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25
Structural Cut Elimination
 Proceedings of the Tenth Annual Symposium on Logic in Computer Science
, 1995
"... We present new proofs of cut elimination for intuitionistic, classical, and linear sequent calculi. In all cases the proofs proceed by three nested structural inductions, avoiding the explicit use of multisets and termination measures on sequent derivations. This makes them amenable to elegant and ..."
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Cited by 64 (8 self)
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We present new proofs of cut elimination for intuitionistic, classical, and linear sequent calculi. In all cases the proofs proceed by three nested structural inductions, avoiding the explicit use of multisets and termination measures on sequent derivations. This makes them amenable to elegant and concise implementations in Elf, a constraint logic programming language based on the LF logical framework. 1 Introduction Gentzen's sequent calculi [Gen35] for intuitionistic and classical logic have been the central tool in many prooftheoretical investigations and applications of logic in computer science such as logic programming or automated theorem proving. The central property of sequent calculi is cut elimination (Gentzen's Hauptsatz) which yields consistency of the logic as a corollary. The algorithm for cut elimination may be interpreted computationally, similarly to the way normalization for natural deduction may be viewed as functional computation. For the case of linear logic, ...
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.
CallbyName, CallbyValue, CallbyNeed, and the Linear Lambda Calculus
, 1994
"... Girard described two translations of intuitionistic logic into linear logic, one where A > B maps to (!A) o B, and another where it maps to !(A o B). We detail the action of these translations on terms, and show that the first corresponds to a callbyname calculus, while the second corresponds t ..."
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Cited by 28 (5 self)
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Girard described two translations of intuitionistic logic into linear logic, one where A > B maps to (!A) o B, and another where it maps to !(A o B). We detail the action of these translations on terms, and show that the first corresponds to a callbyname calculus, while the second corresponds to callbyvalue. We further show that if the target of the translation is taken to be an affine calculus, where ! controls contraction but weakening is allowed everywhere, then the second translation corresponds to a callbyneed calculus, as recently defined by Ariola, Felleisen, Maraist, Odersky, and Wadler. Thus the different calling mechanisms can be explained in terms of logical translations, bringing them into the scope of the CurryHoward isomorphism.
From Action Calculi to Linear Logic
, 1998
"... . Milner introduced action calculi as a framework for investigating models of interactive behaviour. We present a typetheoretic account of action calculi using the propositionsastypes paradigm; the type theory has a sound and complete interpretation in Power's categorical models. We go on to give ..."
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Cited by 19 (7 self)
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. Milner introduced action calculi as a framework for investigating models of interactive behaviour. We present a typetheoretic account of action calculi using the propositionsastypes paradigm; the type theory has a sound and complete interpretation in Power's categorical models. We go on to give a sound translation of our type theory in the (type theory of) intuitionistic linear logic, corresponding to the relation between Benton's models of linear logic and models of action calculi. The conservativity of the syntactic translation is proved by a modelembedding construction using the Yoneda lemma. Finally, we briefly discuss how these techniques can also be used to give conservative translations between various extensions of action calculi. 1 Introduction Action calculi arose directly from the ßcalculus [MPW92]. They were introduced by Milner [Mil96], to provide a uniform notation for capturing many calculi of interaction such as the ßcalculus, the calculus, models of distribut...
The Structure of Exponentials: Uncovering the Dynamics of Linear Logic Proofs
 Computational Logic and Proof Theory
, 1993
"... . We construct the exponential graph of a proof ß in (second order) linear logic, an artefact displaying the interdependencies of exponentials in ß. Within this graph superfluous exponentials are defined, the removal of which is shown to yield a correct proof ß . with essentially the same set of r ..."
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Cited by 18 (6 self)
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. We construct the exponential graph of a proof ß in (second order) linear logic, an artefact displaying the interdependencies of exponentials in ß. Within this graph superfluous exponentials are defined, the removal of which is shown to yield a correct proof ß . with essentially the same set of reductions. Applications to intuitionistic and classical proofs are given by means of reductionpreserving embeddings into linear logic. The last part of the paper puts things the other way round, and defines families of linear logics in which exponential dependencies are ruled by a given graph. We sketch some work in progress and possible applications. 1 Introduction An exponential "!", "?" in a linear proof is superfluous if we can remove it and obtain a proof that (1) is still correct, and (2) has the same dynamics as the original one. If we can get rid of an exponential in a linear proof, we know that the subproof introducing it (by a L? or a R! rule) will endure no nonlinear handling (...
On the unity of duality
 Special issue on “Classical Logic and Computation
, 2008
"... Most type systems are agnostic regarding the evaluation strategy for the underlying languages, with the value restriction for ML which is absent in Haskell as a notable exception. As type systems become more precise, however, detailed properties of the operational semantics may become visible becaus ..."
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Cited by 12 (2 self)
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Most type systems are agnostic regarding the evaluation strategy for the underlying languages, with the value restriction for ML which is absent in Haskell as a notable exception. As type systems become more precise, however, detailed properties of the operational semantics may become visible because properties captured by the types may be sound under one strategy but not the other. For example, intersection types distinguish between callbyname and callbyvalue functions, because the subtyping law (A → B) ∩ (A → C) ≤ A → (B ∩ C) is unsound for the latter in the presence of effects. In this paper we develop a prooftheoretic framework for analyzing the interaction of types with evaluation order, based on the notion of polarity. Polarity was discovered through linear logic, but we propose a fresh origin in Dummett’s program of justifying the logical laws through alternative verificationist or pragmatist “meaningtheories”, which include a bias towards either introduction or elimination rules. We revisit Dummett’s analysis using the tools of MartinLöf’s judgmental method, and then show how to extend it to a unified polarized logic, with Girard’s “shift ” connectives acting as intermediaries. This logic safely combines intuitionistic and dual intuitionistic reasoning principles, while simultaneously admitting a focusing interpretation for the classical sequent calculus. Then, by applying the CurryHoward isomorphism to polarized logic, we obtain a single programming language in which evaluation order is reflected at the level of types. Different logical notions correspond directly to natural programming constructs, such as patternmatching, explicit substitutions, values and callbyvalue continuations. We give examples demonstrating the expressiveness of the language and type system, and prove a basic but modular type safety result. We conclude with a brief discussion of extensions to the language with additional effects and types, and sketch the sort of explanation this can provide for operationallysensitive typing phenomena. 1
A Pragmatic Interpretation Of Substructural Logics
"... Following work by Dalla Pozza and Garola [2, 3] on a pragmatic interpretation of intuitionistic and deontic logics, which has given evidence of their compatibility with classical semantics, we present sequent calculus system ILP formalizing the derivation of assertive judgements and obligations from ..."
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Cited by 11 (8 self)
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Following work by Dalla Pozza and Garola [2, 3] on a pragmatic interpretation of intuitionistic and deontic logics, which has given evidence of their compatibility with classical semantics, we present sequent calculus system ILP formalizing the derivation of assertive judgements and obligations from mixed contexts of assertions and obligations and we prove the cutelimination theorem for it. For the formalization of reallife normative systems it is essential to consider inferences from mixed contexts of assertions and obligations, and also of assertions justi able relatively to a given state of information and obligations valid in a given normative system. In order to provide a formalization of the notion of causal implication and its interaction with obligations, the sequents of ILP have two areas in the antecedent, expressing the relevant and the ordinary intuitionistic consequence relation, respectively. To provide a pragmatic interpretation of reasoning with the lin...
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.
Kripke semantics and proof systems for combining intuitionistic logic and classical logic. Submitted
, 2011
"... We combine intuitionistic logic and classical logic into a new, firstorder logic called Polarized Intuitionistic Logic. This logic is based on a distinction between two dual polarities which we call red and green to distinguish them from other forms of polarization. The meaning of these polarities ..."
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Cited by 3 (3 self)
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We combine intuitionistic logic and classical logic into a new, firstorder logic called Polarized Intuitionistic Logic. This logic is based on a distinction between two dual polarities which we call red and green to distinguish them from other forms of polarization. The meaning of these polarities is defined modeltheoretically by a Kripkestyle semantics for the logic. Two proof systems are also formulated. The first system extends Gentzen’s intuitionistic sequent calculus LJ. In addition, this system also bears essential similarities to Girard’s LC proof system for classical logic. The second proof system is based on a semantic tableau and extends Dragalin’s multipleconclusion version of intuitionistic sequent calculus. We show that soundness and completeness hold for these notions of semantics and proofs, from which it follows that cut is admissible in the proof systems and that the propositional fragment of the logic is decidable. 1