Recursiveness Research Articles

Nominal system T

This paper introduces a new recursion principle for inductive data modulo α-equivalence of bound names. It makes use of Odersky-style local names when recursing over bound names. It is formulated in an extension of Gödel's System T with names that can be tested for equality, explicitly swapped in expressions and restricted to a lexical scope. The new recursion principle is motivated by the nominal sets notion of "α-structural recursion", whose use of names and associated freshness side-conditions in recursive definitions formalizes common practice with binders. The new Nominal System T presented here provides a calculus of total functions that is shown to adequately represent α-structural recursion while avoiding the need to verify freshness side-conditions in definitions and computations. Adequacy is proved via a normalization-by-evaluation argument that makes use of a new semantics of local names in Gabbay-Pitts nominal sets.

“A princesa que tudo via”: processos cognitivos e criativos na alfabetização

The article reflects theoretical foundations on the cognitive processes in literacy. 'A princesa que tudo via' is a metaphor, a tool that facilitates the understanding of the human cognition and the acquisition of literacy. It is a procedure that allows us to consider the narrative elements of the text in analogy to recent studies on human cognition and creativity. Such narratives configure themselves in guiding principles for the creation of strategies for literacy, in triggering and operating, within the same tune, the bio-cultural dimensions of human cognition. The principle of recursion allows thinking of both the autonomy and creativity of the subject activated by the culture, by its symbolic, imaginary and cultural dimensions, which pervade the experience of the human being, triggering in him, the subjective states that make the cognition possible. In a complex perspective, the principles abstracted from the tale allow to see the eco-self-dependent feature of the constitution of the subject of knowledge. The creativity emerges from the dynamic dependence-autonomy implicit in learning that must be considered by the teacher in the strategies for literacy teaching.

The Paradigm Recursion: Is It More Accessible When Introduced in Middle School?

Recursion is a programming paradigm as well as a problem solving strategy thought to be very challenging to grasp for university students. This article outlines a pilot study, which expands the age range of students exposed to the concept of recursion in computer science through instruction in a series of interesting and engaging activities. In this study, a small number of students (n = 9) aged 11 to 13 years, were presented with a new and unique recursion curriculum involving hands-on experiences over a seven-week period at the University of Victoria, Canada. The curriculum was comprised of a series of progressively challenging recursion activities—roughly based upon the ideas of ‘Computer Science Unplugged’ (Bell, Witten, & Fellows, 2009)—and included programming applications with MicroWorlds EX, a programming language based on LOGO. Through this engagement, an increased number of students recognized and understood the concepts covered. We hypothesize that through experiences for youth with activities such as those outlined here, the number of students who understand fundamental computer science applications and who might potentially pursue computer science in post-secondary education will increase. We hypothesis further that through an earlier encounter of “challenging” concepts the learning and understanding of those will become easier at the university level. In this paper, the curriculum, classroom experiences, preliminary, largely descriptive and qualitative results and next steps in the research are discussed.

Two extensions of system F with (co)iteration and primitive (co)recursion principles

This paper presents two extensions of the second order polymorphic lambda calculus, system F, with monotone (co)inductive types supporting (co)iteration, primitive (co)recursion and inversion principles as primitives. One extension is inspired by the usual categorical approach to programming by means of initial algebras and final coalgebras; whereas the other models dialgebras, and can be seen as an extension of Hagino's categorical lambda calculus within the framework of parametric polymorphism. The systems are presented in Curry-style, and are proven to be terminating and type-preserving. Moreover their expressiveness is shown by means of several programming examples, going from usual data types to lazy codata types such as streams or infinite trees.

Evidence and argumentation: A reply to Everett (2009)

Evidence and argumentation:A reply to Everett (2009) Andrew Nevins, David Pesetsky, and Cilene Rodrigues This discussion note is a brief response to Everett (2009, E09), which was a reply to our assessment (Nevins, Pesetsky, & Rodrigues 2009, NP&R) of Everett's (2005, CA) earlier claims about the Amazonian language Pirahã. An important (and somewhat hidden) feature of E09 is a set of new empirical assertions presented in defense of CA's hypothesis that Pirahã lacks embedding. We argue that these new claims have not been supported by appropriate evidence; but if they are correct nonetheless, they WEAKEN rather than strengthen the case against embedding. We conclude with a discussion of comparable questions of argumentation and evidence that arise elsewhere in E09, and the relevance of these issues to the public discussion of CA's claims. 1. Goals Of This Reply One of our reasons for undertaking the project reported in NP&R was the extraordinary attention that Everett's claims about Pirahã have received in the popular press—and the equally extraordinary significance that has been attributed to them. The New Scientist (March 18, 2006) suggested, for example, that Pirahã might constitute 'the final nail in the coffin for Noam Chomsky's hugely influential theory of universal grammar'; and the Chicago Tribune (June 10, 2007), under the headline 'Shaking language to the core', reported that Everett had 'fired a volley straight at the theory when he reported that the Brazilian tribe he was studying didn't use recursives [sic]'. More recently, the Times of London (October 24, 2008) has characterized Everett's claim that 'Pirahã lack the grammatical principle of recursion' as an 'astonishing find'. If the conclusions in NP&R are correct, of course, Pirahã presents us with no nail, no coffin, no volley, and no astonishing find. Much of NP&R was devoted to an examination of the claims and conclusions about Pirahã language and culture that were put forward in CA, the article that triggered the publicity.1 We argued that many of CA's claims were not coherent as stated; that the logic by which factual evidence was said to support them was flawed; and also that CA failed to discuss a wealth of published counterevidence (indeed, failed to acknowledge even the existence of this counter evidence). Finally, we argued, the linguistic evidence presented in CA not only fails to support claims of Pirahã exceptionality, but actually suggests THE OPPOSITE: that Pirahã fits straightforwardly into the known typology of human languages. In the field of linguistics, as in other fields, evidence does matter—as does the logic of the argumentation that links evidence to hypothesis. It is precisely in this domain that linguists can contribute to the Pirahã discussion in ways that reporters cannot. Since very few of the press reports on Pirahã had even raised the key questions of evidence and argumentation, [End Page 671] we considered it important to try to put these issues on the agenda at large, a task rendered particularly timely by the continuing public interest. Similar considerations prompt us to offer this brief response to Everett's reply (E09) that accompanied the publication of NP&R. At the same time, we do not believe that Language readers would welcome a point-by-point discussion of all the claims and counterclaims taken up in E09. For example, although E09 mischaracterizes many arguments and claims from NP&R, these errors can be easily found by simply comparing E09's citations of NP&R with what we actually wrote, so we see little reason to correct them here.2 We also believe that the most effective replies to E09's comments on such issues as the proper meaning of 'universal grammar' can already be found in the sections of NP&R to which E09's comments respond—so once again, we do not rehearse these issues a second time. Instead, we focus in this commentary on one aspect of E09 that is less obvious, but of some significance nonetheless, precisely because of its direct bearing on questions of evidence and argumentation. Everett represents E09 as demonstrating that CA 'is essentially correct in its description of the facts and that it corrects errors...

A tensão entre cálculo e classificação como dinâmica da exclusão social

As the formal logic informs us, calculus and algorithm are two faces of the same coin. Nevertheless, algorithm suffers a forcing of calculus’s properties when it deals with irreducibly ontological entities trying their formalization through the classification. If, on one hand, the requirements of calculus produce classification, on the other hand, they shock against this one. In the contemporaneous economic dynamics, calculus tends to exclude from the productive processes the rigidity of classification (and the subjects involved in it), but it promotes the classificatory logic in the market and public services. Obsolete in the productive life, classification is requested in the private life, and it inserts the individual who is socially disqualified into a mechanism that confirms and amplifies at every single moment the negative value of his condition. Key words: social inequality, poverty self-reproduction, recursivity of the social processes.

Sequences of real functions on [formula omitted] in constructive reverse mathematics

We give an overview of the role of equicontinuity of sequences of real-valued functions on [ 0 , 1 ] and related notions in classical mathematics, intuitionistic mathematics, Bishop’s constructive mathematics, and Russian recursive mathematics. We then study the logical strength of theorems concerning these notions within the programme of Constructive Reverse Mathematics. It appears that many of these theorems, like a version of Ascoli’s Lemma, are equivalent to fan-theoretic principles.

A gentle introduction to mutual recursion

Recursion is an important topic in computer science curricula. It is related to the acquisition of competences regarding problem decomposition, functional abstraction and the concept of induction. In comparison with direct recursion, mutual recursion is considered to be more complex. Consequently, it is generally addressed superficially in CS1/2 programming courses and textbooks. We show that, when a problem is approached appropriately, not only can mutual recursion be a powerful tool, but it can also be easy to understand and fun. This paper provides several intuitive and attractive algorithms that rely on mutual recursion, and which have been designed to help strengthen students' ability to decompose problems and apply induction. Furthermore, we show that a solution based on mutual recursion may be easier to design, prove and comprehend than other solutions based on direct recursion. We have evaluated the use of these algorithms while teaching recursion concepts. Results suggest that mutual recursion, in comparison with other types of recursion, is not as hard as it seems when: (1) determining the result of a (mathematical) function call, and, most importantly, (2) designing algorithms for solving simple problems.

A class of $${\Sigma _{3}^{0}}$$ modular lattices embeddable as principal filters in $${\mathcal{L}^{\ast }(V_{\infty })}$$

Let I 0 be a a computable basis of the fully effective vector space V ∞ over the computable field F. Let I be a quasimaximal subset of I 0 that is the intersection of n maximal subsets of the same 1-degree up to *. We prove that the principal filter $${\mathcal{L}^{\ast}(V,\uparrow )}$$ of V = cl(I) is isomorphic to the lattice $${\mathcal{L}(n, \overline{F})}$$ of subspaces of an n-dimensional space over $${\overline{F}}$$ , a $${\Sigma _{3}^{0}}$$ extension of F. As a corollary of this and the main result of Dimitrov (Math Log 43:415–424, 2004) we prove that any finite product of the lattices $${(\mathcal{L}(n_{i}, \overline{F }_{i}))_{i=1}^{k}}$$ is isomorphic to a principal filter of $${\mathcal{ L}^{\ast}(V_{\infty})}$$ . We thus answer Question 5.3 “What are the principal filters of $${\mathcal{L}^{\ast}(V_{\infty}) ?}$$ ” posed by Downey and Remmel (Computable algebras and closure systems: coding properties, handbook of recursive mathematics, vol 2, pp 977–1039, Stud Log Found Math, vol 139, North-Holland, Amsterdam, 1998) for spaces that are closures of quasimaximal sets.

On the computing power of fuzzy Turing machines

We work with fuzzy Turing machines ( FTMs) and we study the relationship between this computational model and classical recursion concepts such as computable functions, recursively enumerable (r.e.) sets and universality. FTMs are first regarded as acceptors. It has recently been shown by J. Wiedermann that these machines have more computational power than classical Turing machines. Still, the context in which this formulation is valid has an unnatural implicit assumption. We settle necessary and sufficient conditions for a language to be r.e., by embedding it in a fuzzy language recognized by a FTM. We do the same thing for n-r.e. set. It is shown that there is no universal fuzzy machine, and “universality” is analyzed for smaller classes of FTMs. We argue for a definition of computable fuzzy function, when FTMs are understood as transducers. It is shown that, in this case, our notion of computable fuzzy function coincides with the classical one.

On extremal solutions of inclusion problems with applications to game theory

A recursion principle, generalized iteration methods and the axiom of choice are applied to prove the existence of extremal fixed points of set-valued mappings in posets, extremal solutions of an inclusion problem, and extremal Nash equilibria for a normal-form game.

Doppler ultrasound wall removal based on the spatial correlation of wavelet coefficients

In medical Doppler ultrasound systems, a high-pass filter is commonly used to reject echoes from the vessel wall. However, this leads to the loss of the information from the low velocity blood flow. Here a spatially selective noise filtration algorithm cooperating with a threshold denoising based on wavelets coefficients is applied to estimate the wall clutter. Then the blood flow signal is extracted by subtracting the wall clutter from the mixed signal. Experiments on computer simulated signals with various clutter-to-blood power ratios indicate that this method achieves a lower mean relative error of spectrum than the high-pass filtering and other two previously published separation methods based on the recursive principle component analysis and the irregular sampling and iterative reconstruction, respectively. The method also performs well when applied to in vivo carotid signals. All results suggest that this approach can be implemented as a clutter rejection filter in Doppler ultrasound instruments.

Exploring recursion with fibonacci numbers

This paper addresses the relationship between recursion and combinatorial problems, which may benefit teaching recursion in CS1/2 courses. Problems whose solutions are related to Fibonacci numbers are particularly interesting, since some can be decomposed by using different strategies, which may be based on the numerous Fibonacci identities or even on the concept of mutual recursion.

The mathematics of infinity: a guide to great ideas

Preface. 1. Elementary Set Theory. 1.1 Sets. 1.2 Cartesian Products. 1.3 Power Sets. 1.4 Something From Nothing. 1.5 Indexed Families of Sets. 2. Functions. 2.1 Functional Preliminaries. 2.2 Images and Preimages. 2.3 One-to-one and Onto . 2.4 Bijections. 2.5 Inverse Functions. 3. Counting Infinite Sets. 3.1 Finite Sets. 3.2 Hilbert's Infinite Hotel. 3.3 Equivalent Sets and Cardinality. 4. Infinite Cardinals. 4.1 Countable Sets. 4.2 Uncountable Sets. 4.3 Two Infinities. 4.4 Power Sets. 4.5 The Arithmetic of Cardinals. 5. Well Ordered Sets. 5.1 Successors of Elements. 5.2 The Arithmetic of Ordinals. 5.4 Magnitude versus Cardinality. 6. Inductions and Numbers. 6.1 Mathematical Induction. 6.2 Transfinite Induction. 6.3 Mathematical Recursion . 6.4 Number Theory. 6.5 The Fundamental Theorem of Arithmetic. 6.6 Perfect Numbers. 7. Prime Numbers. 7.1 Prime Number Generators. 7.2 The Prime Number Theorem. 7.3 Products of Geometric Series. 7.4 The Riernann Zeta Function. 7.5 Real Numbers. 8. Logic and Meta-Mathematics. 8.1 The Collection of All Sets. 8.2 Other Than True or False. Bibliography. Index.

Recursions for multivariate compound phase variables

We show how to generalize the result given in [Eisele, K.-Th., 2006. Recursions for compound phase distributions. Insurance: Math. Econom. 38, 149–156] to the multivariate case, i.e. we find a Panjer-like recursion principle for the distribution of a multivariate compound phase variable. Recursion formulas and procedures for the bivariate case are given in detail. We give a possible application for agricultural risks and calculate concrete examples via a VB-program.

Generalized functions with pseudobounded support in constructive mathematics

In this paper we define pseudoboundedness for support of a distribution which is weaker than boundedness in Bishop's constructive mathematics. We prove in Bishop's framework that a distribution (sequentially continuous linear functional on the space D ( R ) of test functions) with pseudobounded support is a sequentially continuous linear functional on the space E ( R ) of infinitely differentiable functions on R . We also show that the following three propositions can be proved in classical mathematics, Brouwer's intuitionistic mathematics and constructive recursive mathematics of Markov's school, but cannot be in Bishop's framework: every sequentially continuous linear functional on E ( R ) is bounded on E ( R ) ; every bounded distribution with pseudobounded support is bounded on E ( R ) ; every sequentially continuous linear functional on E ( R ) is a distribution with compact support.

On local non‐compactness in recursive mathematics

AbstractA metric space is said to be locally non‐compact if every neighborhood contains a sequence that is eventually bounded away from every element of the space, hence contains no accumulation point. We show within recursive mathematics that a nonvoid complete metric space is locally non‐compact iff it is without isolated points.The result has an interesting consequence in computable analysis: If a complete metric space has a computable witness that it is without isolated points, then every neighborhood contains a computable sequence that is eventually computably bounded away from every computable element of the space. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Consistency of the theory of contexts

The Theory of Contexts is a type-theoretic axiomatization aiming to give a metalogical account of the fundamental notions of variable and context as they appear in Higher Order Abstract Syntax. In this paper, we prove that this theory is consistent by building a model based on functor categories. By means of a suitable notion of forcing, we prove that this model validates Classical Higher Order Logic, the Theory of Contexts, and also (parametrised) structural induction and recursion principles over contexts. Our approach, which we present in full detail, should also be useful for reasoning on other models based on functor categories. Moreover, the construction could also be adopted, and possibly generalized, for validating other theories of names and binders.

Inductive inference of approximations for recursive concepts

This paper provides a systematic study of inductive inference of indexable concept classes in learning scenarios where the learner is successful if its final hypothesis describes a finite variant of the target concept, i.e., learning with anomalies. Learning from positive data only and from both positive and negative data is distinguished. The following learning models are studied: learning in the limit, finite identification, set-driven learning, conservative inference, and behaviorally correct learning. The attention is focused on the case that the number of allowed anomalies is finite but not a priori bounded. However, results for the special case of learning with an a priori bounded number of anomalies are presented, too. Characterizations of the learning models with anomalies in terms of finite tell-tale sets are provided. The observed varieties in the degree of recursiveness of the relevant tell-tale sets are already sufficient to quantify the differences in the corresponding learning models with anomalies. Finally, a complete picture concerning the relations of all models of learning with and without anomalies mentioned above is derived.

On the Foundations of Constructive Mathematics – Especially in Relation to the Theory of Continuous Functions

We discuss the foundations of constructive mathematics, including recursive mathematics and intuitionism, in relation to classical mathematics. There are connections with the foundations of physics, due to the way in which the different branches of mathematics reflect reality. Many different axioms and their interrelationship are discussed. We show that there is a fundamental problem in BISH (Bishop’s school of constructive mathematics) with regard to its current definition of ‘continuous function’. This problem is closely related to the definition in BISH of ‘locally compact’. Possible approaches to this problem are discussed. Topology seems to be a key to understanding many issues. We offer several new simplifying axioms, which can form bridges between the various branches of constructive mathematics and classical mathematics (‘reuniting the antipodes’). We give a simplification of basic intuitionistic theory, especially with regard to so-called ‘bar induction’. We then plead for a limited number of axiomatic systems, which differentiate between the various branches of mathematics. Finally, in the appendix we offer BISH an elegant topological definition of ‘locally compact’, which unlike the current definition is equivalent to the usual classical and/or intuitionistic definition in classical and intuitionistic mathematics, respectively.