Topic
Simple set
About: Simple set is a research topic. Over the lifetime, 66 publications have been published within this topic receiving 644 citations.
Papers published on a yearly basis
Papers
TL;DR: It is shown that one can solve Post's Problem by constructing generic sets in the usual set theoretic framework applied to tiny universes, and this method leads to a new class of recursively enumerable sets: r.e. generic sets.
Abstract: We show that one can solve Post's Problem by constructing generic sets in the usual set theoretic framework applied to tiny universes. This method leads to a new class of recursively enumerable sets: r.e. generic sets. All r.e. generic sets are low and simple and therefore of Turing degree strictly between 0 and 0'. Further they supply the first example of a class of low recursively enumerable sets which are automorphic in the lattice S of recursively enumerable sets with inclusion. We introduce the notion of a promptly simple set. This describes the essential feature of r.e. generic sets with respect to automorphism constructions. It is obvious that constructions of generic sets in set theory and certain con- structions of recursively enumerable (r.e) sets have something in common. It is typical for the construction of a generic set U that a sentence 0 about U is finally made true (i.e. M(U) I= 0) if one has unboundedly often the chance to make it true during the construction of U (i.e. Vp 3q E G}) are of incomparable Turing degree. The sentences 3x((G)o (x) # {e}(G) 1(x)) and 3x ((G)1(x) # {e}(G)o (x)) are finally made true for every e where one has infinitely often the chance to make them true without filling up the com- plement of (G)O, (G)1. Then we cannot have (G)o = {e}(G)1 because such an e
54 citations
1 Dec 1993
TL;DR: For finite binary images on an n-dimensional Cartesian grid, appropriate definitions of simple set are given which satisfy all the axioms and verification that a parallel reduction operator for n- dimensional binary images preserves the topology of all possible input images may be achievable by checking only a finite number of cases.
Abstract: Loosely speaking, a simple set of a finite binary image is a set of 1s whose deletion `preserves topology.' This concept can be made precise in different (and inequivalent) ways. Ronse established results which imply that, for finite 2-D binary images on a Cartesian grid and three different definitions of simple set, a set S of 1s is simple if every subset of S that lies in a 2- point by 2-point square is simple. In fact this is a special case of a general result which applies to arbitrary finite binary images -- not just 2-D images on a Cartesian grid -- and any definition of simple set which satisfies three axioms stated in this paper. For finite binary images on an n-dimensional Cartesian grid, we give appropriate definitions of simple set which satisfy all the axioms. When these definitions of simple set are used, verification that a parallel reduction operator for n-dimensional binary images preserves the topology of all possible input images may be achievable by checking only a finite number of cases.
51 citations
TL;DR: The main contribution of this paper is a characterisation of the non-trivial simple sets composed of exactly two voxels, such sets being called minimal simple pairs.
Abstract: Preserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. This paper constitutes an introduction to the study of non-trivial simple sets in the framework of cubical 3-D complexes. A simple set has the property that the homotopy type of the object in which it lies is not changed when the set is removed. The main contribution of this paper is a characterisation of the non-trivial simple sets composed of exactly two voxels, such sets being called minimal simple pairs.
32 citations
TL;DR: A recursive oracles is constructed relative to which a simple set exists for NP, showing that the time bounds are not a crucial hypothesis; bounds on the way in which the oracle is accessible, namely, the number of queries and/or thenumber of nondeterministic steps, are shown to be the fundamental hypothesis.
Abstract: Relativizations of complexity classes in which simple sets exist are considered. A recursive oracle is constructed relative to which a simple set exists for NP. Some other general theorems are proven, showing that the time bounds are not a crucial hypothesis; bounds on the way in which the oracle is accessible, namely, the number of queries and/or the number of nondeterministic steps, are shown to be the fundamental hypothesis. As a result, simple sets are shown to exist in many different relativized complexity classes.
29 citations
TL;DR: In this article, it was shown that for every coinfinite r.i.d. set A there is a complete set B such that L * (A) ≃ eff L* (B) and that every simple set is automorphic (in E*) to complete sets.
Abstract: We show that for every coinfinite r.e. set A there is a complete r.e. set B such that L* (A) ≃ eff L* (B) and that every promptIy simple set is automorphic (in E*) to a complete set
27 citations
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| Year | Papers |
|---|---|
| 2021 | 1 |
| 2017 | 2 |
| 2016 | 2 |
| 2015 | 2 |
| 2014 | 1 |
| 2013 | 3 |