A focused proof system provides a normal form to cut-free proofs in which the application of invertible and non-invertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cut-free 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.

Abstract. A focused proof system provides a normal form to cut-free proofs that structures the application of invertible and non-invertible 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.

Higher-order representations of objects such as programs, proofs, formulas and types have become important to many symbolic computation tasks. Systems that support such representations usually depend on the implementation of an intensional view of the terms of some variant of the typed lambda calculus. New notations have been proposed for the lambda calculus that provide an excellent basis for realizing such implementations. There are, however, several choices in the actual deployment of these notations the practical consequences of which are not currently well understood. We attempt to develop such an understanding here by examining the impact on performance of different combinations of the features afforded by such notations. Amongst the facets examined are the treatment of bound variables, eagerness and laziness in substitution and reduction, the ability to merge different structure traversals into one and the virtues of annotations on terms that indicate their dependence on variables bound by external abstractions. We complement qualitative assessments with experiments conducted by executing programs in a language that supports an intensional view of lambda terms while varying relevant aspects of the implementation of the language. Our study provides insights into the preferred approaches to representing and reducing lambda terms and also exposes characteristics of computations that have a somewhat unanticipated effect on performance.

Communicated by P.J. Scott MSC: 03F52 03B47 03B20 Keywords: Focused proof systems Unity of logic Linear logic a b s t r a c tWe present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical, intuitionistic, and multiplicative-additive 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 classical-linear hybrid logics.

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