Iraj Kalantari

A Data Structure and an Algorithm for the Nearest Point Problem

In this paper we present a tree structure for storing points from a normed space whose norm is effectively computable. We then give an algorithm for finding the nearest point from the tree to a given query point. Our algorithm searches the tree and uses the triangle inequality to eliminate searching of the entirety of some branches of the tree whenever a certain predicate is satisfied. Our data structure uses 0(n) for storage. Empirical data which we have gathered suggest that the expected complexity for preprocessing and the search time are, respectively, 0(nlogn) and 0(logn).

A basic family of iteration functions for polynomial root finding and its characterizations

Abstract Let p(x) be a polynomial of degree n⩾2 with coefficients in a subfield K of the complex numbers. For each natural number m⩾2, let Lm(x) be the m×m lower triangular matrix whose diagonal entries are p(x) and for each j=1,…,m−1, its jth subdiagonal entries are p j (x) j! . For i=1,2, let Lmi)(x) be the matrix obtained from Lm(x) by deleting its first i rows and its last i columns. L1(1)(x)≡1. Then, the function Bm(x)=x−p(x) det (L m−1 (1) (x)) det (L m (1) (x)) defines a fixed-point iteration function having mth order convergence rate for simple roots of p(x). For m=2 and 3, Bm(x) coincides with Newtons and Halleys, respectively. The function Bm(x) is a member of S(m,m+n−2), where for any M⩾m, S(m,M) is the set of all rational iteration functions g(x) ∈ K(x) such that for all roots θ of p(x), then g(x)=θ+∑i=mMγi(x)(θ−x)i, with γi(x) ∈ K(x) and well-defined at any simple root θ. Given g ∈ S(m,M), and a simple root θ of p(x), gi(θ)=0, i=1, …, m−1 and the asymptotic constant of convergence of the corresponding fixed-point iteration is γ m (θ) = (−1)g m (θ) m! . For Bm(x) we obtain γ m (θ)= (−1) m det (L m+1 (2) (θ)) det (L m (1)(θ)) . If all roots of p(x) are simple, Bm(x) is the unique member of S(m,m + n − 2). By making use of the identity 0 = ∑ i=0 n [p (i) (x) i!] (θ − x) i , we arrive at two recursive formulas for constructing iteration functions within the S(m,M) family. In particular, the family of Bm(x) can be generated using one of these formulas. Moreover, the other formula gives a simple scheme for constructing a family of iteration functions credited to Euler as well as Schroder, whose mth order member belongs to S(m,mn), m>2. The iteration functions within S(m,M) can be extended to any arbitrary smooth function f, with the uniform replacement of p(j) with f(j) in g as well as in γm(θ).

Simplicity in effective topology

The recursion-theoretic study of mathematical structures other than ω is now a field of much activity. Analysis and algebra are two such structures which have been studied for their effective contents. Studies in analysis began with the work of Specker on nonconstructive proofs in analysis [16] and in Lacombes inspiring notes on relevant notions of recursive analysis [8]. Studies in algebra originated in the work of Frolich and Shepherdson on effective extensions of explicit fields [1] and in Rabins study of computable fields [15]. Equipped with the richness of modern techniques in recursion theory, Metakides and Nerode [11]–[13] began investigating the effective content of vector spaces and fields; these studies have been extended by Kalantari, Remmel, Retzlaff, Shore and others. Kalantari and Retzlaff [5] began a foundational inquiry into effectiveness in topological spaces. They consider a topological space X with a countable basis ⊿ for the topology. The space is fully effective , that is, the basis elements are coded into ω and the operation of intersection of basis elements and the relation of inclusion among them are both computable. Similar to , the lattice of recursively enumerable (r.e.) subsets of ω , the collection of r.e. open subsets of X forms a lattice ℒ(X) under the usual operations of union and intersection.

Point-free topological spaces, functions and recursive points; filter foundation for recursive analysis. I

Abstract In this paper we develop a point-free approach to the study of topological spaces and functions on them, establish platforms for both and present some findings on recursive points. (The effectivization of the functions on our spaces and related results are presented in a sequel.) In the first sections of the paper, we obtain conditions under which our approach leads to the generation of ideal objects (points) with which mathematicians work. Next, we apply the effective version of our approach to the real numbers, and make exact connections to the classical approach to recursive reals. In the succeeding sections of the paper, we introduce machinery to produce functions on topological spaces and find succinct conditions which will be effectivized in our sequel.

Degrees of Recursively Enumerable Topological Spaces

In [5], Metakides and Nerode introduced the study of recursively enumerable (r.e.) substructures of a recursively presented structure. The main line of study presented in [5] is to examine the effective content of certain algebraic structures. In [6], Metakides and Nerode studied the lattice of r.e. subspaces of a recursively presented vector space. This lattice was later studied by Kalantari, Remmel, Retzlaff and Shore. Similar studies have been done by Metakides and Nerode [7] for algebraically closed fields, by Remmel [10] for Boolean algebras and by Metakides and Remmel [8] and [9] for orderings. Kalantari and Retzlaff [4] introduced and studied the lattice of r.e. subsets of a recursively presented topological space. Kalantari and Retzlaff considered X , a topological space with ⊿, a countable basis. This basis is coded into integers and with the help of this coding, r.e. subsets of ω give rise to r.e. subsets of X . The notion of “recursiveness” of a topological space is the natural next step which gives rise to the question of what should be the “degree” of an r.e. open subset of X ? It turns out that any r.e. open set partitions ⊿; into four sets whose Turing degrees become central in answering the question raised above. In this paper we show that the degrees of the elements of the partition of ⊿ imposed by an r.e. open set can be “controlled independently” in a sense to be made precise in the body of the paper. In [4], Kalantari and Retzlaff showed that given A any r.e. set and any r.e. open subset of X , there exists an r.e. open set ℋ which is a subset of and is dense in (in a topological sense) and in which A is coded. This shows that modulo a nowhere dense set, an r.e. open set can become as complicated as desired. After giving the general technical and notational machinery in §1, and giving the particulars of our needs in §2, in §3 we prove that the set ℋ described above could be made to be precisely of degree of A . We then go on and establish various results (both existential and universal) on the mentioned partitioning of ⊿. One of the surprising results is that there are r.e. open sets such that every element of partitioning of ⊿ is of a different degree. Since the exact wording of the results uses the technical definitions of these partitioning elements, we do not summarize the results here and ask the reader to examine §3 after browsing through §§1 and 2.

A blend of methods of recursion theory and topology

Abstract This paper is a culmination of our new foundations for recursive analysis through recursive topology as reported in Kalantari and Welch (Ann Pure Appl. Logic 93 (1998) 125; 98 (1999) 87). While in those papers we developed groundwork for an approach to point free analysis and applied recursion theory, in this paper we blend techniques of recursion theory with those of topology to establish new findings. We present several new techniques different from existing ones which yield interesting results. Incidental to our work is a unifying explanation of various schools of study for recursive analysis.

Recursive and nonextendible functions over the reals; filter foundation for recursive analysis.II

Abstract In this paper we continue our work of Kalantari and Welch (1998). There we introduced machinery to produce a point-free approach to points and functions on topological spaces and found conditions for both which lend themselves to effectivization. While we studied recursive points in that paper, here, we present two useful classes of recursive functions on topological spaces, apply them to the reals, and find precise accounting for the nature of the properties of some examples that exist in the literature. We end with a construction of a recursive function on a small subset of the unit interval which is strongly nonextendible.

Maximal Zone Diagrams and their Computation

The notion of the zone diagram of a finite set of points in the Euclidean plane is an interesting and rich variation of the classical Voronoi diagram, introduced by Asano, Matousek, Tokuyama. Here, we define the more inclusive notion of a maximal zone diagram. The proof of existence of maximal zone diagrams depends on less restrictive initial conditions and is thus conveniently established via Zorns lemma in contrast to the use of fixed-point theory in proving the existence of a unique zone diagram. A zone diagram is a particular maximal zone diagram satisfying a unique dominance property. We give a characterization for maximal zone diagrams which allows recognition of maximality of certain subsets called subzone diagrams, as well as that of their iterative improvement toward maximality. Maximal zone diagrams offer their own interesting theoretical and computational challenges.

Effective topological spaces. II: A hierarchy

This paper (which is the continuation of the preceding paper [7]) is an investigation of definability hierarchies on effective topological spaces. An open subset U of an effective space X is definable iff there is a parameter free definition φ of U so that the atomic predicate symbols of φ are recursively open relations on X. The complexity of a definable open set may be identified with the quantifier complexity of its definition. For example, a set U is an ∃∃∀∃-set if it has an ∃∃∀∃ parameter free definition using only recursively open predicate symbols. Since X is not equipped with a natural pairing apparatus such a U need not be an ∃∀∃-set. Let Σ denote the class of all ∃-sets, ∃∃-sets, ∃∃∃-sets etc. We show that an open set is in Σ iff it is equivalent modulo a nowhere dense set to a recursively enumerable open set (such sets are said to be essentially recursively enumerable or e.r.e.). Thus Σ = e.r.e. Indeed we show the existence of a universal Σ-set as well as the existence of universal sets for higher levels of the definability hierarchy.

Effective topological spaces I: a definability theory

On cree une theorie de la definissabilite pour des espaces topologiques basee sur la notion presente de recurrence sur des espaces topologiques effectifs. On utilise cette definissabilite pour montrer Σ=recursivement enumerable et decouvrir une hierarchie. Lien entre forcage et definissabilite