JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III Year DIGITAL SYSTEM DESIGN Previous Question Paper (July - 2023 - R16)

Two states, S_i and S_j, of a sequential machine are defined as equivalent if and only if for every possible input sequence, the machine produces the same output sequence regardless of whether it starts in state S_i or S_j. If two states are equivalent, they are indistinguishable in terms of their future behavior.

This concept is fundamental to machine minimization because it allows for the identification and merging of redundant states. By replacing a set of equivalent states with a single representative state, the total number of states in the state table can be reduced, leading to a simpler, more efficient circuit implementation with fewer flip-flops and reduced combinational logic complexity.

The process of simplifying an incompletely specified machine involves finding the minimum number of states required to represent the same input-output behavior. Since some transitions or outputs are unspecified (don't-care conditions), the states are considered compatible rather than equivalent.

1. Identify Compatible States: Two states are compatible if, for every input, their outputs are identical whenever both are specified, and their next states are either identical or also compatible.
2. Find Maximal Compatibles: Construct a compatibility graph or table to identify all sets of states that are pairwise compatible. A set is a maximal compatible if no other state can be added to it while maintaining pairwise compatibility.
3. Form a Minimum Closed Cover: Select a minimum number of maximal compatibles such that every state of the original machine is included in at least one chosen compatible (covering) and the set of chosen compatibles is closed under the next-state transition (closure).

This simplification is typically performed using the Partitioning Method or by constructing a Compatibility Table to systematically eliminate redundant states.