EXERCISES |
|
|
|
|
|
|
149 |
|
TABLE 7.9. Monoid of the Machine That Recognizes 010 |
|
|
|
|||||
|
|
|
|
|
|
|
|
|
|
[ ] |
[0] |
[1] |
[00] |
[01] |
[10] |
[11] |
[010] |
|
|
|
|
|
|
|
|
|
[ ] |
[ ] |
[0] |
[1] |
[00] |
[01] |
[10] |
[11] |
[010] |
[0] |
[0] |
[00] |
[01] |
[00] |
[00] |
[010] |
[00] |
[00] |
[1] |
[1] |
[10] |
[11] |
[10] |
[10] |
[11] |
[11] |
[10] |
[00] |
[00] |
[00] |
[00] |
[00] |
[00] |
[00] |
[00] |
[00] |
[01] |
[01] |
[010] |
[00] |
[010] |
[010] |
[00] |
[00] |
[010] |
[10] |
[10] |
[10] |
[10] |
[10] |
[10] |
[10] |
[10] |
[10] |
[11] |
[11] |
[11] |
[11] |
[11] |
[11] |
[11] |
[11] |
[11] |
[010] |
[010] |
[010] |
[010] |
[010] |
[010] |
[010] |
[010] |
[010] |
|
|
|
|
|
|
|
|
|
node β in the tree. The tree must eventually stop growing because there are only a finite number of transition functions. Every input sequence has the same effect as one of the solid black nodes in Figure 7.8. These nodes provide a complete set of representatives for the monoid of the machine.
Therefore, the monoid of the machine that recognizes the sequence 010 contains only eight elements: [ ], [0], [1], [00], [01], [10], [11], and [010], out of a possible 256 transition functions between states. Its table is given in Table 7.9.
For further reading on the mathematical structure of finite-state machines and automata see Hopcroft et al. [18], Kolman [20], or Stone [22].
EXERCISES
Are the structures described in Exercises 7.1 to 7.13 semigroups or monoids or neither? Give the identity of each monoid.
7.1.(N, gcd).
7.2.(Z, [), where a[b = a.
7.3.(R, ), where x y = x2 + y2.
7.4.(R, ), where x y = 3 x3 + y3.
7.5.(Z3, −).
7.6.(R, | |), where | | is the absolute value.
7.7.(Z, max), where max (m, n) is the larger of m and n.
7.8.(Z, ), where x y = x + y + xy.
7.9.(S, gcd), where S = {1, 2, 3, 4, 5, 6}.
7.10.(X, max), where X is the set of real-valued functions on the unit interval [0,1] and if f, g X, then max (f, g) is the function on X defined by
max(f, g)(x) = max(f (x), g(x)).
7.11. (T , lcm) where T = {1, 2, 4, 5, 10, 20}.
150 |
7 MONOIDS AND MACHINES |
7.12.The set of all relations on a set X, where the composition of two relations R and S is the relation RS defined by xRSz if and only if for some y X, xRy and ySz.
7.13.({a, b, c}, ), where the table for is given in Table 7.10.
TABLE 7.10
|
a |
b |
c |
|
|
|
|
a |
a |
b |
c |
b |
b |
a |
a |
c |
c |
a |
a |
|
|
|
|
Write out the tables for the monoids and semigroups described in Exercises 7.14 to 7.17.
7.14.(S, gcd), where S = {1, 2, 3, 4, 6, 8, 12, 24}.
7.15.(T , gcd), where T = {1, 2, 3, 4}.
7.16.(XX , Ž ), where X = {1, 2, 3}.
7.17.({e, c, c2, c3, c4}, ·), where multiplication by c is indicated by an arrow in Figure 7.9.
e |
|
c |
|
c2 |
c3 |
|
|
|
|
|
|
|
|
c4
Figure 7.9
7.18.Find all the commutative monoids on the set S = {e, a, b} with identity e.
7.19.Are all the elements of the free semigroup generated by {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} simply the nonnegative integers written in the base 10?
7.20.A submonoid of a monoid (M, ·) is identity and such that x · y N , for all of the monoid given in Exercise 7.17.
a subset N of M containing the x, y N . Find all the submonoids
7.21.Prove that there is a monoid isomorphism between (FM({a}), ) and (N, +).
7.22.(Representation theorem for monoids) Prove that any monoid (M, ) is isomorphic to a submonoid of (MM , Ž ). This gives a representation of any monoid as a monoid of transformations.
7.23.Prove that any cyclic monoid is either isomorphic to (N, +) or is isomorphic to a monoid of the form shown in Figure 7.1, for some values of k and m.
7.24.(Morphism theorem for monoids) Let f : (M, ) → (N, ·) be a morphism of monoids. Let R be the relation on M defined by m1Rm2 if and only if
EXERCISES |
151 |
f (m1) = f (m2). Prove that the quotient monoid (M/R, ) is isomorphic to the submonoid (Imf, ·) of (N, ·). (See Exercise 7.20.)
7.25.An automorphism of a monoid M is an isomorphism from M to itself. Prove that the set of all automorphisms of a monoid M forms a group under composition.
7.26.A machine has three states, s1, s2, and s3 and two input symbols, α and β. The effect of the input symbols on the states is given by Table 7.11. Draw the state diagram and find the monoid of this machine.
TABLE 7.11
Initial |
Next State |
|
|
|
|
State |
h(α) |
h(β) |
|
|
|
s1 |
s1 |
s1 |
s2 |
s3 |
s1 |
s3 |
s2 |
s1 |
7.27. Prove that every finite monoid is the monoid of some finite-state machine.
For Exercises 7.28 to 7.30, draw state diagrams of machines with the given input set, I, that will recognize the given sequence.
7.28. |
1101, where I |
= {0, 1}. |
7.29. 0101, where I = {0, 1}. |
7.30. |
2131, where I |
= {1, 2, 3}. |
|
Which of the relations described in Exercises 7.31 to 7.34 are congruence relations on the monoid (N, +)? Find the quotient monoid when the relation is a congruence relation.
7.31. |
aRb if a − b is even. |
7.32. |
aRb if a > b. |
7.33. |
aRb if a = 2r b for some r Z. |
7.34. |
aRb if 10|(a − b). |
The machines in Tables 7.12, 7.13, and 7.14 have state set S = {s1 , s2 , s3 } and input set I = {0 , 1 }.
7.35. Draw the table of the monoid of the machine defined by Table 7.12.
TABLE 7.12
Initial |
Next State |
|
|
|
|
State |
h(0) |
h(1) |
|
|
|
s1 |
s2 |
s1 |
s2 |
s1 |
s2 |
s3 |
s3 |
s2 |
152 |
7 MONOIDS AND MACHINES |
7.36. Draw the table of the monoid of the machine defined by Table 7.13.
TABLE 7.13
Initial |
Next State |
|
|
|
|
State |
h(0) |
h(1) |
|
|
|
s1 |
s2 |
s1 |
s2 |
s3 |
s1 |
s3 |
s3 |
s2 |
7.37.Find the number of elements in the monoid of the machine defined by Table 7.14.
TABLE 7.14
Initial |
Next State |
|
|
|
|
State |
h(0) |
h(1) |
|
|
|
s1 |
s2 |
s1 |
s2 |
s3 |
s3 |
s3 |
s1 |
s1 |
7.38.Find the number of elements in the semigroup of the machine, given by Figure 7.3, that controls the elevator.
7.39.Find the monoid of the machine in Figure 7.10.
a, b |
b |
s1 |
a s2 |
g |
g |
|
g |
s3
a, b
Figure 7.10
7.40.A serial adder, illustrated in Figure 7.11, is a machine that adds two numbers in binary form. The two numbers are fed in together, one digit at a time, starting from the right end. Their sum appears as the output. The machine has input symbols 00, 01, 10, and 11, corresponding to the rightmost digits of the numbers. Figure 7.12 gives the state diagram of such a machine, where the symbol “sij /j ” indicates that the machine is in state sij and emits an output j . The carry digit is the number i of the state sij . Find the monoid of this machine.
EXERCISES |
153 |
Serial adder
Figure 7.11
00 |
|
|
00 |
10, 01 |
s00 |
0 |
10, 01 |
s01 |
1 |
|
11 |
11 |
|
|
|
|
|
||
10, 01 |
|
00 |
00 |
11 |
|
10, 01 s11 |
|||
s10 |
0 |
|
1 |
|
|
|
11 |
|
|
Figure 7.12. State diagram of the serial adder.
The circuits in Exercises 7.41 to 7.44 represent the internal structures of some finite-state machines constructed from transistor circuits. These circuits are controlled by a clock, and the rectangular boxes denote delays of one time unit. The input symbols are 0 and 1 and are fed in at unit time intervals. The internal states of the machines are described by the contents of the delays. Draw the state diagram and find the elements in the semigroup of each machine.
7.41.
AND |
Delay |
|
y |
||
|
7.42. |
|
|
NAND |
Delay |
|
y1 |
||
|
||
OR |
Delay |
|
y2 |
||
|
7.43.
|
|
Delay |
|
|
AND |
|
Delay |
|
|
|
|
|
|
|
|
||||
|
|
y1 |
|
|
|
y2 |
|||
|
|
|
|
|
|
||||
|
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
7.44. |
Delay |
Delay |
|
NOR |
|||
y1 |
y2 |
154 |
7 MONOIDS AND MACHINES |
7.45.In the spring, a plant bud has to have the right conditions in order to develop. One particular bud has to have a rainy day followed by two warm days, without being interrupted by cool or freezing days, in order to develop. Furthermore, if a freezing day occurs after the bud has developed, the bud dies. Draw a state diagram for such a bud using the input symbols R, W , C, F to stand for rainy, warm, cool, and freezing days, respectively. What is the number of elements in the resulting monoid of this bud?
7.46.A dog can either be passive, angry, frightened, or angry and frightened, in which case he bites. If you give him a bone, he becomes passive. If you remove one of his bones, he becomes angry, and, if he is already frightened, he will bite you. If you threaten him, he becomes frightened, but, if he is already angry, he will bite. Write out the table of the monoid of the dog.