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Combining inference and disinference rules with enumeration for model building (Extended Abstract)
, 1997
"... ) Ricardo Caferra and Nicolas Peltier Laboratory LEIBNIZIMAG 46, Avenue F'elix Viallet 38031 Grenoble Cedex FRANCE Ricardo.Caferra@imag.fr, Nicolas.Peltier@imag.fr Phone: (33) (0)4 76 57 46 59 1 Introduction The possibility of systematic model building in firstorder logic exists at least since t ..."
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) Ricardo Caferra and Nicolas Peltier Laboratory LEIBNIZIMAG 46, Avenue F'elix Viallet 38031 Grenoble Cedex FRANCE Ricardo.Caferra@imag.fr, Nicolas.Peltier@imag.fr Phone: (33) (0)4 76 57 46 59 1 Introduction The possibility of systematic model building in firstorder logic exists at least since the introduction of the tableaux method (Hintikka, Beth, Smullyan,: : : ), approximately 40 years ago. Some striking results in interactive model building have been obtained less than 20 years ago [21]. But it is only since less than 10 years that results on model building are regularly published [15, 7, 8, 5, 12, 13, 20, 22]. One important difference must be underlined between proof calculi and model building methods. Different calculi for firstorder logic (resolution, tableaux, connection method (matings), : : : ) are potentially able to prove the same class of theorems. In papers on model building methods what is emphasized in general is the class of models that the methods can (or cannot...
Automated Model Building as Future Research Topic
"... . Zabel, and N. Peltier [CZ91, CZ92, CP95b] and J. Slaney [Sla92, Sla93]. An earlier approach by S. Winker [Win82], although practically relevant and successful, did not define a general algorithmic method. Tammet's approach, like ours, is based on resolution decision procedures. His method of finit ..."
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. Zabel, and N. Peltier [CZ91, CZ92, CP95b] and J. Slaney [Sla92, Sla93]. An earlier approach by S. Winker [Win82], although practically relevant and successful, did not define a general algorithmic method. Tammet's approach, like ours, is based on resolution decision procedures. His method of finite Model Building applies to the monadic and Ackermann class and is based on the termination of an ordering refinement. In the resulting model description the interpretation of the function symbols is given completely, but the interpretation of predicate symbols is only partial. Moreover, Tammet uses narrowing and works with equations on the object language level. In [FL95] Model Building is based on termination sets for hyperresolution (which yield other decision classes). Finite Model Building is performed as postprocessing step and is based on the transformation of Herbrand models; it does not use equality reasoning but filtration. Caferra and Zabel define an equational extension
Automated Equational Deduction with Otter
, 1995
"... Contents 1 Introduction 1 2 Otter and MACE 3 2.1 Otter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.1.1 Notes on Otter Proof Notation : : : : : : : : : : : : : : : 3 2.2 MACE : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 Test Chapter 3 3 Lattices a ..."
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Contents 1 Introduction 1 2 Otter and MACE 3 2.1 Otter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.1.1 Notes on Otter Proof Notation : : : : : : : : : : : : : : : 3 2.2 MACE : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 Test Chapter 3 3 Lattices and Latticelike Structures 9 4 The Rule (gL) 23 4.1 Problems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 23 4.2 Sample Figures : : : : : : : : : : : : : : : : : : : : : : : : : : : : 44 5 Quasigroups 51 6 Semigroups 57 6.1 A Conjecture of Padmanabhan : : : : : : : : : : : : : : : : : : : 57 7 Groups 69 7.1 SelfDual Bases for Group Theory : : : : : : : : : : : : : : : : : 69 8 TC and RC 73 9 Problems not yet placed in the proper chapter 83 iii iv CONTENTS List
Automated ModelBuilding for FirstOrderLogic with Equality
"... Introduction Although automated model builing is a new branch in the field of automated deduction, the central problem is quite old and attracted attention by prominent scientists in the last three centuries. The aim of the new logic mathesis universalis formulated by G. W. Leibniz (1646 1716) i ..."
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Introduction Although automated model builing is a new branch in the field of automated deduction, the central problem is quite old and attracted attention by prominent scientists in the last three centuries. The aim of the new logic mathesis universalis formulated by G. W. Leibniz (1646 1716) in the late 17 th century was a threefold one: (i) Construction of a universal alphabetcalled characteristica universalis, whose characters should allow the representation of all other conceptions. (ii) Evolving of a calculus ratiocinator, a calculus provided to deal with the aforementioned universal characters in a purely mechanical way. And eventually (iii) an algorithmars iudicandito decide for arbitrary sentences, given through the tokens of the mathesis universalis, their validity. Leibniz' main vision was the use of the calculus ratiocinator to eliminate misleadi
Automated Deduction in Equational Logic and Geometry
, 1995
"... Algebras, pages 263 297. Pergamon Press, Oxford, U.K., 1970. [24] K. Kunen. Single axioms for groups. J. Automated Reasoning, 9(3):291308, 1992. [25] H. Lakser, R. Padmanabhan, and C. R. Platt. Subdirect decomposition of P/lonka sums. Duke Math. J., 39(3):485488, 1972. [26] A. I. Mal'cev. ..."
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Algebras, pages 263 297. Pergamon Press, Oxford, U.K., 1970. [24] K. Kunen. Single axioms for groups. J. Automated Reasoning, 9(3):291308, 1992. [25] H. Lakser, R. Padmanabhan, and C. R. Platt. Subdirect decomposition of P/lonka sums. Duke Math. J., 39(3):485488, 1972. [26] A. I. Mal'cev. Uber die Einbettung von assoziativen Systemen Gruppen I. Mat. Sbornik, 6(48):331336, 1939. [27] B. Mazur. Arithmetic on curves. Bull. AMS, 14:207259, 1986. [28] J. McCharen, R. Overbeek, and L. Wos. Complexity and related enhancements for automated theoremproving programs. Computers and Math. Applic., 2:116, 1976. [29] J. McCharen, R. Overbeek, and L. Wos. Problems and experiments for and with automated theoremproving programs. IEEE Trans. on Computers, C25(8):773782, August 1976. [30] W. McCune. Automated discovery of new axiomatizations of the left group and right group calculi. J. Automated Reasoning, 9(1):124, 1992. [31] W. McCune. Single axioms for groups and Abelian g...
Automated reasoning: Real uses and . . .
"... An automated reasoning program has provided invaluable assistance in answering certain previously open questions in mathematics and in formal logic. These questions would not have been answered, at least by those who obtained the results, were it not for the program's contribution. Others have used ..."
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An automated reasoning program has provided invaluable assistance in answering certain previously open questions in mathematics and in formal logic. These questions would not have been answered, at least by those who obtained the results, were it not for the program's contribution. Others have used such a program to design logic circuits, many of which proved superior (with respect to transistor count) to the existing designs, and to validate the design of other circuits. These successes establish the value of an automated reasoning program for research and suggest the value for practical applications. We thus conclude that the field of automated reasoning is on the verge of becoming one of the more significant branches of computer science. Further, we conclude that the field has already advanced from stage 1, that of potential usefulness, to stage 2, that of actual usefulness. To pass to stage 3, that of wide acceptance and use, requires, among other things, easy access to an automated reasoning program and an understanding of the various aspects of automated reasoning. In fact, an automated reasoning program is available that is portable and can be run on relatively inexpensive machines. Moreover, a system exists for producing a reasoning program tailored to given specifications. As for the requirement of understanding the aspects of automated reasoning, much research remainsâ€”research aided by access to a reasoning program. A large obstacle has thus been removed, permitting many to experiment with and find uses for a computer program that can be relied upon as a most valuable automated reasoning assistant.
Elsevier Logic and artificial intelligence
, 1989
"... Nilsson, N.J., Logic and artificial intelligence, Artificial Intelligence 47 (1990) 3156. The theoretical foundations of the logical approach to artificial intelligence are presented. Logical languages are widely used for expressing the declarative knowledge needed in artificial intelligence system ..."
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Nilsson, N.J., Logic and artificial intelligence, Artificial Intelligence 47 (1990) 3156. The theoretical foundations of the logical approach to artificial intelligence are presented. Logical languages are widely used for expressing the declarative knowledge needed in artificial intelligence systems. Symbolic logic also provides a clear semantics for knowledge representation languages and a methodology for analyzing and comparing deductive inference techniques. Several observations gained from experience with the approach are discussed. Finally, we confront some challenging problems for artificial intelligence and describe what is being done in an attempt to solve them. 1.