={anbn: n≥0}, (not a regular language) • Thus, a linear grammar may generate a language that is not accepted by any finite automaton. S=>ε S=>aB=>aSb=>ab S=>aB=>aSb=>aaBb=>aaSbb=>aabb

  10. NON-LINEAR ???

  11. Linearity of Grammar-Contd.. • Right linear grammar: VT*V VT* • Left linear grammar: VVT* VT*

  12. ASSIGNMENT NO.2 • For each of the following three languages on ∑={a,b}, draw a DFA that accepts it: (6 points) • All strings that have no b’s (note that it includes ε). • All strings with atleast two a’s and any number of b’s. • All strings with atmost two a’s and any number of b’s. • Due Date: 28th March 2013.

More Related