Fall 2012 -- Prof. Claude Crépeau

COMP-330
Theory of Computation
Fall 2012 -- Prof. Claude Crépeau
Lec. 9 : Context-Free
Grammars
1

COMP 330 Fall 2012:
Lectures Schedule
1. Introduction
1.5. Some basic mathematics
2. Deterministic finite automata
+Closure properties,
3. Nondeterministic finite automata
4. Determinization+Kleene’s theorem
5-6. Regular Expressions, GNFA and
Regular Languages
6-7. The pumping lemma
8. Minimization+Myhill-Nerode theorem
9. Context-free languages
13. MIDTERM
15. Pushdown automata
16. Parsing
17. The pumping lemma for CFLs
18. Introduction to computability
19. Models of computation
Basic computability theory
20. Reducibility, undecidability and Rice’s theorem
21. Undecidable problems about CFGs
22. Post Correspondence Problem
23. Validity of FOL is RE / Gödel’s and Tarski’s thms
24. Universality / The recursion theorem
25. Degrees of undecidability
26. Introduction to complexity
2

Context-Free Grammars
Let’s call the following grammar G 1 :
A 0A1
A B
B #
Derivation of a string “000#111” :
A0A100A11000A111000B111000#111.
3

Definition of CFG
Variables
A, B, C, ⟨TERM⟩, ⟨EXPR⟩
Alphabet (of terminals)
0,1,#
Substitution Rules
Start Variable
A 0A1
⟨EXPR⟩ ⟨TERM⟩
A
(left-hand side of the first substitution rule)
4

Definition of CFG
5

Parse Tree
6

Definition of CFL
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
uAvuwv.
We say that u derives v ( u * v ) if u=v or if
uu 1 u 2 ...u k v, k≥0.
The language of G is { w∈∑* | S * w }.
7

Context-Free Grammars
Formally, grammar G 1 :
V = {A,B}
∑ = {0,1,#}
R = {A 0A1, A B, B #}
S = A
L(G 1 ) = { 0 n #1 n | n≥0 }.
8

Example of CFG
9

Example of CFG
⟨ARTICLE⟩ a | the
means
⟨ARTICLE⟩ a
⟨ARTICLE⟩ the
9

Regular Operations :
Kleene’s theorem (CFG)
18

Regular Operations :
Kleene’s theorem (CFL)
CFLs
19

Kleene’s
theorem (CFL)
20

Kleene’s
theorem (CFL)
Let G A =(V A ,∑,R A ,S A ) be a CFG generating L A and
G B =(V B ,∑,R B ,S B ) be a CFG generating L B (V A ∩V B =∅).
20

Kleene’s
theorem (CFL)
Let G A =(V A ,∑,R A ,S A ) be a CFG generating L A and
G B =(V B ,∑,R B ,S B ) be a CFG generating L B (V A ∩V B =∅).
Consider
G U =( {S U }∪V A ∪V B ,∑,{S U S A | S B }∪R A ∪R B ,S U ).
20

Kleene’s
theorem (CFL)
Let G A =(V A ,∑,R A ,S A ) be a CFG generating L A and
G B =(V B ,∑,R B ,S B ) be a CFG generating L B (V A ∩V B =∅).
Consider
G U =( {S U }∪V A ∪V B ,∑,{S U S A | S B }∪R A ∪R B ,S U ).
L U = L A ∪ L B .
20

Regular Operations :
Kleene’s theorem (CFL)
CFLs
21

Kleene’s
theorem (CFL)
22

Kleene’s
theorem (CFL)
Let G A =(V A ,∑,R A ,S A ) be a CFG generating L A and
G B =(V B ,∑,R B ,S B ) be a CFG generating L B (V A ∩V B =∅).
22

Kleene’s
theorem (CFL)
Let G A =(V A ,∑,R A ,S A ) be a CFG generating L A and
G B =(V B ,∑,R B ,S B ) be a CFG generating L B (V A ∩V B =∅).
Consider
G C =( {S C }∪V A ∪V B ,∑,{S C S A S B }∪R A ∪R B ,S C ).
22

Kleene’s
theorem (CFL)
Let G A =(V A ,∑,R A ,S A ) be a CFG generating L A and
G B =(V B ,∑,R B ,S B ) be a CFG generating L B (V A ∩V B =∅).
Consider
G C =( {S C }∪V A ∪V B ,∑,{S C S A S B }∪R A ∪R B ,S C ).
L C = L A ∘L B .
22

Regular Operations :
Kleene’s theorem (CFL)
CFLs
23

Kleene’s
theorem (CFL)
24

Kleene’s
theorem (CFL)
Let G A =(V A ,∑,R A ,S A ) be a CFG generating L A .
24

Kleene’s
theorem (CFL)
Let G A =(V A ,∑,R A ,S A ) be a CFG generating L A .
Consider
G S =( {S S }∪V A ,∑,{S S E | S A S S }∪R A ,S S ).
24

Kleene’s
theorem (CFL)
Let G A =(V A ,∑,R A ,S A ) be a CFG generating L A .
Consider
G S =( {S S }∪V A ,∑,{S S E | S A S S }∪R A ,S S ).
L S = (L A )*.
24

Construction tools
(and Reductions)
CFLs are closed under union, concatenation and
star. If there exists a CFL C s. t. either A*=A’,
A∪C=A’, A∘C=A’ (but not complement nor
intersection) or any combinations of these
operations then A’ is a CFL as long as A is.
( If A’ is NON-CFL then so is A. )
25

Construction tools
Constructing a CFG for a regular language L:
M = (Q={q 0 ,q 1 ,...,q k },∑,δ,q 0 ,F) is converted to
G = (V={R 0 ,R 1 ,...,R k },∑,R,S=R 0 ) where
R contains rule R i aR j for each δ(q i ,a) = q j
in M, and rule R i E for each accept-state
q i ∈F.
R 0 is the start variable.
26

extra EXAMPLE of CFG
27

extra EXAMPLE of CFG
28

Ambiguity in CFGs
29

Leftmost Derivation
A derivation is Leftmost if every time a variable
is substituted, it is always the leftmost variable.
E
X
A
M
PLE
30

Ambiguity
A string w is derived ambiguously by a CFG
G if it has two or more distinct leftmost
derivations. Grammar G is ambigious if it
generates some string ambiguously.
31

Ambiguity
Ambiguity is not desirable in CFG because it
may lead to unexpected interpretations of a
string, for instance in the context of arithmetic
expressions or programming languages.
However, some languages are inherently
ambiguous, meaning that all grammars
genrating this language must be ambiguous.
example : {a i b j c k | i=j or j=k}
32

Ambiguous version of
example 2.4
G 5
33

Ambiguous CFG
34

Noam Chomsky
Chomsky Normal Form
35

Chomsky Normal Form
36

Chomsky Normal Form
37

157
38

Proof:
39

Proof:
First, we add a new start variable S 0 and the
rule S 0 S, where S was the origial start
variable.
39

Chomsky Normal Form
40

Second, we take care of all E-rules. We
remove an E-rule A E, where A is not the
start variable.
41

Second, we take care of all E-rules. We
remove an E-rule A E, where A is not the
start variable.
Then for each occurrence of A on the righthand
side of a rule we add a new rule with
that occurrence deleted.
41

Second, we take care of all E-rules. We
remove an E-rule A E, where A is not the
start variable.
Then for each occurrence of A on the righthand
side of a rule we add a new rule with
that occurrence deleted.
Accordingly, each rule R A is replaced by
R E unless it has been already removed.
41

Chomsky Normal Form
42

Third, we handle all unit rules by removing
each unit rule A B.
43

Third, we handle all unit rules by removing
each unit rule A B.
In consequence whenever B u appears, we
add the rule A u unless this is a unit rule
previously removed.
43

Chomsky Normal Form
44

Finally, we convert all remaining rules as
follows: A u 1 u 2 ...u k for k>2, where each u i
is a variable or terminal with a series of
rules A u 1 A 1 , A 1 u 2 A 2 ,..., A k-2 u k-1 u k
where each A i is a new variable.
45

Finally, we convert all remaining rules as
follows: A u 1 u 2 ...u k for k>2, where each u i
is a variable or terminal with a series of
rules A u 1 A 1 , A 1 u 2 A 2 ,..., A k-2 u k-1 u k
where each A i is a new variable.
When k=2, and A u 1 u 2 , we may replace
any terminal u i by a variable U i and the rule
U i u i.
45

Chomsky Normal Form
46

COMP-330
Theory of Computation
Fall 2012 -- Prof. Claude Crépeau
Lec. 9 : Context-Free
Grammars
47