1. Lecture 8 NFA Subset Construction & Epsilon Transitions
CSCE 355 Foundations of Computation
Lecture 8 NFA Subset Construction & Epsilon Transitions
Topics:
Regular expressions
Thompson Construction
Examples Thompson Construction
June 10, 2015

2. TEST 1 – Post Mortem – Problems due Thursday
Last Time: Readings 2.3
Mutual Induction Proof revisited
Languages denoted by regular expressions
Examples
Ruby Regular Expressions
TEST 1 – Post Mortem – Problems due Thursday
#6 Prove the number of nodes in an m-ary tree is at most
1+m+m^2 … m^h and leave as a sum.
New: Readings section
Author’s Website Solutions Online
ε-NFA  NFA
ε-NFA  DFA subset construction
Regular Expressions
Relations Machines and Regular expressions

3. Figure 2.20 - ε-NFA for fixed decimals
Given the transition table
Draw the transition Diagram
Compute the Eclosure for each state ε
+,- .
0,1,2,…9 q0
{q1} Ф q1
{q2}
{q1,q4} q2 q3
{q5}
{q3} q4 q5

5. Delta-hat ( δ ) with ε-transitions

6. Eliminating ε-transitions

9. Regular Expressions DFAs recognize languages NFAs recognize languages
Regular expressions denote languages – so that we can write the description of a language as the combination of less complex languages
Examples:
r = a*b* L(r)={w in {a,b}*| all a’s come before any b}
r= (0+1)*000(0+1)* L(r) = {w | w contains 000}

10. Grep
Unix utility
man grep
man –k regexp

11. Recursive Definition of Reg Expr
Definition of regular expressions over an alphabet Σ
Base cases:
if a ε Σ then A is a regular expression and denotes L(a) = { a }
ε is a regular expression and denotes L(ε) = { ε }
A variable, usually a capital such as L, and of course the L(L) = L
Recursive definition
If r and s are regular expressions denoting the languages L(r) and L(s) then
rs is a regular expression denoting L(rs) = L(r)L(s)
r+s is a regular expression denoting L(r+s) = L(r) L(s)
r* is a regular expression denoting [L(r*)] = [L(r)] *
(r) is a regular expression denoting L( (r) ) = L(r)

12. Thompson Construction
Based on recursive (inductive) definition of regular expressions
We describe NFAs (with epsilon moves) that recognize the base cases.
Then assuming we have NFAs for smaller expressions r and s we construct NFAs for
r + s rs r*

13. Recursive Definition of Reg Expr
Definition of regular expressions over an alphabet Σ
Base cases:
if a ε Σ then A is a regular expression and denotes L(a) = { a }
ε is a regular expression and denotes L(ε) = { ε }
A variable, usually a capital such as L, and of course the L(L) = L
Recursive definition
If r and s are regular expressions denoting the languages L(r) and L(s) then
rs is a regular expression denoting L(r)L(s)
r+s is a regular expression denoting L(r) L(s)
r* is a regular expression denoting [L(r)]* =

14. Thompson Construction Base cases

15. Thompson Construction Recursive cases

16. Thompson Construction Recursive cases

17. Thompson Construction Examples

20. 2008 Sample test 1 outline Proof Techniques Inductive proof
mutual induction proof
Given DFA
Transition diagram
Input “abaa”
L(M)
Give DFA for L
NFA
NFA for L
NFA  DFA (Subset)
εNFA
εNFA for L
ε-closure (ECLOSE in text)
εNFA  DFA (Subset)
Ruby regular expressions

21. Design DFA that accepts Language L
Example
For a DFA D how do you prove L(D) = L ?

22. Mutual Induction Proof
Define three statements for a mutual induction proof that could help in proving that L(M) = L ={x ε {0, 1}* | that x has a number of zeroes divisible by 3 }

25. HW solutions 2.5.1 State\input ε a b c p Φ { p } { q } { r } q *r
ε-closure{p} = {p}

26. 2.5.3 a,b
The set of strings consisting of zero or more a’s followed by zero or more b’s followed by zero or more c’s
The set of strings consisting of either 01 repeated one or more times or 010 repeated one of more times

27. 2.3.2 Subset NFA without ε NFA DFA 1 { p } { q, s } { q } 1 p
State\input 1
{ p }
{ q, s }
{ q }
State\input 1 p
{ q, s }
{ q } *q
{ r }
{q, r} r
{ s }
{ p } *s ϕ

28. 2.3.4 Done before
The set of strings over {0,1, … 9} such that the final digit has appeared before
The set of strings over {0,1, … 9} such that the final digit has not appeared before
The set of strings of 0’s and 1’s such that there are two 0’s separated by a number of positions that is a multiple of 4. Note 0 is allowable multiple.

29. Tenth symbol from the right is a ‘1’

32. Pop Quiz Induction proof

35. Homework:
2.5.1
2.5.3 a,b
Regular expressions