Abstract
We present methods for automatically proving theorems in theories axiomatized by a set of Horn clauses. These methods address both deductive and inductive reasoning. They are based on the concept of simplification and require minimal human interaction.
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References
R. Aubin, Mechanizing structural induction, Theor. Comp. Sci. 9(1979)329–362.
R.S. Boyer and J.S. Moore,A Computational Logic (Academic Press, New York, 1979).
R.M. Burstall, Proving properties of programs by structural induction, Comp. J. 12(1969)41–48.
N. Dershowitz, Termination of rewriting, J. Symb. Comp. 3(1987)69–116.
S.J. Garland and J.V. Guttag, An overview of LP, the Larch Prover,Proc. 3rd Conf. on Rewriting Techniques and Applications, Chapel Hill, NC, ed. N. Dershowitz, Lecture Notes in Computer Science 355 (Springer, 1989) pp. 137–151.
J. Hsiang and M. Rusinowitch, Proving refutational completeness of theorem proving strategies: The transfinite semantic tree method, J. ACM 38(1991)559–587.
J. Hsiang and M. Rusinowitch, On word problem in equational theories,Proc. 14th Int. Colloq. on Automata, Languages and Programming, Karlsruhe, ed. Th. Ottmann, Lecture Notes in Computer Science 267 (Springer, 1987) pp. 54–71.
G. Huet and J.-M. Hullot, Proofs by induction in equational theories with constructors, J. Comp. Syst. Sci. 25(1982)239–266. Preliminary version inProc. 21st IEEE Symp. on Foundations of Computer Science (1980).
J.-P. Jouannaud and E. Kounalis, Proof by induction in equational theories without constructors,Proc. 1st IEEE Symp. on Logic in Computer Science, Cambridge, MA (1986) pp. 358–366.
S. Kamin and J.-J. Lévy, Attempts for generalizing the recursive path ordering, unpublished manuscript (1980).
S. Kaplan, Simplifying conditional term rewriting systems: Unification, termination and confluence, J. Symb. Comp. 4(1987295–334.
S. Kaplan and J.-L. Rémy, Completion algorithms for conditional rewriting systems, in:Resolution of Equations in Algebraic Structures, Volume 2: Rewriting Techniques, eds. H. Aït-Kaci and M. Nivat (Academic Press, 1989) pp. 141–170.
D.E. Knuth and P.B. Bendix, Simple word problems in universal algebras, in:Computational Problems in Abstract Algebra, ed. J. Leech (Pergamon Press, Oxford, 1970).
E. Kounalis, Testing for inductive (co)-reducibility,Proc. 15th CAAP, Copenhagen, ed. A. Arnold, Lecture Notes in Computer Science 431 (Springer, 1990).
E. Kounalis and M. Rusinowitch, On word problems in Horn theories,Proc. 9th Int. Conf. on Automated Deduction, Argonne, IL, eds. E. Lusk and E. Overbeek, Lecture Notes in Computer Science 310 (Springer, 1988).
D.S. Lankford, Canonical algebraic simplifications, Technical Report, Louisiana Tech. University (1975).
D.R. Musser, On proving inductive properties of abstract data types,Proc. 7th ACM Symp. on Principles of Programming Languages (1980) pp. 154–162.
P. Padawitz,Computing in Horn Clause Theories (Springer, 1988).
G. Peterson, A technique for establishing completeness results in theorem proving with equality, SIAM J. Comp. 12(1983)82–100.
D. Plaisted, Semantic confluence tests and completion methods, Inf. Contr. 65(1985)182–215.
U.S. Reddy, Term rewriting induction,Proc. 10th Int. Conf. on Automated Deduction, Kaiserslautern, ed. M.E. Stickel, Lecture Notes in Computer Science 449 (Springer, 1990) pp. 162–177.
G.A. Robinson and L.T. Wos, Paramodulation and first-order theorem proving, in:Machine Intelligence, eds. B. Meltzer and D. Mitchie (Edinburgh University Press, 1969) pp. 135–150.
J.A. Robinson, A machine-oriented logic based on the resolution principle, J. ACM 12(1965)23–41.
M. Rusinowitch, Functionally reflexive axioms and linear strategies: A counterexample, Assoc. Autom. Reasoning Newsletter, No. 8 (April, 1987).
M. Rusinowitch,Démonstration Automatique-Techniques de Réécriture (InterEditions, 1989).
M. Rusinowitch, Theorem-proving with resolution and superposition, J. Symb. Comp. 11(1991)21–49. Extended version of: Theorem-proving with resolution and superposition; An extension of the Knuth and Bendix procedure to a complete set of inference rules,Proc. Int. Conf. on Fifth Generation Computer Systems, Tokyo (Springer, 1988).
L. Wos, G.A. Robinson, D.F. Carso and L. Shalla, The concept of demodulation in theorem proving, J. ACM 14(1967)698–709.
H. Zhang, Reduction, superposition and induction: Automated reasoning in an equational logic, Ph.D. Thesis, Department of Computer Science, Rensselaer Polytechnis Institute, Troy, NY (1988).
H. Zhang, D. Kapur and M.S. Krishnamoorthy, A mechanizable induction principle for equational specifications,Proc. 9th Int. Conf. on Automated Deduction, Argonne, IL, eds. E. Lusk and R. Overbeek, Lecture Notes in Computer Science 310 (Springer, 1988) pp. 162–181.
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Kounalis, E., Rusinowitch, M. Reasoning with conditional axioms. Ann Math Artif Intell 15, 125–149 (1995). https://doi.org/10.1007/BF01534452
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DOI: https://doi.org/10.1007/BF01534452