Permutability in proof terms for intuitionistic sequent calculus with cuts

This paper gives a comprehensive and coherent view on permutability in the intuitionistic sequent calculus with cuts. Specifically we show that, once permutability is packaged into appropriate global reduction procedures, it organizes the internal structure of the system and determines fragments with computational interest, both for the computation-as-proof-normalization and the computation-as-proof-search paradigms. The vehicle of the study is a lambda-calculus of multiary proof terms with generalized application, previously developed by the authors (the paper argues this system represents the simplest fragment of ordinary sequent calculus that does not fall into mere natural deduction). We start by adapting to our setting the concept of normal proof, developed by Mints, Dyckhoff, and Pinto, and by defining natural proofs, so that a proof is normal iff it is natural and cut-free. Natural proofs form a subsystem with a transparent Curry-Howard interpretation (a kind of formal vector notation for lambda-terms with vectors consisting of lists of lists of arguments), while searching for normal proofs corresponds to a slight relaxation of focusing (in the sense of LJT). Next, we define a process of permutative conversion to natural form, and show that its combination with cut elimination gives a concept of normalization for the sequent calculus. We derive a systematic picture of the full system comprehending a rich set of reduction procedures (cut elimination, flattening, permutative conversion, normalization, focalization), organizing the relevant subsystems and the important subclasses of cut-free, normal, and focused proofs.Partially financed by FCT through project UID/MAT/00013/2013, and by COST action CA15123 EUTYPES. The first and the last authors were partially financed by Fundação para a Ciência e a Tecnologia (FCT) through project UID/MAT/00013/2013. The first author got financial support by the COST action CA15123 EUTYPES.info:eu-repo/semantics/publishedVersio

Characterization Of Nitrogen Fixation (Nif) Genes From Paenibacilus Polymyxa [QR89.7. Y19 2007 f rb].

Paenibacillus polymyxa adalah sejenis bakteria Gram positif yang berupaya menurunkan dinitrogen (N2) kepada ammonia. Satu fragmen nifH separa telah diamplifikasi dengan menggunakan sepasang primer degenerat. Paenibacilus polymyxa is a Gram positive bacterium capable of converting dinitrogen (N2) to ammonia. A partial nifH fragment was amplified by using a pair of nifH degenerate primers

Contexts in Lambda Calculus

Klop, J.W. [Promotor]Vrijer, R.C. de [Copromotor

Demonstration of improved sensitivity of echo atom interferometers to gravitational acceleration

We have developed two configurations of an echo interferometer that rely on standing wave excitation of a laser-cooled sample of rubidium atoms. Both configurations are sensitive to acceleration along the axis of excitation. For a two-pulse configuration, the signal from the interferometer is modulated at the recoil frequency and exhibits a sinusoidal frequency chirp as a function of pulse spacing. In comparison, for a three-pulse stimulated echo configuration, the signal is observed without recoil modulation and exhibits a modulation at a single frequency as a function of pulse spacing. The three-pulse configuration is less sensitive to effects of vibrations and magnetic field curvature, leading to a longer experimental timescale. For both configurations of the interferometer, we show that a measurement of acceleration with a statistical precision of 0.5% can be realized by analyzing the shape of the echo envelope, which has a temporal duration of a few microseconds. Using the two-pulse interferometer, we obtain measurements of acceleration that are statistically precise to 6 parts per million on a 25 ms timescale. In comparison, using the three-pulse interferometer, we obtain measurements of acceleration that are statistically precise to 75 parts per billion on a timescale of 70 ms. The inhomogeneous field of a magnetized vacuum chamber limited the experimental timescale and resulted in prominent systematic effects. Extended timescales and improved signal-to-noise ratio observed in recent echo experiments using a non-magnetic vacuum chamber suggest that echo techniques are suitable for a high precision measurement of gravitational acceleration g. We discuss methods for reducing systematic effects and improving the signal-to-noise ratio. Simulations suggest that an optimized experiment with improved vibration isolation that utilizes atoms selected in the magnetic sublevel mF = 0 state can result in measurements of g precise to 0.5 parts per billion with a timescale of 300 ms

A Framework for Defining Logical Frameworks

In this paper, we introduce a General Logical Framework, called GLF, for defining Logical Frameworks, based on dependent types, in the style of the well known Edinburgh Logical Framework LF. The framework GLF features a generalized form of lambda abstraction where beta-reductions fire provided the argument satisfies a logical predicate and may produce an n-ary substitution. The type system keeps track of when reductions have yet to fire. The framework GLF subsumes, by simple instantiation, LF as well as a large class of generalized constrained-based lambda calculi, ranging from well known restricted lambda calculi, such as Plotkin's call-by-value lambda calculus, to lambda calculi with patterns. But it suggests also a wide spectrum of completely new calculi which have intriguing potential as Logical Frameworks. We investigate the metatheoretical properties of the calculus underpinning GLF and illustrate its expressive power. In particular, we focus on two interesting instantiations of GLF. The first is the Pattern Logical Framework (PLF), where applications fire via pattern-matching in the style of Cirstea, Kirchner, and Liquori. The second is the Closed Logical Framework (CLF) which features, besides standard beta-reduction, also a reduction which fires only if the argument is a closed term. For both these instantiations of GLF we discuss standard metaproperties, such as subject reduction, confluence and strong normalization. The GLF framework is particularly suitable, as a metalanguage, for encoding rewriting logics and logical systems, where rules require proof terms to have special syntactic constraints, e.g. logics with rules of proof, in addition to rules of derivations, such as, e.g., modal logics, and call-by-value lambda calculus