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This paper shows some of the differences and similarities between computation and hypercomputation, the similarities relating to the complexity of propositional computation and the differences being the propositions that can be decided computationally or hypercomputationally. The methods used are ordinal Turing machines with infinitely long programs and diagonalization out of computing complexity classes. The main results are the characterization of inequalities of run time complexities of serial, indeterministic serial and parallel computers and hypercomputers and the specification of a hierarchy of hypercomputers that can hypercompute the truths of all propositions in the standard class model of set theory, the von Neumann hierarchy of pure sets. [ABSTRACT FROM AUTHOR]

Highlights • We present an algorithm that transforms context-free grammars into a single set of supercombinators. • We evaluate our algorithm with the use of 62,008 grammar samples obtained from Groningen Meaning Bank. • We have found the limit of supercombinator set, which in case of our sample set is a sequence of Catalan numbers. • We show the way how to identify the most common structures of input grammars. Abstract We present an algorithm that transforms context-free grammars into a non-redundant set of supercombinators. This set contains interconnected lambda calculus' supercombinators that are enriched by grammar operations. The resulting set is scalable and it can be extended with new supercombinators created from grammars. We describe this algorithm in detail and then we apply it on 62,008 grammar samples in order to find out the properties and limits of acquired supercombinator set. We show that this set has a maximum theoretical limit of possible supercombinators. That limit is the sequence of Catalan numbers. We show that in some cases we are able to reach that limit if we use large enough input data source and we limit the size of supercombinators permitted into the final set. We also describe another benefit of our algorithm, which is the identification of most reoccurring structures in the input set. [ABSTRACT FROM AUTHOR]

We show that the only polynomial sets with a generating function of the form F ( x t − R ( t ) ) and satisfying a three-term recursion relation are the monomial set and the rescaled ultraspherical, Hermite, and Chebyshev polynomials of the first kind. [ABSTRACT FROM AUTHOR]

We give solutions to two of the questions in a paper by Brendle, Brooke-Taylor, Ng and Nies. Our examples derive from a 2014 construction by Khan and Miller as well as new direct constructions using martingales. At the same time, we introduce the concept of i.o. subuniformity and relate this concept to recursive measure theory. We prove that there are classes closed downwards under Turing reducibility that have recursive measure zero and that are not i.o. subuniform. This shows that there are examples of classes that cannot be covered with methods other than probabilistic ones. It is easily seen that every set of hyperimmune degree can cover the recursive sets. We prove that there are both examples of hyperimmune-free degree that can and that cannot compute such a cover. [ABSTRACT FROM AUTHOR]

In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Graßmannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace designs. We construct a 2- ( 6 , 3 , 78 ) 5 design by computer, which corresponds to a halving LS 5 [ 2 ] ( 2 , 3 , 6 ) . The application of the new recursion method to this halving and an already known LS 3 [ 2 ] ( 2 , 3 , 6 ) yields two infinite two-parameter series of halvings LS 3 [ 2 ] ( 2 , k , v ) and LS 5 [ 2 ] ( 2 , k , v ) with integers v ≥ 6 , v ≡ 2 ( mod 4 ) and 3 ≤ k ≤ v − 3 , k ≡ 3 ( mod 4 ) . Thus in particular, two new infinite series of nontrivial subspace designs with t = 2 are constructed. Furthermore as a corollary, we get the existence of infinitely many nontrivial large sets of subspace designs with t = 2 . [ABSTRACT FROM AUTHOR]

Hjorth and Nies proposed notions of randomness corresponding to the higher recursion setting. In particular, they defined the notion of Π 1 1 -Martin-Löf randomness. In this article we present examples of Π 1 1 -Martin-Löf random reals which are obtained as measures of Π 1 1 open sets. [ABSTRACT FROM AUTHOR]

We elaborate a recursion-theoretic method to define real numbers which are random to any jump of the halting problem. This is obtained by generalizing a result of V. Becher and G. Chaitin dealing with the class of cofinite sets. As in the work of V. Becher and S. Grigorieff (but this time in the context of open sets in Cantor space) appeal to completeness phenomena replaces machine arguments. [ABSTRACT FROM AUTHOR]

Using dimension-like invariant base dimension I of a space by a normal base, the inductive dimensions of the product of compacta in Filippov's example are computed. [ABSTRACT FROM AUTHOR]

Abstract: The main topic of the present work is the relation that a set X is strongly hyperimmune-free relative to Y. Here X is strongly hyperimmune-free relative to Y if and only if for every partial X-recursive function p there is a partial Y-recursive function q such that every a in the domain of p is also in the domain of q and satisfies , that is, p is majorised by q. For X being hyperimmune-free relative to Y, one also says that X is pmaj-reducible to Y . It is shown that between degrees not above the Halting problem the pmaj-reducibility coincides with the Turing reducibility and that therefore only recursive sets have a strongly hyperimmune-free Turing degree while those sets Turing above the Halting problem bound uncountably many sets with respect to the pmaj-relation. So pmaj-reduction is identical with Turing reduction on a class of sets of measure 1. Moreover, every pmaj-degree coincides with the corresponding Turing degree. The pmaj-degree of the Halting problem is definable with the pmaj-relation. [Copyright &y& Elsevier]

Abstract: We study the strength of , Rainbow Ramsey Theorem for colorings of triples, and prove that implies neither nor . To this end, we establish some recursion theoretic properties of cohesive sets and rainbows for colorings of pairs. We show that every sequence (2-bounded coloring of pairs) admits a cohesive set (infinite rainbow) of non-PA Turing degree; and that every -recursive sequence (2-bounded coloring of pairs) admits a cohesive set (infinite rainbow). [Copyright &y& Elsevier]

Abstract: Franek et al. recently described four constructions for large sets of -intersecting Steiner triple systems of order (STS ) [F. Franek, M.J. Grannell, T.S. Griggs, A. Rosa, On the large sets of -intersecting Steiner triple systems of order , Des. Codes Cryptogr. 26 (2002) 243–256]. In this study we focus on large sets of -intersecting STS . Some recursive constructions are presented that involve three-wise balanced designs. By applying these constructions we obtain some new infinite classes of such large sets, and the large sets of -intersecting KTS are used to produce some new large sets of Kirkman triple systems. [Copyright &y& Elsevier]

Abstract: This article introduces the notions of prolongations and prolongational limit sets of control systems. It is shown how the prolongational limit sets play in studying recursive and dispersive concepts of control systems. [Copyright &y& Elsevier]

Abstract: An () over a -set is a collection of pairwise disjoint resolvable (almost resolvable) Mendelsohn triple systems of order . In this paper several direct construction methods for OLRMTSs and OLARMTSs are presented and then applied to produce some new orders; the smallest unknown for , the smallest unknown for , and some other small designs are displayed; a few new existence families are also obtained by known recursive constructions. [Copyright &y& Elsevier]

Abstract: We use discrete Morse theory to determine the Möbius function of generalized factor order. Ordinary factor order on the Kleene closure of a set is the partial order defined by letting if contains as a subsequence of consecutive letters. Generalized factor order takes into account a partial order on the alphabet , that is, whenever contains a subsequence such that for each , in . Using Babson and Hersh’s application of Robin Forman’s discrete Morse theory to poset order complexes, we are able to give a recursive formula for the Möbius function in the case where each element of covers a unique letter in . [Copyright &y& Elsevier]

Abstract: Let denote a graph formed from unlabelled graphs and a labelled graph replacing every vertex of by the graph and joining the vertices of with all the vertices of those of whenever . For unlabelled graphs , let stand for the class of all graphs taken over all possible orderings of . A prime inductive class of graphs, , is said to be a set of all graphs, which can be produced by recursive applying of where is a graph from a fixed set of prime graphs and are either graphs from the set of prime graphs or graphs obtained in the previous steps. Similar inductive definitions for cographs, -trees, series–parallel graphs, Halin graphs, bipartite cubic graphs or forbidden structures of some graph classes were considered in the literature (Batagelj (1994) Drgas-Burchardt et al. (2010) and Hajós (1961) ). This paper initiates a study of prime inductive classes of graphs giving a result, which characterizes, in their language, the substitution closed induced hereditary graph classes. Moreover, for an arbitrary induced hereditary graph class it presents a method for the construction of maximal induced hereditary graph classes contained in and substitution closed. The main contribution of this paper is to give a minimal forbidden graph characterization of induced hereditary prime inductive classes of graphs. As a consequence, the minimal forbidden graph characterization for some special induced hereditary prime inductive graph classes is given There is also offered an algebraic view on the class of all prime inductive classes of graphs of the type . [Copyright &y& Elsevier]

Abstract: A -disjoint path cover of a graph is defined as a set of internally vertex-disjoint paths connecting given sources and sinks in such a way that every vertex of the graph is covered by a path in the set. In this paper, we analyze the -disjoint path cover of recursive circulant under the condition that at most faulty vertices and/or edges are removed. It is shown that when , has a -disjoint path cover (of one-to-one type) joining any pair of two distinct source and sink for arbitrary and subject to . In addition, it is proven that when , has a -disjoint path cover (of unpaired many-to-many type) joining any two disjoint sets of sources and sinks for arbitrary and satisfying , in which sources and sinks are freely matched. In particular, the mentioned bounds and of the two cases are shown to be optimal. [Copyright &y& Elsevier]

Abstract: An implicit characterization of the class is given, without using any minimization scheme. This is the first purely recursion-theoretic formulation of . [Copyright &y& Elsevier]

Abstract: Given a set of vertices of a connected graph , the metric representation of a vertex of with respect to is the vector , where , denotes the distance between and . is a resolving set for if for every pair of distinct vertices of , . The metric dimension of , , is the minimum cardinality of any resolving set for . Let and be two graphs of order and , respectively. The corona product is defined as the graph obtained from and by taking one copy of and copies of and joining by an edge each vertex from the th-copy of with the th-vertex of . For any integer , we define the graph recursively from as . We give several results on the metric dimension of . For instance, we show that given two connected graphs and of order and , respectively, if the diameter of is at most two, then . Moreover, if and the diameter of is greater than five or is a cycle graph, then . [Copyright &y& Elsevier]

Abstract: It has been previously observed that for many -learnable computable families of computably enumerable (c.e. for short) sets all their computable numberings are evidently -equivalent, i.e. are equivalent with respect to reductions computable in the halting problem. We show that this holds for all -learnable computable families of c.e. sets, and prove that, in general, the converse is not true. In fact there is a computable family of c.e. sets such that all computable numberings of are computably equivalent and is not -learnable. Moreover, we construct a computable family of c.e. sets which is not -learnable though all of its computable numberings are -equivalent. We also give a natural example of a computable -learnable family of c.e. sets which possesses non--equivalent computable numberings. So, for the computable families of c.e. sets, the properties of -learnability and -equivalence of all computable numberings are independent. [Copyright &y& Elsevier]

Abstract: This work initiates a systematic investigation into the matrix forms of the Pascal triangle as mathematical objects in their own right. The present paper is especially devoted to the so-called G-matrices, i.e. the set of the twelve triangular matrix forms that can be derived from the Pascal triangle expanded to the level . For , the G-matrix set reduces to a set of four distinct matrices. The twelve G-matrices are defined and the classic Pascal recursion is reformulated for each of the twelve G-matrices. Three sets of matrix transformations are then introduced to highlight different relations between the twelve G-matrices and for generating them from appropriately chosen subsets. [Copyright &y& Elsevier]

Abstract: A new axiomatic system OST of operational set theory is introduced in which the usual language of set theory is expanded to allow us to talk about (possibly partial) operations applicable both to sets and to operations. OST is equivalent in strength to admissible set theory, and a natural extension of OST is equivalent in strength to ZFC. The language of OST provides a framework in which to express “small” large cardinal notions—such as those of being an inaccessible cardinal, a Mahlo cardinal, and a weakly compact cardinal—in terms of operational closure conditions that specialize to the analogue notions on admissible sets. This illustrates a wider program whose aim is to provide a common framework for analogues of large cardinal notions that have appeared in admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics, and systems of recursive ordinal notations that have been used in proof theory. [Copyright &y& Elsevier]

The sets of five numbers picked in the North Carolina Cash- 5 Lottery game were identical on July 9th and 11th, 2007. This coincidence was the topic of a local television station news story on July 12 in which I played a minor role. This article documents the coincidence, my interactions with the television re- porter seeking to understand how likely, or unlikely, the coincidence was, and some afterthoughts, including an analysis of the likelihood of matching sets of numbers. [ABSTRACT FROM AUTHOR]

This paper presents a novel and mathematically rigorous Bayes' recursion for tracking a target that generates multiple measurements with state dependent sensor field of view and clutter. Our Bayesian formulation is mathematically well-founded due to our use of a consistent likelihood function derived from random finite set theory. It is established that under certain assumptions, the proposed Bayes' recursion reduces to the cardinalized probability hypothesis density (CPHD) recursion for a single target. A particle implementation of the proposed recursion is given. Under linear Gaussian and constant sensor field of view assumptions, an exact closed-form solution to the proposed recursion is derived, and efficient implementations are given. Extensions of the closed-form recursion to accommodate mild nonlinearities are also given using linearization and unscented transforms. [ABSTRACT FROM AUTHOR]

This paper presents a sound and complete category-theoretic notion of models for Linear Abadi & Plotkin Logic [Birkedal et al., 2006], a logic suitable for reasoning about parametricity in combination with recursion. A subclass of these called parametric LAPL structures can be seen as an axiomatization of domain theoretic models of parametric polymorphism, and we show how to solve general (nested) recursive domain equations in these. Parametric LAPL structures constitute a general notion of model of parametricity in a setting with recursion. In future papers we will demonstrate this by showing how many different models of parametricity and recursion give rise to parametric LAPL structures, including Simpson and Rosolini's set theoretic models [Rosolini and Simpson, 2004], a syntactic model based on Lily [Pitts, 2000, Bierman et al., 2000] and a model based on admissible pers over a reflexive domain [Birkedal et al., 2007]. [ABSTRACT FROM AUTHOR]

Abstract: The concept of recursive coalgebra of a functor was introduced in the 1970s by Osius in his work on categorical set theory to discuss the relationship between wellfounded induction and recursively specified functions. In this paper, we motivate the use of recursive coalgebras as a paradigm of structured recursion in programming semantics, list some basic facts about recursive coalgebras and, centrally, give new conditions for the recursiveness of a coalgebra based on comonads, comonad-coalgebras and distributive laws of functors over comonads. We also present an alternative construction using countable products instead of cofree comonads. [Copyright &y& Elsevier]

We establish a first step towards a “Rice theorem” for tilings: for non-trivial sets, it is undecidable to know whether two different tile sets produce the same tilings of the plane. Then, we study quasiperiodicity functions associated with tilings. This function is a way to measure the regularity of tilings. We prove that, not only almost all recursive functions can be obtained as quasiperiodicity functions, but also, a function which overgrows any recursive function. [Copyright &y& Elsevier]

We prove the existence of a perfect Π01 set of reals such that the uniform join of any of its infinite subsets is above 0′. We explain a couple of ways in which this demonstrates the optimality of the Low Basis Theorem. [Copyright &y& Elsevier]