FINDING THE LIMIT OF INCOMPLETENESS I

In this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem ($\textsf {G1}$ for short). We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ (i.e., T is weaker than $\mathbf {R}$ w.r.t. interpretation) and $\textsf {G1}$ holds for T? In this paper, we show that there are many such theories based on Jeřábek’s work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle $, we can construct a r.e. theory $U_{\langle A,B\rangle }$ such that $U_{\langle A,B\rangle }$ is weaker than $\mathbf {R}$ w.r.t. interpretation and $\textsf {G1}$ holds for $U_{\langle A,B\rangle }$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf {0}< \mathbf {d}<\mathbf {0}^{\prime }$, there is a theory T with Turing degree $\mathbf {d}$ such that $\textsf {G1}$ holds for T and T is weaker than $\mathbf {R}$ 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 $\textsf {G1}$ holds.

NON-DENSITY OF GENERIC M-REDUCIBILITY FOR C.E. SETS

Generic-case approach to algorithmic problems was suggested by I. Kapovich, A. Myasnikov, V. Shpilrain and P. Schupp in 2003. This approach studies behavior of an al-gorithm on typical (almost all) inputs and ignores the rest of inputs. C. Jockusch and P. Schupp in 2012 began the study of generic computability in the context of classical computability theory. In particular, they defined a generic analog of Turing reducibility. A. Rybalov in 2018 introduced a generic analog of classical m-reducibility. In this paper we study the generic m-reducibility for c.e. sets and prove that unlike classical m-reducibility, generic m-reducibility does not have the density property for c.e. sets.

m-topped degrees

This chapter assesses m-topped degrees. The notion of m-topped degrees comes from a general study of the interaction between Turing reducibility and stronger reducibilities among c.e. sets. For example, this study includes the contiguous degrees. A c.e. Turing degree d is m-topped if it contains a greatest degree among the many one degrees of c.e. sets in d. Such degrees were constructed in Downey and Jockusch. The dynamics of the cascading phenomenon occurring in the construction of m-topped degrees strongly resemble the dynamics of the embedding of the 1–3–1 lattice in the c.e. degrees. Similar dynamics occurred in the original construction of a noncomputable left–c.e. real with only computable presentations, which was discussed in the previous chapter.

Turing reducibility and Turing degrees

A reducibility is a relation of comparative computational complexity (which can be made precise in various non-equivalent ways) between mathematical objects of appropriate sorts. Much of recursion theory concerns such relations, initially between sets of natural numbers (in so-called classical recursion theory), but later between sets of other sorts (in so-called generalized recursion theory). This article considers only the classical setting. Also Turing first defined such a relation, now called Turing- (or just T-) reducibility; probably most logicians regard it as the most important such relation. Turing- (or T-) degrees are the units of computational complexity when comparative complexity is taken to be T-reducibility.

Continuous higher randomness

We investigate the role of continuous reductions and continuous relativization in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with respect to van Lambalgen’s theorem and the Miller–Yu/Levin theorem. We study lowness for continuous relativization of randomness, and show the equivalence of the higher analogues of the different characterizations of lowness for Martin-Löf randomness. We also characterize computing higher [Formula: see text]-trivial sets by higher random sequences. We give a separation between higher notions of randomness, in particular between higher weak 2-randomness and [Formula: see text]-randomness. To do so we investigate classes of functions computable from Kleene’s [Formula: see text] based on strong forms of the higher limit lemma.

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 relationsR,S, a componentwise reducibility is defined byR≤S⇔ ∃f∀x, y[x R y↔f(x)S f(y)].Here,fis taken from a suitable class of effective functions. For us the relations will be on natural numbers, andfmust be computable. We show that there is a${\rm{\Pi }}_1^0$-complete equivalence relation, but no${\rm{\Pi }}_k^0$-complete fork≥ 2. We show that${\rm{\Sigma }}_k^0$preorders arising naturally in the above-mentioned areas are${\rm{\Sigma }}_k^0$-complete. This includes polynomial timem-reducibility on exponential time sets, which is${\rm{\Sigma }}_2^0$, almost inclusion on r.e. sets, which is${\rm{\Sigma }}_3^0$, and Turing reducibility on r.e. sets, which is${\rm{\Sigma }}_4^0$.