stateminimization

State minimization is a process in automata theory and computer science used to reduce the number of states in a finite automaton while preserving its recognized language. This is often done to optimize the performance of systems that rely on finite state machines, such as parsers, controllers, and circuit designs.

The goal of state minimization is to find an equivalent automaton with the fewest possible states. An automaton is considered equivalent to another if it accepts the exact same set of input strings. Minimization is typically applied to deterministic finite automata (DFAs), although algorithms also exist for non-deterministic finite automata (NFAs).

A common algorithm for DFA state minimization is the partition refinement method. This approach starts by grouping states based on whether they are accepting or non-accepting. Then, it iteratively refines these partitions by further subdividing groups of states based on the transitions they make to states in other partitions. If two states transition to states in different partitions for the same input symbol, they are distinguishable and must remain in separate partitions. This process continues until no further refinement is possible. The resulting partitions represent the states of the minimal automaton, where each partition corresponds to a single state. States within the same partition are considered equivalent.

Another related concept is indistinguishability. Two states in an automaton are indistinguishable if, for any input string, starting from either state leads to the same outcome (either both reach an accepting state or both reach a non-accepting state). State minimization algorithms essentially identify and merge indistinguishable states.