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26
A New Deconstructive Logic: Linear Logic
, 1995
"... The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different a ..."
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Cited by 102 (11 self)
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The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's , FD ([9, 11, 27, 31]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of `programmingwithproofs' ([22, 23]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for nonadditive proof nets, to be precise) using appropriate embeddings (socalled decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making. A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these `deconstructi...
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 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
Computation with classical sequents
 MATHEMATICAL STRUCTURES OF COMPUTER SCIENCE
, 2008
"... X is an untyped continuationstyle formal language with a typed subset which provides a CurryHoward isomorphism for a sequent calculus for implicative classical logic. X can also be viewed as a language for describing nets by composition of basic components connected by wires. These features make X ..."
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Cited by 16 (16 self)
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X is an untyped continuationstyle formal language with a typed subset which provides a CurryHoward isomorphism for a sequent calculus for implicative classical logic. X can also be viewed as a language for describing nets by composition of basic components connected by wires. These features make X an expressive platform on which algebraic objects and many different (applicative) programming paradigms can be mapped. In this paper we will present the syntax and reduction rules for X and in order to demonstrate the expressive power of X, we will show how elaborate calculi can be embedded, like the λcalculus, Bloo and Rose’s calculus of explicit substitutions λx, Parigot’s λµ and Curien and Herbelin’s λµ ˜µ.
From proofs to focused proofs: a modular proof of focalization in linear logic
 CSL 2007: Computer Science Logic, volume 4646 of LNCS
, 2007
"... dale.miller at inria.fr saurin at lix.polytechnique.fr Abstract. Probably the most significant result concerning cutfree sequent calculus proofs in linear logic is the completeness of focused proofs. This completeness theorem has a number of proof theoretic applications — e.g. in game semantics, Lu ..."
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Cited by 16 (6 self)
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dale.miller at inria.fr saurin at lix.polytechnique.fr Abstract. Probably the most significant result concerning cutfree sequent calculus proofs in linear logic is the completeness of focused proofs. This completeness theorem has a number of proof theoretic applications — e.g. in game semantics, Ludics, and proof search — and more computer science applications — e.g. logic programming, callbyname/value evaluation. Andreoli proved this theorem for firstorder linear logic 15 years ago. In the present paper, we give a new proof of the completeness of focused proofs in terms of proof transformation. The proof of this theorem is simple and modular: it is first proved for MALL and then is extended to full linear logic. Given its modular structure, we show how the proof can be extended to larger systems, such as logics with induction. Our analysis of focused proofs will employ a proof transformation method that leads us to study how focusing and cut elimination interact. A key component of our proof is the construction of a focalization graph which provides an abstraction over how focusing can be organized within a given cutfree proof. Using this graph abstraction allows us to provide a detailed study of atomic bias assignment in a way more refined that is given in Andreoli’s original proof. Permitting more flexible assignment of bias will allow this completeness theorem to help establish the completeness of a number of other automated deduction procedures. Focalization graphs can be used to justify the introduction of an inference rule for multifocus derivation: a rule that should help us better understand the relations between sequentiality and concurrency in linear logic. 1
A unified sequent calculus for focused proofs
 In LICS: 24th Symp. on Logic in Computer Science
, 2009
"... Abstract—We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical logic, intuitionistic logic, and multiplicativeadditive linear logic are derived as fragments of LKU by increasing the sensitivity of specialize ..."
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Cited by 13 (7 self)
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Abstract—We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical logic, intuitionistic logic, and multiplicativeadditive linear logic are derived as fragments of LKU by increasing the sensitivity of specialized structural rules to polarity information. We develop a unified, streamlined framework for proving cutelimination in the various fragments. Furthermore, each sublogic can interact with other fragments through cut. We also consider the possibility of introducing classicallinear hybrid logics. KeywordsProof theory; focused proof systems; linear logic I.
From X to π; representing the classical sequent calculus
"... Abstract. We study the πcalculus, enriched with pairing and nonblocking input, and define a notion of type assignment that uses the type constructor →. We encode the circuits of the calculus X into this variant of π, and show that all reduction (cutelimination) and assignable types are preserved. ..."
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Cited by 12 (12 self)
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Abstract. We study the πcalculus, enriched with pairing and nonblocking input, and define a notion of type assignment that uses the type constructor →. We encode the circuits of the calculus X into this variant of π, and show that all reduction (cutelimination) and assignable types are preserved. Since X enjoys the CurryHoward isomorphism for Gentzen’s calculus LK, this implies that all proofs in LK have a representation in π.
Confluence and Preservation of Strong Normalisation in an Explicit Substitutions Calculus
, 1995
"... Explicit substitutions calculi are formal systems that implement fireduction by means of an internal substitution operator. Thus, in that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. This feature is useful, f ..."
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Cited by 10 (0 self)
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Explicit substitutions calculi are formal systems that implement fireduction by means of an internal substitution operator. Thus, in that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. This feature is useful, for instance, to represent incomplete proofs in type based proof systems. The oe calculus of explicit substitutions proposed by Abadi, Cardelli, Curien and L'evy gives an elegant way to deal with management of variable names and substitutions of calculus. However, oe does not preserve strong normalisation of calculus and it is not a confluent system. Typed variants of oe without composition are strongly normalising but not confluent, while variants with composition are confluent but do not preserve strong normalisation. Neither of them enjoys both properties. In this paper we propose the i  calculus an we present the full proofs of its main properties. This is, as far as we know, the...
A Focused Approach to Combining Logics
, 2010
"... We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical, intuitionistic, and multiplicativeadditive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structura ..."
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Cited by 7 (5 self)
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We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical, intuitionistic, and multiplicativeadditive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structural rules to polarity information. We identify a general set of criteria under which cut elimination holds in such fragments. From cut elimination we derive a unified proof of the completeness of focusing. Furthermore, each sublogic can interact with other fragments through cut. We examine certain circumstances, for example, in which a classical lemma can be used in an intuitionistic proof while preserving intuitionistic provability. We also examine the possibility of defining classicallinear hybrid logics.
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