83 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
Finding the limit of incompleteness I
In this paper, we examine the limit of applicability of G\"{o}del's first
incompleteness theorem ( for short). We first define the notion " holds for the theory ". This paper is motivated by the following
question: can we find a theory with a minimal degree of interpretation for
which holds. To approach this question, we first examine the following
question: is there a theory such that Robinson's interprets
but does not interpret (i.e. is weaker than
w.r.t. interpretation) and holds for ? In this paper,
we show that there are many such theories based on Je\v{r}\'{a}bek's work using
some model theory. We prove that for each recursively inseparable pair , we can construct a r.e. theory such that
is weaker than w.r.t. interpretation and
holds for . As a corollary, we answer a
question from Albert Visser. Moreover, we prove that for any Turing degree
, there is a theory with Turing
degree such that holds for and is weaker than
w.r.t. Turing reducibility. As a corollary, based on Shoenfield's
work using some recursion theory, we show that there is no theory with a
minimal degree of Turing reducibility for which holds.Comment: 18 pages. Accepted and to appear in Bulletin of Symbolic Logi
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