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Complexity and Computability Theory I

This lecture, led by Rina Zviel-Girshin and Lea Epstein, explores the fundamental concepts of context-free grammars (CFGs) in computability theory. It covers key topics such as the structure of grammars, rewriting rules, and the definition of languages generated by CFGs. Ambiguity in grammars is discussed along with examples illustrating the parsing of strings using production rules. Additionally, the session provides insights into how CFGs relate to natural language grammar, making it essential for students aiming to understand computational models.

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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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