State Reduction and Assignment

State Reduction and Assignment in Sequential Circuits

State reduction and state assignment are techniques used in the design of digital sequential circuits, particularly finite state machines (FSMs). These techniques help simplify the state space and make the design more efficient by reducing the number of states or optimizing how those states are represented. These methods are crucial in minimizing the hardware resources required for the implementation of FSMs.

1. State Reduction

State reduction is the process of minimizing the number of states in a finite state machine (FSM) without changing its behavior or functionality. This reduction is important because fewer states typically lead to simpler, faster, and less resource-intensive designs.

The goal of state reduction is to identify and merge equivalent states. Two states are considered equivalent if, from each state, the outputs and the transitions to other states are the same for the same inputs.

Steps in State Reduction:

1. Construct the State Table:

   • Start by writing the state transition table or state diagram for the FSM. This table includes the current states, inputs, next states, and outputs.

2. Identify Equivalent States:

   • Compare each pair of states to see if they behave the same for all inputs. If two states have the same outputs and lead to equivalent states under all possible inputs, they are equivalent.
   • The comparison can be done systematically using a state equivalence table.

3. Group Equivalent States:

   • Once equivalent states are identified, group them together. These groups will be combined into a single state, effectively reducing the number of states.

4. Redraw the State Diagram:

   • After reducing the states, redraw the state diagram, merging the equivalent states into one. This new diagram will have fewer states and represent the same behavior as the original.

Example of State Reduction:

Let’s consider a simple FSM with the following state table:

Present State	Input 0	Input 1	Output
A	B	C	0
B	A	C	0
C	B	C	1

In this case:

• States A and B behave identically: both transition to state B on input 0 and to state C on input 1, and both have output 0.
• Therefore, states A and B can be merged into a single state.

After reduction, the state table becomes:

Present State	Input 0	Input 1	Output
AB	C	C	0
C	B	C	1

This reduced FSM has fewer states and performs the same behavior.

Types of State Equivalence:

• Equivalent States: States that have the same behavior under all inputs and outputs.
• Incomparable States: States that behave differently for at least one input. These must be kept separate in the reduced FSM.

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2. State Assignment

State assignment refers to the process of assigning binary codes (or state labels) to the states of an FSM. It’s essential for the implementation of the FSM in hardware, particularly when using flip-flops to represent the states.

The goal of state assignment is to choose binary codes that minimize the hardware resources (such as flip-flops and logic gates) required for the implementation. A good state assignment can minimize the number of bits required for the state registers, simplify the logic between the states, and reduce the complexity of the design.

Types of State Assignment:

1. Unrestricted State Assignment:

   • This is the simplest form of state assignment, where each state is assigned a unique binary code with no consideration for minimizing the design complexity. Typically, the states are assigned consecutive binary values.
   • For example, for four states, the assignment might be:
     • State A = 00
     • State B = 01
     • State C = 10
     • State D = 11

2. One-Hot Encoding:

   • In one-hot encoding, each state is assigned a single "1" in a binary vector, with all other bits being "0". This is an efficient method when the number of states is small and when quick state transitions are necessary.
   • For example, with four states, the assignment could be:
     • State A = 0001
     • State B = 0010
     • State C = 0100
     • State D = 1000

   In this case, only one flip-flop is set to "1" at any given time. One-hot encoding can simplify the transition logic but requires more flip-flops (one per state).

3. Gray Code Assignment:

   • In Gray code assignment, the states are assigned binary codes in such a way that only one bit changes between consecutive states. This minimizes the number of state transitions in certain applications, especially in counters.
   • For example, for four states in Gray code:
     • State A = 00
     • State B = 01
     • State C = 11
     • State D = 10

   Gray code assignment helps minimize the number of changes in the flip-flop outputs during state transitions.

4. Binary Assignment (Optimal):

   • In binary assignment, states are mapped directly to binary numbers. This approach works well for smaller FSMs, but when there are many states, the binary assignment can result in complex transition logic.
   • The goal in optimal binary state assignment is to minimize the logic required for state transitions and reduce the hardware implementation complexity.

Guidelines for Effective State Assignment:

1. Minimize Transition Logic:

   • Ideally, choose state assignments that minimize the number of bits that change between states for a given input. Fewer bit transitions lead to simpler transition logic.

2. Use Gray Code for Cyclic or Counting FSMs:

   • In cyclic FSMs, such as counters, Gray code assignment is ideal because it reduces the likelihood of multiple bit changes between states.

3. Use One-Hot Encoding for Simplicity:

   • If the number of states is small and fast state transitions are required, one-hot encoding may be the best choice.

4. Avoid Long Sequences of Transitions:

   • Avoid assigning states in such a way that multiple bits change between consecutive states. This can lead to unnecessary complexity in the state transition logic.

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Combining State Reduction and State Assignment

To design an efficient FSM, state reduction and state assignment should be done in combination:

1. First, apply state reduction to minimize the number of states. This step helps eliminate redundant states and simplifies the FSM.
2. Then, assign binary codes to the reduced states. Choose an optimal state assignment technique based on the desired hardware implementation.

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Example: State Reduction and Assignment

Suppose we have an FSM with the following state diagram:

Present State	Input	Next State
A	0	B
A	1	C
B	0	A
B	1	C
C	0	A
C	1	B

• Step 1: State Reduction:

  • Upon inspection, we notice that states B and C are symmetric. Both have the same transitions for the inputs 0 and 1, making them equivalent. Thus, we can merge them into a single state.

  After reduction, we have two states:

  • State A (original A)
  • State BC (merged B and C)

• Step 2: State Assignment:

  • We now assign binary codes to the two states:
    • State A = 0
    • State BC = 1

The simplified state table for the reduced FSM is:

Present State	Input	Next State
0	0	1
0	1	1
1	0	0
1	1	0

Thus, we have reduced the FSM from 3 states to 2 states, simplifying the design and reducing the hardware resources required.

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Conclusion

State reduction and state assignment are critical techniques in the design of efficient sequential circuits, especially finite state machines. State reduction helps minimize the number of states, while state assignment involves choosing optimal binary codes for those states to simplify the transition logic. By carefully applying both techniques, we can reduce the complexity of the FSM, improve performance, and minimize the hardware resources needed for the implementation.