Recursively Defined Research Articles

Códigos de Reed-Muller

The Reed-Muller codes were discovered by David Eugene Muller and decoded by Irving Stoy Reed in 1954. Such codes belong to the linear code family and are widely used nowadays, mainly for their simple and efficient decoding algorithm. There are several ways to define Reed-Muller codes. In this work, we present, in a clear and simple way, a recursive definition for all Reed-Muller codes of order r \in N, denoted by R (r, m), where 0 <= r <= m, m \in N. Using this definition, we show the main parameters: length, number of elements and minimum distance of first-order Reed-Muller codes, R(1, m) for all m \in N. In addition, we present an application of the first-order codes in a National Aeronautics and Space Administration (NASA) space program.

Unit Sphere-Constrained and Higher Order Interpolations in Laplace’s Method of Initial Orbit Determination

This paper explores an alternative interpolation approach for the line-of-sight measurements in Laplace’s method for angles-only initial orbit determination (IOD). The classical implementation of the method uses Lagrange polynomials to interpolate three or more unit line-of-sight (LOS) vectors from a ground-based or orbiting site to an orbiting target. However, such an approach does not guarantee unit magnitude of the interpolated line-of-sight path except at the measurement points. The violation of this constraint leads to unphysical behavior in the derivatives of the interpolated line-of-sight history, which can lead to poor initial orbit determination (IOD) performance. By adapting a spherical interpolation method used in the field of computer graphics, we can obtain an interpolated line-of-sight history that is always unit norm. This new spherical interpolation method often leads to significant performance improvements in Laplace’s IOD method, with comparable robustness to measurement noise as the traditional interpolation methods. This paper also demonstrates the benefit of interpolating through more than three data points, which is easily enabled up to an arbitrary number of measurements by recursive definitions of the interpolations.

Elementary inductive dichotomy: Separation of open and clopen determinacies with infinite alternatives

We introduce a new axiom called inductive dichotomy, a weak variant of the axiom of inductive definition, and analyze the relationships with other variants of inductive definition and with related axioms, in the general second order framework, including second order arithmetic, second order set theory and higher order arithmetic. By applying these results to the investigations on the determinacy axioms, we show the following. (i) Clopen determinacy is consistency-wise strictly weaker than open determinacy in these frameworks, except second order arithmetic; this is an enhancement of Schweber–Hachtman separation of open and clopen determinacy into the consistency-wise separation. (ii) Hausdorff–Kuratowski hierarchy of differences of opens is faithfully reflected by the hierarchy of consistency strengths of corresponding parameter-free determinacies in the aforementioned frameworks; this result is valid also in second order arithmetic only except clopen determinacy.

HR-SQL: Extending SQL with hypothetical reasoning and improved recursion for current database systems

In this work we present a formalization, backed with a contrasted implementation, of a relational database language (called HR-SQL) that extends SQL in two aspects. On the one hand, including non-linear and mutual recursion. On the other hand, including hypothetical relations and queries. Regarding expressiveness, HR-SQL allows a novel form of hypothetical reasoning, completely integrated with recursive definitions. In addition, aggregate functions are added. Regarding formalization, the extended language is founded on a stratified fixpoint semantics based on logic programming techniques. We include results of the existence of such fixpoints. An algorithm that transforms a database containing hypothetical definitions into an equivalent one without hypothesis and its correctness proof are presented. Regarding the implementation, we introduce here a system that incorporates the aforementioned benefits and enhances former implementations in several areas. The current HR-SQL system is targeted to several state-of-the-art relational database systems, and could be used with any SQL-based system with minor modifications in the implementation.

Enhancing Magic Sets with an Application to Ontological Reasoning

AbstractMagic sets are a Datalog to Datalog rewriting technique to optimize query answering. The rewritten program focuses on a portion of the stable model(s) of the input program which is sufficient to answer the given query. However, the rewriting may introduce new recursive definitions, which can involve even negation and aggregations, and may slow down program evaluation. This paper enhances the magic set technique by preventing the creation of (new) recursive definitions in the rewritten program. It turns out that the new version of magic sets is closed for Datalog programs with stratified negation and aggregations, which is very convenient to obtain efficient computation of the stable model of the rewritten program. Moreover, the rewritten program is further optimized by the elimination of subsumed rules and by the efficient handling of the cases where binding propagation is lost. The research was stimulated by a challenge on the exploitation of Datalog/dlv for efficient reasoning on large ontologies. All proposed techniques have been hence implemented in the dlv system, and tested for ontological reasoning, confirming their effectiveness.

수학적 귀납법 단원의 과제 설계를 통한 교사 공동체의 학습: 공동학습형 연구공동체의 활동 사례를 중심으로

Mathematical induction has been known to be difficult topic for didactic transposition, especially for tasks designs, due to the formality of reasoning structures and expressions embedded in them. This study organized a co-learning inquiry community with teachers and researchers to overcome these difficulties and to lead the learning of teacher community. A series of activities consisting of tasks design, implementation, and reflection were conducted. Then the learning of teacher community was analyzed by describing which knowledge elements were developed. As a result, teacher community drew a proof-planning activity as a didactic strategy and designed tasks that enable students to recognize the recursibility of the proposition function as they rediscover the roles of inductive definitions of sequence and operations. Through this, teacher community developed the knowledge of the logical structure of mathematical induction, knowledge of students understanding and difficulties about learning mathematical induction, and knowledge of didactic strategies which support the learning process of mathematical induction.

On the Meaning of Transition System Specifications

Transition System Specifications provide programming and specification languages with a semantics. They provide the meaning of a closed term as a process graph: a state in a labelled transition system. At the same time they provide the meaning of an n-ary operator, or more generally an open term with n free variables, as an n-ary operation on process graphs. The classical way of doing this, the closed-term semantics, reduces the meaning of an open term to the meaning of its closed instantiations. It makes the meaning of an operator dependent on the context in which it is employed. Here I propose an alternative process graph semantics of TSSs that does not suffer from this drawback. Semantic equivalences on process graphs can be lifted to open terms conform either the closed-term or the process graph semantics. For pure TSSs the latter is more discriminating. I consider five sanity requirements on the semantics of programming and specification languages equipped with a recursion construct: compositionality, applied to n-ary operators, recursion and variables, invariance under $\alpha$-conversion, and the recursive definition principle, saying that the meaning of a recursive call should be a solution of the corresponding recursion equations. I establish that the satisfaction of four of these requirements under the closed-term semantics of a TSS implies their satisfaction under the process graph semantics.

A Recursive Definition of Goodness of Space for Bridging the Concepts of Space and Place for Sustainability

Conceived and developed by Christopher Alexander through his life’s work, The Nature of Order, wholeness is defined as a mathematical structure of physical space in our surroundings. Yet, there was no mathematics, as Alexander admitted then, that was powerful enough to capture his notion of wholeness. Recently, a mathematical model of wholeness, together with its topological representation, has been developed that is capable of addressing not only why a space is good, but also how much goodness the space has. This paper develops a structural perspective on goodness of space (both large- and small-scale) in order to bridge two basic concepts of space and place through the very concept of wholeness. The wholeness provides a de facto recursive definition of goodness of space from a holistic and organic point of view. A space is good, genuinely and objectively, if its adjacent spaces are good, the larger space to which it belongs is good, and what is contained in the space is also good. Eventually, goodness of space, or sustainability of space, is considered a matter of fact rather than of opinion under the new view of space: space is neither lifeless nor neutral, but a living structure capable of being more living or less living, or more sustainable or less sustainable. Under the new view of space, geography or architecture will become part of complexity science, not only for understanding complexity, but also for making and remaking complex or living structures.

Direct spectra of Bishop spaces and their limits

We apply fundamental notions of Bishop set theory (BST), an informal theory that complements Bishop's theory of sets, to the theory of Bishop spaces, a function-theoretic approach to constructive topology. Within BST we develop the notions of a direct family of sets, of a direct spectrum of Bishop spaces, of the direct limit of a direct spectrum of Bishop spaces, and of the inverse limit of a contravariant direct spectrum of Bishop spaces. Within the extension of Bishop's informal system of constructive mathematics BISH with inductive definitions with rules of countably many premises, we prove the fundamental theorems on the direct and inverse limits of spectra of Bishop spaces and the duality principle between them.

Tone Association and Output Locality in Non-Linear Structures

This paper offers a computational characterization of tone-to-TBU association processes using a restricted least-fixed point logic. Crucially, least fixed point logics allow recursive definitions which capture output-oriented processes. The added requirement that these definitions are quantifier-free ensures that they are inherently local, a restriction that is well-motivated for phonological processes in general. The typology developed here distinguishes between possible and impossible tone mappings, capturing a wider range of attested tone mappings (left-to-right, right-to-left, edge-in, quality-sensitive) than previous rule-based or optimization approaches, while also explaining why certain unattested mapping patterns (for example center-out association) are impossible. This thus represents a strong first approximation of a definition for output-based local functions over non-linear structures

Equivalents of the finitary non-deterministic inductive definitions

We present statements equivalent to some fragments of the principle of non-deterministic inductive definitions (NID) by van den Berg (2013), working in a weak subsystem of constructive set theory CZF. We show that several statements in constructive topology which were initially proved using NID are equivalent to the elementary and finitary NIDs. We also show that the finitary NID is equivalent to its binary fragment and that the elementary NID is equivalent to a variant of NID based on the notion of biclosed subset. Our result suggests that proving these statements in constructive topology requires genuine extensions of CZF with the elementary or finitary NID.

Excitation-Current-Dependent Magnetoinductance in Inductive Elements Containing CoFeSiB Ribbons Annealed in Transverse Magnetic Field

An important issue for the practical application of magnetic field elements is to select an appropriate operating condition. The effect of excitation parameters on the magnetoinductance effect has been investigated in inductive elements consisting of a magnetically annealed Co67Fe4Si14.5B14.5 amorphous ribbon wound with a single coil loaded with an excitation current (Iac). The influence of the frequency on the magnetoinductance effect is closely related to the amplitude of Iac. With an increase of the amplitude of Iac from 1.0 mA to 4.0 mA, the curve of the maximum magnetoinductance ratio (ηmax) versus frequency changes from a monotonic reduction to show a peak of about 2450% at 32.1 kHz. The local average sensitivity (Sloc) remains almost the same across a wider magnetic field range from 0 A m−1 to 32 A m−1 under an excitation current of 4.0 mA at 2.1 kHz. These results demonstrate that both the magnetic field sensitivity and linear measurement range of such inductive elements closely depend on the parameters of Iac, which is discussed herein from the viewpoint of the magnetization dynamics and definition of inductance.

Oriented Coloring on Recursively Defined Digraphs

Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented graph G = ( V , A ) is a partition of the vertex set V into k independent sets such that all the arcs linking two of these subsets have the same direction. The oriented chromatic number of an oriented graph G is the smallest k such that G allows an oriented k-coloring. Deciding whether an acyclic digraph allows an oriented 4-coloring is NP-hard. It follows that finding the chromatic number of an oriented graph is an NP-hard problem, too. This motivates to consider the problem on oriented co-graphs. After giving several characterizations for this graph class, we show a linear time algorithm which computes an optimal oriented coloring for an oriented co-graph. We further prove how the oriented chromatic number can be computed for the disjoint union and order composition from the oriented chromatic number of the involved oriented co-graphs. It turns out that within oriented co-graphs the oriented chromatic number is equal to the length of a longest oriented path plus one. We also show that the graph isomorphism problem on oriented co-graphs can be solved in linear time.

An Indian Quasi-Fregean Theory of Number

A very interesting account of the reference of number words in classical Indian philosophy was given by Maheśa Chandra (1836–1906) in his Brief Notes on the Modern Nyāya System of Philosophy and its Technical Terms (BN), a primer on Navya-Nyāya terminology and doctrines. Despite its English title, BN is a Sanskrit work. The section on “number” (saṃkhyā) provides an exposition of a theory of number which can account for both the adjectival and the substantival use of number words in Sanskrit. According to D. H. H. Ingalls (1916–1999), some ideas about the reference of number words in BN are close to the Frege–Russell theory of natural number. Ingalls’s comparison refers to a concept of number in Navya-Nyāya which is related to the things numbered via the so-called “circumtaining relation” (paryāpti). Although there is no theory of sets in Navya-Nyāya, Navya-Naiyāyikas do have a realist theory of properties (dharma) and their theory of number is a theory of properties as constituents of empirical reality, anchored to their system of ontological categories. As shown by George Bealer, properties can serve the same purpose as sets in the Frege–Russell theory of natural number. In the present paper, we attempt a formal reconstruction of Maheśa Chandra’s exposition of the Navya-Nyāya theory of number, which accounts for its affinity to George Bealer’s neo-Fregean analysis. As part of our appraisal of the momentousness and robustness of the “circumtaining” concept of number, we show that it can be cast into a precise recursive definition of natural number and we prove property versions of Peano’s axioms from this definition.

Completeness and expressiveness of pointer program verification by separation logic

Reynolds' separation logical system for pointer program verification is investigated. This paper proves its completeness theorem that states that every true asserted program is provable in the logical system. In order to prove the completeness, this paper shows the expressiveness theorem that states the weakest precondition of every program and every assertion can be expressed by some assertion. This paper also introduces an extension of the assertion language with inductive definitions and proves the soundness theorem, the expressiveness theorem, and the completeness theorem.

Determinacy and monotone inductive definitions

We prove the equivalence of the determinacy of $$\Sigma^0_3$$ (effectively Gδσ) games with the existence of a β-model satisfying the axiom of $$\Pi^2_1$$ monotone induction, answering a question of Montalban [8]. The proof is tripartite, consisting of (i) a direct and natural proof of $$\Sigma^0_3$$ determinacy using monotone inductive operators, including an isolation of the minimal complexity of winning strategies; (ii) an analysis of the convergence of such operators in levels of G¨odel’s L, culminating in the result that the nonstandard models isolated by Welch [18] satisfy $$\Pi^2_1$$ monotone induction; and (iii) a recasting of Welch’s [17] Friedman-style game to show that this determinacy yields the existence of one of Welch’s nonstandard models. Our analysis in (iii) furnishes a description of the degree of $$\Pi^2_1$$ -correctness of the minimal β-model of $$\Pi^2_1$$ monotone induction.

Smart Design for Sustainable Neighborhood Development

Abstract This study proposes the Smart Design method to support the design decision making in the sustainable neighborhood development with multiple objectives. Instead of the “creative design” approach in the scenario making in traditional PSS and recent Geodesign frameworks, the Smart Design method applies the optimization algorithms to search for optimal design solutions in the design space. It integrates the design thinking, computational performance modeling and optimization techniques to efficiently and effectively approximate optimal designs. This method is applied to a hypothetical residential neighborhood design case study with three sustainability objectives: to maximize FAR, to minimize building energy use, and to minimize outdoor human discomfort. Based on the form parameterization, the Nondominated Sorting Genetic Algorithm II (NSGA-II) algorithm is utilized to guide the evolution of the neighborhood design throughout 80 generations, with neighborhood performance modeling tools. The Smart Design method is able to identify 38 representative design solutions as Pareto optimal which are equally optimal. Those solutions set a basis for discussions and negotiations among stake holders to make design decisions with the three objectives. Further research will be focused on addressing the challenges such as recursive objective definitions, parametrization of complex forms, quantification of performances and optimization uncertainties, from simple cases to more realistic and complex designs for sustainable neighborhood development.

A flexible type system for the small Veblen ordinal

We introduce and analyze two theories for typed (accessible part) inductive definitions and establish their proof-theoretic ordinal to be the small Veblen ordinal $$\vartheta \Omega ^\omega $$ . We investigate on the one hand the applicative theory $$\mathsf {FIT}$$ of functions, (accessible part) inductive definitions, and types. It includes a simple type structure and is a natural generalization of S. Feferman’s system $$\mathrm {QL}(\mathsf {F}_\mathsf {0}\text {-}\mathsf {IR}_{N})$$ . On the other hand, we investigate the arithmetical theory $$\mathsf {TID}$$ of typed (accessible part) inductive definitions, a natural subsystem of $$\mathsf {ID}_1$$ , and carry out a wellordering proof within $$\mathsf {TID}$$ that makes use of fundamental sequences for ordinal notations in an ordinal notation system based on the finitary Veblen functions. The essential properties for describing the ordinal notation system are worked out.

Induction and explanatory definitions in mathematics

In this paper, I argue that there are cases of explanatory induction in mathematics. To do so, I first introduce the notion of explanatory definition in the context of mathematical explanation. A large part of the paper is dedicated to introducing and analyzing this notion of explanatory definition and the role it plays in mathematics. After doing so, I discuss a particular inductive definition in advanced mathematics— $${ CW}$$ -complexes—and argue that it is explanatory. With this, we see that there are cases of explanatory induction.

Bisimulation as path type for guarded recursive types

In type theory, coinductive types are used to represent processes, and are thus crucial for the formal verification of non-terminating reactive programs in proof assistants based on type theory, such as Coq and Agda. Currently, programming and reasoning about coinductive types is difficult for two reasons: The need for recursive definitions to be productive, and the lack of coincidence of the built-in identity types and the important notion of bisimilarity. Guarded recursion in the sense of Nakano has recently been suggested as a possible approach to dealing with the problem of productivity, allowing this to be encoded in types. Indeed, coinductive types can be encoded using a combination of guarded recursion and universal quantification over clocks. This paper studies the notion of bisimilarity for guarded recursive types in Ticked Cubical Type Theory, an extension of Cubical Type Theory with guarded recursion. We prove that, for any functor, an abstract, category theoretic notion of bisimilarity for the final guarded coalgebra is equivalent (in the sense of homotopy type theory) to path equality (the primitive notion of equality in cubical type theory). As a worked example we study a guarded notion of labelled transition systems, and show that, as a special case of the general theorem, path equality coincides with an adaptation of the usual notion of bisimulation for processes. In particular, this implies that guarded recursion can be used to give simple equational reasoning proofs of bisimilarity. This work should be seen as a step towards obtaining bisimilarity as path equality for coinductive types using the encodings mentioned above.