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PartitionBased Logical Reasoning for FirstOrder and Propositional Theories
 Artificial Intelligence
, 2000
"... In this paper we provide algorithms for reasoning with partitions of related logical axioms in propositional and firstorder logic (FOL). We also provide a greedy algorithm that automatically decomposes a set of logical axioms into partitions. Our motivation is twofold. First, we are concerned with ..."
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Cited by 52 (8 self)
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In this paper we provide algorithms for reasoning with partitions of related logical axioms in propositional and firstorder logic (FOL). We also provide a greedy algorithm that automatically decomposes a set of logical axioms into partitions. Our motivation is twofold. First, we are concerned with how to reason e#ectively with multiple knowledge bases that have overlap in content. Second, we are concerned with improving the e#ciency of reasoning over a set of logical axioms by partitioning the set with respect to some detectable structure, and reasoning over individual partitions. Many of the reasoning procedures we present are based on the idea of passing messages between partitions. We present algorithms for reasoning using forward messagepassing and using backward messagepassing with partitions of logical axioms. Associated with each partition is a reasoning procedure. We characterize a class of reasoning procedures that ensures completeness and soundness of our messagepassing ...
Theorem proving with structured theories (full report
, 2001
"... Motivated by the problem of query answering over multiple structured commonsense theories, we exploit graphbased techniques to improve the efficiency of theorem proving for structured theories. Theories are organized into subtheories that are minimally connected by the literals they share. We prese ..."
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Cited by 26 (5 self)
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Motivated by the problem of query answering over multiple structured commonsense theories, we exploit graphbased techniques to improve the efficiency of theorem proving for structured theories. Theories are organized into subtheories that are minimally connected by the literals they share. We present messagepassing algorithms that reason over these theories using consequence finding, specializing our algorithms for the case of firstorder resolution, and for batch and concurrent theorem proving. We provide an algorithm that restricts the interaction between subtheories by exploiting the polarity of literals. We attempt to minimize the reasoning within each individual partition by exploiting existing algorithms for focused incremental and general consequence finding. Finally, we propose an algorithm that compiles each subtheory into one in a reduced sublanguage. We have proven the soundness and completeness of all of these algorithms. 1
Deduction Systems Based on Resolution
, 1991
"... A general theory of deduction systems is presented. The theory is illustrated with deduction systems based on the resolution calculus, in particular with clause graphs. This theory distinguishes four constituents of a deduction system: ffl the logic, which establishes a notion of semantic entailmen ..."
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Cited by 19 (0 self)
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A general theory of deduction systems is presented. The theory is illustrated with deduction systems based on the resolution calculus, in particular with clause graphs. This theory distinguishes four constituents of a deduction system: ffl the logic, which establishes a notion of semantic entailment; ffl the calculus, whose rules of inference provide the syntactic counterpart of entailment; ffl the logical state transition system, which determines the representation of formulae or sets of formulae together with their interrelationships, and also may allow additional operations reducing the search space; ffl the control, which comprises the criteria used to choose the most promising from among all applicable inference steps. Much of the standard material on resolution is presented in this framework. For the last two levels many alternatives are discussed. Appropriately adjusted notions of soundness, completeness, confluence, and Noetherianness are introduced in order to characterize...
A maximalliteral unit strategy for Horn clauses
 In Proc. CTRS90
, 1991
"... A new positiveunit theoremproving procedure for equational Horn clauses is presented. It uses a term ordering to restrict paxamodulation to potentially maximal sides of equations. Completeness is shown using proof orderings. 1. ..."
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Cited by 16 (0 self)
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A new positiveunit theoremproving procedure for equational Horn clauses is presented. It uses a term ordering to restrict paxamodulation to potentially maximal sides of equations. Completeness is shown using proof orderings. 1.
OrderingBased Strategies for Horn Clauses
, 1991
"... Two new theoremproving procedures for equational Horn clauses are presented. The largest literal is selected for paramodulation in both strategies, except that one method treats positive literals as larger than negative ones and results in a unit strategy. Both use term orderings to restrict paramo ..."
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Cited by 11 (3 self)
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Two new theoremproving procedures for equational Horn clauses are presented. The largest literal is selected for paramodulation in both strategies, except that one method treats positive literals as larger than negative ones and results in a unit strategy. Both use term orderings to restrict paramodulation to potentially maximal sides of equations and to increase the amount of allowable simplification (demodulation). Completeness is shown using proof orderings.
The Complexity of Automated Reasoning
, 1989
"... This thesis explores the relative complexity of proofs produced by the automatic theorem proving procedures of analytic tableaux, linear resolution, the connection method, tree resolution and the DavisPutnam procedure. It is shown that tree resolution simulates the improved tableau procedure and th ..."
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Cited by 9 (0 self)
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This thesis explores the relative complexity of proofs produced by the automatic theorem proving procedures of analytic tableaux, linear resolution, the connection method, tree resolution and the DavisPutnam procedure. It is shown that tree resolution simulates the improved tableau procedure and that SLresolution and the connection method are equivalent to restrictions of the improved tableau method. The theorem by Tseitin that the DavisPutnam Procedure cannot be simulated by tree resolution is given an explicit and simplified proof. The hard examples for tree resolution are contradictions constructed from simple Tseitin graphs.
A Precondition Prover for Analogy
 BioSystems
, 1990
"... We describe here a prover PC that normally acts as an ordinary theorem prover, but which returns a "precondition" when it is unable to prove the given formula. If F is the formula attempted to be proved and PC returns the precondition Q, then (Q \Gamma! F ) is a theorem (that PC can prove). This pr ..."
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Cited by 8 (0 self)
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We describe here a prover PC that normally acts as an ordinary theorem prover, but which returns a "precondition" when it is unable to prove the given formula. If F is the formula attempted to be proved and PC returns the precondition Q, then (Q \Gamma! F ) is a theorem (that PC can prove). This prover, PC, uses a ProofPlan. In its simplest mode, when there is no proofplan, it acts like ordinary Abduction. We show here how this method can be used to derive certain proofs by analogy. To do this, it uses a proofplan from a given guiding proof to help construct the proof of a similar theorem, by "debugging" (automatically) that proofplan. We show here the analogy proofs of a few simple example theorems and one hard pair, Ex4 and Ex4L. The given proofplan for Ex4 is used by the system to prove automatically Ex4; and that same proofplan is then used to prove Ex4L, during which the proofplan is "debugged" (automatically). These two examples are similar to two other, more difficult, t...
Deciding the guarded fragments by resolution
 Journal of Symbolic Computation
, 2003
"... The guarded fragment is a fragment of firstorder logic that has been introduced for two main reasons: First, to explain the good computational and logical behavior of propositional modal logics. Second, to serve as a breeding ground for wellbehaved process logics. In this paper we give resolution ..."
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Cited by 4 (2 self)
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The guarded fragment is a fragment of firstorder logic that has been introduced for two main reasons: First, to explain the good computational and logical behavior of propositional modal logics. Second, to serve as a breeding ground for wellbehaved process logics. In this paper we give resolutionbased decision procedures for the guarded fragment and for the loosely guarded fragment (sometimes also called pairwise guarded fragment). By constructing an implementable decision procedure for the guarded fragment and for the loosely guarded fragment, we obtain an effective procedure for deciding modal logics that can be embedded into these fragments. The procedures have been implemented in the theorem prover Bliksem. 1.
The Control Layer in Open Mechanized Reasoning Systems: Annotations and Tactics
, 2000
"... We are interested in developing a methodology for integrating mechanized reasoning systems such as Theorem Provers, Computer Algebra Systems, and Model Checkers. Our approach is to provide a framework for specifying mechanized reasoning systems and to use specifications as a starting point for integ ..."
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Cited by 3 (1 self)
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We are interested in developing a methodology for integrating mechanized reasoning systems such as Theorem Provers, Computer Algebra Systems, and Model Checkers. Our approach is to provide a framework for specifying mechanized reasoning systems and to use specifications as a starting point for integration. We build on top of the work presented in Giunchiglia et al. (1994) which introduces the notion of Open Mechanized Reasoning Systems (OMRS) as a specification framework for integrating reasoning systems. An OMRS specification consists of three components: the logic component, the control component, and the interaction component. In this paper we focus on the control level. We propose to specify the control component by first adding control knowledge to the data structures representing the logic by means of annotations and then by specifying proof strategies via tactics. To show the adequacy of the approach we present and discuss a structured specification of constraint contextual rewriting as a set of cooperating specialized reasoning modules.