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Complexity and Computability Theory I

Complexity and Computability Theory I. Lecture #9 Instructor: Rina Zviel-Girshin Lea Epstein. Overview. Grammars Example Context-free grammars Examples Ambiguity. Grammar. Another computational model. A member in the family of rewriting systems.

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Complexity and Computability Theory I

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  1. Complexity and Computability Theory I Lecture #9 Instructor: Rina Zviel-Girshin Lea Epstein

  2. Overview • Grammars • Example • Context-free grammars • Examples • Ambiguity Rina Zviel-Girshin @ASC

  3. Grammar • Another computational model. • A member in the family of rewriting systems. • The computation is by rewriting a string. • We start with an empty string and rewrite the string according to the grammar until we have an output. • All possible outputs of a grammar is the language of the grammar. Rina Zviel-Girshin @ASC

  4. The origin • The origin of the name grammar for this computational model is in natural languages, where grammar is a collection of rules. • This collection defines what is legal in the language and what is not.   Rina Zviel-Girshin @ASC

  5. The grammar computational model • In the same manner the grammar computational model is primarily a collection of: • rules of rewriting, • rules how to build strings that are in the language, • structural rules for the language. Rina Zviel-Girshin @ASC

  6. Some facts • The grammar consists of a collection ofrules over an alphabet  and a set of variables (usually denoted by capital letters of the Latin alphabet). • Every grammar has a start symbol also called a start variable (usually denoted by S). • Every grammar has at least one rule. Rina Zviel-Girshin @ASC

  7. notation • We will use the notation  in grammar rules.  • What does it mean: ()? • It means : •  can be replaced by  •  constructs  •  produces  •  rewrites to  •  reduces to  Rina Zviel-Girshin @ASC

  8. Example of a grammar • ={a,b,c} • The following grammar generates all strings over . SaS (add a) SbS (add b) ScS (add c) S (delete S) Rina Zviel-Girshin @ASC

  9. w =aacb production • How can the word w=aacb be produced? • SaS • We used the SaS production because w starts with a and the only rule that starts with a is SaS. • From S that remains we need to produce w'=acb. • SaSaaS • We used the SaS production because w' also starts with a and the only rule that starts with a is SaS. Rina Zviel-Girshin @ASC

  10. w =aacb production (cont.) • From S we need to produce w''=cb. • SaSaaSaacS • We used the ScS production because w'' starts with c and the only rule that starts with c is ScS. • From S we need to produce b. • SaSaaSaacSaacbS • We used the SbS production to produce b. Rina Zviel-Girshin @ASC

  11. w =aacb production (cont.) • But S is still remaining in final production. We want to delete it. We will use the rule S to delete S. • SaSaaSaacSaacbSaacb • So we managed to produce w using the rules of the grammar. Rina Zviel-Girshin @ASC

  12. Parsing • What we did is called parsing a word w accordingly to a given grammar. • To parse a word or sentence means to break it into parts that confirm to a given grammar. • We can represent the same production sequence by a parse tree or derivation tree. • Each node in the tree is either letter or variable. • Only a variable node can have children. Rina Zviel-Girshin @ASC

  13. Parsing w=aacb Rina Zviel-Girshin @ASC

  14. Parsing w=aacb • Or a step by step derivation: Rina Zviel-Girshin @ASC

  15. Parsing w=aacb (cont.) Rina Zviel-Girshin @ASC

  16. Context-free grammar A context-free grammar (CFG) G is a 4-tuple (V, , S, R), where 1. V is a finite set called the variables 2.  is a finite set, disjoint from V, called the terminals 3. S is a start symbol 4. R is a finite set of production rules, with each rule being a variable and a string of variables and terminals: ab, aV and b(VU)* Rina Zviel-Girshin @ASC

  17. uAv yields uwv • If u, v and w are strings of variables and terminals and Aw is a rule of the grammar, we say that uAv yields uwv, written uAvuwv. • We write u*w if there exists a sequence u1, u2, ..uk, k0 and uu1u2...w. Rina Zviel-Girshin @ASC

  18. notation We also use the following notations:  means derives in one step + means derives in one or more steps * means derives in zero or more steps Rina Zviel-Girshin @ASC

  19. The language of the grammar • The language of the grammar is L(G) = {w* | w* and S * w} • The language generated by CFG is called a context-free language (CFL). Rina Zviel-Girshin @ASC

  20. Is the following definition correct? • The language of the grammar is L(G) = {w* | w* and S + w} • Yes. • Because a derivation in zero steps derivation produces only S. • S is not a string over *, so can't belong to L. Rina Zviel-Girshin @ASC

  21. Examples over ={0,1} • Construct a grammar for the following language L = {0,00,1} • G = (V={S},={0,1},S, R) where R: S  0 S  00 S  1 or S  0 | 00 | 1 Rina Zviel-Girshin @ASC

  22. Examples over ={0,1} • Construct a grammar for the following language L = {0n1n |n0} • G = (V={S},={0,1},S, R) where R: S0S1 S or S0S1 |  Rina Zviel-Girshin @ASC

  23. Examples over ={0,1} • Construct a grammar for the following language L = {0n1n |n1} • G = (V={S},={0,1},S, R) where R: S  0S1 | 01 Rina Zviel-Girshin @ASC

  24. Examples over ={0,1} • Construct a grammar for the following language L = {0*1+} • G = (V={S,B},={0,1},S, R) where R: S 0S | 1B B 1B |  Rina Zviel-Girshin @ASC

  25. Examples over ={0,1} • Construct a grammar for the following language L = {02i+1 | i0} • G = (V={S},={0,1},S, R) where R: S 0 | 00S Rina Zviel-Girshin @ASC

  26. Examples over ={0,1} • Construct a grammar for the following language L = {0i+11i | i0} • G = (V={S},={0,1},S, R) where R: S 0 | 0S1 Rina Zviel-Girshin @ASC

  27. Examples over ={0,1} • Construct a grammar for the following language L = {w| w* and |w|mod 2=1} • G = (V={S},={0,1},S, R) where R: S 0 | 1| 1S1| 0S0 |1S0 | 0S1 Rina Zviel-Girshin @ASC

  28. Examples over ={0,1} • Construct a grammar for the following language L = {0n1n |n1} {1n0n | n0} •  G = (V={S,A,B},={0,1},S, R) where R: S  A | B A  0A1 | 01 B  1B0 |  Rina Zviel-Girshin @ASC

  29. From a grammar to a CFL • Give a description of L(G) for the following grammar: S  0S0 | 1 • L(G) = {0n10n|n0} Rina Zviel-Girshin @ASC

  30. From a grammar to a CFL • Give a description of L(G) for the following grammar: S  0S0 | 1S1 | # • L(G) = {The subset of all palindromes over ={0,1} with # in the middle} or • L(G) = {w#wR| w*} Rina Zviel-Girshin @ASC

  31. From a grammar to a CFL • Give a description of L(G) for the following grammar: S  0A | 0B A1S B1 • L(G) = {(01)n |n1 } Rina Zviel-Girshin @ASC

  32. From a grammar to a CFL • Give a description of L(G) for the following grammar: S  0S11 | 0 • L(G) = {0 n+112n |n1 } Rina Zviel-Girshin @ASC

  33. From a grammar to a CFL • Give a description of L(G) for the following grammar: S  E | NE N  D | DN D  0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 E  0 | 2 | 4 | 6 • L(G) = {w | w represents an even octal number } Rina Zviel-Girshin @ASC

  34. From a grammar to a CFL • Give a description of L(G) for the following grammar: S  N.N | -N.N N  D | DN D  0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 • L(G) = {w | w represents a rational number (that has a finite representation) } Rina Zviel-Girshin @ASC

  35. Question • Can any finite language be constructed by a grammar?  Yes. Proof: • Let L={wi | in and wi*} be a finite language over . • We construct the following grammar: Sw1 Sw2 .. Swn Rina Zviel-Girshin @ASC

  36. Question (cont.) • The grammar derives all n words of L. • The grammar is finite (n production rules). • The grammar syntax is correct. Rina Zviel-Girshin @ASC

  37. Ambiguity • The ability of grammar to generate the same string in several ways is called ambiguity. • That means that the string have different parse trees and may have different meanings.  • A grammar is ambiguous if there exists a string w that has at least two different parse trees. Rina Zviel-Girshin @ASC

  38. Example • The string 3+4*5 can be produced in several ways:  EE+E | E*E | T T0|1|2|..|9  Rina Zviel-Girshin @ASC

  39. Example (cont.) • So if we use this grammar to produce a programming language then we will have several computations of 3+4*5. • There is no precedence of * over the +. • This language will be impossible to use because the user won't know which computation compiler uses. • Two possible results: 35 or 23. Rina Zviel-Girshin @ASC

  40. The conclusion • The conclusion: • programming languages should have a unique interpretation or • the grammar of the programming language would be unambiguous. Rina Zviel-Girshin @ASC

  41. Any Questions? Rina Zviel-Girshin @ASC

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