1. CSCI 432 Computer Science Theory
Context Free Grammars
CSCI 432 Computer Science Theory

2. Why Study CFGs What is "grammar"?
What does grammar have to do with computing?
Examples of grammar in computing?

3. Terminology S → bA A → aA A → e Those are three rules or productions.
The starting rule is usually the first rule listed.
S and A are nonterminals.
a and b are terminals.
Only a single nonterminal appears on the left side of a rule.

4. Example
S → bA 1
A → aA 2
A → e 3
Suppose we want to generate the word "baaa" using this grammar. S
bA 1
baA 2
baaA 2
baaaA 2
baaa 3
textbook page 70

5. Non Context Free Grammar
aAb → acb
How the rule for A is used depends on whether or not it is surrounded by an a and a b. Thus, that rule is not free of context.

6. Nondeterminism B → bB B → b
We cannot determine which version of B to use to generate the desired word.
We have to try possibilities until we generate the desired word.
Thus, using CFGs to generate words is nondeterministic.
textbook page 72

7. Example 1 L(G) = {anbn} S → aSb S → e S aSb aaSbb aaaSbbb aaabbb So…
Can a CFG represent a
non-regular language?
L(G) = {anbn}
S → aSb
S → e S
aSb
aaSbb
aaaSbbb
aaabbb
All regular languages
can be represented with
a CFG.
textbook example 3.1.1

8. Example 2 a*b* S → aS S → Sb S → e
So how would we generate the word aaabb?
S S
aS Sb
aaS aSb
aaaS aaSb
aaaSb aaSbb
aaaSbb aaaSbb
aaabb aaabb
These 3 rules could also
be written as:
S → aS | Sb | e
textbook example 3.1.2

9. Example 3
What does this grammar generate? S → aSa | bSb | e For example S aSa aaSaa aabSbaa aabaSabaa aabaabaa
textbook example 3.1.3

10. Example 4
Binary string with even number of 0's. 1 S → 1S 2 S → 0A0S 3 S → e 4 A → 1A 5 A → e Try to generate the string S
0A0S 2
01A0S 4
011A0S 4
0111A0S 4 ,5

11. Practice
Binary string with even number of 0's. 1 S → 1S 2 S → 0A0S 3 S → e 4 A → 1A 5 A → e Generate the string 11010 S
1S 1
11S 1
110A0S 2
1101A0S 4
11010S 3

12. Practice
Use the following grammar to produce the string "the dog eats the bone". 1 S → NP VP 2 NP → the N 3 VP → V NP 4 V → eats | buries 5 N → dog | bone

13. Grammatically Correct ≠ Meaningful
We could also use this grammar to generate the string "the bone buries the dog". 1 S → NP VP 2 NP → the N 3 VP → V NP 4 V → eats | buries 5 N → dog | bone

14. Using Grammars
How do we use a grammar to check the validity of data or a source code file?
Bad Answer : generate lots of possibilities until you generate a match. Then you know the data matches the grammar.
Good Answer : Parsing.