Abstract
In the last years there has been a renewed of interest for methods based upon principles similar to those of matrix reduction. In this paper we first present matrix reduction and we characterize some of its features with respect to resolution. We then define an abstraction of connexion graphs for matrix reduction and, using this abstraction we introduce an implementation technique and a search of proofs which is separated into two parts: a plan search and a plan validation. The plan abstracts from the arguments of literals. Given a plan, a set of equations in the terms is determined. The existence of a solution of this set of equations validates the plan. The plan and the solution define a refutation of the initial set of clauses. When no solution exists, a subset of the equations is isolated which is responsible for the fact. After backtracking, future plans will never use this subset.
These ideas has been used for implementing the method of matrix reduction in Pascal on a microcomputer (LSI11). The separation into plan search and plan validation can be used (and is particularly adapted) for a parallel implementation.
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References
ANDREWS P.B.: “Theorem proving via general mating”, JACM, Vol.28 N∘2, April 1981 (193–214).
BIBEL W.: “Tautology testing with a generalized matrix reduction method”, Theoretical Computer Science 8, 1979 (31–44).
BIBEL W.: “On matrices with connections”, JACM, Vol.28 N∘4, October 1981 (633–645).
CHANG C. and LEE R.: “Symbolic logic and mechanical theorem proving”, Academic Press, 1973.
CHANG C. and SLAGLE J.R.: “Using rewriting rules for connection graphs to prove theorems”, Artificial Intelligence 12, 1979 (159–180).
DAVIS M.: “Eliminating the irrelevant from mechanical proofs”, Symposia in Applied Mathematics Vol.15, 1963 (15–30).
HENSCHEN L.J.: “Theorem proving by covering expressions”, JACM, Vol.26, N∘3, July 1979 (385–400).
HUET G.: “Résolution d'équations dans les langages d'ordre 1,2,...,ω”, Thèse d'état, Université Paris VII, 1976.
KLEENE S.: “Mathematical logic”, John Wiley and Sons, 1967.
KOWALSKI R.: “Logic for problem solving”, North-Holland 1979.
LOVELAND D.W.: “Mechanical theorem proving by model elimination”, JACM, Vol.15, N∘2, April 1968 (236–251).
LOVELAND D.W.: “A unified view of some linear Herbrand procedures”, JACM, Vol.19, N∘2, April 1972 (366–384).
LOVELAND D. W.: “Automated theorem proving: A Logical basis”, North-Holland, 1978.
PLAISTED D.A.: “Theorem proving with abstraction”, Artificial Intelligence 16, 1981 (47–108).
PRAWITZ D.: “An improved proof procedure”, Theoria 26, 1960 (102–139).
PRAWITZ D.: “Advances and problems in mechanical proof procedures”, Machine Intelligence 4, Edinburgh Univ. Press 1969, (59–71).
PRAWITZ D.: “A proof procedure with matrix reduction”, Symposium on Automatic Demonstration-Springer Verlag 1970 (207–214).
ROBINSON J.A.: “A machine oriented logic based on the resolution principle”, JACM, Vol.12, N∘1, January 1965 (23–41).
ROBINSON J.A.: “Logic: form and function, the mechanization of deductive reasoning”, University Press Edinburgh, 1979.
ROUSSELL Ph.: “Prolog: Manuel de Référence et d'Utilisation”, Groupe d'Intelligence Artificielle, Université d'Aix Marseille, Luminy, Sept. 1975.
SAYA H.: “Complétude de la stratégie du support dans la méthode de réduction matricielle”, R.R. Imag N∘93, Novembre 1977.
SAYA H. and CAFERRA R.: “A structure sharing technique for matrices and substitutions in Prawitz's theorem proving method”, R.R. IMAG N∘101, Decem. 1977.
SAYA H. et CAFERRA R.: “Représentation compacte de matrices et traitement de l'égalité formelle dans la méthode de Prawitz de démonstration automatique”, Congrès AFCET 1978, tome 2 (371–381).
SHOSTAK R.E.: “A graph-theoretic view of resolution theorem proving”, TR-20-74 Center of Research in Computing Technology-Harvard University 1974.
SICKEL S.: “A search technique for clause Interconnectivity Graphs”, IEEE Transactions on Computers, Vol.C-25 N∘8 August 1976 (823–835).
SICKEL S.: “Variable range restrictions in resolution theorem proving”, Machine Intelligence 8-Ellis Horwood Ltd 1977 (73–85).
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© 1982 Springer-Verlag Berlin Heidelberg
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Caferra, R. (1982). Proof by matrix reduction as plan + validation. In: Loveland, D.W. (eds) 6th Conference on Automated Deduction. CADE 1982. Lecture Notes in Computer Science, vol 138. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0000067
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DOI: https://doi.org/10.1007/BFb0000067
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