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 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

Unifying Classical and Intuitionistic Logics for Computational Control

International audienceWe 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

Kripke Semantics and Proof Systems for Combining Intuitionistic Logic and Classical Logic

International audienceWe 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 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 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

Processing and Property Investigation of Single-Walled Carbon Nanotube (SWNT) Buckypaper/Epoxy Resin Matrix Nanocomposites

Abstract Due to their extraordinary high modulus and strength, single-walled carbon nanotubes (SWNTs) are considered by many researchers as the most promising reinforcement materials for use in high performance composites. However, fabricating SWNT-reinforced composites with good tube dispersion and high SWNT loading is a great challenge since SWNTs have a strong tendency to form bundles or ropes and rapidly increase the viscosity during processing. In this research, a new method was developed to fabricate nanocomposites with preformed SWNT networks and high tube loading. SWNTs were first dispersed in water-based suspension with the aid of surfactant and sonication. Through a filtration process, SWNTs were fabricated into thin membranes called buckypapers to form networks of SWNT ropes. The tube/resin impregnation of the produced buckypaper was realized by infiltrating acetone diluted epoxy resin (Epon 862/EPI Cure W system) along the thickness direction. A hot press molding process was used for curing to produce the final nanocomposites of multiple layer buckypapers with high SWNT loading (up to 39 wt%). Dynamic mechanical analysis (DMA) results show that the storage moduli of the resulting nanocomposites were as high as 15 Gpa. The DMA results also indicate that the nanotubes had a strong influence on the composites damping properties. The through thickness permeability of the resulting buckypapers with nanoscale pore structure was also measured. AFM and SEM observations show that the SWNTs have a good dispersion in the buckypaper and nanocomposites. The research results show that the proposed buckypaper/resin infiltration approach is capable of fabricating nanocomposites with controllable nanostructure and high SWNT loading, which are important for developing high performance nanotube-based composites.

Explicit Substitutions for Contextual Type Theory

In this paper, we present an explicit substitution calculus which distinguishes between ordinary bound variables and meta-variables. Its typing discipline is derived from contextual modal type theory. We first present a dependently typed lambda calculus with explicit substitutions for ordinary variables and explicit meta-substitutions for meta-variables. We then present a weak head normalization procedure which performs both substitutions lazily and in a single pass thereby combining substitution walks for the two different classes of variables. Finally, we describe a bidirectional type checking algorithm which uses weak head normalization and prove soundness.Comment: In Proceedings LFMTP 2010, arXiv:1009.218

Normalization by Evaluation for Call-by-Push-Value and Polarized Lambda-Calculus

We observe that normalization by evaluation for simply-typed lambda-calculus with weak coproducts can be carried out in a weak bi-cartesian closed category of presheaves equipped with a monad that allows us to perform case distinction on neutral terms of sum type. The placement of the monad influences the normal forms we obtain: for instance, placing the monad on coproducts gives us eta-long beta-pi normal forms where pi refers to permutation of case distinctions out of elimination positions. We further observe that placing the monad on every coproduct is rather wasteful, and an optimal placement of the monad can be determined by considering polarized simple types inspired by focalization. Polarization classifies types into positive and negative, and it is sufficient to place the monad at the embedding of positive types into negative ones. We consider two calculi based on polarized types: pure call-by-push-value (CBPV) and polarized lambda-calculus, the natural deduction calculus corresponding to focalized sequent calculus. For these two calculi, we present algorithms for normalization by evaluation. We further discuss different implementations of the monad and their relation to existing normalization proofs for lambda-calculus with sums. Our developments have been partially formalized in the Agda proof assistant

A Lambda Term Representation Inspired by Linear Ordered Logic

We introduce a new nameless representation of lambda terms inspired by ordered logic. At a lambda abstraction, number and relative position of all occurrences of the bound variable are stored, and application carries the additional information where to cut the variable context into function and argument part. This way, complete information about free variable occurrence is available at each subterm without requiring a traversal, and environments can be kept exact such that they only assign values to variables that actually occur in the associated term. Our approach avoids space leaks in interpreters that build function closures. In this article, we prove correctness of the new representation and present an experimental evaluation of its performance in a proof checker for the Edinburgh Logical Framework. Keywords: representation of binders, explicit substitutions, ordered contexts, space leaks, Logical Framework.Comment: In Proceedings LFMTP 2011, arXiv:1110.668

Normalization by evaluation for call-by-push-value and polarized lambda calculus

We observe that normalization by evaluation for simply-typed lambda-calculus with weak coproducts can be carried out in a weak bi-cartesian closed category of presheaves equipped with a monad that allows us to perform case distinction on neutral terms of sum type. The placement of the monad influences the normal forms we obtain: for instance, placing the monad on coproducts gives us eta-long beta-pi normal forms where pi refers to permutation of case distinctions out of elimination positions. We further observe that placing the monad on every coproduct is rather wasteful, and an optimal placement of the monad can be determined by considering polarized simple types inspired by focalization. Polarization classifies types into positive and negative, and it is sufficient to place the monad at the embedding of positive types into negative ones. We consider two calculi based on polarized types: pure call-by-push-value (CBPV) and polarized lambda-calculus, the natural deduction calculus corresponding to focalized sequent calculus. For these two calculi, we present algorithms for normalization by evaluation. We further discuss different implementations of the monad and their relation to existing normalization proofs for lambda-calculus with sums. Our developments have been partially formalized in the Agda proof assistant

Controlled nanostructure and high loading of single-walled carbon nanotubes reinforced polycarbonate composite

Abstract This paper presents an effective technique to fabricate thermoplastic nanocomposites with high loading of well-dispersed single-walled carbon nanotubes (SWNTs). SWNT membranes were made from a multi-step dispersion and filtration method, and then impregnated with polycarbonate solution to make thermoplastic nanocomposites. High loading of nanotubes was achieved by controlling the viscosity of polycarbonate solution. SEM and AFM characterization results revealed the controlled nanostructure in the resultant nanocomposites. Dynamic mechanical property tests indicated that the storage modulus of the resulting nanocomposites at 20 wt% nanotubes loading was improved by a factor of 3.4 compared with neat polycarbonate material. These results suggest the developed approach is an effective way to fabricate thermoplastic nanocomposites with good dispersion and high SWNT loading