1. Language Recognition
MSU CSE 260

2. Outline Introduction Regular Expressions & Regular Sets
Kleene’s Theorem
Regular Sets and Regular Grammars
Examples
More powerful Types of Machines
Exercise 10.4

3. Introduction
The sets (languages) recognized by FAs are those formed from:
null (empty) set ,
empty string (), and
singleton strings
by taking (in arbitrary order):
concatenations,
unions,
Kleene Closures.
Regular languages

4. Regular Expressions
Definition. The regular expressions over a set I are defined recursively by:
the symbol  is a regular expression;
the symbol  is a regular expression;
the symbol x is a regular expression whenever x  I.
the symbols (AB), (AB), and A* are regular expressions whenever A and B are regular expressions.
Examples. The following are regular expressions over I={0,1}: 10*, (10)*, 001, 0(01)*, (0*1)*
Note. AB is sometimes represented as A+B.
Precedence rules: Kleene closure(*), concatenation, union

5. Regular Sets Each regular expression represents a (regular) set.
Reg. Exp. Set represented
 the empty set 
 the set {}
x the set {x}
(AB) the concatenation of the sets represented by A and by B
(AB) the union of the sets represented by A and by B
A* the Kleene closure of the set represented by A
Examples. 10*: strings consisting of a 1 followed by any number of 0s
(10)*: any arbitrary combinations of string 10 (including )
0  01: the set consisting of the two strings 0 and 01
0(0  1)*: the set of strings which begin with 0

6. Kleene’s Theorem
Kleene’s Theorem. A set is regular if and only if it is recognized by a finite-state automaton.
Proof. The proof is done by construction of NFAs for sets represented by regular expressions.

7. NFAs for Basic Regular Sets
recognizes 
Start s0
recognizes {}
Start s0
The NFA is easier to build and understand than the equivalent FA. a
recognizes {a}
Start s0 s1

8. NFA for Concatenation
Let A be recognized by MA=(SA, I, fA, sA, FA), and
B recognized by MB=(SB, I, fB, sB, FB)
MAB f MA MB
Start f i sA sB i
The NFA is easier to build and understand than the equivalent FA.
A transition to final state in MA produces a transition to sB.
A transition from sB in MB produces a transition from sAB=sA if sA is a final state.
Start state of MAB is sAB=sA, which is final if both sA and sB are final.
Final states of MAB include all final states of MB.

9. NFA for Union Let A be recognized by MA=(SA, I, fA, sA, FA), and
B recognized by MB=(SB, I, fB, sB, FB) i MA i sA
MAB
Start
sAB MB
The NFA is easier to build and understand than the equivalent FA. i sB i
Start state of MAB is sAB, which is final if sA or sB is final.
Final states of MAB are the final states of MA and MB.

10. NFA for Kleene Closure Let A be recognized by MA=(SA, I, fA, sA, FA).
Start i
MA*
sA* sA i
The NFA is easier to build and understand than the equivalent FA.
Transitions from sA produce transitions from sA* and all final states of MA.
Start state of MA* is sA*, which is a final state.
Final states of MA* include all final states of MA.

11. NFA for a Set Generated by a Regular Grammar
Theorem. A set generated by a regular grammar is regular.
Let G = (V, T, S, P) be a regular grammar.
Then the corresponding NFA is M = (S, I, f, s0, F):
S contains a state sA for each nonterminal symbol A of G, and an additional (final) state sF.
Start state is s0 = sS.
Transitions are formed from the productions:
A  aB produces a transition from sA to sB on input a
A  a produces a transition from sA to sF on input a
F also includes s0 if P includes production S  .

12. Example
Let G=(V,T,S,P) where V={0,1,A,S}, T={0,1}, and the productions in P are:
S1A, S0, S, A0A, A1A, A1
The corresponding NFA is M=(S, I, f, s0, F):
S={sS, sA, sF}, s0=sS, F={sF}, I={0,1}, and state diagram
0,1 1 sA
Start sS 1 sF

13. Regular Grammar Generating a Regular Set
Theorem. If a set is regular, then there is a regular grammar that generates it.
Let M = (S, I, f, s0, F) be the NFA recognizing this set.
Then the regular grammar generating it is G = (V, T, S, P):
Assign a nonterminal symbol As to each state s of S;
Assign a terminal symbol to each input symbol in I;
Start symbol S corresponds to start state s0.
Productions are formed from the transitions:
A transition from s to t on input a produces As  aAt
A transition from s to a final state on input a produces As  a
P also includes production S   if L(M).

14. Example Let NFA M=(S, I, f, s0, F), where:
S={s0, s1, s2}, I={0,1}, F={s0, s2}, and state diagram: s1
Start s0 1 1 1 s2
The corresponding regular grammar G=(V,T,S,P)
V={S=A0, A1, A2, 0, 1}, T={0, 1}, and productions in P:
S0A1, S1A2, A10A1, A11A2, A20A1,
A21A2, A11, A21, S1, S

15. More powerful Types of Machines
The set {0n1n | n = 0, 1, 2, …} is not regular; has no FA.
The limitation of FAs is their finite memory.
More powerful models of computation:
Pushdown automata
Include all FA’s features, plus a stack for unlimited memory.
Recognize languages generated by context-free grammars.
Example: {0n1n | n=0, 1, 2, …}
Linear bounded automata
Context-sensitive languages. Example: {0n1n2n | n = 0, 1, 2,…}
Turing machines
Include FA’s features and a tape (infinite in both directions) for read and write. ►All languages generated by grammars.

16. Exercise 10.4