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Controlled Integrations of the Cut Rule into Connection Tableau Calculi
"... In this paper techniques are developed and compared which increase the inferential power of tableau systems for classical firstorder logic. The mechanisms are formulated in the framework of connection tableaux, which is an amalgamation of the connection method and the tableau calculus, and a genera ..."
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Cited by 61 (3 self)
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In this paper techniques are developed and compared which increase the inferential power of tableau systems for classical firstorder logic. The mechanisms are formulated in the framework of connection tableaux, which is an amalgamation of the connection method and the tableau calculus, and a generalization of model elimination. Since connection tableau calculi are among the weakest proof systems with respect to proof compactness, and the (backward) cut rule is not suitable for the firstorder case, we study alternative methods for shortening proofs. The techniques we investigate are the folding up and the folding down operation. Folding up represents an efficient way of supporting the basic calculus, which is topdown oriented, with lemmata derived in a bottomup manner. It is shown that both techniques can also be viewed as controlled integrations of the cut rule. In order to remedy the additional redundancy imported into tableau proof procedures by the new inference rules, we develop and apply an extension of the regularity condition on tableaux and the mechanism of antilemmata which realizes a subsumption concept on tableaux. Using the framework of the theorem prover SETHEO, we have implemented three new proof procedures which overcome the deductive weakness of cutfree tableau systems. Experimental results demonstrate the superiority of the systems with folding up over the cutfree variant and the one with folding down.
Analytic tableaux
 Automated Deduction: A Basis for Applications, volume I, chapter 1
, 1998
"... The aim of this chapter is twofold: first, introducing the basic concepts of analytic tableaux and, secondly, presenting stateoftheart techniques for using nonclausal tableaux in automated deduction. ..."
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Cited by 17 (9 self)
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The aim of this chapter is twofold: first, introducing the basic concepts of analytic tableaux and, secondly, presenting stateoftheart techniques for using nonclausal tableaux in automated deduction.
Investigating a general hierarchy of polynomially decidable classes of CNF's based on short treelike resolution proofs
, 1999
"... We investigate a hierarchy Gk (U ; S) of classes of conjunctive normal forms, recognizable and SATdecidable in polynomial time, with special emphasize on the corresponding hardness parameter hU ;S (F ) for clausesets F (the first level of inclusion). At level 0 an (incomplete, polytime) oracl ..."
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Cited by 16 (10 self)
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We investigate a hierarchy Gk (U ; S) of classes of conjunctive normal forms, recognizable and SATdecidable in polynomial time, with special emphasize on the corresponding hardness parameter hU ;S (F ) for clausesets F (the first level of inclusion). At level 0 an (incomplete, polytime) oracle U for unsatisfiability detection and an oracle S for satisfiability detection is used. The hierarchy from [Pretolani 96] is improved in this way with respect to strengthened satisfiability handling, simplified recognition and consistent relativization. Also a hierarchy of canonical polytime reductions with Unitclause propagation at the first level is obtained. General methods for upper and lower bounds on hU ;S (F ) are developed and applied to a number of wellknown examples. hU ;S (F ) admits several different characterizations, including the space complexity of treelike resolution and the use of pebble games as in [Esteban, Tor'an 99]. Using for S the class of linearly sat...
Completeness for Linear Regular Negation Normal Form Inference Systems
, 1997
"... . Completeness proofs that generalize the AndersonBledsoe excess literal argument are developed for calculi other than resolution. A simple proof of the completeness of regular, connected tableaux for formulas in conjunctive normal form (CNF) is presented. These techniques also provide completeness ..."
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Cited by 8 (3 self)
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. Completeness proofs that generalize the AndersonBledsoe excess literal argument are developed for calculi other than resolution. A simple proof of the completeness of regular, connected tableaux for formulas in conjunctive normal form (CNF) is presented. These techniques also provide completeness results for some inference mechanisms that do not rely on clause form. In particular, the completeness of regular, connected tableaux for formulas in negation normal form (NNF), and the completeness of NCresolution under a linear restriction, are established. Keywords: Logic for Artificial Intelligence, completeness, tableau method, resolution, nonclausal inference, negation normal form ? This research was supported in part by the National Science Foundation under grants CCR9404338 and CCR9504349. 1 Introduction Robinson developed semantic tree arguments to provide completeness proofs for resolution and related inference systems; these arguments are not entirely transparent. The exc...
On AntiLinks
, 1994
"... . The concept of antilink is defined, and useful equivalencepreserving operations on propositional formulas based on antilinks are introduced. These operations eliminate a potentially large number of subsumed paths in a negation normal form formula. The operations have linear time complexity in ..."
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Cited by 7 (7 self)
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. The concept of antilink is defined, and useful equivalencepreserving operations on propositional formulas based on antilinks are introduced. These operations eliminate a potentially large number of subsumed paths in a negation normal form formula. The operations have linear time complexity in the size of that part of the formula containing the antilink. These operations are useful for prime implicant/implicate algorithms because most of the computational effort in such algorithms is spent on subsumption checks. 1. Introduction Many algorithms have been proposed to compute the prime implicates of propositional boolean formula. Most algorithms [1,2,3,4,14] assume that the input is either in conjunctive normal form (CNF) or in disjunctive normal form (DNF). Other algorithms [10] require the input to be a conjunction of DNF formulas. In [12], a set of techniques for finding the prime implicates of formulas in negation normal form (NNF) is proposed. Those techniques are based on dis...
A Disjunctive Positive Refinement of Model Elimination and its Application to Subsumption Deletion
 Journal of Automated Reasoning
, 1995
"... The Model Elimination (ME) calculus is a refutational complete, goaloriented calculus for firstorder clause logic. In this paper, we introduce a new variant called disjunctive positive ME (DPME); it improves on Plaisted's positive refinement of ME in that reduction steps are allowed only with p ..."
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Cited by 5 (0 self)
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The Model Elimination (ME) calculus is a refutational complete, goaloriented calculus for firstorder clause logic. In this paper, we introduce a new variant called disjunctive positive ME (DPME); it improves on Plaisted's positive refinement of ME in that reduction steps are allowed only with positive literals stemming from disjunctive clauses.
Tableaux, Path Dissolution, and Decomposable Negation Normal Form for Knowledge Compilation
"... Decomposable negation normal form (DNNF) was developed primarily for knowledge compilation. Formulas in DNNF are linkless, in negation normal form (NNF), and have the property that atoms are not shared across conjunctions. Full dissolvents are linkless NNF formulas that do not in general have the la ..."
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Cited by 2 (0 self)
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Decomposable negation normal form (DNNF) was developed primarily for knowledge compilation. Formulas in DNNF are linkless, in negation normal form (NNF), and have the property that atoms are not shared across conjunctions. Full dissolvents are linkless NNF formulas that do not in general have the latter property. However, many of the applications of DNNF can be obtained with full dissolvents. Two additional methods — regular tableaux and semantic factoring — are shown to produce equivalent DNNF. A class of formulas is presented on which earlier DNNF conversion techniques are necessarily exponential; path dissolution and semantic factoring handle these formulas in linear time.
Semantic Tableaux with Ordering Restrictions
 Proc. 12th Conference on Automated Deduction CADE, Nancy/France, LNAI 814
, 1994
"... . The aim of this paper is to make restriction strategies based on orderings of the Herbrand universe available for semantic tableaulike calculi as well. A marriage of tableaux and ordering restriction strategies seems to be most promising in applications where generation of counter examples is ..."
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Cited by 2 (2 self)
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. The aim of this paper is to make restriction strategies based on orderings of the Herbrand universe available for semantic tableaulike calculi as well. A marriage of tableaux and ordering restriction strategies seems to be most promising in applications where generation of counter examples is required. In this paper, starting out from semantic trees, we develop a formal tool called refutation graphs, which (i) serves as a basis for completeness proofs of both resolution and tableaux, and (ii) is compatible with socalled Aordering restrictions. The main result is a firstorder ground tableau procedure complete for Aordering restrictions. Introduction In recent years one could observe a kind of renaissance of tableaurelated methods in automated theorem proving after the field has been dominated by resolution approaches for many years 2 . Tableaux are easy to adjust to nonclassical logics, and they have already a number of advantages for classical firstorder logic that...