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.
If $\mathcal{N}$ is a proper Polish metric space and $\mathcal{M}$ is any countable dense submetric space of $\mathcal{N}$, then the Scott rank of $\mathcal{N}$ in the natural first order language of metric spaces is countable and in fact at most $ω_1^{\mathcal{M}} + 1$, where $ω_1^{\mathcal{M}}$ is the Church-Kleene ordinal of $\mathcal{M}$ (construed as a subset of $ω$) which is the least ordinal with no presentation on $ω$ computable from $\mathcal{M}$. If $\mathcal{N}$ is a rigid Polish metric space and $\mathcal{M}$ is any countable dense submetric space, then the Scott rank of $\mathcal{…
Given a countable scattered linear order $L$ of Hausdorff rank $α< ω_1$ we show that it has a $d\text{-}Σ_{2α+1}$ Scott sentence. Ash calculated the back and forth relations for all countable well-orders. From this result we obtain that this upper bound is tight, i.e., for every $α< ω_1$ there is a linear order whose optimal Scott sentence has this complexity. We further show that for all countable $α$ the class of Hausdorff rank $α$ linear orders is $\pmb Σ_{2α+2}$ complete.
The Scott rank of a countable structure is a measure, coming from the proof of Scott's isomorphism theorem, of the complexity of that structure. The Scott spectrum of a theory (by which we mean a sentence of $\mathcal{L}_{ω_1 ω}$) is the set of Scott ranks of countable models of that theory. In $ZFC + PD$ we give a descriptive-set-theoretic classification of the sets of ordinals which are the Scott spectrum of a theory: they are particular $\boldsymbolΣ^1_1$ classes of ordinals. Our investigation of Scott spectra leads to the resolution (in $ZFC$) of a number of open problems about Scott ranks…
We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $ω$ and that non-standard models of true arithmetic must have Scott rank greater than $ω$. Other than that there are no restrictions. By giving a reduction via $Δ^{\mathrm{in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $ω$-jump of models of an arbitrary completion $T$ of $\mathrm{PA}$ we show that every countable ordinal $α>ω$ is realized as the Scott rank of a model of $T$.
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 …
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 fin…
An analog of Nadel's effective bound for the continuous Scott rank of metric structures, developed by Ben Yaacov, Doucha, Nies, and Tsankov, will be established: Let $\mathscr{L}$ be a language of continuous logic with code $\hat{\mathscr{L}}$. Let $Ω$ be a weak modulus of uniform continuity with code $\hatΩ$. Let $\mathcal{D}$ be a countable $\mathscr{L}$-pre-structure. Let $\bar{\mathcal{D}}$ denote the completion structure of $\mathcal{D}$. Then $\mathrm{SR}_Ω(\bar{D}) \leq ω_1^{\hat{\mathscr{L}}\oplus\hatΩ\oplus\mathcal{D}}$, the Church-Kleene ordinal relative to $\hat{\mathscr{L}}\oplus\h…
We define and investigate HC-forcing invariant formulas of set theory, whose interpretations in the hereditarily countable sets are well behaved under forcing extensions. This leads naturally to a notion of cardinality ||Phi|| for sentences Phi of $L_{ω_1,ω}$, which counts the number of sentences of $L_{\infty,ω}$ that, in some forcing extension, become a canonical Scott sentence of a model of Phi. We show this cardinal bounds the complexity of (Mod(Phi), iso), the class of models of Phi with universe omega, by proving that (Mod(Phi),iso) is not Borel reducible to (Mod(Psi),iso) whenever ||Psi…
We give several new examples of computable structures of high Scott rank. For earlier known computable structures of Scott rank $ω_1^{CK}$, the computable infinitary theory is $\aleph_0$-categorical. Millar and Sacks asked whether this was always the case. We answer this question by constructing an example whose computable infinitary theory has non-isomorphic countable models. The standard known computable structures of Scott rank $ω_1^{CK}+1$ have infinite indiscernible sequences. We give two constructions with no indiscernible ordered triple.
We study two complexity notions of groups - a computable Scott sentence and the index set of a group. Finding the exact complexity of one of them usually involves finding the complexity of the other, but this is not the case sometimes. J. Knight et al. determined the complexity of index sets of various structures. In this paper, we focus on finding the complexity of computable Scott sentences and index sets of various groups, generalizing methods that was previously used by J. Knight et al. We found computable Scott sentences for various different groups or class of groups, including nilpotent…
A countable structure is said to be extendible if it has the same Scott sentence as some uncountable structure. Rigid structures are not extendible. We give an example of an extendible model with a rigid elementary extension.
We give effective versions of some results on Scott sentences. We show that if $\mathcal{A}$ has a computable $Π_α$ Scott sentence, then the orbits of all tuples are defined by formulas that are computable $Σ_β$ for some $β<α$. (This is an effective version of a result of Montalbán.) We show that if a countable structure $\mathcal{A}$ has a computable $Σ_α$ Scott sentence and one that is computable $Π_α$, then it has one that is computable $d$-$Σ_β$ for some $β< α$. (This is an effective version of a result of A. Miller.) We also give an effective version of a result of D. Miller. Using the no…
We study the complexity of automatic structures via well-established concepts from both logic and model theory, including ordinal heights (of well-founded relations), Scott ranks of structures, and Cantor-Bendixson ranks (of trees). We prove the following results: 1) The ordinal height of any automatic well- founded partial order is bounded by ω^ω; 2) The ordinal heights of automatic well-founded relations are unbounded below the first non-computable ordinal; 3) For any computable ordinal there is an automatic structure of Scott rank at least that ordinal. Moreover, there are automatic structu…
We develop general machinery to cast the class of potential canonical Scott sentences of an infinitary sentence $Φ$ as a class of structures in a related language. From this, we show that $Φ$ has a Borel complete expansion if and only if $S_\infty$ divides $Aut(M)$ for some countable model $M\models Φ$. Using this, we prove that for theories $T_h$ asserting that $\{E_n\}$ is a countable family of cross cutting equivalence relations with $h(n)$ classes, if $h(n)$ is uniformly bounded then $T_h$ is not Borel complete, providing a converse to Theorem~2.1 of \cite{LU}.
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…
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 complexity of uncountable structures; we establish basic properties of this reducibility, and study it in the context of {\em generic presentability}, the existence of a copy of the structure in every extension by a given forcing. We show that every forcing notion making $ω_2$ countable generically presents some countable structure with no copy in the ground …
We consider the conjugacy problem for the automorphism groups of a number of countable homogeneous structures. In each case we find the precise complexity of the conjugacy relation in the sense of Borel reducibility.
Given a countable o-minimal theory T, we characterize the Borel complexity of isomorphism for countable models of T up to two model-theoretic invariants. If T admits a nonsimple type, then it is shown to be Borel complete by embedding the isomorphism problem for linear orders into the isomorphism problem for models of T. This is done by constructing models with specific linear orders in the tail of the Archimedean ladder of a suitable nonsimple type. If the theory admits no nonsimple types, then we use Mayer's characterization of isomorphism for such theories to compute invariants for countabl…
We study the complexity of the classification problem for countable models of set theory (ZFC). We prove that the classification of arbitrary countable models of ZFC is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well-founded models of ZFC.
We give, for each countable ordinal $ξ\geq 1$, an example of a ${\bfΔ}^0_2$ countable union of Borel rectangles that cannot be decomposed into countably many ${\bfΠ}^0_ξ$ rectangles. In fact, we provide a graph of a partial injection with disjoint domain and range, which is a difference of two closed sets, and which has no ${\bfΔ}^0_ξ$-measurable countable coloring.
We consider the classification problem for several classes of countable structures which are "vertex-transitive", meaning that the automorphism group acts transitively on the elements. (This is sometimes called homogeneous.) We show that the classification of countable vertex-transitive digraphs and partial orders are Borel complete. We identify the complexity of the classification of countable vertex-transitive linear orders. Finally we show that the classification of vertex-transitive countable tournaments is properly above $E_0$ in complexity.
We decide the Borel complexity of the conjugacy problem for automorphism groups of countable homogeneous digraphs. Many of the homogeneous digraphs, as well as several other homogeneous structures, have already been addressed in previous articles. In this article we complete the program, and establish a dichotomy theorem that this complexity is either the minimum or the maximum among relations which are classifiable by countable structures. We also discuss the possibility of extending our results beyond graphs to more general classes of countable homogeneous structures.
Hjorth has introduced a Scott analysis for general Polish group actions, and has asked whether his notion of rank satisfies a boundedness principle similar to the one of Scott rank - namely, the orbit equivalence relation is Borel if and only if Hjorth ranks are bounded. We present the principles of Hjorth analysis and Hjorth rank, and answer Hjorth's question positively. From that we get a positive answer to a conjecture due to Hjorth - for every limit ordinal $α$ , the set of elements whose orbit is of complexity less than $α$ is a Borel set. We then show Nadel's theorem for Hjorth rank - th…
We show that if A is a linear order then Th(A) is either $\aleph_0$-categorical or Borel complete (in the sense of Friedman and Stanley). We generalize this; if A has countably many unary predicates attached, then Th(A) is $\aleph_0$-categorical, has finitely many countable models (at least three), is Borel equivalent to equality on the reals, is Borel equivalent to "countable sets of reals," or is Borel complete. Furthermore, each of these cases corresponds in a natural way to a count of models of all sizes, up to back-and-forth equivalence. All these cases are possible and we compute precise…
We prove that for a countable, commutative ring $R$, the class of countable $R$-modules either has only countably many isomorphism types, or else it is Borel complete. The machinery gives a succinct proof of the Borel completeness of TFAB, the class of torsion-free abelian groups. We also prove that for any countable ring $R$, both the class of left $R$-modules endowed with an endomorphism and the class of left $R$-modules with four named submodules are Borel complete.
In this note, we analyze the classification problem for compact metrizable $G$-ambits for a countable discrete group $G$ from the point of view of descriptive set theory. More precisely, we prove that the topological conjugacy relation on the standard Borel space of compact metrizable $G$-ambits is Borel for every countable discrete group $G$.
The complexity of bisimilarity on pointmass processes
ICC UBA & CONICET（UBA计算中心与阿根廷国家科学研究委员会）；Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación. Centro de Investigación y Estudios de Matemática (CIEM-FaMAF)（科尔多瓦国立大学数学、天文学、物理与计算学院。数学研究与评估中心）；Conicet（阿根廷国家科学研究委员会）
We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of the Hjorth's conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube hom…
We study the Borel complexity of sets of normal numbers in several numeration systems. Taking a dynamical point of view, we offer a unified treatment for continued fraction expansions and base $r$ expansions, and their various generalisations: generalised Lüroth series expansions and $β$-expansions. In fact, we consider subshifts over a countable alphabet generated by all possible expansions of numbers in $[0,1)$. Then normal numbers correspond to generic points of shift-invariant measures. It turns out that for these subshifts the set of generic points for a shift-invariant probability measur…
We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We construct a large family of II$_1$ factors whose Cartan subalgebras up to unitary conjugacy are not classifiable by countable structures, providing the first such examples. Additionally, we construct examples of II$_1$ factors whose Cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures. Finally, we show directly that the Cartan subalgebras of the hyperfinite II$_1$ factor up to unitary conjugacy are not classifiable by countable structures, and d…
