Complexity and Computability Theory I

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# 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.

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## Complexity and Computability Theory I

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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

Rina Zviel-Girshin @ASC

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 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
• 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
• 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

Example of a grammar
• ={a,b,c}
• The following grammar generates all strings over .

S (delete S)

Rina Zviel-Girshin @ASC

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.)
• 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 (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
• 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

Rina Zviel-Girshin @ASC

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

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Parsing w=aacb (cont.)

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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
• 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
• 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?
• 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 | 

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

S  0S0 | 1

• L(G) = {0n10n|n0}

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

S  0S11 | 0

• L(G) = {0 n+112n |n1 }

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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 }

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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) }

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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

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Question (cont.)
• The grammar derives all n words of L.
• The grammar is finite (n production rules).
• The grammar syntax is correct.

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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
• The string 3+4*5 can be produced in several ways:

EE+E | E*E | T

T0|1|2|..|9

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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:
• programming languages should have a unique interpretation

or

• the grammar of the programming language would be unambiguous.

Rina Zviel-Girshin @ASC

Any Questions?

Rina Zviel-Girshin @ASC