535 research outputs found
The Church Problem for Countable Ordinals
A fundamental theorem of Buchi and Landweber shows that the Church synthesis
problem is computable. Buchi and Landweber reduced the Church Problem to
problems about ω-games and used the determinacy of such games as one of
the main tools to show its computability. We consider a natural generalization
of the Church problem to countable ordinals and investigate games of arbitrary
countable length. We prove that determinacy and decidability parts of the
Bu}chi and Landweber theorem hold for all countable ordinals and that its full
extension holds for all ordinals < \omega\^\omega
Synthesis of Finite-state and Definable Winning Strategies
Church\u27s Problem asks for the construction of a procedure which,
given a logical specification on sequence pairs, realizes
for any input sequence an output sequence such that
satisfies . McNaughton reduced Church\u27s Problem to a problem about two-player-games.
B"uchi and Landweber gave a solution for
Monadic Second-Order Logic of Order () specifications in terms of finite-state strategies.
We consider two natural generalizations of the Church problem to
countable ordinals: the first deals with finite-state strategies;
the second deals with -definable strategies. We investigate
games of arbitrary countable length and prove the computability of
these generalizations of Church\u27s problem
Generalized Effective Reducibility
We introduce two notions of effective reducibility for set-theoretical
statements, based on computability with Ordinal Turing Machines (OTMs), one of
which resembles Turing reducibility while the other is modelled after Weihrauch
reducibility. We give sample applications by showing that certain (algebraic)
constructions are not effective in the OTM-sense and considerung the effective
equivalence of various versions of the axiom of choice
Infinite computations with random oracles
We consider the following problem for various infinite time machines. If a
real is computable relative to large set of oracles such as a set of full
measure or just of positive measure, a comeager set, or a nonmeager Borel set,
is it already computable? We show that the answer is independent from ZFC for
ordinal time machines (OTMs) with and without ordinal parameters and give a
positive answer for most other machines. For instance, we consider, infinite
time Turing machines (ITTMs), unresetting and resetting infinite time register
machines (wITRMs, ITRMs), and \alpha-Turing machines for countable admissible
ordinals \alpha
Decreasing Diagrams for Confluence and Commutation
Like termination, confluence is a central property of rewrite systems. Unlike
for termination, however, there exists no known complexity hierarchy for
confluence. In this paper we investigate whether the decreasing diagrams
technique can be used to obtain such a hierarchy. The decreasing diagrams
technique is one of the strongest and most versatile methods for proving
confluence of abstract rewrite systems. It is complete for countable systems,
and it has many well-known confluence criteria as corollaries.
So what makes decreasing diagrams so powerful? In contrast to other
confluence techniques, decreasing diagrams employ a labelling of the steps with
labels from a well-founded order in order to conclude confluence of the
underlying unlabelled relation. Hence it is natural to ask how the size of the
label set influences the strength of the technique. In particular, what class
of abstract rewrite systems can be proven confluent using decreasing diagrams
restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find
that two labels suffice for proving confluence for every abstract rewrite
system having the cofinality property, thus in particular for every confluent,
countable system.
Secondly, we show that this result stands in sharp contrast to the situation
for commutation of rewrite relations, where the hierarchy does not collapse.
Thirdly, investigating the possibility of a confluence hierarchy, we
determine the first-order (non-)definability of the notion of confluence and
related properties, using techniques from finite model theory. We find that in
particular Hanf's theorem is fruitful for elegant proofs of undefinability of
properties of abstract rewrite systems
The Lost Melody Phenomenon
A typical phenomenon for machine models of transfinite computations is the
existence of so-called lost melodies, i.e. real numbers such that the
characteristic function of the set is computable while itself is
not (a real having the first property is called recognizable). This was first
observed by J. D. Hamkins and A. Lewis for infinite time Turing machine, then
demonstrated by P. Koepke and the author for s. We prove that, for
unresetting infinite time register machines introduced by P. Koepke,
recognizability equals computability, i.e. the lost melody phenomenon does not
occur. Then, we give an overview on our results on the behaviour of
recognizable reals for s. We show that there are no lost melodies for
ordinal Turing machines or ordinal register machines without parameters and
that this is, under the assumption that exists, independent of
. Then, we introduce the notions of resetting and unresetting
-register machines and give some information on the question for which
of these machines there are lost melodies
A Conversation on Divine Infinity and Cantorian Set Theory
This essay is written as a drama that opens with Aristotle, St. Augustine of Hippo, St. Thomas Aquinas, and Nicholas of Cusa debating the nature and reality of infinity, introducing historical concepts such as potential, actual, and divine infinity. Georg Cantor, founder of set theory, then gives a lecture on set theory and transfinite numbers. The lecture concludes with a discussion of the theological motivations and implications of set theory and Cantor\'s absolute infinity. The paradoxes inherent in analyzing absolute infinity seem to provide a useful analogy for understanding God\'s unknowable nature and the divine relation to creation
Computational reverse mathematics and foundational analysis
Reverse mathematics studies which subsystems of second order arithmetic are
equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main
philosophical application of reverse mathematics proposed thus far is
foundational analysis, which explores the limits of different foundations for
mathematics in a formally precise manner. This paper gives a detailed account
of the motivations and methodology of foundational analysis, which have
heretofore been largely left implicit in the practice. It then shows how this
account can be fruitfully applied in the evaluation of major foundational
approaches by a careful examination of two case studies: a partial realization
of Hilbert's program due to Simpson [1988], and predicativism in the extended
form due to Feferman and Sch\"{u}tte.
Shore [2010, 2013] proposes that equivalences in reverse mathematics be
proved in the same way as inequivalences, namely by considering only
-models of the systems in question. Shore refers to this approach as
computational reverse mathematics. This paper shows that despite some
attractive features, computational reverse mathematics is inappropriate for
foundational analysis, for two major reasons. Firstly, the computable
entailment relation employed in computational reverse mathematics does not
preserve justification for the foundational programs above. Secondly,
computable entailment is a complete relation, and hence employing it
commits one to theoretical resources which outstrip those available within any
foundational approach that is proof-theoretically weaker than
.Comment: Submitted. 41 page
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