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Lecture#14-15

Lecture#14-15. Grammar. A grammar G = (V, T, S, P) consists of the following quadruple: a set V of variables (non-terminal symbols), including a starting symbol S  NT a set T of terminals (same as an alphabet, ) A start symbol S  V a set P of production rules Example: S  aS | A

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Lecture#14-15

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  1. Lecture#14-15 Unit-2

  2. Grammar • A grammar G = (V, T, S, P) consists of the following quadruple: • a set V of variables (non-terminal symbols), including a starting symbol S  NT • a set T of terminals (same as an alphabet, ) • A start symbol S  V • a set P of production rules • Example: S  aS | A A bA | λ Unit-2

  3. Derivation • Strings are “derived” from a grammar • Example of a derivation S  aS  aaS  aaA  aabA  aab • At each step, a nonterminal is replaced by the sentential form on the right-hand side of a rule (a sentential form can contain nonterminals and/or terminals) • Automata recognize languages; grammars generate languages Unit-2

  4. Regular grammar • A grammar is said to be right-linear if all productions are of the form AxB or Ax, where A and B are variables and x is a string of terminals. (This means that if there is a variable on the right side of the production rule, then it is the rightmost element in the rule.) • A grammar is said to be left-linear if all productions are of the form ABx or Ax • A regular grammar is either right-linear or left-linear. Unit-2

  5. Linear grammar • A grammar can be linear without being right- or left-linear. • A linear grammar is a grammar in which at most one variable can occur on the right side of any production rule, without any restriction on the position of the variable. • Example: S  aS | A A Ab | λ Unit-2

  6. Another formalism for regular languages • Every regular grammar generates a regular language, and every regular language can be generated by a regular grammar. • A regular grammar is a simpler, special-case of a context-free grammar • The regular languages are a proper subset of the context-free languages Unit-2

  7. Exercises • Find a regular grammar that generates the language on  = {a,b} consisting of all strings with no more than three a’s. Unit-2

  8. Exercises • Find a regular grammar that generates the language on  = {a,b} consisting of all strings with no more than three a’s S  bS | aA | λ A  bA | aB | λ B  bB | aC | λ C  bC | λ Unit-2

  9. Exercises • Find a regular grammar that generates the language consisting of even-length strings over {a,b}. Unit-2

  10. Exercises • Find a regular grammar that generates the language consisting of even-length strings over {a,b}. S  aaS | abS | baS | bbS | λ Unit-2

  11. Programming languages • Programming languages are context-free, but not regular • Programming languages have the following features that require infinite “stack memory” • matching parentheses in algebraic expressions • nested if .. then .. else statements, and nested loops • block structure Unit-2

  12. Exercise • Given a grammar, you should be able to say what language it generates • Use set notation to define the language generated by the following grammars 1) S  aaSB | λ B  bB | b 2) S  aSbb | A A  cA | c Unit-2

  13. Exercise S  aaSB | λ B  bB | b It helps to list some of the strings that can be formed: S  aaSB  aaB  aab S  aaSB  aaB  aabB  aabb S  aaSB  aaB  aabB  aabbB  aabbb S  aaSB  aaB  aabB  aabbB  aabbbB  aabbbb S  aaSB  aaaaSBB  aaaaBB  aaaaBb  aaaabb S  aaSB  aaaaSBB  aaaaBB  aaaaBbB  aaaaBbb  aaaabbb What is the pattern? L = {(aa)nbnb*} Unit-2

  14. Exercise • Given a language, you should be able give a grammar that generates it. • For example, give a regular (right-linear) grammar for the language consisting of all strings over {a, b, c} that begin with a, contain exactly two b’s, and end with cc. Unit-2

  15. Exercise • Give a regular (right-linear) grammar for the language consisting of all strings over {a, b, c} that begin with a, contain exactly two b’s, and end with cc: S  aA A  bB | aA | cA B  bC | aB | cB C  aC | cC | cD D  c Unit-2

  16. Derivation Given the grammar, S  aaSB | λ B  bB | b the string aab can be derived in two different ways: S  aaSB  aaB  aab S  aaSB  aaSb  aab Unit-2

  17. Leftmost (rightmost) derivation In a leftmost derivation, the leftmost nonterminal is replaced at each step. In a rightmost derivation, the rightmost nonterminal is replaced at each step. Many derivations are neither leftmost nor rightmost. Unit-2

  18. Theorem 3.3 • Every language generated by a right-linear grammar is regular. • Proof: • Specify a procedure for automatically constructing an NFA that mimics the derivations of a right-linear grammar. Unit-2

  19. Theorem 3.3 • Justification: The sentential forms produced by a right linear grammar have exactly one variable, which occurs as the rightmost symbol. Assume that our grammar has a production rule D dE and that, during the derivation of a string, there is a step wcD  wcdE We can construct an NFA which has states D and E, and an arc labeled d from D to E. NFAs can be converted to DFAs. All languages accepted by DFAs are regular. Unit-2

  20. Theorem 3.3 Construction: For each variable Vi in the grammar there will be a state in the automaton labeled Vi. The initial state of the automaton will be labeled V0 and will correspond to the S variable in the grammar. For each production rule Vi a1a2…amVj the automaton will have transitions such that δ*(Vi, a1a2…am) = Vj For each production rule Vi a1a2…am the automaton will have transitions such that δ*(Vi, a1a2…am) = Vfinal Unit-2

  21. Theorem 3.3 Construct an NFA that accepts the language generated by the grammar: S  aA convert to: V0 aV1 A abS | b V1 abV0 | b a b V0 V1 Vf b a Unit-2

  22. Theorem 3.4 • Every regular language can be generated by a right-linear grammar. • Proof: • Generate a DFA for the language. • Specify a procedure for automatically constructing a right-linear grammar from the DFA. Unit-2

  23. Theorem 3.4 • Given a regular language L, let M = (Q, , δ, q0, F) be a DFA that accepts L. Let Q = {q0, q1, …, qn} and  = {a1, a2, …, am}. • Construct the grammar G = (V, T, S, P) with: V = {q0, q1, …, qn} T = {a1, a2, …, am} S = q0. P = {} initially. • P, the set of production rules, is constructed as follows: Unit-2

  24. Theorem 3.4 • For each transition δ(qi, aj) = qk in the transition table of M, add to P the production: qi ajqk • If qk is in F, then add to P the production: qk λ Unit-2

  25. Example • Construct a right-linear grammar for the language L = L(aab*a) • First, build an NFA for L: a a a q0 q1 q2 qf b Unit-2

  26. Example, cont. a a a q0 q1 q2 qf b P = {} initially. Add to P a rule for each transition in the NFA: q0 aq1 q1  aq2 q2  bq2 q2  aqf Since qf is in F, add to P the production: qf  λ Unit-2

  27. Example a a a q0 q1 q2 qf b Now P = You can convert to normal grammar notation: {q0 aq1 S  aA q1  aq2 A  aB q2  bq2 B  bB q2  aqf B  aC qf  λ } C  λ Unit-2

  28. Theorem 3.5 A language L is regular if and only if there exists a left-linear grammar G such that L = L(G). Proof: The strategy here is a little tricky. We describe an algorithm to construct a right-linear grammar that generates the reverse of all the strings generated by the left-linear grammar. Unit-2

  29. Theorem 3.5 Given any left-linear grammar we can construct from it an right-linear grammar G’ by replacing productions of the form: A  Bv with A  vRB and A  v with A  vR Since L(G’) is generated by a right-linear grammar, it is regular. It can be demonstrated that L(G) = (L(G’))R. It can be proven that the reverse of any regular language is also regular (see exercise 12, section 2.3 in the Linz text). Hence, L is regular. Unit-2

  30. Theorem 3.6 A language L is regular if and only if there exists a regular grammar G such that L = L(G). Proof: Combine our definition of regular grammars, which includes the statement, “A regular grammar is either right-linear or left-linear”, with theorems 3.4 and 3.5 Unit-2

  31. 3 ways of specifying regular languages Regular expressions DFA NFA Regular grammars describe accept Regular languages generate Unit-2

  32. THANK YOU Unit-2

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