arXiv subjects
Publications and source records attributed to Matthew Harrison-Trainor.
We give a new and effective classification of all Borel Wadge classes of subsets of Baire space. This relies on the true stage machinery originally developed by Montalbán. We use this machinery to give a new proof of Louveau and Saint-Raymond's separation theorem for Borel Wadge classes. This gives a proof of Borel Wadge determinacy in the subsystem $\text{ATR}_0+Π^1_1$-I of second-order arithmetic.
We present the true stages machinery and illustrate its applications to descriptive set theory. We use this machinery to provide new proofs of the Hausdorff-Kuratowski and Wadge theorems on the structure of ${\mathbf Δ}^0_ξ$, Louveau and Saint-Raymond's separation theorem, and Louveau's separation theorem.
The $Ω$ numbers-the halting probabilities of universal prefix-free machines-are known to be exactly the Martin-L{ö}f random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-L{ö}f random left-c.e. real $α$, a universal prefix-free machine U whose halting probability is $α$. We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real $α$, one cannot uniformly produce a left-c.e. real $β$ such that $α$ -- $β$ is neither left-c.e. nor right-c.e.
Qualitative and quantitative approaches to reasoning about uncertainty can lead to different logical systems for formalizing such reasoning, even when the language for expressing uncertainty is the same. In the case of reasoning about relative likelihood, with statements of the form $φ\succsimψ$ expressing that $φ$ is at least as likely as $ψ$, a standard qualitative approach using preordered preferential structures yields a dramatically different logical system than a quantitative approach using probability measures. In fact, the standard preferential approach validates principles of reasoning that are incorrect from a probabilistic point of view. However, in this paper we show that a natural modification of the preferential approach yields exactly the same logical system as a probabilistic approach--not using single probability measures, but rather sets of probability measures. Thus, the same preferential structures used in the study of non-monotonic logics and belief revision may be used in the study of comparative probabilistic reasoning based on imprecise probabilities.
An oracle $A$ is low-for-speed if it is unable to speed up the computation of a set which is already computable: if a decidable language can be decided in time $t(n)$ using $A$ as an oracle, then it can be decided without an oracle in time $p(t(n))$ for some polynomial $p$. The existence of a set which is low-for-speed was first shown by Bayer and Slaman who constructed a non-computable computably enumerable set which is low-for-speed. In this paper we answer a question previously raised by Bienvenu and Downey, who asked whether there is a minimal degree which is low-for-speed. The standard method of constructing a set of minimal degree via forcing is incompatible with making the set low-for-speed; but we are able to use an interesting new combination of forcing and full approximation to construct a set which is both of minimal degree and low-for-speed.
Every countable structure has a sentence of the infinitary logic $\mathcal{L}_{ω_1 ω}$ which characterizes that structure up to isomorphism among countable structures. Such a sentence is called a Scott sentence, and can be thought of as a description of the structure. The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure. We begin with an introduction to the area, with short and simple proofs where possible, followed by a survey of recent advances.
A set is introreducible if it can be computed by every infinite subset of itself. Such a set can be thought of as coding information very robustly. We investigate introreducible sets and related notions. Our two main results are that the collection of introreducible sets is $Π^1_1$-complete, so that there is no simple characterization of the introreducible sets; and that every introenumerable set has an introreducible subset.
In an election in which each voter ranks all of the candidates, we consider the head-to-head results between each pair of candidates and form a labeled directed graph, called the margin graph, which contains the margin of victory of each candidate over each of the other candidates. A central issue in developing voting methods is that there can be cycles in this graph, where candidate $\mathsf{A}$ defeats candidate $\mathsf{B}$, $\mathsf{B}$ defeats $\mathsf{C}$, and $\mathsf{C}$ defeats $\mathsf{A}$. In this paper we apply the central limit theorem, graph homology, and linear algebra to analyze how likely such situations are to occur for large numbers of voters. There is a large literature on analyzing the probability of having a majority winner; our analysis is more fine-grained. The result of our analysis is that in elections with the number of voters going to infinity, margin graphs that are more cyclic in a certain precise sense are less likely to occur.
When classes of structures are not first-order definable, we might still try to find a nice description. There are two common ways for doing this. One is to expand the language, leading to notions of pseudo-elementary classes, and the other is to allow infinite conjuncts and disjuncts. In this paper we examine the intersection. Namely, we address the question: Which classes of structures are both pseudo-elementary and $\mathcal{L}_{ω_1 ω}$-elementary? We find that these are exactly the classes that can be defined by an infinitary formula that has no infinitary disjunctions.
The rate of randomness (or dimension) of a string $σ$ is the ratio $C(σ)/|σ|$ where $C(σ)$ is the Kolmogorov complexity of $σ$. While it is known that a single computable transformation cannot increase the rate of randomness of all sequences, Fortnow, Hitchcock, Pavan, Vinodchandran, and Wang showed that for any $0<α<β<1$, there are a finite number of computable transformations such that any string of rate at least $α$ is turned into a string of rate at least $β$ by one of these transformations. However, their proof only gives very loose bounds on the correspondence between the number of transformations and the increase of rate of randomness one can achieve. By translating this problem to combinatorics on (hyper)graphs, we provide a tight bound, namely: Using $k$ transformations, one can get an increase from rate $α$ to any rate $β< kα/(1+(k-1)α)$, and this is optimal.
A problem is a multivalued function from a set of \emph{instances} to a set of \emph{solutions}. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness property if every computable instance of the problem admits a solution relative to which the property holds. For example, cone avoidance is the ability, given a non-computable set $A$ and a computable instance of a problem $\mathsf{P}$, to find a solution relative to which $A$ is still non-computable. In this article, we compare relativized versions of computability-theoretic notions of preservation which have been studied in reverse mathematics, and prove that the ones which were not already separated by natural statements in the literature actually coincide. In particular, we prove that it is equivalent to admit avoidance of 1 cone, of $ω$ cones, of 1 hyperimmunity or of 1 non-$Σ^0_1$ definition. We also prove that the hierarchies of preservation of hyperimmunity and non-$Σ^0_1$ definitions coincide. On the other hand, none of these notions coincide in a non-relativized setting.
An abelian group $A$ is said to be cancellable if whenever $A \oplus G$ is isomorphic to $A \oplus H$, $G$ is isomorphic to $H$. We show that the index set of cancellable rank 1 torsion-free abelian groups is $Π^0_4$ $m$-complete, showing that the classification by Fuchs and Loonstra cannot be simplified. For arbitrary non-finitely generated groups, we show that the cancellation property is $Π^1_1$ $m$-hard; we know of no upper bound, but we conjecture that it is $Π^1_2$ $m$-complete.
A computable structure $\mathcal{A}$ has degree of categoricity $\mathbf{d}$ if $\mathbf{d}$ is exactly the degree of difficulty of computing isomorphisms between isomorphic computable copies of $\mathcal{A}$. Fokina, Kalimullin, and Miller showed that every degree d.c.e. in and above $\mathbf{0}^{(n)}$, for any $n < ω$, and also the degree $\mathbf{0}^{(ω)}$, are degrees of categoricity. Later, Csima, Franklin, and Shore showed that every degree $\mathbf{0}^{(α)}$ for any computable ordinal $α$, and every degree d.c.e. in and above $\mathbf{0}^{(α)}$ for any successor ordinal $α$, is a degree of categoricity. We show that every degree c.e. in and above $\mathbf{0}^{(α)}$, for $α$ a limit ordinal, is a degree of categoricity. We also show that every degree c.e. in and above $\mathbf{0}^{(ω)}$ is the degree of categoricity of a prime model, making progress towards a question of Bazhenov and Marchuk.
Let $\mathscr{L}$ be a recursive language. Let $S(\mathscr{L})$ be the set of $\mathscr{L}$-structures with domain $ω$. Let $Φ: {}^ω2 \rightarrow S(\mathscr{L})$ be a $Δ_1^1$ function with the property that for all $x,y \in {}^ω2$, $ω_1^x = ω_1^y$ if and only if $Φ(x) \approx_{\mathscr{L}} Φ(y)$. Then there is some $x \in {}^ω2$ so that $\mathrm{SR}(Φ(x)) = ω_1^x + 1$.
Universality has been an important concept in computable structure theory. A class $\mathcal{C}$ of structures is universal if, informally, for any structure, of any kind, there is a structure in $\mathcal{C}$ with the same computability-theoretic properties as the given structure. Many classes such as graphs, groups, and fields are known to be universal. This paper is about the class of finitely generated groups. Because finitely generated structures are relatively simple, the class of finitely generated groups has no hope of being universal. We show that finitely generated groups are as universal as possible, given that they are finitely generated: for every finitely generated structure, there is a finitely generated group which has the same computability-theoretic properties. The same is not true for finitely generated fields. We apply the results of this investigation to quasi Scott sentences.
We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and $p$-adically closed valued fields. We give an effectiveness condition, related to Hensel's lemma, on a valued field which is necessary and sufficient to extend the valuation to any algebraic extension. We show that there is a computable formally $p$-adic field which does not embed into any computable $p$-adic closure, but we give an effectiveness condition on the divisibility relation in the value group which is sufficient to find such an embedding. By checking that algebraically closed valued fields and $p$-adically closed valued fields of infinite transcendence degree have the Mal'cev property, we show that they have computable dimension $ω$.
A set $A$ is coarsely computable with density $r \in [0,1]$ if there is an algorithm for deciding membership in $A$ which always gives a (possibly incorrect) answer, and which gives a correct answer with density at least $r$. To any Turing degree $\mathbf{a}$ we can assign a value $Γ_T(\mathbf{a})$: the minimum, over all sets $A$ in $\mathbf{a}$, of the highest density at which $A$ is coarsely computable. The closer $Γ_T(\mathbf{a})$ is to $1$, the closer $\mathbf{a}$ is to being computable. Andrews, Cai, Diamondstone, Jockush, and Lempp noted that $Γ_T$ can take on the values $0$, $1/2$, and $1$, but not any values in strictly between $1/2$ and $1$. They asked whether the value of $Γ_T$ can be strictly between $0$ and $1/2$. This is the Gamma question. Replacing Turing degrees by many-one degrees, we get an analogous question, and the same arguments show that $Γ_m$ can take on the values $0$, $1/2$, and $1$, but not any values strictly between $1/2$ and $1$. We will show that for any $r \in [0,1/2]$, there is an $m$-degree $\mathbf{a}$ with $Γ_m(\mathbf{a}) = r$. Thus the range of $Γ_m$ is $[0,1/2] \cup \{1\}$. Benoit Monin has recently announced a solution to the Gamma question for Turing degrees. Interestingly, his solution gives the opposite answer: the only possible values of $Γ_T$ are $0$, $1/2$, and $1$.
Scott showed that for every countable structure $\mathcal{A}$, there is a sentence of the infinitary logic $\mathcal{L}_{ω_1ω}$, called a Scott sentence for $\mathcal{A}$, whose models are exactly the isomorphic copies of $\mathcal{A}$. Thus, the least quantifier complexity of a Scott sentence of a structure is an invariant that measures the complexity "describing" the structure. Knight et al.~have studied the Scott sentences of many structures. In particular, Knight and Saraph showed that a finitely generated structure always has a $Σ^0_3$ Scott sentence. We give a characterization of the finitely generated structures for whom the $Σ^0_3$ Scott sentence is optimal. One application of this result is to give a construction of a finitely generated group where the $Σ^0_3$ Scott sentence is optimal.