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Ioannis Souldatos

Publications and source records attributed to Ioannis Souldatos.

14 recordsLinked to original sources
arXiv2025

On the Computability of Finding Capacity-Achieving Codes

This work studies the problem of constructing capacity-achieving codes from an algorithmic perspective. Specifically, we prove that there exists a Turing machine which, given a discrete memoryless channel $p_{Y|X}$, a target rate $R$ less than the channel capacity $C(p_{Y|X})$, and an error tolerance $ε> 0$, outputs a block code $\mathcal{C}$ achieving a rate at least $R$ and a maximum block error probability below $ε$. The machine operates in the general case where all transition probabilities of $p_{Y|X}$ are computable real numbers, and the parameters $R$ and $ε$ are rational. The proof builds on Shannon's channel coding theorem and relies on an exhaustive search approach that systematically enumerates all codes of increasing block length until a valid code is found. This construction is formalized using the theory of recursive functions, yielding a $μ$-recursive function $\mathrm{FindCode} : \mathbb{N}^3 \rightharpoonup \mathbb{N}$ that takes as input appropriate encodings of $p_{Y|X}$, $R$, and $ε$, and, whenever $R < C(p_{Y|X})$, outputs an encoding of a valid code. By Kleene's normal form theorem, which establishes the computational equivalence between Turing machines and $μ$-recursive functions, we conclude that the problem is solvable by a Turing machine. This result can also be extended to the case where $ε$ is a computable real number, while we further discuss an analogous generalization of our analysis when $R$ is computable as well. We note that the assumptions that the probabilities of $p_{Y|X}$, as well as $ε$ and $R$, are computable real numbers cannot be further weakened, since computable reals constitute the largest subset of $\mathbb{R}$ representable by algorithmic means.

cs.IT↗
arXiv2022

A Lower Bound for the Hanf Number for Joint Embedding

In [13] the authors show that if $μ$ is a strongly compact cardinal, $K$ is an Abstract Elementary Class (AEC) with $LS(K)<μ$, and $K$ satisfies joint embedding (amalgamation) cofinally below $μ$, then $K$ satisfies joint embedding (amalgamation) in all cardinals $\ge μ$. The question was raised if the strongly compact upper bound was optimal. In this paper we prove the existence of an AEC $K$ that can be axiomatized by an $\mathcal{L}_{ω_1,ω}$-sentence in a countable vocabulary, so that if $μ$ is the first measurable cardinal, then (1) $K$ satisfies joint embedding cofinally below $μ$ ; (2) $K$ fails joint embedding cofinally below $μ$; and (3) $K$ satisfies joint embedding above $μ$. Moreover, the example can be generalized to an AEC $K^χ$ axiomatized in $\mathcal{L}_{χ^+, ω}$, in a vocabulary of size $χ$, such that (1)-(3) hold with $μ$ being the first measurable above $χ$. This proves that the Hanf number for joint embedding is contained in the interval between the first measurable and the first strongly compact. Since these two cardinals can consistently coincide, the upper bound from [13] is consistently optimal. This is also the first example of a sentence whose joint embedding spectrum is (consistently) neither an initial nor an eventual interval of cardinals. By Theorem 3.26, it is consistent that for any club $C$ on the first measurable $μ$, JEP holds exactly on $\lim C$ and everywhere above $μ$.

math.LO↗
arXiv2021

Non-absoluteness of Hjorth's Cardinal Characterization

In [5], Hjorth proved that for every countable ordinal $α$, there exists a complete $\mathcal{L}_{ω_1,ω}$-sentence $ϕ_α$ that has models of all cardinalities less than or equal to $\aleph_α$, but no models of cardinality $\aleph_{α+1}$. Unfortunately, his solution does not yield a single $\mathcal{L}_{ω_1,ω}$-sentence $ϕ_α$, but a set of $\mathcal{L}_{ω_1,ω}$-sentences, one of which is guaranteed to work. It was conjectured in [9] that it is independent of the axioms of ZFC which of these sentences has the desired property. In the present paper, we prove that this conjecture is true. More specifically, we isolate a diagonalization principle for functions from $ω_1$ to $ω_1$ which is a consequence of the Bounded Proper Forcing Axiom (BPFA) and then we use this principle to prove that Hjorth's solution to characterizing $\aleph_2$ in models of BPFA is different than in models of CH. In addition, we show that large cardinals are not needed to obtain this independence result by proving that our diagonalization principle can be forced over models of CH.

math.LO↗
arXiv2020

Kurepa trees and spectra of $\mathcal{L}_{ω_1,ω}$-sentences

We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a \emph{single} $\mathcal{L}_{ω_1,ω}$-sentence $ψ$ that codes Kurepa trees to prove the consistency of the following: (1) The spectrum of $ψ$ is consistently equal to $[\aleph_0,\aleph_{ω_1}]$ and also consistently equal to $[\aleph_0,2^{\aleph_1})$, where $2^{\aleph_1}$ is weakly inaccessible. (2) The amalgamation spectrum of $ψ$ is consistently equal to $[\aleph_1,\aleph_{ω_1}]$ and $[\aleph_1,2^{\aleph_1})$, where again $2^{\aleph_1}$ is weakly inaccessible. This is the first example of an $\mathcal{L}_{ω_1,ω}$-sentence whose spectrum and amalgamation spectrum are consistently both right-open and right-closed. It also provides a positive answer to a question in [18]. (3) Consistently, $ψ$ has maximal models in finite, countable, and uncountable many cardinalities. This complements the examples given in [1] and [2] of sentences with maximal models in countably many cardinalities. (4) $2^{\aleph_0}<\aleph_{ω_1}<2^{\aleph_1}$ and there exists an $\mathcal{L}_{ω_1,ω}$-sentence with models in $\aleph_{ω_1}$, but no models in $2^{\aleph_1}$. This relates to a conjecture by Shelah that if $\aleph_{ω_1}<2^{\aleph_0}$, then any $\mathcal{L}_{ω_1,ω}$-sentence with a model of size $\aleph_{ω_1}$ also has a model of size $2^{\aleph_0}$. Our result proves that $2^{\aleph_0}$ can not be replaced by $2^{\aleph_1}$, even if $2^{\aleph_0}<\aleph_{ω_1}$.

math.LO↗
arXiv2018

Complete $\mathcal{L}_{ω_1,ω}$-Sentences with Maximal Models in Multiple Cardinalities

In [BKS15] examples of incomplete sentences are given with maximal models in more than one cardinality. The question was raised whether one can find similar examples of complete sentences. In this paper we give examples of complete $L_{ω_1,ω}$-sentences with maximal models in more than one cardinality. From (homogeneous) characterizability of $κ$ we construct sentences with maximal models in $κ$ and in one of $κ^+, κ^ω, 2^κ$ and more. Indeed, consistently we find sentences with maximal models in uncountably many distinct cardinalities.

math.LO↗
arXiv2017

Non-Absoluteness of Model Existence at $\aleph_ω$

In [FHK13], the authors considered the question whether model-existence of $L_{ω_1,ω}$-sentences is absolute for transitive models of ZFC, in the sense that if $V \subseteq W$ are transitive models of ZFC with the same ordinals, $φ\in V$ and $V\models "φ\text{ is an } L_{ω_1,ω}\text{-sentence}"$, then $V \models "φ\text{ has a model of size } \aleph_α"$ if and only if $W \models "φ\text{ has a model of size } \aleph_α"$. From [FHK13] we know that the answer is positive for $α=0,1$ and under the negation of CH, the answer is negative for all $α>1$. Under GCH, and assuming the consistency of a supercompact cardinal, the answer remains negative for each $α>1$, except the case when $α=ω$ which is an open question in [FHK13]. We answer the open question by providing a negative answer under GCH even for $α=ω$. Our examples are incomplete sentences. In fact, the same sentences can be used to prove a negative answer under GCH for all $α>1$ assuming the consistency of a Mahlo cardinal. Thus, the large cardinal assumption is relaxed from a supercompact in [FHK13] to a Mahlo cardinal. Finally, we consider the absoluteness question for the $\aleph_α$-amalgamation property of $L_{ω_1,ω}$-sentences (under substructure). We prove that assuming GCH, $\aleph_α$-amalgamation is non-absolute for $1<α<ω$. This answers a question from [SS]. The cases $α=1$ and $α$ infinite remain open. As a corollary we get that it is non-absolute that the amalgamation spectrum of an $L_{ω_1,ω}$-sentence is empty.

math.LO↗
arXiv2017

Characterizing the powerset by a complete (Scott) sentence

This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing the work from http://arxiv.org/abs/1007.2426v1. A cardinal $κ$ is characterized by a Scott sentence $ϕ_M$, if $ϕ_M$ has a model of size $κ$, but no model of $κ^+$. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if $\aleph_β$ is characterized by a Scott sentence, then $2^{\aleph_{β+β_1}}$ is (homogeneously) characterized by a Scott sentence, for all $0<β_1<ω_1$. So, the answer to the above question is positive, except the case $β_1=0$ which remains open. As a consequence we derive that if $α\leβ$ and $\aleph_β$ is characterized by a Scott sentence, then $\aleph_{α+α_1}^{\aleph_{β+β_1}}$ is also characterized by a Scott sentence, for all $α_1<ω_1$ and $0<β_1<ω_1$. Whence, depending on the model of ZFC, we see that the class of characterizable and homogeneously characterizable cardinals is much richer than previously known. Several open questions are also mentioned at the end.

math.LO↗
arXiv2016

Hanf Number for Scott Sentences of Computable Structures

The Hanf number for a set $S$ of sentences in $L_{ω_1,ω}$ (or some other logic) is the least infinite cardinal $κ$ such that for all $φ\in S$, if $φ$ has models in all infinite cardinalities less than $κ$, then it has models of all infinite cardinalities. S-D. Friedman asked what is the Hanf number for Scott sentences of computable structures. We show that the value is $\beth_{ω_1^{CK}}$. The same argument proves that $\beth_{ω_1^{CK}}$ is the Hanf number for Scott sentences of hyperarithmetical structures.

math.LO↗
arXiv2015

On automorphisms groups of structures of countable cofinality

In [2] Su Gao proves that the following are equivalent for a countable $M$ (cf. theorem 1.2 too): (I)There is an uncountable model of the Scott sentence of $M$. (II) There exists some $j\in \overline{Aut(M)}\setminus Aut(M)$, where $\overline{Aut(M)}$ is the closure of $Aut(M)$ under the product topology in $ω^ω$. (III) There is an $L_{ω_1,ω}$- elementary embedding $j$ from $M$ to itself such that $range(j)\subset M$. We generalize his theorem to all cardinals $κ$ of of cofinality $ω$ (cf. theorem 4.2). The following are equivalent: (I$^*$) There is a model of the Scott sentence of $M$ of size $κ^+$. (II$^*$) For all $α<β<κ^+$, there exist functions $j_{β,α}$ in $\overline{Aut(M)}^{T}\setminus Aut(M)$, such that for $α< β<γ<κ^+$, \begin{equation}(*) j_{γ,β}\circ j_{β,α}=j_{γ,α},\end{equation} where $\overline{Aut(M)}^{T}$ is the closure of $Aut(M)$ under the product topology in $κ^κ$. (III$^*$) For every $β<κ^+$, there exist $L_{\infty,κ}^{fin}$- elementary embeddings (cf. definition 2.5) $(j_α)_{α<β}$ from $M$ to itself such that $α_1<α_2\Rightarrow range(j_{α_1})\subset range(j_{α_2})$. Theorem 4.2 holds both for countable and uncountable $κ$. Condition (*) in (II$^*$), which does not appear in the countable case, can not be removed when $κ$ is uncountable (cf. theorem 4.5). Condition (II$^*$) imply the existence of at least $κ^ω$ automorphisms of $M$ (cf. corollary 4.6). It is unknown to the author whether a purely topological proof of corollary 4.6 exists.

math.LO↗
arXiv2015

The Joint Embedding Property and Maximal Models

We introduce the notion of a `pure` Abstract Elementary Class to block trivial counterexamples. We study classes of models of bipartite graphs and show: Main Theorem (cf. Theorem 3.5.2 and Corollary 3.5.6): If $(λ_i : i \le α<\aleph_1)$ is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whose models satisfy JEP$(<λ_0)$, there is an $L_{ω_1,ω}$ -sentence $ψ$ whose models form a pure AEC and (1) The models of $ψ$ satisfy JEP$(<λ_0)$, while JEP fails for all larger cardinals and AP fails in all infinite cardinals. (2) There exist $2^{λ_i^+}$ non-isomorphic maximal models of $ψ$ in $λ_i^+$, for all $i \le α$, but no maximal models in any other cardinality; and (3) $ψ$ has arbitrarily large models. In particular this shows the Hanf number for JEP and the Hanf number for maximality for pure AEC with Lowenheim number $\aleph_0$ are at least $\beth_{ω_1}$. We show that although AP$(κ)$ for each $κ$ implies the full amalgamation property, JEP$(κ)$ for each κdoes not imply the full joint embedding property. We show the main combinatorial device of this paper cannot be used to extend the main theorem to a complete sentence.

math.LO↗
arXiv2012

Notes on cardinals that are characterizable by a complete (Scott) sentence

This is part I of a study on cardinals that are characterizable by Scott sentences. Building on [3], [6] and [1] we study which cardinals are characterizable by a Scott sentence $ϕ$, in the sense that $ϕ$ characterizes $κ$, if $ϕ$ has a model of size $κ$, but no models of size $κ^+$. We show that the set of cardinals that are characterized by a Scott sentence is closed under successors, countable unions and countable products (cf. theorems 2.3, 3.4, and corollary 3.6). We also prove that if $\aleph_α$ is characterized by a Scott sentence, at least one of $\aleph_alpha$ and $\aleph_alpha^+$ is homogeneously characterizable (cf. definition 1.3 and theorem 2.9). Based on Shelah's [8], we give counterexamples that characterizable cardinals are not closed under predecessors, or cofinalities.

math.LO↗
arXiv2010

Linear Orderings and Powers of Characterizable Cardinal

The current paper answers an open question of abs/1007.2426 We say that a countable model M characterizes an infinite cardinal kappa, if the Scott sentence of M has a model in cardinality kappa, but no models in cardinality kappa plus. If M is linearly ordered by <, we will say that the linear ordering (M,<) characterizes kappa. It is known that if kappa is characterizable, then kappa plus is characterizable by a linear ordering. Also, if kappa is characterizable by a dense linear ordering with an increasing sequence of size kappa, then 2^kappa is characterizable. We show that if kappa is homogeneously characterizable, then kappa is characterizable by a dense linear ordering, while the converse fails. The main theorems are: 1) If kappa>2^lambda is a characterizable cardinal, lambda is characterizable by a dense linear ordering and lambda is the least cardinal such that kappa^lambda>kappa, then kappa^lambda is also characterizable, 2) if aleph_alpha and kappa^(aleph_alpha) are characterizable cardinals, then the same is true for kappa^(aleph_(alpha+beta)), for all countable beta. Combining these two theorems we get that if kappa>2^(aleph_alpha) is a characterizable cardinal, aleph_alpha is characterizable by a dense linear ordering and aleph_alpha is the least cardinal such that kappa^(aleph_alpha)>kappa, then for all beta<alpha+omega_1, kappa^(aleph_beta) is characterizable. Also if kappa is a characterizable cardinal, then kappa^(aleph_alpha) is characterizable, for all countable alpha.

math.LO↗
arXiv2010

Independently Axiomatizable L_{omega_1,omega} Theories

In partial answer to a question posed by Arnie Miller (http://www.math.wisc.edu/~miller/res/problem.pdf) and X. Caicedo, we obtain sufficient conditions for an L_{omega_1,omega} theory to have an independent axiomatization. As a consequence we obtain two corollaries: The first, assuming Vaught's Conjecture, every L_{omega_1,omega} theory in a countable language has an independent axiomatization. The second, this time outright in ZFC, every intersection of a family of Borel sets can be formed as the intersection of a family of independent Borel sets.

math.LO↗