Download CS3012: Formal Languages and Compilers

CS3518
Exercises
CS3518
Finite State Automata
1.
Questions about FSAs that accept a language
Describe in words the languages accepted by each of the following FSAs:
1
0
1
1
0
1
0
(i )
Answer (i):
strings of 0's and 1's with an even number of 0's
Answer (ii):
strings of 0's and 1's, starting with a 10, and no pairs of 0's
Answer (iii):
strings of 0's and 1's with either a pair of 0's or a pair of 1's
1
(i i)
CS3518
2.
Exercises
Give a formal definition for the FSAs which define the following languages
(i.e. specify Q, I, F, T, E):
Answer: (I show the diagram, but you can easily convert to QIFTE)
a
a
a,b
b
a,b
b

a
a

b
a
b
a
(ii)
3.
b
b
(iii)
Give a formal definition for a FSA accepting the infinite set of strings representing numbers divisible
by 2. (i.e. 0,2,4,6,8,10,12,….)
Answer:
Just check the last digit, make sure it’s even (need two states)
Let E=0,2,4,6,8
Let O=1,3,5,7,9
Then QIFTE=({1,2},{1},{2},{O,E},{(1,O,1),(1,E,2),(2,O,1),(2,E,2)})
4.
Give a formal definition for a FSA accepting the infinite set of strings representing numbers divisible
by 3. (this one is tricky, try (on paper) doing a division of 3 into some large number x, and see
if the process you are using to deal with each new digit of x could be automated by a machine.)
Answer:
(rough definition)
3 states 0,1,2
from 0 to 0: label 0,3,6,9
from 0 to 1: label 1,4,7
from 0 to 2: label 2,5,8
from 1 to 0: label 2,5,8
from 1 to 1: label 0,3,6,9
from 1 to 2: label 1,4,7
from 2 to 0: label 1,4,7
from 2 to 1: label 2,5,8
from 2 to 2: label 0,3,6,9
CS3518
5.
Exercises
Give a formal definition for a FSA accepting the set of strings over {a, b} which contain an even
number of a's and an even number of b's. (this one is tricky, try doing it manually for some random
string of a’s and b’s. What do you need to remember as you’re going through the string? How many
states will the FSA need, to remember this?)
Answer:
6.
Specification of a suitable Mealy Machine (a diagram can capture the same information):
Q = {q0,q1,q2,q3}
I = q0
T = {a,b}
E = {(q0,a,q1), (q0,b,q0), (q1,a,q2), (q1,b,q0), (q2,a,q2), (q2,b,q3), (q3,a,q1), (q3,b,q0)}
Gamma = {1} (or any alphabet containing 1 as one of its symbols)
O = The function that assigns the empty string to every combination of a state and an input symbol, with
one exception, namely: (q2,b,1).
Some things to note:
-- q2 is the state in which the automaton has just read aa
-- In q2, when reading a, loop back onto q2 (think of an input such as aaaab)
-- In q3, when reading a, go back to q1 (the state that remembers that one a has been read)
7. Specification of a suitable Moore Machine (a diagram can capture the same information):
Q={q0,q1,q2,q3}
I=q0
T={0,1}
E={(q0,0,q1),(q0,1,q2),(q1,0,q3),(q1,1,q1),(q2,0,q1),(q2,1,q2),(q3,0,q3),(q3,1,q1)}
Gamma={0,1}
O={(q0,lambda),(q1,1),(q2,0),(q3,0),}
8. Yes, this does make sense.
Moore Machine: Q,I,T,E as for NDFSAs (hence possibly several initial states in I; E may be a relation
that’s not functional in its first two arguments). One might even allow O to be a relation that’s not
functional.
Mealy Machine: Q,I,T,E as for NDFSAs. One might even allow O to be a relation that’s not functional in
its first two arguments.
9. Moore and Mealy Machines can handle the empty input in slightly different ways:
Moore Machine: In most examples, the initial state q0 will have empty output, but suppose we have
(q0,b), where b is a symbol of T. This will cause the machine to output b when confronted with the empty
string.
Mealy Machine: Since output in Mealy Machines is associated with an input symbol, and Mealy
Machines do not allow the empty string as an input (check the definition!) they are unable to output b.
CS3518
Exercises
Notice how the notion of equivalence (between Moore Machines and Mealy Machines) in the slides
discounts this difference between the two kinds of machines.
10. Have a browse. One standard example is a vending machine, where a given sequence of coins is
associated with a product coming out of the machine (and maybe change as well).
11. (Optional.) Have a browse. Among the web pages that you may find instructive are
http://en.wikipedia.org/wiki/Hidden_Markov_model
http://mlg.eng.cam.ac.uk/zoubin/papers/ijprai.pdf
Once again, this is optional..