scholarly journals Distribution of Variables in Lambda-Terms with Restrictions on De Bruijn Indices and De Bruijn Levels

10.37236/8579 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Bernhard Gittenberger ◽  
Isabella Larcher
Keyword(s):  
The Other ◽  
Bounded Number ◽  
Mean And Variance ◽  
De Bruijn ◽  
De Bruijn Indices ◽  
Strange Phenomenon ◽  
Normally Distributed

We consider two special subclasses of lambda-terms that are restricted by a bound on the number of abstractions between a variable and its binding lambda, the so-called De-Bruijn index, or by a bound on the nesting levels of abstractions, i.e., the number of De Bruijn levels, respectively. We show that the total number of variables is asymptotically normally distributed for both subclasses of lambda-terms with mean and variance asymptotically equal to $Cn$ and $\tilde{C}n$, respectively, where the constants $C$ and $\tilde{C}$ depend on the bound that has been imposed. For the class of lambda-terms with bounded De Bruijn index we derive closed formulas for the constant. For the other class of lambda-terms that we consider, namely lambda-terms with a bounded number of De Bruijn levels, we show quantitative and distributional results on the number of variables, as well as abstractions and applications, in the different De Bruijn levels and thereby exhibit a so-called "unary profile" that attains a very interesting shape.  Our results give a combinatorial explanation of an earlier discovered strange phenomenon exhibited by the counting sequence of this particular class of lambda-terms. 

scholarly journals The Impact Of Financial Crisis On Islamic And Conventional Indices Of The GCC Countries

2015 ◽  
Vol 31 (2) ◽  
pp. 357 ◽  
Author(s):  
Hela Miniaoui ◽  
Hameedah Sayani ◽  
Anissa Chaibi
Keyword(s):  
Financial Crisis ◽  
The Other ◽  
Garch Models ◽  
Use Of Data ◽  
Market Index ◽  
Gcc Countries ◽  
Mean And Variance ◽  
The Mean ◽  
The Impact ◽  
Study Performance

<p>We study performance of Islamic and conventional indices of the Gulf Cooperation Council (GCC) countries in the wake of financial crisis of 2008 and test whether Islamic indices were less risky than conventional indices. We make use of data of the six GCC markets as well as the Dow Jones Islamic Market Index GCC. The mean and variance of each of the indices are analyzed based on augmented GARCH models. The results show that the financial crisis impacted on the mean returns of Bahrain, the other indices remained unaffected. The financial crisis, however, impacted volatility in three GCC markets (Kuwait, Bahrain, and the UAE), while the impact on the remaining markets (Saudi Arabia, Oman, and Qatar) and the Islamic index was insignificant. More interestingly, we show that the Islamic index did not exhibit lower volatility than its conventional counterparts.</p>

scholarly journals A unified treatment of syntax with binders

2012 ◽  
Vol 22 (4-5) ◽  
pp. 614-704 ◽  
Author(s):  
NICOLAS POUILLARD ◽  
FRANÇOIS POTTIER
Keyword(s):  
Natural Number ◽  
Data Structures ◽  
Transformation Function ◽  
Natural Numbers ◽  
Programming Error ◽  
Free Variables ◽  
De Bruijn ◽  
Abstract Interface ◽  
Representation Techniques ◽  
De Bruijn Indices

AbstractAtoms and de Bruijn indices are two well-known representation techniques for data structures that involve names and binders. However, using either technique, it is all too easy to make a programming error that causes one name to be used where another was intended. We propose an abstract interface to names and binders that rules out many of these errors. This interface is implemented as a library in Agda. It allows defining and manipulating term representations in nominal style and in de Bruijn style. The programmer is not forced to choose between these styles: on the contrary, the library allows using both styles in the same program, if desired. Whereas indexing the types of names and terms with a natural number is a well-known technique to better control the use of de Bruijn indices, we index types with worlds. Worlds are at the same time more precise and more abstract than natural numbers. Via logical relations and parametricity, we are able to demonstrate in what sense our library is safe, and to obtain theorems for free about world-polymorphic functions. For instance, we prove that a world-polymorphic term transformation function must commute with any renaming of the free variables. The proof is entirely carried out in Agda.

Quantitative Refinement of Calibrated14C Distributions

1994 ◽  
Vol 41 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Glenn P. Biasi ◽  
Ray Weldon
Keyword(s):  
Sedimentation Rate ◽  
Quantitative Data ◽  
San Andreas Fault ◽  
Bayes Theorem ◽  
The Other ◽  
Historical Information ◽  
Extra Information ◽  
San Andreas ◽  
Mean And Variance ◽  
The Bayesian Approach

AbstractA new method is presented for using known ordering or other relationships between14C samples to reduce14C dating uncertainty. The order of sample formation is often known from, for example, stratigraphic superposition, dendrochronology, or crosscutting field relations. Constraints such as a minimum time between dates and limits from historical information are also readily included. Dendrochronologically calibrated calendric date histograms initially represent each date. The method uses Bayes theorem and the relational constraints to upweight date ranges in each date distribution consistent with the other date distributions and the constraints, and downweight unlikely portions. The reweighted date distributions retain all dating possibilities present in the initial calibrated date distributions, but each date in the result now reflects the extra information such as ordering supplied through the constraints. In addition, one may add information incrementally, and thus analyze systematically its effect on all the date distributions. Thus, the method can be used to assess the consistency of the quantitative data at hand. The Bayesian approach also uses the empirical calibrated date distributions directly, so information is not lost prematurely by summarized dates to a mean and variance or "confidence intervals." The approach is illustrated with data from two densely sampled paleoseismic sites on the San Andreas Fault in southern California. An average reduction in14C date distribution variance of 59% is achieved using ordering information alone, and 85% is achieved by also applying sedimentation rate constraints and historical information.

Pure Type Systems with de Bruijn Indices

2002 ◽  
Vol 45 (2) ◽  
pp. 187-201 ◽  
Author(s):  
F. Kamareddine
Keyword(s):  
Type Systems ◽  
Pure Type ◽  
Pure Type Systems ◽  
De Bruijn ◽  
De Bruijn Indices

The Descriptive Use of Absolute Differences Between Pairs of Scores With a Common Mean and Variance

1994 ◽  
Vol 19 (2) ◽  
pp. 103-110 ◽  
Author(s):  
Kenneth O. McGraw ◽  
S. P. Wong
Keyword(s):  
Common Mean ◽  
Mean And Variance ◽  
The Absolute ◽  
Degree Of Similarity ◽  
Normally Distributed

Researchers concerned with the degree of similarity in pairs of scores sharing a common mean and variance can express similarity in terms of the absolute differences between the scores. This article discusses the distribution of absolute differences between pairs of normally distributed scores having a common mean and variance and illustrates procedures for calculating moments and areas within this distribution.

Pure type systems with explicit substitutions

2015 ◽  
Vol 25 ◽  
Author(s):  
DANIEL FRIDLENDER ◽  
MIGUEL PAGANO
Keyword(s):  
Type Systems ◽  
Pure Type ◽  
Explicit Substitution ◽  
Explicit Substitutions ◽  
Pure Type Systems ◽  
De Bruijn ◽  
New Formulation ◽  
De Bruijn Indices ◽  
Meta Theory

AbstractWe introduce a new formulation of pure type systems (PTSs) with explicit substitution and de Bruijn indices and formally prove some of its meta-theory. Using techniques based on Normalisation by Evaluation, we prove that untyped conversion can be typed for predicative PTSs. Although this equivalence was settled by Siles and Herbelin for the conventional presentation of PTSs, we strongly conjecture that our proof method can also be applied to PTSs with η.

scholarly journals Proof-relevant π-calculus: a constructive account of concurrency and causality

2017 ◽  
Vol 28 (9) ◽  
pp. 1541-1577 ◽  
Author(s):  
ROLY PERERA ◽  
JAMES CHENEY
Keyword(s):  
Formal Definition ◽  
Meta Data ◽  
Transition Relation ◽  
Π Calculus ◽  
De Bruijn ◽  
Target States ◽  
Definition Of ◽  
De Bruijn Indices ◽  
Proof Terms

We present a formalisation in Agda of the theory of concurrent transitions, residuation and causal equivalence of traces for the π-calculus. Our formalisation employs de Bruijn indices and dependently typed syntax, and aligns the ‘proved transitions’ proposed by Boudol and Castellani in the context of CCS with the proof terms naturally present in Agda's representation of the labelled transition relation. Our main contributions are proofs of the ‘diamond lemma’ for the residuals of concurrent transitions and a formal definition of equivalence of traces up to permutation of transitions.In the π-calculus, transitions represent propagating binders whenever their actions involve bound names. To accommodate these cases, we require a more general diamond lemma where the target states of equivalent traces are no longer identical, but are related by abraidingthat rewires the bound and free names to reflect the particular interleaving of events involving binders. Our approach may be useful for modelling concurrency in other languages where transitions carry meta-data sensitive to particular interleavings, such as dynamically allocated memory addresses.

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