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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

Complexity and Computability Theory I

Lecture #9

Instructor: Rina Zviel-Girshin

Lea Epstein

overview
Overview
  • Grammars
  • Example
  • Context-free grammars
  • Examples
  • Ambiguity

Rina Zviel-Girshin @ASC

grammar
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

the origin
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

the grammar computational model
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

some facts
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

notation
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

example of a grammar
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

w aacb production
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

w aacb production cont
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

w aacb production cont1
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

parsing
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

parsing w aacb
Parsing w=aacb

Rina Zviel-Girshin @ASC

parsing w aacb1
Parsing w=aacb
  • Or a step by step derivation:

Rina Zviel-Girshin @ASC

parsing w aacb cont
Parsing w=aacb (cont.)

Rina Zviel-Girshin @ASC

context free grammar
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

uav yields uwv
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

notation1
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

the language of the grammar
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

is the following definition correct
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

examples over 0 1
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

examples over 0 11
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

examples over 0 12
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

examples over 0 13
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

examples over 0 14
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

examples over 0 15
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

examples over 0 16
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

examples over 0 17
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

from a grammar to a cfl
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

from a grammar to a cfl1
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

from a grammar to a cfl2
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

from a grammar to a cfl3
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

from a grammar to a cfl4
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

from a grammar to a cfl5
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

question
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

question cont
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

ambiguity
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

example
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

example cont
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

the conclusion
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

slide41

Any Questions?

Rina Zviel-Girshin @ASC

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