Chuck Liang

Focusing and polarization in linear, intuitionistic, and classical logics

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 Girards LC and LU proof systems.

Focusing and polarization in intuitionistic logic

A focused proof system provides a normal form to cut-free 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 Girards LC and LU proof systems.

Choices in Representation and Reduction Strategies for Lambda Terms in Intensional Contexts

Abstract 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. Among 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.

Compiler Construction in Higher Order Logic Programming

This paper describes a general method of compiler implementation using higher order abstract syntax and logic programming. A working compiler written in ?Prolog is used to demonstrate this method. Various stages of compilation are formulated as higher order logic programming including parsing and the generation of higher order repr sentations, type checking, intermediate representation in continuatio passing style, and machine-dependent code generation. The performance overhead of using higher order representations is also addressed.

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 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 classicallinear hybrid logics.

Tradeoffs in the Intensional Representation of Lambda Terms

Higher-order representations of objects such as programs, specifications and proofs are important to many metaprogramming and symbolic computation tasks. Systems that support such representations often depend on the implementation of an intensional view of the terms of suitable typed lambda calculi. Refined lambda calculus notations have been proposed that can be used in realizing such implementations. There are, however, choices in the actual deployment of such notations whose practical consequences are not well understood. Towards addressing this lacuna, the impact of three specific ideas is examined:the de Bruijn representation of bound variables, the explicit encoding of substitutions in terms and the annotation of terms to indicate their independence on external abstractions. Qualitative assessments are complemented by experiments over actual computations. The empirical study is based on ?Prolog programs executed using suitable variants of a low level, abstract machine based implementation of this language.

A Unified Sequent Calculus for Focused Proofs

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 multiplicative-additive 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 cut-elimination in the various fragments. Furthermore, each sublogic can interact with other fragments through cut. We also consider the possibility of introducing classical-linear hybrid logics.

Kripke semantics and proof systems for combining intuitionistic logic and classical logic

Abstract We combine intuitionistic logic and classical logic into a new, first-order 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 model-theoretically by a Kripke-style semantics for the logic. Two proof systems are also formulated. The first system extends Gentzenʼs intuitionistic sequent calculus LJ. In addition, this system also bears essential similarities to Girardʼs LC proof system for classical logic. The second proof system is based on a semantic tableau and extends Dragalinʼs multiple-conclusion 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.

Let-Polymorphism and Eager Type Schemes

This paper presents an algorithm for polymorphic type inference involving the let construct of ML in the context of higher order abstract syntax. It avoids the polymorphic closure operation of the algorithm W of Damas and Milner by using a uniform treatment of type variables at the meta-level. The basic technique of the algorithm facilitates the declarative formulation of type inference as goal-directed proof-search in a logical frameworks setting.

Unifying Classical and Intuitionistic Logics for Computational Control

We show that control operators and other extensions of the Curry-Howard isomorphism can be achieved without collapsing all of intuitionistic logic into classical logic. For this purpose we introduce a unified propositional logic using polarized formulas. We define a Kripke semantics for this logic. Our proof system extends an intuitionistic system that already allows multiple conclusions. This arrangement reveals a greater range of computational possibilities, including a form of dynamic scoping. We demonstrate the utility of this logic by showing how it can improve the formulation of exception handling in programming languages, including the ability to distinguish between different kinds of exceptions and constraining when an exception can be thrown, thus providing more refined control over computation compared to classical logic. We also describe some significant fragments of this logic and discuss its extension to second-order logic.