Abstract
Since we are looking for unification algorithms for a large enough class of equational theories, we are interested in syntactic theories because they have a nice decomposition property which provides a very simple unification procedure. A presentation is said resolvent if any equational theorem can be proved using at most one equality step at the top position. A theory which has a finite and resolvent presentation is called syntactic. In this paper we give decidability results about open problems in syntactic theories: unifiability in syntactic theories is not decidable, resolventness of a presentation and syntacticness of a theory are even not semidecidable. Therefore we claim that the condition of syntacticness is too weak to get unification algorithms directly.
This research has been partially supported by ESPRIT Basic Research Action COMPASS #3264, by the GRECO de Programmation of CNRS and by the Fonds National Suisse pour la Recherche Scientifique.
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References
J. Christian. High performance permutative completion. Technical report ACT-AI-303-89, MCC, 1989. PhD thesis.
H. Comon. Unification et disunification. Théories et applications. Thèse d'Université de l'Institut Polytechnique de Grenoble, 1988.
N. Dershowitz and J.-P. Jouannaud. Handbook of Theoretical Computer Science, volume B, chapter 15: Rewrite systems. North-Holland, 1990. Also aś: Research report 478, LRI.
G. Huet and D. Oppen. Equations and rewrite rules: A survey. In R. Book, editor, Formal Language Theory: Perspectives and Open Problems, pages 349–405. Academic Press, New York, 1980.
J.-P. Jouannaud and C. Kirchner. Solving equations in abstract algebras: A rulebased survey of unification. Research report, CRIN, 1990. To appear in Festschrift for Robinson, J.-L. Lassez and G. Plotkin Editors, MIT Press.
J.-P. Jouannaud and H. Kirchner. Completion of a set of rules modulo a set of equations. SIAM Journal of Computing, 15:1155–1194, 1986. Preliminary version in Proceedings 11th ACM Symposium on Principles of Programming Languages, Salt Lake City, 1984.
C. Kirchner. Méthodes et outils de conception systématique d'algorithmes d'unification dans les théories équationnelles. Thèse d'état, Université de Nancy I, 1985.
C. Kirchner. Computing unification algorithms. In Proceeding of the First Symposium on Logic In Computer Science, Boston (USA), pages 206–216, 1986.
C. Kirchner and F. Klay. Syntactic theories and unification. In Proceedings 5th IEEE Symposium on Logic in Computer Science, Philadelphia (Pennsylvania, USA), pages 270–277, 1990.
F. Klay. Undecidable properties of syntactic theories. Rapport interne crin, Centre de Recherche en Informatique de Nancy, 1990.
T. Nipkow. Proof transformations for equational theories. In Proceedings 5th IEEE Symposium on Logic in Computer Science, Philadelphia (Pennsylvania, USA), pages 278–288, 1990.
D. Rémy. Algèbres touffues. Application au typage polymorphe des objets enregistrements dans les langages fonctionnels. Thèse de l'Université de Paris 7, 1990.
J. Siekmann. Unification theory. Journal of Symbolic Computation, 7:207–274, 1989. Special issue on unification. Part one.
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© 1991 Springer-Verlag Berlin Heidelberg
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Klay, F. (1991). Undecidable properties of syntactic theories. In: Book, R.V. (eds) Rewriting Techniques and Applications. RTA 1991. Lecture Notes in Computer Science, vol 488. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-53904-2_92
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DOI: https://doi.org/10.1007/3-540-53904-2_92
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