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Earlier we talked about about finding equivalent
FSMs with fewer states.

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Now we'll develop an approach for finding
such FSMs by looking for two states that that

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can be merged into a single state without
changing the behavior of the FSM in any externally

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distinguishable manner.

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Two states are equivalent if they meet the
following two criteria.

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First, the states must have identical outputs.

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This makes sense: the outputs are visible
to the outside, so if their values differed

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between the two states, that difference would
clearly be externally distinguishable!

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Second, for each combination of input values,
the two states transition to equivalent states.

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Our strategy for deriving an equivalent machine
with fewer states will be to start with our

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original FSM, find pairs of equivalent states
and merge those states.

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We'll keep repeating the process until we
can't find any more equivalent states.

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Let's try this on our ant FSM.

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First we need to find a pair of states that
have the same outputs.

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As it turns out, there's only one such pair:
WALL1 and CORNER, both of which assert the

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turn-right and forward outputs.

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Okay, so let's assume that WALL1 and CORNER
are equivalent and ask if they transition

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to equivalent states for each applicable combination
of input values.

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For these two states, all the transitions
depend only on the value of the R input, so

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we just have to check two cases.

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If R is 0, both states transition to CORNER.

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If R is 1, both states transition to WALL2.

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So both equivalence criteria are satisfied
and we can conclude that the WALL1 and CORNER

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states are equivalent and can be merged.

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This gives us the four-state FSM shown here,
where we've called the single merged state

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WALL1.

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This smaller, equivalent FSM behaves exactly
as the previous 5-state FSM.

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The implementation of the 5-state machine
requires 3 state bits; the implementation

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of the 4-state machine only requires 2 state
bits.

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Reducing the number of state bits by 1 is
huge since it reduces the size of the required

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ROM by half!

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Just as we were able to achieve considerable
hardware savings by minimizing Boolean equations,

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we can do the same in sequential logic by
merging equivalent states.

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Roboant customers are looking forward to the
price cut!

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Let's look at what we'd need to do if we wanted
to implement the FSM using logic gates instead

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a ROM for the combinational logic.

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First we have to build the truth table, entering
all the transitions in the state transition

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diagram.

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We'll start with the LOST state.

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So if the FSM is in this state, the F output
should be 1.

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If both antenna inputs are 0, the next state
is also LOST.

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Assigning the LOST state the encoding 00,
we've captured this information in the first

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row of the table.

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If either antenna is touching, the FSM should
transition from LOST to the rotate-counterclockwise

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state.

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We've given this an encoding of 01.

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There are three combinations of L and R values
that match this transition, so we've added

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three rows to the truth table.

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This takes care of all the transitions from
the LOST state.

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Now we can tackle the transitions from the
rotate-counterclockwise state.

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If either antenna is touching, the next state
is again rotate-counterclockwise.

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So we've identified the matching values for
the inputs and added the appropriate three

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rows to the transition table.

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We can continue in a similar manner to encode
the transitions one-by-one.

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Here's the final table, where we've used don't
cares to reduce the number of rows for presentation.

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Next we want to come up with Boolean equations
for each of the outputs of the combinational

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logic, i.e., the two next-state bits and the
three motion-control outputs.

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Here are the Karnaugh maps for the two next-state
bits.

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Using our K-map skills from Chapter 4, we'll
find a cover of the prime implicants for S1-prime

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and write down the corresponding product terms
in a minimal sum-of-products equation.

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And then do the same for the other next-state
bit.

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We can follow a similar process to derive
minimal sum-of-products expressions for the

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motion-control outputs.

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Implementing each sum-of-products in a straight-forward
fashion with AND and OR gates, we get the

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following schematic for the ant brain.

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Pretty neat!

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Who knew that maze following behavior could
be implemented with a couple of D registers

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and a handful of logic gates?

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There are many complex behaviors that can
be created with surprisingly simple FSMs.

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Early on, the computer graphics folks learned
that group behaviors like swarming, flocking

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and schooling can be modeled by equipping
each participant with a simple FSM.

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So next time you see the massive battle scene
from the Lord of the Rings movie, think of

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many FSMs running in parallel!

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Physical behaviors that arise from simple
interactions between component molecules can

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sometimes be more easily modeled using cellular
automata -

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arrays of communicating FSMS - than by trying
to solve the partial differential equations

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that model the constraints on the molecules'
behavior.

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And here's an idea: what if we allowed the
FSM to modify its own transition table?

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Hmm.

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Maybe that's a plausible model for evolution!

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FSMs are everywhere!

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You'll see FSMs for the rest of your life…