Arithmetical Hierarchy Research Articles

P-m-Mitotic Sets and Arithmetical Hierarchy

Let {0,1}∗be the set of all finite strings of elements from {0,1}, and let Pbe the class of problems recognized by deterministic Turing machines, which run in polynomial time (a problem is simply a subset of {0,1}∗). This article defines the class 𝐏(and shows that 𝐏(is isomorphic to the class 𝐏. Based on the notions of T-mitoticity and T-autoreducibility, K.Ambos-Spies introduced the notions of P-m-mitoticity and P-m-autoreducibility. The notions of P)-m-mitoticity and P)-m-autoreducibility are introduced by analogy with the mentioned notions.The article proves that the index sets {𝑧|𝑊"is P(-m-mitotic} and {z|𝑊"is P(-m-autoreducible} are Σ#-complete.

Complexity of Neural Network Training and ETR: Extensions with Effectively Continuous Functions

The training problem of neural networks (NNs) is known to be ER-complete with respect to ReLU and linear activation functions. We show that the training problem for NNs equipped with arbitrary activation functions is polynomial-time bireducible to the existential theory of the reals extended with the corresponding activation functions. For effectively continuous activation functions (e.g., the sigmoid function), we obtain an inclusion to low levels of the arithmetical hierarchy. Consequently, the sigmoid activation function leads to the existential theory of the reals with the exponential function, and hence the decidability of training NNs using the sigmoid activation function is equivalent to the decidability of the existential theory of the reals with the exponential function, a long-standing open problem. In contrast, we obtain that the training problem is undecidable if sinusoidal activation functions are considered.

Polynomial Time Turing Mitoticity and Arithmetical Hierarchy

Polynomial Time Turing Mitoticity and Arithmetical Hierarchy

Existential Definability over the Subword Ordering

We study first-order logic (FO) over the structure consisting of finite words over some alphabet $A$, together with the (non-contiguous) subword ordering. In terms of decidability of quantifier alternation fragments, this logic is well-understood: If every word is available as a constant, then even the $\Sigma_1$ (i.e., existential) fragment is undecidable, already for binary alphabets $A$. However, up to now, little is known about the expressiveness of the quantifier alternation fragments: For example, the undecidability proof for the existential fragment relies on Diophantine equations and only shows that recursively enumerable languages over a singleton alphabet (and some auxiliary predicates) are definable. We show that if $|A|\ge 3$, then a relation is definable in the existential fragment over $A$ with constants if and only if it is recursively enumerable. This implies characterizations for all fragments $\Sigma_i$: If $|A|\ge 3$, then a relation is definable in $\Sigma_i$ if and only if it belongs to the $i$-th level of the arithmetical hierarchy. In addition, our result yields an analogous complete description of the $\Sigma_i$-fragments for $i\ge 2$ of the pure logic, where the words of $A^*$ are not available as constants.

How to approximate fuzzy sets: mind-changes and the Ershov Hierarchy

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 limit computable 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. In this paper, we combine the Ershov Hierarchy and fuzzy set theory, by introducing and investigating the Fuzzy Ershov Hierarchy.

On bi-infinite and conjugate post correspondence problems

We study two modifications of the Post Correspondence Problem (PCP), namely (1) the bi-infinite version, where it is asked whether there exists a bi-infinite word such that two given morphisms agree on it, and (2) the conjugate version, where we require the images of a solution for two given morphisms are conjugates of each other. For the conjugate PCP we give an undecidability proof by reducing it to the word problem for a special type of semi-Thue systems and for the bi-infinite PCP we give a simple argument that it is in the class Σ20 of the arithmetical hierarchy.

Approximate counting and NP search problems

We study a new class of NP search problems, those which can be proved total using standard combinatorial reasoning based on approximate counting. Our model for this kind of reasoning is the bounded arithmetic theory [Formula: see text] of [E. Jeřábek, Approximate counting by hashing in bounded arithmetic, J. Symb. Log. 74(3) (2009) 829–860]. In particular, the Ramsey and weak pigeonhole search problems lie in the new class. We give a purely computational characterization of this class and show that, relative to an oracle, it does not contain the problem CPLS, a strengthening of PLS. As CPLS is provably total in the theory [Formula: see text], this shows that [Formula: see text] does not prove every [Formula: see text] sentence which is provable in bounded arithmetic. This answers the question posed in [S. Buss, L. A. Kołodziejczyk and N. Thapen, Fragments of approximate counting, J. Symb. Log. 79(2) (2014) 496–525] and represents some progress in the program of separating the levels of the bounded arithmetic hierarchy by low-complexity sentences. Our main technical tool is an extension of the “fixing lemma” from [P. Pudlák and N. Thapen, Random resolution refutations, Comput. Complexity, 28(2) (2019) 185–239], a form of switching lemma, which we use to show that a random partial oracle from a certain distribution will, with high probability, determine an entire computation of a [Formula: see text] oracle machine. The introduction to the paper is intended to make the statements and context of the results accessible to someone unfamiliar with NP search problems or with bounded arithmetic.

Refining the arithmetical hierarchy of classical principles

AbstractWe refine the arithmetical hierarchy of various classical principles by finely investigating the derivability relations between these principles over Heyting arithmetic. We mainly investigate some restricted versions of the law of excluded middle, De Morgan's law, the double negation elimination, the collection principle and the constant domain axiom.

Commutative action logic

Abstract We prove undecidability and pinpoint the place in the arithmetical hierarchy for commutative action logic, i.e. the equational theory of commutative residuated Kleene lattices (action lattices), and infinitary commutative action logic, the equational theory of *-continuous commutative action lattices. Namely, we prove that the former is $\varSigma _1^0$-complete and the latter is $\varPi _1^0$-complete. Thus, the situation is the same as in the more well-studied non-commutative case. The methods used, however, are different: we encode infinite and circular computations of counter (Minsky) machines.

CONSERVATION THEOREMS ON SEMI-CLASSICAL ARITHMETIC

AbstractWe systematically study conservation theorems on theories of semi-classical arithmetic, which lie in-between classical arithmetic $\mathsf {PA}$ and intuitionistic arithmetic $\mathsf {HA}$ . Using a generalized negative translation, we first provide a structured proof of the fact that $\mathsf {PA}$ is $\Pi _{k+2}$ -conservative over $\mathsf {HA} + {\Sigma _k}\text {-}\mathrm {LEM}$ where ${\Sigma _k}\text {-}\mathrm {LEM}$ is the axiom scheme of the law-of-excluded-middle restricted to formulas in $\Sigma _k$ . In addition, we show that this conservation theorem is optimal in the sense that for any semi-classical arithmetic T, if $\mathsf {PA}$ is $\Pi _{k+2}$ -conservative over T, then ${T}$ proves ${\Sigma _k}\text {-}\mathrm {LEM}$ . In the same manner, we also characterize conservation theorems for other well-studied classes of formulas by fragments of classical axioms or rules. This reveals the entire structure of conservation theorems with respect to the arithmetical hierarchy of classical principles.

Freezing, Bounded-Change and Convergent Cellular Automata

This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension, and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-K\r{u}rka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension~1, but also dimension~1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension~ 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension.

ANALYZING CONTENT KNOWLEDGE OF PROSPECTIVE ELEMENTARY SCHOOL TEACHERS ON THE MATERIAL OF FRACTIONAL ARITHMETIC OPERATIONS

Teacher quality is one of the most influential factors in student learning. In-depth material knowledge (content) is very important for prospective teachers who will one day become teachers. Thus, analyzing and describing the Content Knowledge (CK) of prospective teachers needs to be done. The purpose of this study is to describe the CK of prospective elementary school teachers (SD/MI) on the material of fractional arithmetic operations. The subjects of this study were three students from Ar-Raniry State Islamic University of Banda Aceh. This research includes qualitative descriptive research, so that the data were analyzed using qualitative data analysis, namely data reduction, data presentation, and drawing conclusions. Data collection in this study was carried out through tests and combined with interviews so that the instruments used were test questions and interview guidelines. The study results indicate that the content knowledge of the three elementary teacher candidates was still weak, there were teacher candidates who did not understand the arithmetic operation hierarchy and still made carelessness in the concept of fractional arithmetic operations.

Separable Algorithmic Representations of Classical Systems and Their Applications

The main results of the theory of separable algorithmic representations of classical algebraic systems are presented. The most important classes of such systems and their representations in the lower classes of the arithmetic hierarchy - positive and negative - are described. Special attention is paid to the algorithmic, structural and topological properties of separable representations of groups, rings and bodies, as well as to effective analogs of the Maltsev theorem on embedding rings in bodies. The possibilities of using the studied concepts in the framework of theoretical informatics are considered.

PRENEX NORMAL FORM THEOREMS IN SEMI-CLASSICAL ARITHMETIC

AbstractAkama et al. [1] systematically studied an arithmetical hierarchy of the law of excluded middle and related principles in the context of first-order arithmetic. In that paper, they first provide a prenex normal form theorem as a justification of their semi-classical principles restricted to prenex formulas. However, there are some errors in their proof. In this paper, we provide a simple counterexample of their prenex normal form theorem [1, Theorem 2.7], then modify it in an appropriate way which still serves to largely justify the arithmetical hierarchy. In addition, we characterize a variety of prenex normal form theorems by logical principles in the arithmetical hierarchy. The characterization results reveal that our prenex normal form theorems are optimal. For the characterization results, we establish a new conservation theorem on semi-classical arithmetic. The theorem generalizes a well-known fact that classical arithmetic is $\Pi _2$ -conservative over intuitionistic arithmetic.

Is Church’s Thesis Still Relevant?

Abstract The article analyses the role of Church’s Thesis (hereinafter CT) in the context of the development of hypercomputation research. The text begins by presenting various views on the essence of computer science and the limitations of its methods. Then CT and its importance in determining the limits of methods used by computer science is presented. Basing on the above explanations, the work goes on to characterize various proposals of hypercomputation showing their relative power in relation to the arithmetic hierarchy. The general theme of the article is the analysis of mutual relations between the content of CT and the theories of hypercomputation. In the main part of the paper the arguments for abolition of CT caused by the introduction of hypercomputable methods in computer science are presented and critique of these views is presented. The role of the efficiency condition contained in the formulation of CT is stressed. The discussion ends with a summary defending the current status of Church’s thesis within the framework of philosophy and computer science as an important point of reference for determining what the notion of effective calculability really is. The considerations included in this article seem to be quite up-to-date relative to the current state of affairs in computer science.1

Learning theory in the arithmetic hierarchy II

The present work determines the arithmetic complexity of the index sets of u.c.e. families which are learnable according to various criteria of algorithmic learning. Specifically, we prove that the index set of codes for families that are TxtFex $$^a_b$$ -learnable is $$\Sigma _4^0$$ -complete and that the index set of TxtFex $$^*_*$$ -learnable and the index set of TxtFext $$^*_*$$ -learnable families are both $$\Sigma _5^0$$ -complete.

On the complexity of index sets for finite predicate logic programs which allow function symbols

AbstractWe study the recognition problem in the metaprogramming of finite normal predicate logic programs. That is, let $\mathcal{L}$ be a computable first-order predicate language with infinitely many constant symbols and infinitely many $n$-ary predicate symbols and $n$-ary functions symbols for all $n \geq 1$. Then we can effectively list all the finite normal predicate logic programs $Q_0,Q_1,\ldots $ over $\mathcal{L}$. Given some property $\mathcal{P}$ of finite normal predicate logic programs over $\mathcal{L}$, we define the index set $I_{\mathcal{P}}$ to be the set of indices $e$ such that $Q_e$ has property $\mathcal{P}$. We classify the complexity of the index set $I_{\mathcal{P}}$ within the arithmetic hierarchy for various natural properties of finite predicate logic programs. For example, we determine the complexity of the index sets relative to all finite predicate logic programs and relative to certain special classes of finite predicate logic programs of properties such as (i) having no stable models, (ii) having no recursive stable models, (iii) having at least one stable model, (iv) having at least one recursive stable model, (v) having exactly $c$ stable models for any given positive integer $c$, (vi) having exactly $c$ recursive stable models for any given positive integer $c$, (vii) having only finitely many stable models, (viii) having only finitely many recursive stable models, (ix) having infinitely many stable models and (x) having infinitely many recursive stable models.

An intuitionistic formula hierarchy based on high‐school identities

AbstractWe revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal) propositional sequent calculi are formula (i.e., sequent) isomorphisms corresponding to the high‐school identities, we show that one can obtain a more compact variant of a proof system, consisting of non‐invertible proof rules only, and where the invertible proof rules have been replaced by a formula normalization procedure. Moreover, for certain proof systems such as the G4ip sequent calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the non‐invertible proof rules as strict inequalities between exponential polynomials; a careful combinatorial treatment is given in order to establish this fact. Finally, we extend the exponential polynomial analogy to the first‐order quantifiers, showing that it gives rise to an intuitionistic hierarchy of formulas, resembling the classical arithmetical hierarchy, and the first one that classifies formulas while preserving isomorphism.

THE RAMIFIED ANALYTICAL HIERARCHY USING EXTENDED LOGICS

AbstractThe use of Extended Logics to replace ordinary second order definability in Kleene’s Ramified Analytical Hierarchy is investigated. This mirrors a similar investigation of Kennedy, Magidor and Väänänen [11] where Gödel’s universe L of constructible sets is subjected to similar variance. Enhancing second order definability allows models to be defined which may or may not coincide with the original Kleene hierarchy in domain. Extending the logic with game quantifiers, and assuming strong axioms of infinity, we obtain minimal correct models of analysis. A wide spectrum of models can be so generated from abstract definability notions: one may take an abstract Spector Class and extract an extended logic for it. The resultant structure is then a minimal model of the given kind of definability.

Turing Degrees in Refinements of the Arithmetical Hierarchy

We investigate the problem of characterizing proper levels of the fine hierarchy (up to Turing equivalence). It is known that the fine hierarchy exhausts arithmetical sets and contains as a small fragment finite levels of Ershov hierarchies (relativized to ∅n, n < ω), which are known to be proper. Our main result is finding a least new (i.e., distinct from the levels of the relativized Ershov hierarchies) proper level. We also show that not all new levels are proper.