83 research outputs found

    Complexity of equivalence relations and preorders from computability theory

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    We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations R,SR, S, a componentwise reducibility is defined by R\le S \iff \ex f \, \forall x, y \, [xRy \lra f(x) Sf(y)]. Here ff is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and ff must be computable. We show that there is a Π1\Pi_1-complete equivalence relation, but no Πk\Pi k-complete for k2k \ge 2. We show that Σk\Sigma k preorders arising naturally in the above-mentioned areas are Σk\Sigma k-complete. This includes polynomial time mm-reducibility on exponential time sets, which is Σ2\Sigma 2, almost inclusion on r.e.\ sets, which is Σ3\Sigma 3, and Turing reducibility on r.e.\ sets, which is Σ4\Sigma 4.Comment: To appear in J. Symb. Logi

    Finding the limit of incompleteness I

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    In this paper, we examine the limit of applicability of G\"{o}del's first incompleteness theorem (G1\sf G1 for short). We first define the notion "G1\sf G1 holds for the theory TT". This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which G1\sf G1 holds. To approach this question, we first examine the following question: is there a theory TT such that Robinson's R\mathbf{R} interprets TT but TT does not interpret R\mathbf{R} (i.e. TT is weaker than R\mathbf{R} w.r.t. interpretation) and G1\sf G1 holds for TT? 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 A,B\langle A,B\rangle, we can construct a r.e. theory UA,BU_{\langle A,B\rangle} such that UA,BU_{\langle A,B\rangle} is weaker than R\mathbf{R} w.r.t. interpretation and G1\sf G1 holds for UA,BU_{\langle A,B\rangle}. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree 0<d<0\mathbf{0}< \mathbf{d}<\mathbf{0}^{\prime}, there is a theory TT with Turing degree d\mathbf{d} such that G1\sf G1 holds for TT and TT is weaker than R\mathbf{R} 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 G1\sf G1 holds.Comment: 18 pages. Accepted and to appear in Bulletin of Symbolic Logi

    Degrees of members of ∏01 classes

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