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  "body": "\n## Statement\n\nThere is no algorithm that decides, for an arbitrary finite algebra $A$ of finite signature given by its operation tables, whether $A$ is [[finitely based|YGYZAQ12MZ]].\nMore precisely, there is an effective construction that assigns a finite algebra $A(T)$ of finite signature to each [[Turing machine|758EYCN27B]] $T$.\nIt has this property: $T$ halts, started on the empty tape, if and only if $A(T)$ is finitely based.\n\n## Remarks\n\nThe second sentence implies the first, because the [[halting problem|Z7TN2B35B0]] is undecidable.\nSo this theorem gives a negative answer to [[Tarski's finite basis problem|AB22FWBQ0M]].\nMcKenzie published the proof in 1996.\nMcKenzie built his algebra from a simpler finite algebra that encodes the computation of $T$.\nThe proof relates the halting of $T$ to the sizes of the subdirectly irreducible algebras in the variety generated by this simpler algebra.\nMcKenzie also used the simpler algebra to show that it is undecidable whether a finite algebra generates a variety with a finite residual bound.\nWillard later proved that the simpler algebra itself is finitely based if and only if $T$ halts, which gives a shorter route to the theorem.\n\n## Explanations\n\n### L1 human\n\nSome finite algebras have all their laws captured by a finite list, and others do not.\nThe theorem says that no computer program can sort finite algebras into these two kinds.\nIt is proved by hiding a computer program inside a finite algebra so that the algebra has a finite list of laws exactly when the program stops.\n\n### L2 human\n\nThe theorem closes a question that had driven much of the work on finite bases since the 1950s.\nIt shows that results such as Baker's finite basis theorem, which give finite bases under structural hypotheses, cannot be replaced by a general decision procedure.\nThe construction of $A(T)$ links equational logic with residual bounds of varieties, which is also the setting of Park's conjecture.\n",
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