Theory of computation

1. Dr. Hussien Sharaf
Computer Science Department
dr.sharaf@from-masr.com

2. p.434: Regular Languages and non-
regular languages
Dr. Hussien M. Sharaf 2
Language
Defined by
Corresponding
Accepting Machine
Nondeter-minism=
Determinism
Regular expression Finite automaton, transition graph Yes
Context-free
grammar
Pushdown automaton No
Type 0 grammar Turing machine, Post machine,
Pushdown automaton
Yes

3. CFG
 A context-free grammar is a notation for
defining context free languages.
 It is more powerful than finite automata or
RE’s, but still cannot define all possible
languages.
 Useful for nested structures, e.g., parentheses
in programming languages.
 Basic idea is to use “variables” to stand for
sets of strings.
 These variables are defined recursively, in
terms of one another.
Dr. Hussien M. Sharaf 3

4. CFG formal definition
 C =(V, Σ, R, S)
 V: is a finite set of variables.
 Σ: symbols called terminals of the alphabet of the
language being defined.
 S V: a special start symbol.
 R: is a finite set of production rules of the form A→
where A V,  (V Σ)
Dr. Hussien M. Sharaf 4

5. CFG -1
 Define the language { anbn | n > 1}.
 Terminals = {a, b}.
 Variables = {S}.
 Start symbol = S.
 Productions ={
S → ab,
S → aSb }
Summary
S → ab
S → aSb
Dr. Hussien M. Sharaf 5

6. Derivation
 We derive strings in the language of a CFG by starting
with the start symbol, and repeatedly replacing some
variable A by the right side of one of its productions.
 Derivation example for “aabb”
 Using S→ aSb
generates uncompleted string that still has a non-
terminal S.
 Then using S→ ab to replace the inner S
Generates “aabb”
S aSb aabb ……[Successful derivation of aabb]
Dr. Hussien M. Sharaf 6

7. Balanced-parentheses
Prod1 S → (S)
Prod2 S → ()
 Derive the string ((())).
S → (S) …..[by prod1]
→ ((S)) …..[by prod1]
→ ((())) …..[by prod2]
Dr. Hussien M. Sharaf 7

8. Palindrome
 Describe palindrome of a’s and b’s using CGF
 1] S → aS 2] S → bS
 3] S → b 4] S → a
 Derive “baab” from the above grammar.
 S → bS [by 2]
→ baS [by 1]
→ baaS [by 1]
→ baab [by 3]
Dr. Hussien M. Sharaf 8

9. CFG – 2.1
 Describe anything (a+b)* using CGF
1] S → Λ 2] S → Y 3] Y→ aY
4] Y → bY 5] Y →a 6] Y→ b
 Derive “aab” from the above grammar.
 S → aY [by 3]
Y → aaY [by 3]
Y → aab [by 6]
Dr. Hussien M. Sharaf 9

10. CFG – 2.2
1] S → Λ 2] S → Y 3] Y→ aY
4] Y → bY 5] Y →a 6] Y→ b
 Derive “aa” from the above grammar.
 S → aY [by 3]
Y → aa [by 5]
Dr. Hussien M. Sharaf 10

11. Dr. Hussien M. Sharaf 11

12. CFG – 3
 Describe anything (a+b)* using CGF
1] S →aS 2] S → bS 3] S →Λ
 Derive “aab” from the above grammar.
 S → aS [by 1]
S → aaS [by 1]
S → aabS [by 2]
S → aabΛ [by 3] → aab
Dr. Hussien M. Sharaf 12

13. CFG – 4
Describe anything (a+b)*aa(a+b)* using CGF
1] S →XaaX 2] X → aX 3] X → bX
4] X →Λ
Note: rules 2, 3, 4 represents anything (a+b)*
 Derive “baabaab” from the above grammar.
 S → XaaX [by 1] X→ bXaaX [by 3]
X→ bΛaaX [by 4] X → baabX [by 3]
X → baabaX [by 2] X→ baabaaX [by 2]
X→ baabaabX [by 3] X→ baabaabΛ[by 4]
Dr. Hussien M. Sharaf 13

14. CFG – 5
Describe a language that has even number of a’s and even
number of b’s using CGF.
i.e. aababbab
1] S →SS 2] S →BALANCED S
3] S →S BALANCED 4] S →Λ
5] S →UNBALANCED S UNBALANCED
6] BALANCED → aa 7] BALANCED → bb
8] UNBALANCED → ab
9] UNBALANCED → ba
Dr. Hussien M. Sharaf 14

15. CFG – 5
 Derive “aababbab” from the above grammar.
 S → BALANCED S [by 1]
BALANCED→ aa UNBALANCED S UNBALANCED
[by 5]
UNBALANCED → aa ba S UNBALANCED [by 9]
UNBALANCED → aa ba S ab [by 8]
S → aa ba BALANCED S ab [by 3]
UNBALANCED → aa ba bb S ab [by 7]
S → aa ba bb ab [by 4]
→ aababbab
Dr. Hussien M. Sharaf 15

16. CFG – 6
 i.e. {Λ, ab, aaabbb,… }
 S → aSb|Λ
 Derive “aaaabbbb”
 S →aSb→aSb→aaSbb →aaaSbbb
→aaaaSbbbb →aaaabbbb
Dr. Hussien M. Sharaf 16

17. CFG – 7
 i.e. {Λ, ab, abbaabba,… }
 S → aSa| bSb| Λ
 Derive “abbaabba” “abba abba”
 S →aSa→abSba→abbSbba →abbaSabba→ abba abba
→ abbaabba
Dr. Hussien M. Sharaf 17

18. CFG – 8
 i.e. {Λ, abb, ababa,… }
 S → aSa| bSb| a| b| Λ
 Derive “ababa” “ab a ba”
 S →aSa→abSba→abSba →abSba→ ababa
 Note: an
ban
can be represented by S → aSa| b
 But an
ban
bn+1
can not be represented by CFG !
Dr. Hussien M. Sharaf 18

19.  i.e. {Λ, ab, abbaabba,… }
 S → aSa| bSb| Λ Derive abaaba
19
S
S
Λ
aa
S bb
S aa
Dr. Hussien M. Sharaf
CFG – 9

20.  Deduce CFG of addition and parse the
following expression 2+3+5
 1] S→S+S|N
 2] N→1|2|3|4|5|6|7|8|9|0
N1|N2|N3|N4|N5|N6|N7|N8|N9|N0
20
S
S+N
S
+
N
5
S
+
3
N
2
N
Can u make
another parsing
tree ?
Dr. Hussien M. Sharaf
CFG – 10

21. CFG – 11.1
 Deduce CFG of a addition/multiplication and
parse the following expression 2+3*5
 1] S→S+S|S*S|N
 2] N→1|2|3|4|5|6|7|8|9|0
N1|N2|N3|N4|N5|N6|N7|N8|N9|N0
21
S
S*S
S
*
N
5
S
+
3
N
2
N
Can u make
another parsing
tree ?
Dr. Hussien M. Sharaf

22. CFG -11.2 without ambiguity
 Deduce CFG of a addition/multiplication
and parse the following expression 2*3+5
1] S→ Term|Term + S
2] Term → N|N * Term
 3] N→1|2|3|4|5|6|7|8|9|0
N1|N2|N3|N4|N5|N6|N7|N8|N9|N0
22
S
S+N
S
+
N
5
S
*
3
N
2
N
Can u make
another parsing
tree ?
Dr. Hussien M. Sharaf

23. Thank you
Dr. Hussien M. Sharaf 23