Discriminating Lambda-Terms Using Clocked Boehm Trees

As observed by Intrigila, there are hardly techniques available in the lambda-calculus to prove that two lambda-terms are not beta-convertible. Techniques employing the usual Boehm Trees are inadequate when we deal with terms having the same Boehm Tree (BT). This is the case in particular for fixed point combinators, as they all have the same BT. Another interesting equation, whose consideration was suggested by Scott, is BY = BYS, an equation valid in the classical model P-omega of lambda-calculus, and hence valid with respect to BT-equality but nevertheless the terms are beta-inconvertible. To prove such beta-inconvertibilities, we employ `clocked' BT's, with annotations that convey information of the tempo in which the data in the BT are produced. Boehm Trees are thus enriched with an intrinsic clock behaviour, leading to a refined discrimination method for lambda-terms. The corresponding equality is strictly intermediate between beta-convertibility and Boehm Tree equality, the equality in the model P-omega. An analogous approach pertains to Levy-Longo and Berarducci Trees. Our refined Boehm Trees find in particular an application in beta-discriminating fixed point combinators (fpc's). It turns out that Scott's equation BY = BYS is the key to unlocking a plethora of fpc's, generated by a variety of production schemes of which the simplest was found by Boehm, stating that new fpc's are obtained by postfixing the term SI, also known as Smullyan's Owl. We prove that all these newly generated fpc's are indeed new, by considering their clocked BT's. Even so, not all pairs of new fpc's can be discriminated this way. For that purpose we increase the discrimination power by a precision of the clock notion that we call `atomic clock'.Comment: arXiv admin note: substantial text overlap with arXiv:1002.257

Axiomatizing the Quote

We study reflection in the Lambda Calculus from an axiomatic point of view. Specifically, we consider various properties that the quote operator must satisfy as a function on lambda terms. The most important of these is the existence of a definable left inverse, a so-called evaluator for the quote operator. Usually the quote operator encodes the syntax of a given term, and the evaluator proceeds by analyzing the syntax and reifying all constructors by their actual meaning in the calculus. Working in Combinatory Logic, Raymond Smullyan (1994) investigated which elements of the syntax must be accessible via the quote in order for an evaluator to exist. He asked three specific questions, to which we provide negative answers. On the positive side, we give a characterization of quotes which possess all of the desired properties, equivalently defined as being equitranslatable with a standard quote. As an application, we show that Scott\u27s coding is not complete in this sense, but can be slightly modified to be such. This results in a minimal definition of a complete quoting for Combinatory Logic

Gems of Corrado B\"ohm

The main scientific heritage of Corrado B\"ohm consists of ideas about computing, concerning concrete algorithms, as well as models of computability. The following will be presented. 1. A compiler that can compile itself. 2. Structured programming, eliminating the 'goto' statement. 3. Functional programming and an early implementation. 4. Separability in {\lambda}-calculus. 5. Compiling combinators without parsing. 6. Self-evaluation in {\lambda}-calculus

Determinants of Market Participation among Kocho Producers in Hadiya Zone, Southern Region, Ethiopia

Hadiya zone in southern region of Ethiopia has been known for enset farming. kocho is the  major product of enset. In the study area, kocho have been perceived to have high market value, resulting in tradeoffs with staple food. Despite kocho market value, its market participation has not been studied and quantified. Therefore, this paper aims to determine the factors influencing market participation and marketed surplus of kocho. Data was obtained from a sample of 398 households selected using multistage sampling techniques. The data was analyzed using the descriptive statistics and Heckman two-stage model. The results showed that age, sex, access to market information, availability of labour, non/off farm income, farming experience, perception of kocho price, quantity of kocho and livestock holding had significantly influenced market participation decision and extent of marketed surplus. Based on the findings, the study suggest that offering farmers a fair price, encouraging the use of labour saving technology, availing market information services, building farmers’ experience, encouraging farmer’s livestock holding besides farming, and paying attention to female households are needed to increase kocho marketed surplus. Keywords: Heckman two-stage model, market participation, small-scale kocho farmer

Feat: Functional Enumeration of Algebraic Types

In mathematics, an enumeration of a set S is a bijective function from (an initial segment of) the natural numbers to S. We define "functional enumerations" as efficiently computable such bijections. This paper describes a theory of functional enumeration and provides an algebra of enumerations closed under sums, products, guarded recursion and bijections. We partition each enumerated set into numbered, finite subsets. We provide a generic enumeration such that the number of each part corresponds to the size of its values (measured in the number of constructors). We implement our ideas in a Haskell library called testing-feat, and make the source code freely available. Feat provides efficient "random access" to enumerated values. The primary application is property-based testing, where it is used to define both random sampling (for example QuickCheck generators) and exhaustive enumeration (in the style of SmallCheck). We claim that functional enumeration is the best option for automatically generating test cases from large groups of mutually recursive syntax tree types. As a case study we use Feat to test the pretty-printer of the Template Haskell library (uncovering several bugs)

Examining the Role of Early Life Social Conditions on Adult Mortality through Historical Record Linkage: Implications for Contemporary Public Policy

This study examines the effect of early life conditions on adult mortality. The individual, household and community details of children residing in Quebec in 1901 are linked to their subsequent ages at death in late adulthood using the 1901 Canadian Census and Quebec death registers. Preliminary results of logistic regression and Cox proportional hazards regression analyses indicate that childhood poverty status is not significantly associated with risk of death after age forty. Sex, birth cohort, parental literacy status, farm status and number of siblings in the childhood household are found to significantly effect mortality after age 40. Gender differentials in the role of early life conditions are found, with males being affected by a greater number of childhood conditions than females. A forthcoming updated analysis will contain approximately 3-4 times the amount of cases available for the preliminary analyses

Feasibility of field implementation of fortified lentils to improve iron status status of adolescent girls in Bangladesh

The purpose of this crossover study was to establish methodology and logistics to conduct future large-scale community-based efficacy studies. A crossover trial was carried out in which adolescent girls consumed fortified lentil consisting of 2 different cooking preparation styles (thick vs thin) and 3 different lentil ‘portion size’ of raw weight 25 g, 37.5 g and 50 g. The cooked lentils were served 5 days a week over 12 weeks mixed with a standard cup (250 mL) of cooked rice among 100 adolescent girls. Small cotyledon lentil (iron 75-90 ppm) were fortified using NaFeEDTA iron fortificant solution with 1600 ppm of Fe at the Crop Development Centre (CDC), University of Saskatchewan. Visual Analog Scales (VAS) were used to measure the rating of hunger, satiety, and palatability before the serving of the meal, after one bite of the meal and after finishing the meal. In-depth interviews and focus group discussions were carried out with the participants’ mother/guardian to evaluate acceptability and lentil preparation styles at the end of the trial. Mean (SD) age and age of menarche of the adolescent girls were 12.85 (2.00) and 12.06 (0.94) years, respectively. Fifty-four percent heard about iron deficiency anemia, and 48% were of the opinion that eating more iron-rich foods could prevent anemia. The results showed that thick preparation of cooked lentil at 37.5 g portion size had higher positive scores in all VAS variables compared to the thin preparation. Considering the raw amount served, the thick preparation of lentil at 37.5 g would provide 6.9 mg Fe/d. This would cover approximately 86% (and 66% who have started to menstruate) and 46% of the RDA for adolescent girls aged 9-13 years and 14-18 years, respectively. A thick preparation of cooked lentil at 200 g (37.5 g raw lentil amount) would be preferable for a fortified lentil intervention in future human efficacy trials

Computability in constructive type theory

We give a formalised and machine-checked account of computability theory in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant. We first develop synthetic computability theory, pioneered by Richman, Bridges, and Bauer, where one treats all functions as computable, eliminating the need for a model of computation. We assume a novel parametric axiom for synthetic computability and give proofs of results like Rice’s theorem, the Myhill isomorphism theorem, and the existence of Post’s simple and hypersimple predicates relying on no other axioms such as Markov’s principle or choice axioms. As a second step, we introduce models of computation. We give a concise overview of definitions of various standard models and contribute machine-checked simulation proofs, posing a non-trivial engineering effort. We identify a notion of synthetic undecidability relative to a fixed halting problem, allowing axiom-free machine-checked proofs of undecidability. We contribute such undecidability proofs for the historical foundational problems of computability theory which require the identification of invariants left out in the literature and now form the basis of the Coq Library of Undecidability Proofs. We then identify the weak call-by-value λ-calculus L as sweet spot for programming in a model of computation. We introduce a certifying extraction framework and analyse an axiom stating that every function of type ℕ → ℕ is L-computable.Wir behandeln eine formalisierte und maschinengeprüfte Betrachtung von Berechenbarkeitstheorie im Calculus of Inductive Constructions (CIC), der konstruktiven Typtheorie die dem Beweisassistenten Coq zugrunde liegt. Wir entwickeln erst synthetische Berechenbarkeitstheorie, vorbereitet durch die Arbeit von Richman, Bridges und Bauer, wobei alle Funktionen als berechenbar behandelt werden, ohne Notwendigkeit eines Berechnungsmodells. Wir nehmen ein neues, parametrisches Axiom für synthetische Berechenbarkeit an und beweisen Resultate wie das Theorem von Rice, das Isomorphismus Theorem von Myhill und die Existenz von Post’s simplen und hypersimplen Prädikaten ohne Annahme von anderen Axiomen wie Markov’s Prinzip oder Auswahlaxiomen. Als zweiten Schritt führen wir Berechnungsmodelle ein. Wir geben einen kompakten Überblick über die Definition von verschiedenen Berechnungsmodellen und erklären maschinengeprüfte Simulationsbeweise zwischen diesen Modellen, welche einen hohen Konstruktionsaufwand beinhalten. Wir identifizieren einen Begriff von synthetischer Unentscheidbarkeit relativ zu einem fixierten Halteproblem welcher axiomenfreie maschinengeprüfte Unentscheidbarkeitsbeweise erlaubt. Wir erklären solche Beweise für die historisch grundlegenden Probleme der Berechenbarkeitstheorie, die das Identifizieren von Invarianten die normalerweise in der Literatur ausgelassen werden benötigen und nun die Basis der Coq Library of Undecidability Proofs bilden. Wir identifizieren dann den call-by-value λ-Kalkül L als sweet spot für die Programmierung in einem Berechnungsmodell. Wir führen ein zertifizierendes Extraktionsframework ein und analysieren ein Axiom welches postuliert dass jede Funktion vom Typ N→N L-berechenbar ist

A participatory GIS approach to flood risk assessment of informal settlements the case of Cape Town

Includes abstract.Includes bibliographical references.Rural-urban migrations have contributed to the steady increase in the population of Cape Town. Many of the migrants have settled in informal settlements because they cannot afford to rent or buy decent housing. Many of these settlements are however located on marginal and often poorly drained land. Consequently, most of these settlements are prone to flooding after prolonged rainfall. Current flood risk management techniques implemented by the authorities of the Cape Town City Council (CTCC) are ideal for formally planned settlements but are not designed to support informal settlements...This study sought to investigate a methodology that the CTCC could use to improve flood risk assessment