Abstract
Clause Graphs, as they were defined in the 1970s, are graphs representing first order formulas in conjunctive normal form together with the resolution possibilities. The nodes are labelled with literals and the edges (links) connect complementary unifiable literals. This report describes a generalization of this concept, called abstract clause graphs. The nodes of abstract clause graphs are still labelled with literals, the links however connect literals that are “unifiable” relative to a given relation between literals. This relation is not explicitely defined; only certain “abstract” properties are required, for instance the existence of a special purpose unification algorithm is assumed which computes substitutions, the application of which makes the relation hold for two literals.
When instances of already existing literals are added to the graph (e.g. due to resolution or factoring), the links to the new literals are derived from the links of their ancestors. An inheritance mechanism for such links is presented which operates only on the attached substitutions and does not have to unify the literals. This solves a long standing open problem of connection graph calculi: how to inherit links (with several unifiers attached) such that no unifier has to be computed more than once.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bruynooghe, M. ‘The Inheritance of Links in a Connection Graph’. Report CW2 (1975). Applied Mathematics and Programming Division. Katholieke Universiteit Leuven.
Eisinger, N. ‘Subsumption and Connection Graphs’. Proc. of IJCAI-81, Vancouver (1981).
Herold, A. ‘Some Basic Notions of First-Order Unification Theory’. Interner Bericht 15/83, Inst, für Informatik I, Univ. of Karlsruhe, (1983).
Karl Mark G. Raph, ‘The Markgraf Karl Refutation Procedure’. Interner Bericht, Memo-Seki-MK-84-01, FB Informatik, Univ. of Kaiserslautern (1984).
Kowalski, R. ‘A Proof Procedure Using Connection Graphs’. J.ACM 22,4, (1975).
Ohlbach, H.J. ‘Theory Unification in Abstract Clause Graphs’. Interner Bericht, FB. Informatik, Univ. of Kaiserslautern (1985).
Siekmann, J. ‘Universal Unification’ Proc. of CADE-84, Nappa USA. Springer (1984).
Schmidt-Schauss, M. ‘A Many-Sorted Calculus with Polymorphic Functions Based on Resolution and Paramodulation’. Proc. of IJCAI-85, Los Angeles (1985).
Stickel, M.E. ‘Theory Resolution: Building in Non-Equational Theories’. SRI Report, (1983).
Siekmann, J., Wrightson, G. ‘Paramodulated Connection Graphs’. Acta Informatica (1978).
Szabo, P. ‘Unifikationstheorie erster Ordnung’ Dissertation, Inst, für Informatik I, Univ. of Karlsruhe (1982).
Walther, Ch. ‘Elimination of Redundant Links in Extended Connection graphs’. Proc. of GWAI-81, Springer Fachberichte (1981) and Interner Bericht 10/81, University of Karlsruhe.
Walther, Ch. ‘A Many-Sorted Calculus Based on Resolution and Paramodulation’. Interner Bericht 34/82 Inst, für Informatik I, Univ. of Karlsruhe (1982). see also Proc. of IJCAI-8.3, Karlsruhe (1983).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1986 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ohlbach, H.J. (1986). Theory Unification in Abstract Clause Graphs . In: Stoyan, H. (eds) GWAI-85. Informatik-Fachberichte, vol 118. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-71145-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-71145-9_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-16451-7
Online ISBN: 978-3-642-71145-9
eBook Packages: Springer Book Archive