## Finding the Minimal FSM

 A minimal FSM, M, is one that uses the fewest possible states to recognize a language L(M). The intuitive idea is that the elimination from an FSM of all states which are equivalent will result in the minimal FSM.      Roughly speaking, two states are equivalent if they both take you to the same place. The algorithm for finding the minimal FSM involves partitioning the states of the given FSM (recall that a partition of a set is one where each member of the set is in some subset of the set and the subsets are disjoint) into states that satisfy this notion of equivalence.
 The algorithm is illustrated by using it to find the minimal FSM for the FSM shown in the figure in the above right. The main purpose in providing an illustration of this algorithm is to demonstrate the sense in which global information about an FSM is required to construct a minimal FSM.

The table to the right illustrates the steps involved in the algorithm. The first column simply numbers the states (or partitions) of the new machine as they are determined. The second column lists the partition at a particular level in the the partitioning (a partitioning is a set of partitions where each partition is a refinement of the parent partition). The third column displays the transition matrix that provides the basis for determining the next level of partitioning that may be required for the partition shown in the corresponding row of column 2.

The algorithm begins with an initial partition of the states that places all final (or accepting) states in one subset and all non final (or non accepting) states in another subset. These two subsets are shown in column 2 of rows 1 and 2. The rightmost column provides the transition table for each of these subsets. This table is constructed for each state in the partition. These are listed in column 1 of the Transition Tables. The cell entries in columns 2 through 4 of the table are constructed as follows. For the row state and each column input element, determine the partition of which its target state is an element. For example, consider the Transition Table that corresponds to partition 02. q0 is the first state listed. On input element a, the transition is to state q1 which is an element of partition 02. Therefore, 02 is entered in this cell.

After this is done for each state in the partition The Transition Table has been completed. An examination of the Transition Table corresponding to partition 02 immediately reveals which states are in the same equivalence class at this level of the partitioning. Two states are in the same equivalence class if their cell entries are identical. In this and subsequent transition tables differing shades of gray are used to visually group states which are equivalent. The three different shades of gray in the transition table for 02 accent the three equivalence classes that are formed from partition 02. These three partitions are shown in the following rows and labeled 1021, 1022, 1023.

#### Trace of the Algorithm for Identifying the Minimal FSM

States

Partition
Transition Table
for Partition
01 = {qhalt}
State  a  b  #
qhalt
02 = {q0,q1,q2,q3,q4,q5,q6,q7,q8,q9}
State  a  b  #
q0  02
q1  02  02
q2      01
q3  02  02
q4    02
q5      01
q6    02
q7    02
q8    02
q9      01
2 1021 = {q0}
1022 = {q1,q3}
State  a  b  #
q1  1022  1023
q3  1024  1024
3  1023 = {q2,q5,q9}
1024 ={q4,q6,q7,q8}
State  a  b  #
q4    1023
q6    1024
q7    1024
q8    1023
4  210221 = {q1}
5  210222 = {q3}
6 210241 = {q4,q8}
210242 = {q6,q7}
State  a  b  #
q6    210242
q7    210241
7 32102421 = {q6}
8 32102422 = {q7}
 The first number in the label indicates the level of partitioning, the middle numbers in small print refer to the previous partition that is being partitioned, and the last number refers to the number of the new partition within the partitioning. Thus, for label 1022, 1 is the next level of partitioning after 0, 02 is the parent partition, and 2 is the number used to refer to the new partition.      A partition label is shown in red if it is a terminal partition; that is, if it can not be partitioned further. The partitioning continues until no more partitioning is possible. In this case, 8 equivalence classes are discovered implying that the minimum number of states required for an FSM to recognize the language: {ab, aabb,aaabbb} is eight.
 The partitioning that results is also shown as a tree in the figure abovet. Again, the terminal partitions are shown in red and of course appear as the bottom nodes of the tree.

 The resulting minimal machine is shown to the right. The stare names used in this depiction of the minimal FSM do not correspond to the names used in the original machine. The correspondence is between state names and partitions. In this diagram partition 1021 corresponds to q0, 210221 to q1, 1023 to q2, 210222 to q3, 210241 to q4, 3210241 to q5,3210242 to q6,and 01 to qhalt.      Note, that the algorithm that identified this machine examined the entire set of rules; and the algorithm required that the learning be completed. No simple extension of the "Learning FSMs" considered earlier could identify this machine.
 This minimal machine seems to capture the regularity in the input strings better than the machines learned by the "Learning FSMs". To illustrate this, consider the sets of input strings AB = {ab,aabb,aaabbb}, the set that we have examined, as well as the sets A = {aa, aaaa,aaaaaa} and B = {ab,aacc,aaaddd}. The Basic "Learning FSM" (B_LFSM) required 14 control states to recognize AB. It would also require 14 control states to recognize set A and 14 control states to recognize set B. Further, the graph of each of these machines is identical for each of these input sets.      The Extended "Learning FSM" (E_LFSM) required 11 control states to recognize AB. It would also require 11 control states to recognize set B but only 8 control states to recognize set A .The graphs of the machines for AB and B are identical. The minimal machine for AB required 8 control states. A minimal machine for B could also require 8 control states but the graph for these minimal machines would be quite different. And, the minimal machine for set B would require 11 control states; the same as required by the FSM learned by the E_LFSM.  Although each of these types of machines: the FSM learned by the B_LFSM, the FSM learned by the E_LFSM, and the corresponding minimal FSM, can each recognize the same input sets; the way the set is represented in the structure of the machine can be quite different.      What should we require of an "intelligent" learner? Simply to recognize the input strings or to in some sense represent the regularities that might be present in the input set? And, what should we expect of the human learner? Do we simply recognize the input strings or do we typically represent the regularities that we find in the input set? And finally, in what sense does even a minimal FSM represent the regularity? Is it enough to implicitly capture regularities in the form of the machine or should these regularities be represented in a more explicit sense? These are difficult questions to which we will return when we take up human and machine learning.      Finally, what exactly is the regularity that we see in the set, {ab,aabb,aaabbb}? Is a string of a's followed by an equal number of b's? Or, is it the base string ab with 0, 1, or 2 ab strings recursively embedded within the base string? If you recall the particular Turing Machine that was constructed to recognize the strings a**n b**n, you might have noted that this latter regularity seems to be the one implicit in that machine.

#### An 'FSM' that "Learns"

© Charles F. Schmidt