Abstract
In unification theory, equational theories can be classified according to the existence and cardinality of minimal complete solution sets for equation systems. For unitary, finitary, and infinitary theories minimal complete solution sets always exist and are singletons, finite, or possibly infinite sets, respectively. In nullary theories, minimal complete sets do not exist for some equation systems. These classes form the unification hierarchy.
We show that it is not possible to decide where a given equational theory resides in the unification hierarchy. Moreover, it is proved that for some classes this problem is not even recursively enumerable.
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References
Baader, F., ‘The theory of idempotent semigroups is of unification type zero’, J. Automated Reasoning 2, 283–286 (1986).
Baader, F., personal communication, 1989.
Book, R. and Siekmann, J., ‘On unification: equational theories are not bounded’, J. Symbolic Computation 2, 317–324 (1986).
Bürckert, H.-J., Herold, A. and Schmidt-Schauß, M., ‘On equational theories, unification, and decidability’, Proc. 2nd Conference on Rewriting Techniques and Applications, LNCS 256, Springer, 1987, pp. 240–250. Also Kirchner, C. (Ed.), J. Symbolic Computation, Special Issue on Unification, 1989.
Kirchner, C. (Ed.), J. Symbolic Computation, Special Issue on Unification, 1989.
Fages, F. and Huet, G., ‘Complete sets of unifiers and matchers in equational theories’, Theoretical Computer Science 43, 189–200 (1986).
Lyndon, R. C. and Schupp, P. E., Combinatorial Group Theory, Springer, 1977.
Perkins, P., ‘Unsolvable problems for equational theories’, Notre Dame J. Formal Logic 8, 175–185 (1967).
Plotkin, G., ‘Building in equational theories’, Machine Intelligence 7, 73–90 (1972).
Robinson, J. A., ‘A machine-oriented logic based on the resolution principle’, J. ACM 12, 23–41 (1965).
Rotman, J. J., The Theory of Groups, Allyn and Bacon, 1973.
Schmidt-Schauß, M., ‘Unification under associativity and idempotence is of type nullary’, J. Automated Reasoning 2, 277–281 (1986).
Schmidt-Schauß, M., ‘Unification in a combination of arbitrary disjoint equational theories’, in Kirchner, C. (Ed.), J. Symbolic Computation, Special Issue on Unification, 1989.
Siekmann, J. H., Unification and Matching Problems, Ph.D. Thesis, University of Essex, 1978.
Siekmann, J. H., ‘Unification theory. A survey’, in Kirchner, C. (Ed.), J. Symbolic Computation, Special Issue on Unification, 1989.
Tidén, E., ‘Unification in combinations of collapse-free theories with disjoint sets of function symbols’, Proc. 8th Conference on Automated Deduction, LNCS 230, Springer, 1986, 431–450.
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Nutt, W. The unification hierarchy is undecidable. J Autom Reasoning 7, 369–381 (1991). https://doi.org/10.1007/BF00249020
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DOI: https://doi.org/10.1007/BF00249020