Tobias Nipkow
Abstract
This paper examines the unification problem in the class of primal algebras and the varieties they generate. An algebra is called primal if every function on its carrier can be expressed just in terms of the basic operations of the algebra. The two-element Boolean algebra is the simplest nontrivial example: Every truth-function can be realized in terms of the basic connectives, for example, negation and conjunction.
It is shown that unification in primal algebras is unitary, that is, if an equation has a solution, it has a single most general one. Two unification algorithms, based on equation-solving techniques for Boolean algebras due to Boole and Lo¨wenheim, are studied in detail. Applications include certain finite Post algebras and matrix rings over finite fields. The former are algebraic models for many-valued logics, the latter cover in particular modular arithmetic.
Then unification is extended from primal algebras to their direct powers, which leads to unitary unification algorithms covering finite Post algebras, finite, semisimple Artinian rings, and finite, semisimple nonabelian groups.
Finally the fact that the variety generated by a primal algebra coincides with the class of its subdirect powers is used. This yields unitary unification algorithms for the equational theories of Post algebras and p-rings.
- 1 BARRINGER, H., CHENG, J. H., AND JONES, C. B. A logic coveting undefinedness in program proofs. Acta Inf. 21 (1984), 251-269.Google Scholar
- 2 BOOLE, G. The Mathematical Analysis of Logic. Macmillan, New York, 1847. Reprinted: B. Blackwell, London, England, 1948.Google Scholar
- 3 BOTTNER, W., AND SIMONIS, H. Embedding Boolean expressions into logic programming. J. Symb. Comput. 4 (1987), 191-206. Google Scholar
- 4 CHENG, J.H. A logic for partial functions. Ph.D. Thesis. Dept. ofComput. Sci., Univ. Manchester, Manchester, England, 1986.Google Scholar
- 5 COLMERAUER, A. Prolog II Reference Manual and Theoretical Model. Groupe Intelligence Artificielle, Facult6 des Sciences de Luming, Marseilles, France, 1982.Google Scholar
- 6 COMON, H. Sufficient completeness, term rewriting systems and anti-unification. In Proceedings of the 8th International Conference on Automated Deduction. Lecture Notes in Computer Science, vol. 230. Springer-Verlag, New York, 1986, pp. 128-140. Google Scholar
- 7 FOSTER, A. L. Generalized "Boolean" theory of universal algebra. Math. Zeitschr. 59 (1953), 191-199.Google Scholar
- 8 FOSTER, A. L., AND PIXLEY, A. Semi-categorical algebras. I. Semi-primal algebras. Math. Zeitschr. 83 (1964), 147-169.Google Scholar
- 9 GAREY, M. R., AND JOHNSON, O.S. Computers and Intractability. W. H. Freeman and Company, San Francisco. Calif. 1979. Google Scholar
- 10 GARLAND, S. J., AND GUTTAG, J.V. An overview of LP. The larch prover. In Proceedings of the 3rd Conference on Rewriting Techniques and Applications. Lecture Notes in Computer Science, vol. 355. Springer-Vedag, New York, 1989, pp. 137-151. Google Scholar
- 11 GRATZER, G. UniversalAlgebra, 2nd Ed. Springer-Verlag, New York, 1979.Google Scholar
- 12 HERSTEIN, I.N. Noncommutative Rings. The Mathematical Association of America, Washington, DC, 1968.Google Scholar
- 13 HSIANG, J.Refutational theorem proving using term-rewriting systems. Artif. Int. 25 (1985), 255-300. Google Scholar
- 14 KAPUR, D., AND NARENDRAN, P. Matching, unification, and complexity. SIGSAM Bull. 21, 4 (1987), 6-9. Google Scholar
- 15 KNOEBEL, R.A. Simplicity vis-fi-vis functional completeness. Math. Ann. 189 (1970), 299-307.Google Scholar
- 16 LOWENHEIM, L. Uber daN Aufl6sungsproblem im logischen Klassenkalkiil. Sitzungsber. Berl. Math. Gesell. 7 (1908), 89-94.Google Scholar
- 17 MARTIN, U., AND NIPKOW, T. Unification in Boolean rings. In Proceedings of the 8th international Conference on Automated Deduction. Lecture Notes in Computer Science, vol. 230. Springer- Verlag, New York, 1986, pp. 506-513. Google Scholar
- 18 MARTIN, U., AND NIPKOW, T. Unification in Boolean rings. J. Auto. Reas. 4 (1988), 381-396. Google Scholar
- 19 MARTIN, U., AND NIPKOW, T. Boolean unification--The story so far. J. Symb. Comput. 7 (1989), 275-293. Google Scholar
- 20 MARTIN, U., AND NIPKOW, T. Canonical Term Rewriting Systems for p-Rings. Tech Rep. In preparation.Google Scholar
- 21 McCoY, N. H., AND MONTGOMERY, D. A representation of generalized Boolean tings. Duke Math. J. 3 (1937), 455-459.Google Scholar
- 22 POST, E.L. Introduction to a general theory of elementary propositions. Amer. J. Math. 43 ( 1921), 163-185.Google Scholar
- 23 RASIOWA, H. An Algebraic Approach to Non-Classical Logics. North-Holland, Amsterdam, The Netherlands, 1974.Google Scholar
- 24 RATH, H.H. Ein Meta-Interpretierer f'tir Prolog mit Boolescher Unifikation. KAP-MEMO 1/88, SFB 314, Universit/it Karlsruhe, Karlsruhe, West Germany, 1988.Google Scholar
- 25 ROSENBLOOM, P.C. Post algebras. I. Postulates and general theory. Amer. J. Math. 64 (1942), 167-188.Google Scholar
- 26 RUDEANU, S. Boolean Functions and Equations. North-Holland, Amsterdam, The Netherlands, 1974.Google Scholar
- 27 SCHMIDT-SCHAUa, M. Unification in a combination of arbitrary disjoint equational theories. In Proceedings of the 9th International Conference on Automated Deduction. Lecture Notes in Computer Science, vol. 310. Springer-Verlag, New York, 1988, pp. 378-396. Google Scholar
- 28 SCHRODER, E. Vorlesungen fiber die Algebra der Logik. Leipzig, Vol 1, 1890; Vol 2, 1891, 1905; Vol 3, 1895; Reprint 1966, Chelsea, Bronx, New York.Google Scholar
- 29 SIEKMANN, J.H. Unification theory. J. Symb. Comput. 7 (1989), 207-274. Google Scholar
- 30 STONE, M. H. The theory of representation for Boolean algebras. Trans. Amer. Math. Soc. 40 (1936), 37-111.Google Scholar
- 31 TID~N, E. Unification in combinations of collapse-free theories with disjoint function symbols. In Proceedings of the 8th International Conference on Automated Deduction. Lecture Notes in Computer Science, vol. 230. Springer-Verlag, New York, 1986, pp. 431-449. Google Scholar
- 32 WERNER, H. EinJ~hrung in die allgemeine Algebra. Bibliographisches Institut Mannheim/Wien/ Ziirich, 1978.Google Scholar
- 33 YELICK, K.A. Unification in combinations of collapse-free regular theories. J. Symb. Comput. 3 (1987), 153-181. Google Scholar
Index Terms
- Unification in primal algebras, their powers and their varieties
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Reviews
Wayne Scott Snyder
Unification, a fundamental operation in computer science and mathematics, has been studied in a wide variety of settings in recent years. This paper makes a significant contribution to unification theory by studying a particular kind of <__?__Pub Fmt italic>E<__?__Pub Fmt /italic>-unification problem, namely, <__?__Pub Fmt italic>E-unification primal algebras<__?__Pub Fmt /italic>, a class of finite algebras in which every function on the carrier can be expressed as a composition of the basic operations (for example, the Boolean algebra on {<__?__Pub Fmt bold>T<__?__Pub Fmt /bold>,<__?__Pub Fmt bold>F<__?__Pub Fmt /bold>} with operations ¬ and ? <__?__Pub Caret> is primal). The approach taken is to extend the author's previous work with Martin on Boolean unification to arbitrary varieties generated by primal algebras. For the computer scientist, these unification problems are particularly important in that they are decidable (albeit NP-complete) and that when solutions exist, a most general unifier always exists, as in standard unification; this makes implementation of the overall deduction system much simpler. Unification in primal algebras (in the simple case of Boolean algebras) has been used to augment Prolog for the purpose of hardware verification, but promises in the future to be useful in reasoning about various paradigms in computer science, from many-valued logics to modular arithmetic. This theoretical work is well written and rigorous, but it presupposes a fair amount of algebraic maturity on the part of the reader. Although complexity issues are discussed, no practical experience with these algorithms is presented. The author makes a solid contribution to the theory of unification in a setting that could prove to be of significant practical import for automated reasoning about various kinds of logic (including digital logic). This paper should be of interest not only to theoreticians but also to researchers in computer algebra, automated theorem proving, and logic programming.
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