The Type Set of a Variety is Not Computable

Abstract

The type-set of a variety, introduced by D. Hobby and R. McKenzie [6], has proved to be an important invariant of locally finite varieties. It was natural to seek an algorithm which would compute the type-set of V (A), when presented with a finite algebra A with finitely many operations. However, results obtained by J. Berman, E. W. Kiss, P. Pröhle, Á. Szendrei [2] and D. Hobby [5] indicate that such an algorithm must be highly complex. D. Hobby [5] proves that computing the typeset of V (A) is a problem of P -space complete difficulty or worse. In this paper, we prove that computing the type-set of a finitely generated variety is as difficult as it could possibly be. Our results imply that the type-set function is recursively equivalent to the function that outputs the halting character of a Turing machine. If S is any subset of {1, 2, 3, 4, 5}, there exists a finite algebra A of finite type such that typ{V (A)} = S. Thus there are 32 type-sets of finitely generated varieties, and 2 properties of type-sets. If S is any family (equivalently, property) of type-sets, one may ask if there exists an algorithm which would input any finite algebra A of finite type, and output the correct answer to the question “is typ{V (A)} ∈ S?” If such an algorithm exists, then we say that S is a decidable property of typesets. In this paper, we show that for each 2 ≤ i ≤ 5, the property “i ∈ S” is an undecidable property of type-sets S. In contrast, several properties of type-sets

DOI: 10.1142/S0218196701000504

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Cite this paper

@article{McKenzie2001TheTS, title={The Type Set of a Variety is Not Computable}, author={Ralph McKenzie and Japheth Wood}, journal={IJAC}, year={2001}, volume={11}, pages={89-130} }