初始段的可定义性

泛可计算可列集与初始段无前缀复杂度

Universal computably enumerable sets and initial segment prefix-free complexity

我们表明，有图灵完全可计算的任意低的非平凡的初始段前缀自由复杂性的集合。特别地，给定任何具有非平凡无前缀初始段复杂度的可计算可列集$A$，存在图灵完备可计算可列集$B$，其复杂度严格小于$A$的复杂度。另一方面，已知具有平凡的无前缀初始段复杂度的集合不是图灵完备的。此外，我们给出了一个推广的结果，任何有限的可计算可列集$A_i, i<k$与非平凡的初始段前缀自由复杂性。这一点的一个应用对\cite [第11.12节]{rodenisbook}和\cite{MRmerstcdhdtd}中的一个问题给出了否定的答案，该问题要求在c.e.的结构中找到最小对。实数按其初始段无前缀复杂度排序。进一步的结果涉及各种随机性程度的概念。例如，可计算可列实数和可计算可列集合的Solovay度和$K$ -度在初等上是不等价的。此外，基于普通复杂度和无前缀复杂度的随机度在基本上是不等价的;这同样适用于它们的$Δ^0_2$和$Σ^0_1$子结构。

Randomness and initial segment complexity for probability measures

We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure $μ$ on the space of infinite bit sequences is ML absolutely continuous if the non-ML-random bit sequences form a null set with respect to~$μ$. We think of this as a weak randomness notion for measures. We begin with examples, and provide a robustness property related to Solovay tests. Our main work connects our weak randomness notion to the growth of the initial segment complexity for measures~$μ$; the latter is defined as a $μ$-average over the complexity of strings of the same length. W…

Tanaka's Theorem Revisited

Tanaka (1997) proved a powerful generalization of Friedman's self-embedding theorem that states that given a countable nonstandard model $(\mathcal{M},\mathcal{A})$ of the subsystem $\mathrm{WKL}_{0}$ of second order arithmetic, and any element $m$ of $\mathcal{M}$, there is a self-embedding $j$ of $(\mathcal{M},\mathcal{A})$ onto a proper initial segment of itself such that $j$ fixes every predecessor of $m$. Here we extend Tanaka's work by establishing the following results for a countable nonstandard model $(\mathcal{M},\mathcal{A})$ of $\mathrm{WKL}_{0} $ and a proper cut $\mathrm{I}$ of $…

Largest initial segments pointwise fixed by automorphisms of models of set theory

Given a model $\mathcal{M}$ of set theory, and a nontrivial automorphism $j$ of $\mathcal{M}$, let $\mathcal{I}_{\mathrm{fix}}(j)$ be the submodel of $\mathcal{M}$ whose universe consists of elements $m$ of $\mathcal{M}$ such that $j(x)=x$ for every $x$ in the transitive closure of $m$ (where the transitive closure of $m$ is computed within $\mathcal{M}$). Here we study the class $\mathcal{C}$ of structures of the form $\mathcal{I}_{\mathrm{fix}}(j)$, where the ambient model $\mathcal{M}$ satisfies a frugal yet robust fragment of $\mathrm{ZFC}$ known as $\mathrm{MOST}$, and $j(m)=m$ whenever $…

On commutative weak BCK-algebras

The class of weak BCK-algebras is obtained by weakening one of standard BCK axioms. It is known that every weak BCK-algebra is completely determined by the structure of its initial segments. We review several natural classes of commutative weak BCK-algebras, prove that they are equationally definable, and show that the order duals of a multitude of algebras with implication known in the literature in connection with various quantum logics are, in fact, commutative weak BCK-algebras belonging to that or other of these classes. We also characterize initial segments of algebras in each of the cla…