249 research outputs found
Total Representations
Almost all representations considered in computable analysis are partial. We
provide arguments in favor of total representations (by elements of the Baire
space). Total representations make the well known analogy between numberings
and representations closer, unify some terminology, simplify some technical
details, suggest interesting open questions and new invariants of topological
spaces relevant to computable analysis.Comment: 30 page
Renormalization and Computation II: Time Cut-off and the Halting Problem
This is the second installment to the project initiated in [Ma3]. In the
first Part, I argued that both philosophy and technique of the perturbative
renormalization in quantum field theory could be meaningfully transplanted to
the theory of computation, and sketched several contexts supporting this view.
In this second part, I address some of the issues raised in [Ma3] and provide
their development in three contexts: a categorification of the algorithmic
computations; time cut--off and Anytime Algorithms; and finally, a Hopf algebra
renormalization of the Halting Problem.Comment: 28 page
On the Invariance of G\"odel's Second Theorem with regard to Numberings
The prevalent interpretation of G\"odel's Second Theorem states that a
sufficiently adequate and consistent theory does not prove its consistency. It
is however not entirely clear how to justify this informal reading, as the
formulation of the underlying mathematical theorem depends on several arbitrary
formalisation choices. In this paper I examine the theorem's dependency
regarding G\"odel numberings. I introduce deviant numberings, yielding
provability predicates satisfying L\"ob's conditions, which result in provable
consistency sentences. According to the main result of this paper however,
these "counterexamples" do not refute the theorem's prevalent interpretation,
since once a natural class of admissible numberings is singled out, invariance
is maintained.Comment: Forthcoming in The Review of Symbolic Logi
Computable analysis on the space of marked groups
We investigate decision problems for groups described by word problem
algorithms. This is equivalent to studying groups described by labelled Cayley
graphs. We show that this corresponds to the study of computable analysis on
the space of marked groups, and point out several results of computable
analysis that can be directly applied to obtain group theoretical results.
Those results, used in conjunction with the version of Higman's Embedding
Theorem that preserves solvability of the word problem, provide powerful tools
to build finitely presented groups with solvable word problem but with various
undecidable properties. We also investigate the first levels of an effective
Borel hierarchy on the space of marked groups, and show that on many group
properties usually considered, this effective hierarchy corresponds sharply to
the Borel hierarchy. Finally, we prove that the space of marked groups is a
Polish space that is not effectively Polish. Because of this, many of the most
important results of computable analysis cannot be applied to the space of
marked groups. This includes the Kreisel-Lacombe-Schoenfield-Ceitin Theorem and
a theorem of Moschovakis. The space of marked groups constitutes the first
natural example of a Polish space that is not effectively Polish.Comment: 46 pages, Theorem 4.6 was false as stated, it appears now, having
been corrected, as Theorem 5.
Complexity vs Energy: Theory of Computation and Theoretical Physics
This paper is a survey dedicated to the analogy between the notions of {\it
complexity} in theoretical computer science and {\it energy} in physics. This
analogy is not metaphorical: I describe three precise mathematical contexts,
suggested recently, in which mathematics related to (un)computability is
inspired by and to a degree reproduces formalisms of statistical physics and
quantum field theory.Comment: 23 pages. Talk at the satellite conference to ECM 2012, "QQQ Algebra,
Geometry, Information", Tallinn, July 9-12, 201
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