CS 208: Computing Theory

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CS 208: Computing Theory. Assoc. Prof. Dr. Brahim Hnich Faculty of Computer Sciences Izmir University of Economics. Context Free Languages. Context-Free Languages. So far …. Methods for describing regular languages Finite Automata Deterministic Non-deterministic Regular Expressions

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### CS 208: Computing Theory

Assoc. Prof. Dr. Brahim Hnich

Faculty of Computer Sciences

Izmir University of Economics

### Context Free Languages

Context-Free Languages

So far …
• Methods for describing regular languages
• Finite Automata
• Deterministic
• Non-deterministic
• Regular Expressions
• They are all equivalent, and limited
• Cannot some simple languages like {0n1n | n is positive}
• Now, we introduce a more powerful method for describing languages
• Context-free Grammars (CFG)
Are CFGs any useful?
• Extremely useful!
• Artificial Intelligence
• Natural language Processing
• Programming Languages
• specification
• compilation
Example
• This is a CFG which we call G1
• A0A1
• AB
• B#
Example: production rules
• This is a CFG which we call G1
• A0A1
• AB
• B#

Each line is a substitution rules or production rules

Example: variables
• This is a CFG which we call G1
• A0A1
• AB
• B#

A and B are called variables or non-terminals

Example: variables
• This is a CFG which we call G1
• A0A1
• AB
• B#

0,1, and # are called terminals

Example: variables
• This is a CFG which we call G1
• A0A1
• AB
• B#

A is the start variable

Rules
• We use a CFG to describe a language by generating each string of that language
• Write down the start variable
• Pick a variable written down and a production rule that starts with that variable
• Replace that variable with right-hand side of the production rule
• Repeat until no variable remain
Derivations
• This is a CFG which we call G1
• A0A1
• AB
• B#
• Derivations with G1
• A0A10B10#1
• A0A100A1100B1100#11
• A0A100A11000A111000B111000#111
Parse tree
• Parse tree for 0#1 in G1
• A0A10B10#1

A

A

B

1

0

#

Parse tree

Parse tree for 00#11 in G1 A0A100A1100B1100#11

A

A

A

B

1

1

0

0

#

Context-free languages
• All strings generated by a CFG constitute the language of the grammar
• Example: L(G1)={0n#1n | n is positive}
• Any language generated by a context-free grammar is a context-free language
A useful abbreviation
• Production rules
• A  0A1
• A  B
• B  #
• Can be written as
• A  0A1 | B
• B  #
Another example
• CFG G2 describing a fragment of English

<SENTENCE>  <NOUN-PHRASE><VERB-PHRASE>

<NOUN-PHRASE> <CMPLX-NOUN>|<PREP-PHRASE>

<VERB-PHRASE><CMPLX-VERB>|<CMPX-VERB><PREP-PHRASE>

<PREP-PHRASE><PREP><CMPLX-NOUN>

<CMPLX-NOUN><ARTICLE><NOUN>

<CMPLX-VERB><VERB>|<VERB><NOUN-PHRASE>

<ARTICLE> a | the

<NOUN>  boy | girl | flower

<VERB>  touches | likes | sees

<PREP>  with

Another example
• Examples of strings belonging to L(G2)

a boy sees

the boy sees a flower

a girl with a flower likes the boy with a flower

Another example
• Derivation of a boy sees

<SENTENCE>

 <NOUN-PHRASE><VERB-PHRASE>

 <CMPLX-NOUN><VERB-PHRASE>

 <ARTICLE><NOUN> <VERB-PHRASE>

 a <NOUN><VERB-PHRASE>

 a boy <VERB-PHRASE>

 a boy <CMPLX-VERB>

 a boy <VERB>

 a boy sees

Formal definitions
• A context-free grammar is a 4-tuple <V, ∑, R, S> where
• V is a finite set of variables
• ∑is a finite set of terminals
• R is a finite set of rules: each rule is a variable and a finite string of variable and terminals
• S is the start symbol
Formal definitions
• If
• u and v are strings of variable and terminals, and
• A  w is a rule of the grammar,
• Then uAv yields uwv, written uAv  uwv
• We write u * v if
• u = v or
• u u1  …. uk  v
Formal definitions
• The language of grammar G is
• L(G) = {w | S * w}
Example
• Consider G4 =<{S},{(,)},R,S> where R is
• S  (S) | SS | ε
• What is the language of G4?
• Examples: (), (()((())), …
Example
• Consider G4 =<{S},{(,)},R,S> where R is
• S  (S) | SS | ε
• What is the language of G4?
• L(G4) is the set of strings of properly nested parenthesis
Example
• Consider G4 =<{E,T,F},{a,+, x, (, )},R,E> where R is
• E  E + T | T
• T  T X F | F
• F  (E) | a
• What is the language of G4?
• Examples: a+a+a, (a+a) x a
Example
• Consider G4 =<{E,T,F},{a,+, x, (, )},R,E> where R is
• E  E + T | T
• T  T x F | F
• F  (E) | a
• What is the language of G4?
• E stands for expression, T for Term, and F for Factor: so this grammar describes some arithmetic expressions
Ambiguity
• Sometimes a grammar can generate the same string in several different ways!
• This string will have several parse trees
• This is a very serious problem
• Think if a C program can have multiple interpretations?
• If a language has this problem, we say that it is ambiguous
Example
• Consider G5:

<EXPR><EXPR>+<EXPR>|<EXPR>x<EXPR>

|(<EXPR>) | a

G5 is ambiguous because a+axa has two parse tress!

Example
• Consider G5:

<EXPR><EXPR>+<EXPR>|<EXPR>x<EXPR>

|(<EXPR>) | a

G5 is ambiguous because a+axa has two parse tress!

<EXPR>

<EXPR>

<EXPR>

<EXPR>

<EXPR>

a + a x a

Example
• Consider G5:

<EXPR><EXPR>+<EXPR>|<EXPR>x<EXPR>

|(<EXPR>) | a

G5 is ambiguous because a+axa has two parse tress!

<EXPR>

<EXPR>

<EXPR>

<EXPR>

<EXPR>

<EXPR>

<EXPR>

<EXPR>

<EXPR>

<EXPR>

a + a x a

a + a x a

Formal definition: ambiguity
• A string w is generated ambiguously in CFG G if it has two or more different leftmost derivations!
• A derivation is leftmost if at every step the variable being replaced is the leftmost one
• Grammar G is ambiguous if it generates some string ambiguously
Chomsky Normal Form (CNF)
• Every rule has the form
• A  BC
• A  a
• S  ε
• Where S is the start symbol, A, B, and C are any variables – except that B and C may not be the start symbol
Theorem
• Theorem: Any context-free language is generated by a context-free grammar in Chomsky normal form
• How?
• Add new start symbol S0
• Eliminate all rules of the form A  ε
• Eliminate all “unit” rules of the form A  B
• Patch up rules so that grammar denotes the same language
• Convert remaining rules to proper form
Steps to convert any grammar into CNF
• Step1
• Add a new start symbol S0
Steps to convert any grammar into CNF
• Step2: Repeat
• Remove some rule of the form A  ε where A is not the start symbol
• Then, for each occurrence of A on the right-hand side of a rule, we add a new rule with that occurrence deleted
• E.g., if R uAvAu where u and v are strings of variables and terminals
• We add rules: R uvAu, RuAvu, and Ruvu
• For RA add Rε, except if Rε has already been removed
• Until all ε-rules not involving the start symbol have been removed
Steps to convert any grammar into CNF
• Step3: eliminate unit rules
• Repeat
• Remove some rule of the form A  B
• For each Bu, add Au, except if Au has already been removed
• Until all unit rules have been removed
Steps to convert any grammar into CNF
• Step4: convert remaining rules
• Replace each rule A u1u2…uk, where k >2 and each ui is a terminal or a variable with the rules
• Au1A1
• A1u2A2
• A2u3A3
• ….
• Ak-2uk-1uk
• If k=2, we replace any terminal ui in the preceding rules with the new variable Ui and add the rule Uiui
Example
• S  ASA | aB
• A  B | S
• B  b | ε
Example
• Step 1: add new start symbol and new rule
• S0  S
• S  ASA | aB
• A  B | S
• B  b | ε
Example
• Step 2: remove ε-rule B ε
• S0  S
• S  ASA | aB | a
• A  B | S | ε
• B  b
Example
• Step 2: remove ε-rule A ε
• S0  S
• S  ASA | aB | a | SA | AS | S
• A  B | S
• B  b
Example
• Step 3: remove unit rule S S
• S0  S
• S  ASA | aB | a | SA | AS | S
• A  B | S
• B  b
Example
• Step 3: remove unit rule S0 S
• S0  S | ASA | aB | a | SA | AS
• S ASA | aB | a | SA | AS
• A  B | S
• B  b
Example
• Step 3: remove unit rule A B
• S0  ASA | aB | a | SA | AS
• S ASA | aB | a | SA | AS
• A  B | S | b
• B  b
Example
• Step 3: remove unit rule A S
• S0  ASA | aB | a | SA | AS
• S ASA | aB | a | SA | AS
• A  S | b | ASA | aB | a | SA | AS
• B  b
Example
• Step 3: remove unit rule A S
• S0  ASA | aB | a | SA | AS
• S ASA | aB | a | SA | AS
• A  b | ASA | aB | a | SA | AS
• B  b
Example
• Step 4: convert remaining rules
• S0  AA1|UB| a| SA | AS
• S AA1|UB | a | SA | AS
• A  b | AA1 | UB | a | SA | AS
• B  b
• Ua
• A1SA

### Pushdown Automata

Pushdown automata
• Pushdown automat (PDA) are like nondeterministic finite automat but have an extra component called a stack
• Can push symbols onto the stack
• Can pop them (read them back) later
• Stack is potentially unbounded

input

State

control

a

a

b

a

x

y

z

stack

Formal Definition
• A pushdown automaton is a 6-tuple (Q,∑,S, ξ,q0,F), where
• Q is a finite set of states
• ∑ is a finite set of symbols called the alphabet
• S is the stack alphabet
• ξ : Q x ∑ε x Sε P(Q x Sε) is the transition function
• q0 Є Q is the start state
• F ⊆ Q is the set of accept states or final states
Conventions
• Question: when is the stack empty?
• Start by pushing a \$ onto the stack
• If you see it again, stack is empty
• Question: when is input string empty
• Doesn’t matter
• Accepting states accept only if inputs exhausted
Notation
• Transition a,bc means
• Read a from the input
• Pop b from stack
• Push c onto stack
• Meaning of ε transition
• If a = ε , don’t read input
• If b= ε , don’t pop any symbol
• If c= ε , don’t push any symbols
Example
• Recall 0n1n which is not regular
• Consider the following PDA
• For each 0, push it on the stack
• As soon as a 1 is seen, pop a 0 for each 1 read
• Accept if stack is empty when last symbol read
• Reject if stack non-empty, or if input symbol exist, or if 0 read after a 1, etc…
Example

{0n1n| n is positive}

0, ε0

ε,ε\$

1,0 ε

1,0 ε

ε,\$ ε

Example

{aibjck| i=j or i=k}

c,ε ε

b, a  ε

ε,\$ ε

ε,ε ε

ε,ε\$

ε,ε ε

ε, \$ ε

ε,ε ε

a, ε a

b, ε ε

c, a ε

Theorem
• Theorem: A language is context-free if and only some pushdown automaton accepts it
• Proof: we will skip it! (Those interested may read the book)
• Corollary: Every regular language is a context-free language

Context-free

languages

Regular

languages

Conclusions

Context-free grammars

definition

ambiguity

Chomsky normal form

Pushdown automata

definition

Next: Part C;

Computability Theory