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13
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.
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.
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.
Structural focalization
, 2011
"... Focusing, introduced by JeanMarc Andreoli in the context of classical linear logic, defines a normal form for sequent calculus derivations that cuts down on the number of possible derivations by eagerly applying invertible rules and grouping sequences of noninvertible rules. A focused sequent calc ..."
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Cited by 1 (1 self)
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Focusing, introduced by JeanMarc Andreoli in the context of classical linear logic, defines a normal form for sequent calculus derivations that cuts down on the number of possible derivations by eagerly applying invertible rules and grouping sequences of noninvertible rules. A focused sequent calculus is defined relative to some nonfocused sequent calculus; focalization is the property that every nonfocused derivation can be transformed into a focused derivation. In this paper, we present a focused sequent calculus for polarized propositional intuitionistic logic and prove the focalization property relative to a standard presentation of propositional intuitionistic logic. Compared to existing approaches, the proof is quite concise, depending only on the internal soundness and completeness of the focused logic. In turn, both of these properties can be established (and mechanically verified) by structural induction in the style of Pfenning’s structural cut elimination without the need for any tedious and repetitious invertibility lemmas. The proof of cut admissibility for the focused system, which establishes internal soundness, is not particularly novel. The proof of identity expansion, which establishes internal completeness, is the principal contribution of this work. 1
A neutral presentation of synthetic connectives as proof patterns
"... Abstract. It is wellknown that focusing striates a sequent derivation into phases of like polarity where each phase can be seen as inferring a synthetic connective. The calculus of synthetic connectives can be given a uniform presentation by means of neutral proof patterns, with dual polarised inte ..."
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Abstract. It is wellknown that focusing striates a sequent derivation into phases of like polarity where each phase can be seen as inferring a synthetic connective. The calculus of synthetic connectives can be given a uniform presentation by means of neutral proof patterns, with dual polarised interpretations. Permutations of synthetic inferences can be explained by local conditions on proof patterns. Particular focusing systems can be explained as strategic uses of synthetic inferences. 1
A Focused Approach to Combining Logics
"... 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 ..."
Abstract
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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. Key words: focused proof systems, unity of logic, linear logic 1.
(will be inserted by the editor) Logical approximation for program analysis
"... Abstract The abstract interpretation of programs relates the exact semantics of a programming language to a finite approximation of those semantics. In this article, we describe an approach to abstract interpretation that is based in logic and logic programming. Our approach consists of faithfully r ..."
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Abstract The abstract interpretation of programs relates the exact semantics of a programming language to a finite approximation of those semantics. In this article, we describe an approach to abstract interpretation that is based in logic and logic programming. Our approach consists of faithfully representing a transition system within logic and then manipulating this initial specification to create a logical approximation of the original specification. The objective is to derive a logical approximation that can be interpreted as a terminating forwardchaining logic program; this ensures that the approximation is finite and that, furthermore, an appropriate logic programming interpreter can implement the derived approximation. We are particularly interested in the specification of the operational semantics of programming languages in ordered logic, a technique we call substructural operational semantics (SSOS). We show that manifestly sound control flow and alias analyses can be derived as logical approximations of the substructural operational semantics of relevant languages. 1
(will be inserted by the editor) Logical approximation for program analysis
"... Abstract The abstract interpretation of programs relates the exact semantics of a programming language to a finite approximation of those semantics. In this article, we describe an approach to abstract interpretation that is based in logic and logic programming. Our approach consists of faithfully r ..."
Abstract
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Abstract The abstract interpretation of programs relates the exact semantics of a programming language to a finite approximation of those semantics. In this article, we describe an approach to abstract interpretation that is based in logic and logic programming. Our approach consists of faithfully representing a transition system within logic and then manipulating this initial specification to create a logical approximation of the original specification. The objective is to derive a logical approximation that can be interpreted as a terminating forwardchaining logic program; this ensures that the approximation is finite and that, furthermore, an appropriate logic programming interpreter can implement the derived approximation. We are particularly interested in the specification of the operational semantics of programming languages in ordered logic, a technique we call substructural operational semantics (SSOS). We show that manifestly sound control flow and alias analyses can be derived as logical approximations of the substructural operational semantics of relevant languages. 1
(will be inserted by the editor) Logical approximation for program analysis
, 2010
"... Abstract The abstract interpretation of programs relates the exact semantics of a programming language to an approximate semantics that can be effectively computed. We show that, by specifying operational semantics in a specification framework based on bottomup logic programming in ordered logic – ..."
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Abstract The abstract interpretation of programs relates the exact semantics of a programming language to an approximate semantics that can be effectively computed. We show that, by specifying operational semantics in a specification framework based on bottomup logic programming in ordered logic – a technique we call substructural operational semantics (SSOS) – manifestly sound program approximations can be derived by simple and intuitive transformations and approximations of the logic program. As examples, we describe how to derive control flow and alias analyses from the substructural operational semantics of the relevant languages. 1
(will
, 2010
"... This Article is brought to you for free and open access by the School of Computer Science at Research Showcase. It has been accepted for inclusion in ..."
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This Article is brought to you for free and open access by the School of Computer Science at Research Showcase. It has been accepted for inclusion in