T. Moraschini
We prove that the problem of determining whether a finite logical matrix determines an algebraizable logic is complete for EXPTIME. The same result holds for the classes of order algebraizable, weakly algebraizable, equivalential and protoalgebraic logics. Finally, the same problem for the class of truth-equational logic is shown to be hard for EXPTIME.
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A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions consisting of meet-prime logics only.
We review the notion of (finitary) filter pair as a tool for creating and analyzing logics. A filter pair can be seen as a presentation of a logic, given by presenting its lattice of theories as the image of a lattice homomorphism, with certain properties ensuring that the resulting logic is finitary and substitution invariant. Every finitary, substitution invariant logic arises from a filter pair. Particular classes of logics can be characterized as arising from special classes of filter pairs. We consider so-called congruence filter pairs, i.e. filter pairs for which the domain of the lattic…
Scales and the fine structure of K(R). Part III: Scales of minimal complexity
我们获得规模最小的复杂性在$K(\mathbb{R})$使用Levy层次和精细结构理论$K(\mathbb{R})$，也就是说，我们精确地识别这些层次的Levy层次$K(\mathbb{R})$具有规模属性。
Computability theorists have introduced multiple hierarchies to measure the complexity of sets of natural numbers. The Kleene Hierarchy classifies sets according to the first-order complexity of their defining formulas. The Ershov Hierarchy classifies $Δ^0_2$ sets with respect to the number of mistakes that are needed to approximate them. Biacino and Gerla extended the Kleene Hierarchy to the realm of fuzzy sets, whose membership functions range in a complete lattice $L$ (e.g., the real interval $[0; 1]_\mathbb{R}$). In this paper, we combine the Ershov Hierarchy and fuzzy set theory, by intro…
We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in $[0,1]$, we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form $ϕ^\mathcal{M} \leq r$, and the open diagram, which encapsulates strict inequalities of the form $ϕ^\mathcal{M} < r$. We show that the closed and open $Σ_N$ diagrams are $Π^0_{N+1}$ and $Σ_N$ respectively, and that the closed and open $Π_N$…
We survey recent results on the topological complexity of context-free omega-languages which form the second level of the Chomsky hierarchy of languages of infinite words. In particular, we consider the Borel hierarchy and the Wadge hierarchy of non-deterministic or deterministic context-free omega-languages. We study also decision problems, the links with the notions of ambiguity and of degrees of ambiguity, and the special case of omega-powers.
We consider the arithmetic complexity of index sets of uniformly computably enumerable families learnable under different learning criteria. We determine the exact complexity of these sets for the standard notions of finite learning, learning in the limit, behaviorally correct learning and anomalous learning in the limit. In proving the $Σ_5^0$-completeness result for behaviorally correct learning we prove a result of independent interest; if a uniformly computably enumerable family is not learnable, then for any computable learner there is a $Δ_2^0$ enumeration witnessing failure.
This paper establishes some of the fundamental barriers in the theory of computations and finally settles the long-standing computational spectral problem. That is to determine the existence of algorithms that can compute spectra $\mathrm{sp}(A)$ of classes of bounded operators $A = \{a_{ij}\}_{i,j \in \mathbb{N}} \in \mathcal{B}(l^2(\mathbb{N}))$, given the matrix elements $\{a_{ij}\}_{i,j \in \mathbb{N}}$, that are sharp in the sense that they achieve the boundary of what a digital computer can achieve. Similarly, for a Schrödinger operator $H = -Δ+V$, determine the existence of algorithms t…
We produce a class of $ω$-categorical structures with finite signature by applying a model-theoretic construction -- a refinement of the Hrushosvki-encoding -- to $ω$-categorical structures in a possibly infinite signature. We show that the encoded structures retain desirable algebraic properties of the original structures, but that the constraint satisfaction problems (CSPs) associated with these structures can be badly behaved in terms of computational complexity. This method allows us to systematically generate $ω$-categorical templates whose CSPs are complete for a variety of complexity cl…
We study the complexity of approximations to the normalized information distance. We introduce a hierarchy of computable approximations by considering the number of oscillations. This is a function version of the difference hierarchy for sets. We show that the normalized information distance is not in any level of this hierarchy, strengthening previous nonapproximability results. As an ingredient to the proof, we also prove a conditional undecidability result about independence.
One of the central open questions in bounded arithmetic is whether Buss' hierarchy of theories of bounded arithmetic collapses or not. In this paper, we reformulate Buss' theories using free logic and conjecture that such theories are easier to handle. To show this, we first prove that Buss' theories prove consistencies of induction-free fragments of our theories whose formulae have bounded complexity. Next, we prove that although our theories are based on an apparently weaker logic, we can interpret theories in Buss' hierarchy by our theories using a simple translation. Finally, we investigat…
We study the complexity of the space $C^*_p(X)$ of bounded continuous functions with the topology of pointwise convergence. We are allowed to use descriptive set theoretical methods, since for a separable metrizable space $X$, the measurable space of Borel sets in $C^*_p(X)$ (and also in the space $C_p(X)$ of all continuous functions) is known to be isomorphic to a subspace of a standard Borel space. It was proved by A. Andretta and A. Marcone that if $X$ is a $σ$-compact metrizable space, then the measurable spaces $C_p(X)$ and $C^*_p(X)$ are standard Borel and if $X$ is a metrizable analytic…
We show that it is decidable whether a given a regular tree language belongs to the class ${\bf Δ^0_2}$ of the Borel hierarchy, or equivalently whether the Wadge degree of a regular tree language is countable.
Toda proved in 1989 that the (discrete) polynomial time hierarchy, $\mathbf{PH}$, is contained in the class $\mathbf{P}^{#\mathbf{P}}$, namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class $#\mathbf{P}$. This result which illustrates the power of counting is considered to be a seminal result in computational complexity theory. An analogous result in the complexity theory over the reals (in the sense of Blum-Shub-Smale real machines) has been missing so far. In thi…
Foundational Analysis Of The Solvability Complexity Index: The Weihrauch-SCI Intermediate Hierarchy
本文研究了可解性复杂度指数（SCI）的基础理论，探讨了其与Type-2可计算性及Weihrauch可约性的联系，提出Weihrauch-SCI等级概念，并指出原始SCI模型在计算模型中的局限性。
The Alternation Hierarchy of First-Order Logic on Words is Decidable
本文证明了一阶逻辑FO[<]的$Σ_i[<]$片段在正则语言上的可判定性，解决了1971年以来的开放问题，核心是通过多项式闭包技术将可判定分离问题从正簇扩展至其多项式闭包。
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