On subcreative sets and S-reducibility

1974 ◽  
Vol 39 (4) ◽  
pp. 669-677 ◽  
Author(s):  
John T. Gill ◽  
Paul H. Morris
Keyword(s):  
Open Problem ◽  
Halting Problem ◽  
Order Preserving Map ◽  
Turing Reducibility ◽  
Complete Sets ◽  
Special Case

AbstractSubcreative sets, introduced by Blum, are known to coincide with the effectively speedable sets. Subcreative sets are shown to be the complete sets with respect to S-reducibility, a special case of Turing reducibility. Thus a set is effectively speedable exactly when it contains the solution to the halting problem in an easily decodable form. Several characterizations of subcreative sets are given, including the solution of an open problem of Blum, and are used to locate the subcreative sets with respect to the complete sets of other reducibilities. It is shown that q-cylindrification is an order-preserving map from the r.e. T-degrees to the r.e. S-degrees. Consequently, T-complete sets are precisely the r.e. sets whose q-cylindrifications are S-complete.

scholarly journals On composition of formal power series

2002 ◽  
Vol 30 (12) ◽  
pp. 761-770 ◽  
Author(s):  
Xiao-Xiong Gan ◽  
Nathaniel Knox
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Open Problem ◽  
Constant Term ◽  
Radius Of Convergence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Formal Power ◽  
Special Case

Given a formal power seriesg(x)=b0+b1x+b2x2+⋯and a nonunitf(x)=a1x+a2x2+⋯, it is well known that the composition ofgwithf,g(f(x)), is a formal power series. If the formal power seriesfabove is not a nonunit, that is, the constant term offis not zero, the existence of the compositiong(f(x))has been an open problem for many years. The recent development investigated the radius of convergence of a composed formal power series likefabove and obtained some very good results. This note gives a necessary and sufficient condition for the existence of the composition of some formal power series. By means of the theorems established in this note, the existence of the composition of a nonunit formal power series is a special case.

scholarly journals THE INFORMATIONAL CONTENT OF CANONICAL DISJOINT NP-PAIRS

2009 ◽  
Vol 20 (03) ◽  
pp. 501-522 ◽  
Author(s):  
CHRISTIAN GLAßER ◽  
ALAN L. SELMAN ◽  
LIYU ZHANG
Keyword(s):  
Open Problem ◽  
Structural Complexity ◽  
Informational Content ◽  
Proof Systems ◽  
Canonical Pair ◽  
Propositional Proof Systems ◽  
Np Complete ◽  
Complete Sets ◽  
Propositional Proof ◽  
Almost All

We investigate the connection between propositional proof systems and their canonical pairs. It is known that simulations between propositional proof systems translate to reductions between their canonical pairs. We focus on the opposite direction and study the following questions. Q1: For which propositional proof systems f and g does the implication [Formula: see text] hold, and for which does it fail? Q2: For which propositional proof systems of different strengths are the canonical pairs equivalent? Q3: What do (non-)equivalent canonical pairs tell about the corresponding propositional proof systems? Q4: Is every NP-pair (A, B), where A is NP-complete, strongly many-one equivalent to the canonical pair of some propositional proof system? In short, we show that Q1 and Q2 can be answered with 'for almost all', which generalizes previous results by Pudlák and Beyersdorff. Regarding Q3, inequivalent canonical pairs tell that the propositional proof systems are not "very similar," while equivalent, P -inseparable canonical pairs tell that they are not "very different." We can relate Q4 to the open problem in structural complexity that asks whether unions of disjoint NP-complete sets are NP-complete. This demonstrates a new connection between propositional proof systems, disjoint NP-pairs, and unions of disjoint NP-complete sets.

scholarly journals Full rainbow matchings in graphs and hypergraphs

2021 ◽  
pp. 1-19
Author(s):  
Pu Gao ◽  
Reshma Ramadurai ◽  
Ian M. Wanless ◽  
Nick Wormald
Keyword(s):  
Open Problem ◽  
Affirmative Answer ◽  
Simple Graph ◽  
Rainbow Matching ◽  
Special Case

Abstract Let G be a simple graph that is properly edge-coloured with m colours and let \[\mathcal{M} = \{ {M_1},...,{M_m}\} \] be the set of m matchings induced by the colours in G. Suppose that \[m \leqslant n - {n^c}\] , where \[c > 9/10\] , and every matching in \[\mathcal{M}\] has size n. Then G contains a full rainbow matching, i.e. a matching that contains exactly one edge from M i for each \[1 \leqslant i \leqslant m\] . This answers an open problem of Pokrovskiy and gives an affirmative answer to a generalization of a special case of a conjecture of Aharoni and Berger. Related results are also found for multigraphs with edges of bounded multiplicity, and for hypergraphs. Finally, we provide counterexamples to several conjectures on full rainbow matchings made by Aharoni and Berger.

scholarly journals Higher-Order Beta Matching with Solutions in Long Beta-Eta Normal Form

2006 ◽  
Vol 13 (12) ◽  
Author(s):  
Kristian Støvring
Keyword(s):  
Normal Form ◽  
Open Problem ◽  
Higher Order ◽  
Restricted Form ◽  
Matching Equation ◽  
Two Sides ◽  
Special Case

Higher-order matching is a special case of unification of simply-typed lambda-terms: in a matching equation, one of the two sides contains no unification variables. Loader has recently shown that higher-order matching up to beta equivalence is undecidable, but decidability of higher-order matching up to beta-eta equivalence is a long-standing open problem.<br /> <br />We show that higher-order matching up to beta-eta equivalence is decidable if and only if a restricted form of higher-order matching up to beta equivalence is decidable: the restriction is that solutions must be in long beta-eta normal form.

scholarly journals The Rectangle Enclosure and Point-Dominance Problems Revisited

1997 ◽  
Vol 07 (05) ◽  
pp. 437-455 ◽  
Author(s):  
Prosenjit Gupta ◽  
Ravi Janardan ◽  
Michiel Smid ◽  
Bhaskar Dasgupta
Keyword(s):  
Open Problem ◽  
Full Range ◽  
Divide And Conquer ◽  
Independent Interest ◽  
Solution Technique ◽  
Aspect Ratios ◽  
Special Case

We consider the problem of reporting the pairwise enclosures in a set of n axes-parallel rectangles in ℝ 2, which is equivalent to reporting dominance pairs in a set of n points in ℝ 4. Over a decade ago, Lee and Preparata7 gave an O(n log n + k)-time and O(n)-space algorithm for these problems, where k is the number of reported pairs. Since that time, the question of whether there is a faster algorithm has remained an intriguing open problem. In this paper, we give an algorithm which uses O(n + k) space and runs in O(n log n log log n + k log log n) time. Thus, although our result is not a strict improvement over the Lee-Preparata algorithm for the full range of k, it is, nevertheless, the first result since Ref. (6) to make any progress on this long-standing open problem. Our algorithm is based on the divide-and-conquer paradigm. The heart of the algorithm is the solution to a red-blue dominance reporting problem (the "merge" step). We give a novel solution for this problem which is based on the iterative application of a sequence of non-trivial sweep routines. This solution technique should be of independent interest. We also present another algorithm whose bounds match the bounds given in Ref. (6), but which is simpler. Finally, we consider the special case where the rectangles have a small number, α, of different aspect ratios, which is often the case in practice. For this problem, we give an algorithm which runs in O(α n log n + k) time and uses O(n) space.

On Characterizing Injective Sheaves: Corrigendum

1980 ◽  
Vol 32 (6) ◽  
pp. 522-522 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Open Problem ◽  
Boolean Spaces ◽  
Special Case

B. Banaschewski [1] has produced a counterexample to [2, Theorem 6]. As noted in [1, Remark 1], our error occurs in the final paragraph of the purported proof of [2, Theorem 6], for P need not be a subpresheaf of I. Accordingly, it remains an open problem to find an analogue of [2, Proposition 1] in the context of Boolean spaces. We hope that attacks on this problem will be facilitated by the (valid) initial three paragraphs of the argument given for [2, Theorem 6].The following alterations to [2] are in order. Example 5, being a corollary of Theorem 6, remains doubtful, although the special case noted on pp. 1034-1035 is not affected. In Corollary 7, the assertion that j preserves infectives remains doubtful, although the proof for divisibility of j(M)(G) is valid.

scholarly journals On Plain and Hereditary History-Preserving Bisimulation

1999 ◽  
Vol 6 (4) ◽  
Author(s):  
Sibylle B. Fröschle ◽  
Thomas Troels Hildebrandt
Keyword(s):  
General Problem ◽  
Open Problem ◽  
Restricted Class ◽  
Theoretical Terms ◽  
Stepping Stone ◽  
Frontier Zone ◽  
The Difference ◽  
Special Case

We investigate the difference between two well-known notions of<br />independence bisimilarity, history-preserving bisimulation and hereditary<br />history-preserving bisimulation. We characterise the difference between<br />the two bisimulations in trace-theoretical terms, advocating the<br />view that the first is (just) a bisimulation for causality, while the second<br />is a bisimulation for concurrency. We explore the frontier zone between<br />the two notions by defining a hierarchy of bounded backtracking bisimulations. <br />Our goal is to provide a stepping stone for the solution to<br />the intriguing open problem of whether hereditary history-preserving<br />bisimulation is decidable or not. We prove that each of the bounded<br />bisimulations is decidable. However, we also prove that the hierarchy<br />is strict. This rules out the possibility that decidability of the general<br />problem follows directly from the special case. Finally, we give a non-<br />trivial reduction solving the general problem for a restricted class of<br />systems and give pointers towards a full answer.

Extreme self-sacrifice beyond fusion: Moral expansiveness and the special case of allyship

2018 ◽  
Vol 41 ◽  
Author(s):  
Daniel Crimston ◽  
Matthew J. Hornsey
Keyword(s):  
General Theory ◽  
Biological Markers ◽  
Natural Environment ◽  
Relevant Dimension ◽  
Special Case

AbstractAs a general theory of extreme self-sacrifice, Whitehouse's article misses one relevant dimension: people's willingness to fight and die in support of entities not bound by biological markers or ancestral kinship (allyship). We discuss research on moral expansiveness, which highlights individuals’ capacity to self-sacrifice for targets that lie outside traditional in-group markers, including racial out-groups, animals, and the natural environment.

Determination of the phase structure of polymerblends by electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 492-493
Author(s):  
Dr. G. Kaemof
Keyword(s):  
Screw Extruder ◽  
Electron Microscopic ◽  
Thin Sections ◽  
Single Screw Extruder ◽  
Transmission Electron ◽  
The Matrix ◽  
Twin Screw ◽  
Special Case ◽  
Microscopic Investigations

A mixture of polycarbonate (PC) and styrene-acrylonitrile-copolymer (SAN) represents a very good example for the efficiency of electron microscopic investigations concerning the determination of optimum production procedures for high grade product properties.The following parameters have been varied:components of charge (PC : SAN 50 : 50, 60 : 40, 70 : 30), kind of compounding machine (single screw extruder, twin screw extruder, discontinuous kneader), mass-temperature (lowest and highest possible temperature).The transmission electron microscopic investigations (TEM) were carried out on ultra thin sections, the PC-phase of which was selectively etched by triethylamine.The phase transition (matrix to disperse phase) does not occur - as might be expected - at a PC to SAN ratio of 50 : 50, but at a ratio of 65 : 35. Our results show that the matrix is preferably formed by the components with the lower melting viscosity (in this special case SAN), even at concentrations of less than 50 %.

Test Validation Without Measurement

2016 ◽  
Vol 32 (3) ◽  
pp. 204-214 ◽  
Author(s):  
Emilie Lacot ◽  
Mohammad H. Afzali ◽  
Stéphane Vautier
Keyword(s):  
Test Data ◽  
Test Scores ◽  
Scientific Explanation ◽  
Test Validation ◽  
Clinical Tests ◽  
Ethical Perspective ◽  
Power Of Test ◽  
Memory Accuracy ◽  
Social Uses ◽  
Special Case

Abstract. Test validation based on usual statistical analyses is paradoxical, as, from a falsificationist perspective, they do not test that test data are ordinal measurements, and, from the ethical perspective, they do not justify the use of test scores. This paper (i) proposes some basic definitions, where measurement is a special case of scientific explanation; starting from the examples of memory accuracy and suicidality as scored by two widely used clinical tests/questionnaires. Moreover, it shows (ii) how to elicit the logic of the observable test events underlying the test scores, and (iii) how the measurability of the target theoretical quantities – memory accuracy and suicidality – can and should be tested at the respondent scale as opposed to the scale of aggregates of respondents. (iv) Criterion-related validity is revisited to stress that invoking the explanative power of test data should draw attention on counterexamples instead of statistical summarization. (v) Finally, it is argued that the justification of the use of test scores in specific settings should be part of the test validation task, because, as tests specialists, psychologists are responsible for proposing their tests for social uses.

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