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    "summary": "Hopcroft supplied an n log n state-minimization algorithm for complete deterministic finite automata with fixed alphabet, using partition refinement and carefully scheduled splitters.\n\nTwo states of a deterministic automaton are equivalent when every future input word gives the same acceptance outcome. Minimization groups those states together. Starting from accepting and nonaccepting states, the algorithm repeatedly splits groups whose transitions reveal a difference. The useful advance is how splitters are scheduled and represented: repeated work is controlled by smaller-side processing, rather than repeatedly rescanning every pair of states. The inspected report describes inverse transitions, linked lists, correctness and a potential-bound analysis giving n log n time for a fixed input alphabet. This became a reusable way to think about efficient refinement. The automaton must be deterministic and have a transition for every state-symbol pair. Its all-state quotient may still contain states unreachable from a selected starting state; removing those is a separate step. The historical ALGOL tests illustrate the report implementation, while the new finite checks here only test semantic correctness of a modern adaptation.",
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        "description": "The partition-refinement algorithm concerns finite deterministic automata and the specified alphabet/transition representation. Its time bound depends on how the alphabet and transitions are counted. It is not minimization of arbitrary nondeterministic machines or an all-purpose state-space reduction theorem. Later implementation variants retain separate credit.",
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      "statement": "# Hopcroft: efficient partition refinement for finite automaton minimization\n\nHopcroft supplied an n log n state-minimization algorithm for complete deterministic finite automata with fixed alphabet, using partition refinement and carefully scheduled splitters.\n\nTwo states of a deterministic automaton are equivalent when every future input word gives the same acceptance outcome. Minimization groups those states together. Starting from accepting and nonaccepting states, the algorithm repeatedly splits groups whose transitions reveal a difference. The useful advance is how splitters are scheduled and represented: repeated work is controlled by smaller-side processing, rather than repeatedly rescanning every pair of states. The inspected report describes inverse transitions, linked lists, correctness and a potential-bound analysis giving n log n time for a fixed input alphabet. This became a reusable way to think about efficient refinement. The automaton must be deterministic and have a transition for every state-symbol pair. Its all-state quotient may still contain states unreachable from a selected starting state; removing those is a separate step. The historical ALGOL tests illustrate the report implementation, while the new finite checks here only test semantic correctness of a modern adaptation.\n\n## Scientific scope\n\nThe partition-refinement algorithm concerns finite deterministic automata and the specified alphabet/transition representation. Its time bound depends on how the alphabet and transitions are counted. It is not minimization of arbitrary nondeterministic machines or an all-purpose state-space reduction theorem. Later implementation variants retain separate credit.\n\n## People and systems\n\nJohn Hopcroft\n\n\n\n## Original sources\n\n- [Hopcroft — original Stanford technical report STAN-CS-71-190](https://www.cs.cmu.edu/~cdm/resources/Hopcroft1971.pdf)\n- [Hopcroft — original1971 chapter publisher deposit](https://api.crossref.org/works/10.1016%2FB978-0-12-417750-5.50022-1)\n",
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