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Knowledge Representation and Classical Logic
, 2007
"... Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspe ..."
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Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspective of knowledge representation, their formal languages were not sufficiently expressive. On the other hand, most logicians were not concerned about the possibility of automated reasoning; from the perspective of knowledge representation, they were often too generous in the choice of syntactic constructs. In spite of these differences, classical mathematical logic has exerted significant influence on knowledge representation research, and it is appropriate to begin this handbook with a discussion of the relationship between these fields. The language of classical logic that is most widely used in the theory of knowledge representation is the language of firstorder (predicate) formulas. These are the formulas that John McCarthy proposed to use for representing declarative knowledge in his advice taker paper [176], and Alan Robinson proposed to prove automatically using resolution [236]. Propositional logic is, of course, the most important subset of firstorder logic; recent
A focusing inverse method theorem prover for firstorder linear logic
 In Proceedings of CADE20
, 2005
"... Abstract. We present the theory and implementation of a theorem prover forfirstorder intuitionistic linear logic based on the inverse method. The central prooftheoretic insights underlying the prover concern resource management andfocused derivations, both of which are traditionally understood in ..."
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Abstract. We present the theory and implementation of a theorem prover forfirstorder intuitionistic linear logic based on the inverse method. The central prooftheoretic insights underlying the prover concern resource management andfocused derivations, both of which are traditionally understood in the domain of backward reasoning systems such as logic programming. We illustrate how resource management, focusing, and other intrinsic properties of linear connectives affect the basic forward operations of rule application, contraction, and forwardsubsumption. We also present some preliminary experimental results obtained with our implementation.
Imogen: Focusing the Polarized Inverse Method for Intuitionistic Propositional Logic
"... Abstract. In this paper we describe Imogen, a theorem prover for intuitionistic propositional logic using the focused inverse method. We represent finegrained control of the search behavior by polarizing the input formula. In manipulating the polarity of atoms and subformulas, we can often improve ..."
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Cited by 8 (4 self)
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Abstract. In this paper we describe Imogen, a theorem prover for intuitionistic propositional logic using the focused inverse method. We represent finegrained control of the search behavior by polarizing the input formula. In manipulating the polarity of atoms and subformulas, we can often improve the search time by several orders of magnitude. We tested our method against seven other systems on the propositional fragment of the ILTP library. We found that our prover outperforms all other provers on a substantial subset of the library. 1
Efficient Intuitionistic Theorem Proving with the Polarized Inverse Method
"... Abstract. The inverse method is a generic proof search procedure applicable to nonclassical logics satisfying cut elimination and the subformula property. In this paper we describe a general architecture and several highlevel optimizations that enable its efficient implementation. Some of these re ..."
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Abstract. The inverse method is a generic proof search procedure applicable to nonclassical logics satisfying cut elimination and the subformula property. In this paper we describe a general architecture and several highlevel optimizations that enable its efficient implementation. Some of these rely on logicspecific properties, such as polarization and focusing, which have been shown to hold in a wide range of nonclassical logics. Others, such as rule subsumption and recursive backward subsumption apply in general. We empirically evaluate our techniques on firstorder intuitionistic logic with our implementation Imogen and demonstrate a substantial improvement over all other existing intuitionistic theorem provers on problems from the ILTP problem library. 1
Bidirectional Decision Procedures for the Intuitionistic Propositional Modal Logic IS4
"... Abstract. We present a multicontext focused sequent calculus whose derivations are in bijective correspondence with normal natural deductions in the propositional fragment of the intuitionistic modal logic IS4. This calculus, suitable for the enumeration of normal proofs, is the starting point for ..."
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Abstract. We present a multicontext focused sequent calculus whose derivations are in bijective correspondence with normal natural deductions in the propositional fragment of the intuitionistic modal logic IS4. This calculus, suitable for the enumeration of normal proofs, is the starting point for the development of a sequent calculusbased bidirectional decision procedure for propositional IS4. In this system, relevant derived inference rules are constructed in a forward direction prior to proof search, while derivations constructed using these derived rules are searched over in a backward direction. We also present a variant which searches directly over normal natural deductions. Experimental results show that on most problems, the bidirectional prover is competitive with both conventional backward provers using loopdetection and inverse method provers, significantly outperforming them in a number of cases. 1
Magically Constraining the Inverse Method Using Dynamic Polarity Assignment
, 2010
"... Abstract. Given a logic program that is terminating and modecorrect in an idealized Prolog interpreter (i.e., in a topdown logic programming engine), a bottomup logic programming engine can be used to compute exactly the same set of answers as the topdown engine for a given modecorrect query by ..."
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Abstract. Given a logic program that is terminating and modecorrect in an idealized Prolog interpreter (i.e., in a topdown logic programming engine), a bottomup logic programming engine can be used to compute exactly the same set of answers as the topdown engine for a given modecorrect query by rewriting the program and the query using the Magic Sets Transformation (MST). In previous work, we have shown that focusing can logically characterize the standard notion of bottomup logic programming if atomic formulas are statically given a certain polarity assignment. In an analogous manner, dynamically assigning polarities can characterize the effect of MST without needing to transform the program or the query. This gives us a new proof of the completeness of MST in purely logical terms, by using the general completeness theorem for focusing. As the dynamic assignment is done in a general logic, the essence of MST can potentially be generalized to larger fragments of logic. 1
Polarized Inverse Method
"... Abstract. The inverse method is a generic proof search procedure applicable to nonclassical logics satisfying cut elimination and the subformula property. In this paper we describe a general architecture and several highlevel optimizations that enable its efficient implementation. Some of these re ..."
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Abstract. The inverse method is a generic proof search procedure applicable to nonclassical logics satisfying cut elimination and the subformula property. In this paper we describe a general architecture and several highlevel optimizations that enable its efficient implementation. Some of these rely on logicspecific properties, such as polarization and focusing, which have been shown to hold in a wide range of nonclassical logics. Others, such as rule subsumption and recursive backward subsumption apply in general. We empirically evaluate our techniques on firstorder intuitionistic logic with our implementation Imogen and demonstrate a substantial improvement over all other existing intuitionistic theorem provers on problems from the ILTP problem library. 1
Practical Automated Theorem Proving with the Polarized Inverse Method
, 2009
"... Mechanically checked, formal reasoning as envisioned by Leibniz has in the past few decades become a reality. Proof assistants are programs that can rigorously and mechanically check the details of logical arguments. Such formal reasoning has applications to nearly all areas of computer science, fro ..."
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Mechanically checked, formal reasoning as envisioned by Leibniz has in the past few decades become a reality. Proof assistants are programs that can rigorously and mechanically check the details of logical arguments. Such formal reasoning has applications to nearly all areas of computer science, from formal proof in mathematics to specifying and verifying critical properties of hardware, software systems, and security protocols. Twelf is a proof assistant specialized for reasoning about deductive systems such as logics and programming languages. In its domain, it is one of the most powerful tools available. Significant Twelf developments cover many different application areas. For example, in the theory of programming languages, the formalization and proof of type safety for Standard ML represents the first machine verified proofs of important properties of a fullfeatured programming language. Proof carrying authentication frameworks based on Twelf allow for computer security based on formal proof. Proof carrying code and typed assembly languages also based on Twelf allow users to check properties of programs such as memory safety before they are executed. The combination of higher order abstract syntax for representing structures with variable binding, the Elf logic programming interpretation, and the M + 2 metalogic for reasoning about
A logical characterization of Forward and Backward . . .
, 2008
"... The inverse method is a generalization of resolution that can be applied to nonclassical logics. We have recently shown how Andreoli’s focusing strategy can be adapted for the inverse method in linear logic. In this paper we introduce the notion of focusing bias for atoms and show that it gives ris ..."
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The inverse method is a generalization of resolution that can be applied to nonclassical logics. We have recently shown how Andreoli’s focusing strategy can be adapted for the inverse method in linear logic. In this paper we introduce the notion of focusing bias for atoms and show that it gives rise to forward and backward chaining, generalizing both hyperresolution (forward) and SLD resolution (backward) on the Horn fragment. A key feature of our characterization is the structural, rather than purely operational, explanation for forward and backward chaining. A search procedure like the inverse method is thus able to perform both operations as appropriate, even simultaneously. We also present experimental results and an evaluation of the practical benefits of biased atoms for a number of examples from different problem domains.