287,305 research outputs found
The weakness of being cohesive, thin or free in reverse mathematics
Informally, a mathematical statement is robust if its strength is left
unchanged under variations of the statement. In this paper, we investigate the
lack of robustness of Ramsey's theorem and its consequence under the frameworks
of reverse mathematics and computable reducibility. To this end, we study the
degrees of unsolvability of cohesive sets for different uniformly computable
sequence of sets and identify different layers of unsolvability. This analysis
enables us to answer some questions of Wang about how typical sets help
computing cohesive sets.
We also study the impact of the number of colors in the computable
reducibility between coloring statements. In particular, we strengthen the
proof by Dzhafarov that cohesiveness does not strongly reduce to stable
Ramsey's theorem for pairs, revealing the combinatorial nature of this
non-reducibility and prove that whenever is greater than , stable
Ramsey's theorem for -tuples and colors is not computably reducible to
Ramsey's theorem for -tuples and colors. In this sense, Ramsey's
theorem is not robust with respect to his number of colors over computable
reducibility. Finally, we separate the thin set and free set theorem from
Ramsey's theorem for pairs and identify an infinite decreasing hierarchy of
thin set theorems in reverse mathematics. This shows that in reverse
mathematics, the strength of Ramsey's theorem is very sensitive to the number
of colors in the output set. In particular, it enables us to answer several
related questions asked by Cholak, Giusto, Hirst and Jockusch.Comment: 31 page
Mass problems and intuitionistic higher-order logic
In this paper we study a model of intuitionistic higher-order logic which we
call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the
category of sheaves of sets over the topological space consisting of the Turing
degrees, where the Turing cones form a base for the topology. We note that our
Muchnik topos interpretation of intuitionistic mathematics is an extension of
the well known Kolmogorov/Muchnik interpretation of intuitionistic
propositional calculus via Muchnik degrees, i.e., mass problems under weak
reducibility. We introduce a new sheaf representation of the intuitionistic
real numbers, \emph{the Muchnik reals}, which are different from the Cauchy
reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice
principle} and a \emph{bounding principle} where range over Muchnik
reals, ranges over functions from Muchnik reals to Muchnik reals, and
is a formula not containing or . For the convenience of the
reader, we explain all of the essential background material on intuitionism,
sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems,
Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an
English translation of Muchnik's 1963 paper on Muchnik degrees.Comment: 44 page
Cohesive avoidance and arithmetical sets
An open question in reverse mathematics is whether the cohesive principle,
\COH, is implied by the stable form of Ramsey's theorem for pairs,
\SRT^2_2, in -models of \RCA. One typical way of establishing this
implication would be to show that for every sequence of subsets of
, there is a set that is in such that every
infinite subset of or computes an -cohesive set. In this
article, this is shown to be false, even under far less stringent assumptions:
for all natural numbers and , there is a sequence \vec{R}
= \sequence{R_0,...,R_{n-1}} of subsets of such that for any
partition of arithmetical in , there is an
infinite subset of some that computes no set cohesive for . This
complements a number of previous results in computability theory on the
computational feebleness of infinite sets of numbers with prescribed
combinatorial properties. The proof is a forcing argument using an adaptation
of the method of Seetapun showing that every finite coloring of pairs of
integers has an infinite homogeneous set not computing a given non-computable
set
Two kinds of procedural semantics for privative modification
In this paper we present two kinds of procedural semantics for privative modification. We do this for three reasons. The first reason is to launch a tough test case to gauge the degree of substantial agreement between a constructivist and a realist interpretation of procedural semantics; the second is to extend Martin-L ̈f’s Constructive Type Theory to privative modification, which is characteristic of natural language; the third reason is to sketch a positive characterization of privation
P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches
While the P vs NP problem is mainly approached form the point of view of
discrete mathematics, this paper proposes reformulations into the field of
abstract algebra, geometry, fourier analysis and of continuous global
optimization - which advanced tools might bring new perspectives and approaches
for this question. The first one is equivalence of satisfaction of 3-SAT
problem with the question of reaching zero of a nonnegative degree 4
multivariate polynomial (sum of squares), what could be tested from the
perspective of algebra by using discriminant. It could be also approached as a
continuous global optimization problem inside , for example in
physical realizations like adiabatic quantum computers. However, the number of
local minima usually grows exponentially. Reducing to degree 2 polynomial plus
constraints of being in , we get geometric formulations as the
question if plane or sphere intersects with . There will be also
presented some non-standard perspectives for the Subset-Sum, like through
convergence of a series, or zeroing of fourier-type integral for some natural . The last discussed
approach is using anti-commuting Grassmann numbers , making nonzero only if has a Hamilton cycle. Hence,
the PNP assumption implies exponential growth of matrix representation of
Grassmann numbers. There will be also discussed a looking promising
algebraic/geometric approach to the graph isomorphism problem -- tested to
successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure
The Cardinality of an Oracle in Blum-Shub-Smale Computation
We examine the relation of BSS-reducibility on subsets of the real numbers.
The question was asked recently (and anonymously) whether it is possible for
the halting problem H in BSS-computation to be BSS-reducible to a countable
set. Intuitively, it seems that a countable set ought not to contain enough
information to decide membership in a reasonably complex (uncountable) set such
as H. We confirm this intuition, and prove a more general theorem linking the
cardinality of the oracle set to the cardinality, in a local sense, of the set
which it computes. We also mention other recent results on BSS-computation and
algebraic real numbers
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