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32
A Judgmental Analysis of Linear Logic
, 2003
"... We reexamine the foundations of linear logic, developing a system of natural deduction following MartinL of's separation of judgments from propositions. Our construction yields a clean and elegant formulation that accounts for a rich set of multiplicative, additive, and exponential connectives, ext ..."
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Cited by 49 (27 self)
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We reexamine the foundations of linear logic, developing a system of natural deduction following MartinL of's separation of judgments from propositions. Our construction yields a clean and elegant formulation that accounts for a rich set of multiplicative, additive, and exponential connectives, extending dual intuitionistic linear logic but differing from both classical linear logic and Hyland and de Paiva's full intuitionistic linear logic. We also provide a corresponding sequent calculus that admits a simple proof of the admissibility of cut by a single structural induction. Finally, we show how to interpret classical linear logic (with or without the MIX rule) in our system, employing a form of doublenegation translation.
Focusing and Polarization in Linear, Intuitionistic, and Classical Logics
, 2009
"... A focused proof system provides a normal form to cutfree proofs in which the application of invertible and noninvertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cutfree proofs. Within intuitioni ..."
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Cited by 43 (18 self)
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A focused proof system provides a normal form to cutfree proofs in which the application of invertible and noninvertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cutfree proofs. Within intuitionistic and classical logics, there are various different proof systems in the literature that exhibit focusing behavior. These focused proof systems have been applied to both the proof search and the proof normalization approaches to computation. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other intuitionistic proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing. We also use LJF to design a focused proof system LKF for classical logic. Our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in Girard’s LC and LU proof systems.
Focusing the inverse method for linear logic
 Proceedings of CSL 2005
, 2005
"... 1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10 ..."
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Cited by 37 (11 self)
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1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10
Focusing and polarization in intuitionistic logic
 CSL 2007: Computer Science Logic, volume 4646 of LNCS
, 2007
"... dale.miller at inria.fr Abstract. A focused proof system provides a normal form to cutfree proofs that structures the application of invertible and noninvertible inference rules. The focused proof system of Andreoli for linear logic has been applied to both the proof search and the proof normaliza ..."
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Cited by 32 (14 self)
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dale.miller at inria.fr Abstract. A focused proof system provides a normal form to cutfree proofs that structures the application of invertible and noninvertible inference rules. The focused proof system of Andreoli for linear logic has been applied to both the proof search and the proof normalization approaches to computation. Various proof systems in literature exhibit characteristics of focusing to one degree or another. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing. We also use LJF to design a focused proof system for classical logic. Our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in Girard’s LC and LU proof systems. 1
A Concurrent Logical Framework: The Propositional Fragment
, 2003
"... We present the propositional fragment CLF0 of the Concurrent Logical Framework (CLF). CLF extends the Linear Logical Framework to allow the natural representation of concurrent computations in an object language. The underlying type theory uses monadic types to segregate values from computations ..."
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Cited by 31 (3 self)
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We present the propositional fragment CLF0 of the Concurrent Logical Framework (CLF). CLF extends the Linear Logical Framework to allow the natural representation of concurrent computations in an object language. The underlying type theory uses monadic types to segregate values from computations. This separation leads to a tractable notion of definitional equality that identifies computations di#ering only in the order of execution of independent steps. From a logical point of view our type theory can be seen as a novel combination of lax logic and dual intuitionistic linear logic. An encoding of a small Petri net exemplifies the representation methodology, which can be summarized as "concurrent computations as monadic expressions ".
Combining Derivations and Refutations for Cutfree Completeness in BiIntuitionistic Logic
, 2008
"... Biintuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus has recently been shown to fail cutelimination. We present a new cutfree se ..."
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Cited by 7 (0 self)
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Biintuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus has recently been shown to fail cutelimination. We present a new cutfree sequent calculus for biintuitionistic logic, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between intuitionistic implication and dual intuitionistic exclusion, similarly to future and past modalities in tense logic. Our calculus handles this interaction using derivations and refutations as first class citizens. We employ extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of refutations, and rules which compose certain refutations and derivations to form derivations. Automated deduction using terminating backward search is also possible, although this is not our main purpose. 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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Cited by 6 (3 self)
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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
On focusing and polarities in linear logic and intuitionistic logic
, 2006
"... There are a number of cutfree sequent calculus proof systems known that are complete for firstorder intuitionistic logic. Proofs in these different systems can vary a great deal from one another. We are interested in providing a flexible and unifying framework that can collect together important a ..."
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Cited by 6 (2 self)
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There are a number of cutfree sequent calculus proof systems known that are complete for firstorder intuitionistic logic. Proofs in these different systems can vary a great deal from one another. We are interested in providing a flexible and unifying framework that can collect together important aspects of many of these proof systems. First, we suggest that one way to unify these proof systems is to first translate intuitionistic logic formulas into linear logic formulas, then assign a bias (positive or negative) to atomic formulas, and then examine the nature of focused proofs in the resulting linear logic setting. Second, we provide a single focusing proof system for intuitionistic logic and show that these other intuitionistic proof systems can be accounted for by assigning bias to atomic formulas and by inserting certain markers that halt focusing on formulas. Using either approach, we are able to account for proof search mechanisms that allow for forwardchaining (programdirected search), backwardchaining (goaldirected search), and combinations of these two. The keys to developing this kind of proof system for intuitionistic logic involves using Andreoli’s completeness result for focusing proofs and Girard’s notion of polarity used in his LC and LU proof systems. 1
A cutfree sequent calculus for biintuitionistic logic: extended version
, 2007
"... Abstract. Biintuitionistic logic is the extension of intuitionistic logic with a connective dual to implication. Biintuitionistic logic was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus for BiInt has recently been s ..."
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Cited by 5 (1 self)
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Abstract. Biintuitionistic logic is the extension of intuitionistic logic with a connective dual to implication. Biintuitionistic logic was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus for BiInt has recently been shown by Uustalu to fail cutelimination. We present a new cutfree sequent calculus for BiInt, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between implication and its dual, similarly to future and past modalities in tense logic. Our calculus handles this interaction using extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of failed derivation trees. Our simple termination argument allows our calculus to be used for automated deduction, although this is not its main purpose. 1
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 5 (3 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