1 / 41

# Complexity and Computability Theory I - PowerPoint PPT Presentation

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Complexity and Computability Theory I' - beatrice-johnson

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.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Complexity and Computability Theory I

Lecture #9

Instructor: Rina Zviel-Girshin

Lea Epstein

• Grammars

• Example

• Context-free grammars

• Examples

• Ambiguity

Rina Zviel-Girshin @ASC

• 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 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

• 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

• 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

• 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

• ={a,b,c}

• The following grammar generates all strings over .

S (delete S)

Rina Zviel-Girshin @ASC

• 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

• 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

• 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

• 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

Rina Zviel-Girshin @ASC

• Or a step by step derivation:

Rina Zviel-Girshin @ASC

Rina Zviel-Girshin @ASC

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

• 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

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 is

L(G) = {w* | w* and S * w}

• The language generated by CFG is called a context-free language (CFL).

Rina Zviel-Girshin @ASC

• 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}

• 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 ={0,1}

• 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 ={0,1}

• 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 ={0,1}

• 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 ={0,1}

• 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 ={0,1}

• 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 ={0,1}

• 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 ={0,1}

• 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

• Give a description of L(G) for the following grammar:

S  0S0 | 1

• L(G) = {0n10n|n0}

Rina Zviel-Girshin @ASC

• 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

• 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

• Give a description of L(G) for the following grammar:

S  0S11 | 0

• L(G) = {0 n+112n |n1 }

Rina Zviel-Girshin @ASC

• 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

• 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

• 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

• The grammar derives all n words of L.

• The grammar is finite (n production rules).

• The grammar syntax is correct.

Rina Zviel-Girshin @ASC

• 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

• 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

• 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:

• programming languages should have a unique interpretation

or

• the grammar of the programming language would be unambiguous.

Rina Zviel-Girshin @ASC

Rina Zviel-Girshin @ASC