Undecidability of the Horn-clause implication problem

The authors prove that the problem 'given two Horn clauses H/sub 1/=( alpha /sub 1/ V-product alpha /sub 2/ to beta ) and H/sub 2/=( gamma /sub 1/ V-product . . . V-product gamma /sub k/ to delta ), where alpha /sub i/, beta , gamma /sub i/, delta are atomic formulas, decide if H/sub 2/, is a consequence of H/sub 1/' is not recursive. This solves one of the last open decidability problems concerning formulas in pure predicate logic (i.e. without equality symbol). The proof depends on a thorough analysis of derivation trees of one rule of inference with two premisses and one conclusion, and it may have further applications.<<ETX>>

[1]  Jörg Würtz,et al.  Unifying Cycles , 1992, ECAI.

[2]  Wolfgang Bibel,et al.  Cycle Uniication , 1992 .

[3]  Corrado Böhm,et al.  Flow diagrams, turing machines and languages with only two formation rules , 1966, CACM.

[4]  David E. Smith,et al.  Controlling Recursive Inference , 1986, Artif. Intell..

[5]  Jeffrey D. Ullman,et al.  Efficient tests for top-down termination of logical rules , 1988, JACM.

[6]  Jerzy Tiuryn,et al.  The undecidability of the semi-unification problem , 1990, STOC '90.

[7]  Yuri Gurevich,et al.  The Classical Decision Problem , 1997, Perspectives in Mathematical Logic.

[8]  Harry R. Lewis.  Krom Formulas with One Dyadic Predicate Letter , 1976, J. Symb. Log..

[9]  Warren D. Goldfarb,et al.  The decision problem for formulas with a small number of atomic subformulas , 1973, Journal of Symbolic Logic.

[10]  Philippe Devienne,et al.  Halting Problem of One Binary Horn Clause is Undecidable , 1993, STACS.

[11]  Alexander Leitsch.  Implication Algorithms for Classes of Horn Clauses , 1988 .

[12]  Wilhelm Ackermann,et al.  Solvable Cases Of The Decision Problem , 1954 .

[13]  George F. McNulty.  Undecidable Properties of Finite Sets of Equations , 1976, J. Symb. Log..

[14]  Ralph McKenzie.  On Spectra, and the Negative Solution of the Decision Problem for Identities having a Finite Nontrivial Model , 1975, J. Symb. Log..

[15]  Manfred Schmidt-Schauß.  Implication of Clauses is Undecidable , 1988, Theor. Comput. Sci..

[16]  Yuri Gurevich.  The Decision Problem for Standard Classes , 1976, J. Symb. Log..

[17]  B. Dreben,et al.  The decision problem: Solvable classes of quantificational formulas , 1979 .

[18]  Emil L. Post.  A variant of a recursively unsolvable problem , 1946 .

[19]  Warren D. Goldfarb.  The Unsolvability of the Godel Class with Identity , 1984, J. Symb. Log..

[20]  Moshe Y. Vardi,et al.  The Implication Problem for Functional and Inclusion Dependencies is Undecidable , 1985, SIAM J. Comput..

[21]  Emil L. Post.  Recursive Unsolvability of a problem of Thue , 1947, Journal of Symbolic Logic.

[22]  Georg Gottlob,et al.  Subsumption and Implication , 1987, Inf. Process. Lett..

[23]  Peter Perkins.  Unsolvable problems for equational theories , 1967, Notre Dame J. Formal Log..

[24]  George F. McNulty,et al.  The decision problem for equational bases of algebras , 1976 .

[25]  J. Lloyd.  Foundations of Logic Programming , 1984, Symbolic Computation.

[26]  Stål O. Aanderaa.  On the Decision Problem for Formulas in which all Disjunctions are Binary , 1971 .

[27]  Catriel Beeri,et al.  The Implication Problem for Data Dependencies , 1981, ICALP.

[28]  Johann A. Makowsky,et al.  Embedded implicational dependencies and their inference problem , 1981, STOC '81.

[29]  Philippe Devienne.  Weighted Graphs: A Tool for Studying the Halting Problem and Time Complexity in Term Rewriting Systems and Logic Programming , 1990, Theor. Comput. Sci..

[30]  Moshe Y. Vardi.  The implication and finite implication problems for typed template dependencies , 1982, J. Comput. Syst. Sci..

[31]  Maurice Bruynooghe,et al.  A Practical Technique for Detecting Non-terminating Queries for a Restricted Class of Horn Clauses, Using Directed, Weighted Graphs , 1990, ICLP.