1. Finite-state Recognizers

2. Deterministic Recognizers
Treat FSM as a recognizer that classifies input strings into two classes:
strings it accepts and strings it rejects
Finite-state recognizer:
Equivalent to a string of input symbols that enter the machine at successive times
Finite-state control: Moore FSM
States in which output symbol is 1 (0): accepting (rejecting) states
A string is accepted by an FSM: if and only if the state the FSM enters after having read the rightmost symbol is an accepting state
Set of strings recognized by an FSM: all input strings that take the FSM from its starting state to an accepting state

3. Transition Graph
Example: a machine that accepts a string if and only if the string begins and ends with a 1, and every 0 in the string is preceded and followed by at least a single 1
Transition graph: consists of a set of vertices and various directed arcs connecting them
At least one of the vertices is specified as a starting vertex
Arcs are labeled with symbols from the input alphabet
A vertex may have one or more Ii-successors or none
It accepts a string if the string is described by at least one path emanating from a starting vertex and terminating at an accepting vertex
It may be deterministic or non-deterministic

4. Example
Example: 1110 and accepted by the transition graph below, but 100 rejected
Equivalent transition graphs: two or more graphs that recognize the same set of strings
Each graph below accepts a string: if and only if each 1 is preceded by at least two 0’s

5. Graphs Containing -Transitions
-transitions: when no input symbol is used to make the transition
Example: Graph that recognizes a set of strings that start with an even number of 1’s, followed by an even number of 0’s, and end with substring 101

6. Converting Nondeterministic into Deterministic Graphs
Example: Transition graph and its transition table
Successor table and deterministic graph:

7. Theorem
Theorem: Let S be a set of strings that can be recognized by a nondeterministic transition graph Gn. Then S can also be recognized by an equivalent deterministic graph Gd. Moreover, if Gn has p vertices, Gd will have at most 2p vertices

8. Regular Expressions
Example: Sets of strings and the corresponding expression
Graph (a) recognizes set {101}: expression denoted as 101
Graph (b) recognizes set {01,10}: expression =
Graph (c) recognizes {0111,1011}: expression =
Concatenation of and 11
Graph (d) recognizes set { ,1,11,111,1111,…}: expression = 1*

9. Regular Expressions (Contd.)
Example: 01(01)* = …
R* = + R + R2 + R3 + …
Example: Set of strings on {0,1} beginning with a 0 and followed only by 1’s: 01*
Example: Set of strings on {0,1} containing exactly two 1’s: 0*10*10*
Example: Set of all strings on {0,1}:
(0+1)* = …
Example: Set of strings on {0,1} that begin with substring 11: 11(0+1)*
Example: Transition graphs and the sets of strings they recognize

10. Definition and Basic Properties
Let A = {a1,a2,…,ap} be a finite alphabet: then the class of regular expressions over alphabet A is defined recursively as follows:
Any symbol, a1, a2, …, ap alone is a regular expression: as are null string and empty set
If P and Q are regular expressions: then so is their concatenation PQ and their union P+Q
If P is a regular expression: then so is its closure P*
No other expressions are regular: unless they can be generated in a finite number of applications of the above rules
Recognizers for and :

11. Identities + R = R R = R = R = R = R * =
Set of strings that can be described by a regular expression: regular set
Not every set of strings is regular
Set over {0,1}, which consists of k 0’s (for all k), followed by a 1, followed in turn by k 0’s, is not regular: … + 0k10k + …
Requires an infinite number of applications of the union operation
However, certain infinite sums are regular
Set consisting of alternating 0’s and 1’s, starting and ending with a 1: 1(01)*

12. Manipulating Regular Expressions
A regular set may be described by more than one regular expression
Such expressions are called equivalent
Example: Alternating 0’s and 1’s, starting and ending with 1
1(01)* or (10)*1
Let P, Q, and R be regular expressions: then
R + R = R
PQ + PR = P(Q+R); PQ + RQ = (P + R)Q
R*R* = R*
RR* = R*R
(R*)* = R*
+ RR* = R*
(PQ)*P = P(QP)*
(P + Q)* = (P*Q*)* = (P* + Q*)* = P*(QP*)* = (P*Q)*P*
+ (P + Q)*Q = (P*Q)*

13. Examples
Example: Prove that the set of strings in which every 0 is immediately followed by at least two 1’s can be described by both R1 and R2, where
R1 = + 1*(011)*(1*(011)*)*
R2 = ( )*
Proof: R1 = + 1*(011)*(1*(011)*)*
= (1*(011)*)*
= ( )* = R2
Example: Prove the identity
(1 + 00*1) + (1 + 00*1)(0 +10*1)*(0 + 10*1) = 0*1(0 + 10*1)*
Proof: LHS = (1 + 00*1)[ + (0 + 10*1)*(0 + 10*1)]
= [( + 00*)1][ + (0 + 10*1)*(0 + 10*1)]
= 0*1(0 + 10*1)*

14. Transition Graphs Recognizing Regular Sets
Theorem: Every regular expression R can be recognized by a transition graph
Proof:

15. Example
Example: Construct a transition graph recognizing R = (0 + 1(01)*)*

16. Example (Contd.)
Example: Prove that (P + Q)* = P*(QP*)*

17. Informal Techniques
Example: Construct a graph that recognizes P = (01 + (11 + 0)1*0)*11
Graph for Q = (11 + 0)1*0
Graph for P

18. Example
Example: Construct a graph that recognizes R = (1(00)*1 + 01*0)*

19. Regular Sets Corresponding to Transition Graphs
The set of strings that can be recognized by a transition graph (hence, an FSM) is a regular set
Theorem: Let Q, P, and R be regular expressions on a finite alphabet. Then, if P does not contain :
Equation R = Q + RP has a unique solution given by R = QP*
Equation R = Q + PR has a unique solution given by R = P*Q

20. Systems of Equations
Example: Derive the set of strings derived by the following transition graph
A = + A0 + B (1)
B = A0 + B1 + C0 (2)
C = B (3)
Substituting (3) into (2):
B = A0 + B1 + B00 = A0 + B(1 + 00) (4)
From the theorem:
B = A0(1 + 00)* (5)
Substituting (5) into (1):
A = + A0 + A0(1 + 00)*1 = + A(0 + 0(1 + 00)*1) (6)
A = (0 + 0(1 + 00)*1)* = (0 + 0(1 + 00)*1)* (7)
Hence, solution C from (7), (5) and (3):
C = (0 + 0(1 + 00)*1)*0(1 + 00)*0

21. Theorem
Theorem: The set of strings that take an FSM M from an arbitrary state Si to another state Sj is a regular set
Combining the two theorems:
An FSM recognizes a set of strings if and only if it is a regular set
Applications: the correspondence between regular sets and FSMs enables us to determine whether certain sets are regular
Example: Let R denote a regular set on alphabet A that can be recognized by machine M1
Complement R’: set containing all the strings on A that are not contained in R
R’ describes a regular set: since it can be recognized by a machine M2, which is obtained from M1 by complementing the output values associated with the states of M1

22. Examples Example: Let P&Q represent the intersection of sets P and Q
Prove P&Q is regular
Since P’ and Q’ are regular:
P’ + Q’ is regular
Hence, (P’ + Q’)’ is regular
Since P&Q = (P’ + Q’)’: P&Q is regular
Regular expressions containing complementation, intersection, union, concatenation, closure: extended regular expressions
Example: Consider the set of strings on {0,1} s.t. no string in the set contains three consecutive 0’s
Set can be described by: [(0 + 1)*000(0 + 1)*]’
More complicated expression if complementation not used:
( )*( )

23. Example
Example: Let M be an FSM whose input/output alphabet is {0,1}. Assume the machine has a designated starting state. Let z1z2…zn denote the output sequence produced by M in response to input sequence x1x2…xn. Define a set SM, which consists of all the strings w s.t.
w = z1x1z2x2…znxn for any x1x2…xn in (0 + 1)*. Prove that SM is regular.
Given the state diagram of M: replace each directed arc with two directed arcs and a new state, as shown in the figure
Retain the original starting state: designate all the original states as accepting states
The resulting nondeterministic graph recognizes SM: thus SM is regular

24. Example (Contd.)
Example (contd.): Derive SN for machine N shown below

25. Two-way Recognizers
Two-way recognizer (or two-way machine): consists of a finite-state control coupled through a head to a tape
Initially: the finite-state control is in its designated starting state, with its head scanning the leftmost square of the tape
The machine then proceeds to read the symbols of the tape: one at a time
In each cycle of computation: the machine examines the symbol currently scanned by the head, shifts the head one square to the right or left, and then enters a new (not necessarily distinct) state
If the machine eventually moves off the tape on the right end entering an accepting state: the tape is accepted by the machine
A machine can reject a tape: either by moving off its right end while entering a rejecting state or by looping within the tape
Null string can be represented either by: the absence of an input tape or by a completely blank tape
A machine accepts if and only if: its starting state is an accepting state

26. Example
Example: A two-way machine recognizing set 100*

27. Convenience of Using Two-way Machines
Two-way machines are as powerful as one-way machines w.r.t the class of tapes they can recognize
However, for some computations: it is convenient to use two-way machines since they may require fewer states
Example: Consider the two-way machine shown in the table, which accepts a tape if and only if it contains at least three 1’s and at least two 0’s
The minimal one-way machine that is equivalent to the two-way machine has 12 states: since it must examine the tapes for the appropriate number of 0’s and 1’s simultaneously