78 research outputs found
Complexity of equivalence relations and preorders from computability theory
We study the relative complexity of equivalence relations and preorders from
computability theory and complexity theory. Given binary relations , a
componentwise reducibility is defined by R\le S \iff \ex f \, \forall x, y \,
[xRy \lra f(x) Sf(y)]. Here is taken from a suitable class of effective
functions. For us the relations will be on natural numbers, and must be
computable. We show that there is a -complete equivalence relation, but
no -complete for .
We show that preorders arising naturally in the above-mentioned
areas are -complete. This includes polynomial time -reducibility
on exponential time sets, which is , almost inclusion on r.e.\ sets,
which is , and Turing reducibility on r.e.\ sets, which is .Comment: To appear in J. Symb. Logi
Computable de Finetti measures
We prove a computable version of de Finetti's theorem on exchangeable
sequences of real random variables. As a consequence, exchangeable stochastic
processes expressed in probabilistic functional programming languages can be
automatically rewritten as procedures that do not modify non-local state. Along
the way, we prove that a distribution on the unit interval is computable if and
only if its moments are uniformly computable.Comment: 32 pages. Final journal version; expanded somewhat, with minor
corrections. To appear in Annals of Pure and Applied Logic. Extended abstract
appeared in Proceedings of CiE '09, LNCS 5635, pp. 218-23
On the Relation Between Representations and Computability
Computability and decidability are intimately linked problems which have
interested computer scientists and mathematicians for a long time, especially
during the last century. Work performed by Turing, Church, Godel, Post,
Kleene and other authors considered the questions "What is computable?"
and "What is an algorithm?". Very important results with plenty of implica-
tions were obtained, such as the halting theorem [12], the several solutions to
the Entscheidungsproblem [12, 5], the Church-Turing thesis [12] or Godel's
incompleteness theorem. Further work was performed on topics which as of
today have remained purely theoretical but which have o ered us a great
understanding of computability and related questions. Some of this work in-
cludes the one related to degrees of recursive unsolvability [1] [7] and Rice's
theorem [11].
Several formalisms were described and compared, some of the most im-
portant ones being Turing machines and -calculus. These formalisms were
mathematical constructions which allowed the study of the concept of com-
putation or calculation and all of its related questions.
We have found that an often ignored detail and, as we show, important
aspect of computability is related to representation. In particular, we show
that the computability of an abstract problem can only be considered once
a choice of representation has been made. We inquire to what extent this
is essential and what e ects it may have and in what manner. We o er a
wide discussion on its implications, a formalisation of these considerations
and some important results deriving from these formalisations. In particular,
the main result of the work is a proof that computably enumerable repre-
sentations cannot be strictly stronger or weaker than other representations.
We also discuss the Church-Turing thesis with particular interest, inquiring
about its deep meaning and the actual facts and false assumptions related
to it.
Furthermore, we consider the relationship between representation and the
so-called representation degrees and the degrees of recursive unsolvability de-
rived from the concept of oracle machine. We show that these two concepts
o er parallel hierarchies which are very similar in their construction but quite
di erent in their essential meaning and properties.La computabilidad y la decidibilidad son problemas estrechamente relacionados que han interesado ampliamente a informáticos y matemáticos, especialmente a lo largo del ultimo siglo. Los trabajos realizados por Turing,
Church, Godel, Post, Kleene y otros autores se planteaban las preguntas
"Qué es computable?" y "Qué es un algoritmo?". Se lograron muchos
resultados importantes con multitud de implicaciones, como el teorema de
la parada [12], la solución al Entscheidungsproblem [12, 5], la hipótesis de
Church-Turing [12] o el teorema de incompletidud de Godel. Gran cantidad
del trabajo posterior se realizó en relación a otros temas que han permanecido
hasta hoy en el campo de la teorÃa pero que nos han permitido entender en
mayor medida la computabilidad y problemas relacionados. Por ejemplo, el
relacionado con los grados de indecibilidad [1] [7] y el teorema de Rice [11].
Varios formalismos fueron descritos y comparados, algunos de los más
importantes son las máquinas de Turing y el cálculo lambda. Estos formalismos constituÃan construcciones matemáticas que permitÃan el estudio del
concepto de computación o cálculo y todas las preguntas relacionadas.
Un aspecto comúnmente ignorado y relevante de la computabilidad está
relacionado con la representación. En particular, percatamos que la com-
putabilidad de un problema abstracto sólo puede ser considerada una vez se
ha producido una elección de representación. Nos preguntamos hasta qué
punto esto es esencial y qué efectos puede tener y de qué manera. Ofrecemos
una amplia discusión sobre sus implicaciones, una formalización de estas consideraciones y algunos resultados importantes derivados de las mismas. En
particular, el resultado principal del trabajo es una demostración de que las
representaciones computacionalmente enumerables no pueden ser más fuertes
o más débiles que otras. Realizamos una discusión especialmente enfrascada
en relación a la tesis de Church-Turing, su significado más profundo y los
hechos y falacias que giran en torno a ella.
Además, consideramos la relación existente entre la representación y los
llamados grados de representación, y los grados de indecibilidad derivados del
concepto de máquina oráculo. Demostramos que estos dos conceptos ofrecen jerarquÃas paralelas con una construcción muy similar pero notablemente
distintas en su significado esencial y sus propiedades
First Order Theories of Some Lattices of Open Sets
We show that the first order theory of the lattice of open sets in some
natural topological spaces is -equivalent to second order arithmetic. We
also show that for many natural computable metric spaces and computable domains
the first order theory of the lattice of effectively open sets is undecidable.
Moreover, for several important spaces (e.g., , , and the
domain ) this theory is -equivalent to first order arithmetic
Spectral Representation of Some Computably Enumerable Sets With an Application to Quantum Provability
We propose a new type of quantum computer which is used to prove a spectral
representation for a class F of computable sets. When S in F codes the theorems
of a formal system, the quantum computer produces through measurement all
theorems and proofs of the formal system. We conjecture that the spectral
representation is valid for all computably enumerable sets. The conjecture
implies that the theorems of a general formal system, like Peano Arithmetic or
ZFC, can be produced through measurement; however, it is unlikely that the
quantum computer can produce the proofs as well, as in the particular case of
F. The analysis suggests that showing the provability of a statement is
different from writing up the proof of the statement.Comment: 12 pages, LaTeX2e, no figure
Computably Based Locally Compact Spaces
ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in
which the topology on a space is treated, not as an infinitary lattice, but as
an exponential object of the same category as the original space, with an
associated lambda-calculus. In this paper, this is shown to be equivalent to a
notion of computable basis for locally compact sober spaces or locales,
involving a family of open subspaces and accompanying family of compact ones.
This generalises Smyth's effectively given domains and Jung's strong proximity
lattices. Part of the data for a basis is the inclusion relation of compact
subspaces within open ones, which is formulated in locale theory as the
way-below relation on a continuous lattice. The finitary properties of this
relation are characterised here, including the Wilker condition for the cover
of a compact space by two open ones. The real line is used as a running
example, being closely related to Scott's domain of intervals. ASD does not use
the category of sets, but the full subcategory of overt discrete objects plays
this role; it is an arithmetic universe (pretopos with lists). In particular,
we use this subcategory to translate computable bases for classical spaces into
objects in the ASD calculus.Comment: 70pp, LaTeX2e, uses diagrams.sty; Accepted for "Logical Methods in
Computer Science" LMCS-2004-19; see http://www.cs.man.ac.uk/~pt/ASD for
related papers. ACM-class: F.4.
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