State Reduction and State Assignment

To illustrate the process of state reduction and state assignment first we have to know the concepts of the state diagram, state table, and state equation. In this article, we are going to learn all the topics related to state reduction and assignment.

What is State Diagram?

The state graph or state diagram is a pictorial representation of the relationships between the present state, the input state, the next state, and the output state of a sequential circuit i.e. A state diagram is a graphical representation of a sequential circuit's behavior.

Example: Consider an excitation table of J-K flip-flop 

The state diagram of the above table is 

What is State Table?

Even though the behavior of a sequential circuit can be conveniently described using a state diagram, for its implementation the information contained in the state diagram is to be translated into a state table. The tabular form of the state diagram is the state table. The present state, the next state, and the output are the three sections of the diagram.

The State Table of JK flip-flop is                                         

State equation: Q_n+1 = Q_n bar J + Q_n K bar

What is State Reduction?

The state reduction technique generally prevents the addition of duplicate states. The reduction in redundant states reduces the number of flip-flops and logic gates,  reducing the cost of the final circuit. Two states are said to be equivalent if every possible set of inputs generates exactly the same output and the same next state. When two states are equal, one of them can be eliminated without changing the input-output relationship. The state reduction algorithm is applied in the state table to reduce equivalent states.

Advantages of State Reduction

• Reduced Complexity: Lesser number of states result in a reduced complexity of circuits with lesser flip-flops and logic gates.
• Cost Efficiency: Overall cost of the circuit is diminished because of the decision taken for the simplified design.
• Improved Performance: Increased throughput among other features such as low power consumption.

Disadvantages of State Reduction

• Design Time: State reduction may take more time to analyze and compute the data as compared to analyzing the parameter trajectory directly.
• Potential Information Loss: It thus requires precaution so that none of the important information is overlooked during the reduction process.

What is State Assignment?

State assignment refers to the process of assigning binary values to the states of a sequential machine. The binary values should be given to the states in such a way that flip-flop input functions may be implemented with a minimum number of logic gates.

Types of State Assignment Techniques

• Binary Assignment: Gives binary numbers to states on one-to-one basis. This is clear and properly executable yet it may sometimes lead to the creation of an utmost low logical condition.
• Gray Code Assignment: It utilizes Gray codes to bring about minimum change of transition between the states hence reducing the chances of getting glitches in the logic.
• One-Hot Assignment: Conveys each value that specifies a state by means of a specific position of the bits, in which only the one is set to 1. This makes the combinational logic simpler but at a cost of more numbers of flip-flops.

State Assignment rules are as follows

Rule 1: States having the same next state for a given input condition should have assignments that can be grouped into logically adjacent cells in a K-map. 

Rule 2: States that are the next states of a single state should have assignments that can be grouped into logically adjacent cells in a K-map.

Example 1: To explain the concept of state reduction let us consider the state table as 

The state diagram for the above state table is 

Step1: First here we are supposed to identify two or more similar states in the next state and output state. In the above table if we observe states of e and g are having the same next state and output values for all combinations of input i.e. X=0 and X=1.

So eliminate the g state in the state table and wherever g is present replace it with e. Because e and g both are the same i.e. e=g.  

Step 2: Again check if any two states have similar values or not. If any two states have the same next state and output then eliminate one state. 

Here d and f are having the same next state value and output. So eliminate f and wherever f is present replace it with d. Because both are the same d=f 

Step 3: Further observe if any similar states are present or not. The states c and e are having same next states but they are having different outputs. So we can not consider it a reduction state. 

Step 4:  If you observed the state table, the states are represented by using the alphabet. We can not proceed further if we are having alphabets, so, assigning binary numbers to alphabets is called a state assignment.  

To assign binary numbers to the state we have to consider the minimum number of bits. 

The codes must contain n bits for a circuit with m states, where 2ⁿ >= m. In this case, each state requires 2³ >=5=>3 bits to be represented. With three bits, there are eight possible combinations, of which five can be used to represent the states. 

Step 5: Replacing the alphabets with binary numbers.  

Advantages of State Assignment

• Minimized Logic Gates: Pre allocation of states on the hardware component can also have an impact of minimizing the number of logic gates in use.
• Simplified Implementation: Some of the state assignment techniques such as Gray coding and one-hot encoding are beneficial in simplifying the circuit design and also improving on the errors that might be present.

Disadvantages of State Assignment

• Increased Flip-Flops: Some of these techniques such as one-hot encoding may need more flip-flops as compared to other techniques.
• Complexity in Assignment: Deciding on which state should be assigned to an AP can be a challenging task and requires sometimes even a huge amount of time.

Conclusion

State reduction and state assignment are very useful in the design of the sequential circuits. State reduction enable fewer numbers of states hence easier in the design of the circuit than state assignment which optimizes the binary form of the states in order to reduce complexity of the circuit. By doing so, these techniques will enable designers to come up with better and more efficient cost-effective digital systems.