Related papers: The structural complexity of models of arithmetic
We describe arithmetic computations in terms of operations on some well known free algebras (S1S, S2S and ordered rooted binary trees) while emphasizing the common structure present in all them when seen as isomorphic with the set of…
We propose $\omega$MSO$\Join$BAPA, an expressive logic for describing countable structures, which subsumes and transcends both Counting Monadic Second-Order Logic (CMSO) and Boolean Algebra with Presburger Arithmetic (BAPA). We show that…
A fruitful way of obtaining meaningful, possibly concrete, algorithmically random numbers is to consider a potential behaviour of a Turing machine and its probability with respect to a measure (or semi-measure) on the input space of binary…
Modern data analysis depends increasingly on estimating models via flexible high-dimensional or nonparametric machine learning methods, where the identification of structural parameters is often challenging and untestable. In linear…
In Chapter 3 of his Notes on constructive mathematics, Martin-L{\"o}f describes recursively constructed ordinals. He gives a constructively acceptable version of Kleene's computable ordinals. In fact, the Turing definition of computable…
We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…
Bruyere and Carton lifted the notion of finite automata reading infinite words to finite automata reading words with shape an arbitrary linear order L. Automata on finite words can be used to represent infinite structures, the so-called…
In this paper, we investigate connections between structures present in every generic extension of the universe $V$ and computability theory. We introduce the notion of {\em generic Muchnik reducibility} that can be used to to compare the…
The article focuses on a class of second countable groups assembled from profinite and discrete by elementary operations. We focus on a rank associated with these groups that measure their complexity, the decomposition rank. A collection of…
This paper defines a Mitchell rank for supercompact cardinals. If $\kappa$ is a $\theta$-supercompact cardinal then $o_{\theta-sc}(\kappa) = \sup \{ o_{\theta-sc}(\mu) + 1 \ | \ \mu \in m(\kappa)\}$, where $m(\kappa)$ is the collection of…
A new syntactic characterization of problems complete via Turing reductions is presented. General canonical forms are developed in order to define such problems. One of these forms allows us to define complete problems on ordered…
We consider the foundational relation between arithmetic and set theory. Our goal is to criticize the construction of standard arithmetic models as providing grounds for arithmetic truth (even in a relative sense). Our method is to…
We study the satisfiability of ordering constraint satisfaction problems (CSPs) above average. We prove the conjecture of Gutin, van Iersel, Mnich, and Yeo that the satisfiability above average of ordering CSPs of arity $k$ is…
Given a first-order theory $T$ formulated in the usual language of first-order arithmetic, we say that $T$ is of *restricted complexity* if there is some natural number $n$ and some set $\mathcal A$ of $\Sigma_n$-sentences such that $T$ can…
A tree is pathwise-random if all of its paths are Martin-Lof random. We show that (a) no weakly 2-random real computes a perfect pathwise-random tree; it follows that the class of perfect pathwise-random trees is null, with respect to any…
Let $(M,\scott X) \models \ACA$ be such that $P_\scott X$, the collection of all unbounded sets in $\scott X$, admits a definable complete ultrafilter and let $T$ be a theory extending first order arithmetic coded in $\scott X$ such that…
In this paper we study various versions of extension complexity for polygons through the study of factorization ranks of their slack matrices. In particular, we develop a new asymptotic lower bound for their nonnegative rank, shortening the…
This paper presents a new representation of natural numbers and discusses its consequences for computability and computational complexity. The paper argues that the introduction of the first Peano axiom in the traditional definition of…
Arithmetic complexity is considered simpler to understand than Boolean complexity, namely computing Boolean functions via logical gates. And indeed, we seem to have significantly more lower bound techniques and results in arithmetic…
Let S = <s_1, s_2, s_3, ..., s_n> be a given vector of n real numbers. The rank of a real z with respect to S is defined as the number of elements s_i in S such that s_i is less than or equal to z. We consider the following decision…