662 research outputs found
Lattice initial segments of the hyperdegrees
We affirm a conjecture of Sacks [1972] by showing that every countable
distributive lattice is isomorphic to an initial segment of the hyperdegrees,
. In fact, we prove that every sublattice of any
hyperarithmetic lattice (and so, in particular, every countable locally finite
lattice) is isomorphic to an initial segment of . Corollaries
include the decidability of the two quantifier theory of
and the undecidability of its three quantifier theory. The key tool in the
proof is a new lattice representation theorem that provides a notion of forcing
for which we can prove a version of the fusion lemma in the hyperarithmetic
setting and so the preservation of . Somewhat surprisingly,
the set theoretic analog of this forcing does not preserve . On
the other hand, we construct countable lattices that are not isomorphic to an
initial segment of
State Elimination Ordering Strategies: Some Experimental Results
Recently, the problem of obtaining a short regular expression equivalent to a
given finite automaton has been intensively investigated. Algorithms for
converting finite automata to regular expressions have an exponential blow-up
in the worst-case. To overcome this, simple heuristic methods have been
proposed.
In this paper we analyse some of the heuristics presented in the literature
and propose new ones. We also present some experimental comparative results
based on uniform random generated deterministic finite automata.Comment: In Proceedings DCFS 2010, arXiv:1008.127
Minimal bounds and members of effectively closed sets
We show that there exists a non-empty class, with no recursive
element, in which no member is a minimal cover for any Turing degree.Comment: 15 pages, 4 figures, 1 acknowledgemen
Many-one reductions and the category of multivalued functions
Multi-valued functions are common in computable analysis (built upon the Type
2 Theory of Effectivity), and have made an appearance in complexity theory
under the moniker search problems leading to complexity classes such as PPAD
and PLS being studied. However, a systematic investigation of the resulting
degree structures has only been initiated in the former situation so far (the
Weihrauch-degrees).
A more general understanding is possible, if the category-theoretic
properties of multi-valued functions are taken into account. In the present
paper, the category-theoretic framework is established, and it is demonstrated
that many-one degrees of multi-valued functions form a distributive lattice
under very general conditions, regardless of the actual reducibility notions
used (e.g. Cook, Karp, Weihrauch).
Beyond this, an abundance of open questions arises. Some classic results for
reductions between functions carry over to multi-valued functions, but others
do not. The basic theme here again depends on category-theoretic differences
between functions and multi-valued functions.Comment: an earlier version was titled "Many-one reductions between search
problems". in Mathematical Structures in Computer Science, 201
Rethinking the notion of oracle: A link between synthetic descriptive set theory and effective topos theory
We present three different perspectives of oracle. First, an oracle is a
blackbox; second, an oracle is an endofunctor on the category of represented
spaces; and third, an oracle is an operation on the object of truth-values.
These three perspectives create a link between the three fields, computability
theory, synthetic descriptive set theory, and effective topos theory
The physical Church-Turing thesis and the principles of quantum theory
Notoriously, quantum computation shatters complexity theory, but is innocuous
to computability theory. Yet several works have shown how quantum theory as it
stands could breach the physical Church-Turing thesis. We draw a clear line as
to when this is the case, in a way that is inspired by Gandy. Gandy formulates
postulates about physics, such as homogeneity of space and time, bounded
density and velocity of information --- and proves that the physical
Church-Turing thesis is a consequence of these postulates. We provide a quantum
version of the theorem. Thus this approach exhibits a formal non-trivial
interplay between theoretical physics symmetries and computability assumptions.Comment: 14 pages, LaTe
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