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In 1976 S. Eilenberg and M.-P. Schützenberger posed the following diabolical question: if $\mathbf{A}$ is a finite algebraic structure, $Σ$ is the set of all identities true in $\mathbf{A}$, and there exists a finite subset $F$ of $Σ$ such that $F$ and $Σ$ have exactly the same finite models, must there also exist a finite subset $F'$ of $Σ$ such that $F'$ and $Σ$ have exactly the same finite and infinite models? (That is, must the identities of $\mathbf{A}$ be "finitely based"?) It is known that any counter-example to their question (if one exists) must fail to be finitely based in a particularly strange way. In this paper we show that the "inherently nonfinitely based" algebras constructed by Lawrence and Willard from group actions do not fail to be finitely based in this particularly strange way, and so do not provide a counter-example to the question of Eilenberg and Schützenberger. As a corollary, we give the first known examples of inherently nonfinitely based "automatic algebras" constructed from group actions., 15 pages

We show that if a subgroup of the automorphism group of the Fraisse limit of finite Heyting algebras has a countable index, then it lies between the pointwise and setwise stabilizer of some finite set., Revised version with more prose; Comments welcome; 8 pages

We show that the first-order logical theory of the binary overlap-free words (and, more generally, the ${\alpha}$-free words for rational ${\alpha}$, $2 < {\alpha} \leq 7/3$), is decidable. As a consequence, many results previously obtained about this class through tedious case- based proofs can now be proved "automatically", using a decision procedure.

We explain and explore class-theoretic potentialism -- the view that one can always individuate more classes over a set-theoretic universe. We examine some motivations for class-theoretic potentialism, before proving some results concerning the relevant potentialist systems (in particular exhibiting failures of the .2 and .3 axioms). We then discuss the significance of these results for the different kinds of class-theoretic potentialist., (Updated after review.)

We answer several questions about the computable Friedman-Stanley jump on equivalence relations. This jump, introduced by Clemens, Coskey, and Krakoff, deepens the natural connection between the study of computable reduction and its Borel analog studied deeply in descriptive set theory., Comment: 24 pages

CAC for trees is the statement asserting that any infinite subtree of $\mathbb{N}^{, 28 pages

We introduce proper display calculi for intuitionistic, bi-intuitionistic and classical linear logics with exponentials, which are sound, complete, conservative, and enjoy cut elimination and subformula property. Based on the same design, we introduce a variant of Lambek calculus with exponentials, aimed at capturing the controlled application of exchange and associativity. Properness (i.e., closure under uniform substitution of all parametric parts in rules) is the main technical novelty of the present proposal, allowing both for the smoothest proof of cut elimination and for the development of an overarching and modular treatment for a vast class of axiomatic extensions and expansions of intuitionistic, bi-intuitionistic, and classical linear logics with exponentials. Our proposal builds on an algebraic and order-theoretic analysis of linear logic and applies the guidelines of the multi-type methodology in the design of display calculi.

Let us denote by LF the class of all orthomodular lattices (OMLs) that are locally finite (i.e., L in LF provided each finite subset of L generates in L a finite subOML). We first show in this note how one can obtain new locally finite OMLs from the initial ones and enlarge thus the class LF . We find LF considerably large though, obviously, not all OMLs belong to LF . We then study states on the OMLs of LF . We show that local finiteness may to a certain extent make up for distributivity. We for instance show that if L in LF and if for any finite subOML K there is a state s : K to [0, 1] on K, then there is a state on the entire L. We also consider further algebraic and state properties of LF relevant to quantum logic theory., 5 pages, 0 figures

Symmetric and k-cyclic structure of modal pseudocomplemented De Morgan algebras algebras was introduced previously. In this paper, we first present the construction of epimorphims between finite symmetric (or 2-cyclic) modal pseudocomplemented De Morgan algebras. Furthermore, we compute the cardinality of the set of all epimorphism between finite structures. Secondly, we present the construction of finite free algebras on the variety of k-cyclic modal pseudocomplemented De Morgan algebras and display how our computations are in fact generalizations to others in the literature. Our work is strongly based on the properties of epimorphisms and automorphisms and the fact that the variety is finitely generated., arXiv admin note: text overlap with arXiv:2108.01566

There exist two notions of equivalence of behavior between states of a Labelled Markov Process (LMP): state bisimilarity and event bisimilarity. The first one can be considered as an appropriate generalization to continuous spaces of Larsen and Skou's probabilistic bisimilarity, while the second one is characterized by a natural logic. C. Zhou expressed state bisimilarity as the greatest fixed point of an operator $\mathcal{O}$, and thus introduced an ordinal measure of the discrepancy between it and event bisimilarity. We call this ordinal the "Zhou ordinal" of $\mathbb{S}$, $\mathfrak{Z}(\mathbb{S})$. When $\mathfrak{Z}(\mathbb{S})=0$, $\mathbb{S}$ satisfies the Hennessy-Milner property. The second author proved the existence of an LMP $\mathbb{S}$ with $\mathfrak{Z}(\mathbb{S}) \geq 1$ and Zhou showed that there are LMPs having an infinite Zhou ordinal. In this paper we show that there are LMPs $\mathbb{S}$ over separable metrizable spaces having arbitrary large countable $\mathfrak{Z}(\mathbb{S})$ and that it is consistent with the axioms of $\mathit{ZFC}$ that there is such a process with an uncountable Zhou ordinal., v1: 19 pages. v2: role of the logic on Introduction, relation with previous constructions and 1 figure. Many minor corrections. v3: 20 pages. First item in former Lemma 30 was incorrect, but all the main results are still correct. Accepted at Review of Symbolic Logic. We are very grateful for the referee's useful suggestions and detailed reading

We consider various statements that characterize the hyperfinite II$_1$ factors amongst embeddable II$_1$ factors in the non-embeddable situation. In particular, we show that "generically" a II$_1$ factor has the Jung property (which states that every embedding of itself into its ultrapower is unitarily conjugate to the diagonal embedding) if and only if it is self-tracially stable (which says that every such embedding has an approximate lifting). We prove that the enforceable factor, should it exist, has these equivalent properties. Our techniques are model-theoretic in nature. We also show how these techniques can be used to give new proofs that the hyperfinite II$_1$ factor has the aforementioned properties., Comment: 8 pages

Given a continuous and isometric action of a Polish group G on an adequate Polish topometric space $(X,\tau ,\rho )$ and $x \in X$ , we find a necessary and sufficient condition for $\overline {Gx}^{\rho }$ to be co-meagre; we also obtain a criterion that characterizes when such a point exists. This work completes a criterion established in earlier work of the authors.

We study the algorithmic content of Pontryagin - van Kampen duality. We prove that the dualization is computable in the important cases of compact and locally compact totally disconnected Polish abelian groups. The applications of our main results include solutions to questions of Kihara and Ng about presentations of connected Polish spaces, and an unexpected arithmetical characterisation of direct products of solenoid groups among all Polish groups.

In this paper, we characterize the possible cofinalities of the least $��$-strongly compact cardinal. We show that, on the one hand, for any regular cardinal, $��$, that carries a $��$-complete uniform ultrafilter, it is consistent, relative to the existence of a supercompact cardinal above $��$, that the least $��$-strongly compact cardinal has cofinality $��$. On the other hand, provably the cofinality of the least $��$-strongly compact cardinal always carries a $��$-complete uniform ultrafilter., 15 pages

We investigate iterating the construction of $C^{*}$ , the L-like inner model constructed using first order logic augmented with the “cofinality $\omega $ ” quantifier. We first show that $\left (C^{*}\right )^{C^{*}}=C^{*}\ne L$ is equiconsistent with $\mathrm {ZFC}$ , as well as having finite strictly decreasing sequences of iterated $C^{*}$ s. We then show that in models of the form $L[U]$ we get infinite decreasing sequences of length $\omega $ , and that an inner model with a measurable cardinal is required for that.

Given a completely metrizable space $X$, let $\mathfrak{par}(X)$ denote the smallest possible size of a partition of $X$ into Polish spaces, and $\mathfrak{cov}(X)$ the smallest possible size of a covering of $X$ with Polish spaces. Observe that $\mathfrak{cov}(X) \leq \mathfrak{par}(X)$ for every $X$, because every partition of $X$ is also a covering. We prove it is consistent relative to a huge cardinal that the strict inequality $\mathfrak{cov}(X) < \mathfrak{par}(X)$ can hold for some completely metrizable space $X$. We also prove that using large cardinals is necessary for obtaining this strict inequality, because if $\mathfrak{cov}(X) < \mathfrak{par}(X)$ for any completely metrizable $X$, then $0^\dagger$ exists.

We study various combinatorial properties, and the implications between them, for filters generated by infinite-dimensional subspaces of a countable vector space. These properties are analogous to selectivity for ultrafilters on the natural numbers and stability for ordered-union ultrafilters on $\mathrm{FIN}$., 17 pages, 2 figures. Submitted

We define a class of groups equipped with an invariant probability measure, which includes all compact groups and is closed under taking ultraproducts with the induced Loeb measure; in fact, this class also contains the ultraproducts all locally compact unimodular amenable groups. We call the members of this class probability groups and develop the basics of the theory of their measure-preserving actions on probability spaces, including a natural notion of mixing. A short proof reveals that for probability groups mixing implies double recurrence, which generalizes a theorem of Bergelson and Tao proved for ultraproducts of finite groups. Moreover, a quantitative version of our proof gives that $\varepsilon$-approximate mixing implies $3\sqrt{\varepsilon}$-approximate double recurrence. Examples of approximately mixing probability groups are quasirandom groups introduced by Gowers, so the last theorem generalizes and sharpens the corresponding results for quasirandom groups of Bergelson and Tao, as well as of Austin. Lastly, we point out that the fact that the ultraproduct of locally compact unimodular amenable groups is a probability group provides a general alternative to Furstenberg correspondence principle., Added ultraproducts of amenable unimodular locally compact groups as examples of probability groups

Partial combinatory algebras are algebraic structures that serve as generalized models of computation. In this paper, we study embeddings of pcas. In particular, we systematize the embeddings between relativizations of Kleene's models, of van Oosten's sequential computation model, and of Scott's graph model, showing that an embedding between two relativized models exists if and only if there exists a particular reduction between the oracles. We obtain a similar result for the lambda calculus, showing in particular that it cannot be embedded in Kleene's first model., Comment: 31 pages

Our main object of interest is the following notion: we say that a topological space space $X$ is in FinBW($\mathcal{I}$), where $\mathcal{I}$ is an ideal on $\omega$, if for each sequence $(x_n)_{n\in\omega}$ in $X$ one can find an $A\notin\mathcal{I}$ such that $(x_n)_{n\in A}$ converges in $X$. We define an ideal $\mathcal{BI}$ which is critical for FinBW($\mathcal{I}$) in the following sense: Under CH, for every ideal $\mathcal{I}$, $\mathcal{BI}\not\leq_K\mathcal{I}$ ($\leq_K$ denotes the Kat\v{e}tov preorder of ideals) iff there is an uncountable separable space in FinBW($\mathcal{I}$). We show that $\mathcal{BI}\not\leq_K\mathcal{I}$ and $\omega_1$ with the order topology is in FinBW($\mathcal{I}$), for all $\bf{\Pi^0_4}$ ideals $\mathcal{I}$. We examine when FinBW($\mathcal{I}$)$\setminus$FinBW($\mathcal{J}$) is nonempty: we prove under MA($\sigma$-centered) that for $\bf{\Pi^0_4}$ ideals $\mathcal{I}$ and $\mathcal{J}$ this is equivalent to $\mathcal{J}\not\leq_K\mathcal{I}$. Moreover, answering in negative a question of M. Hru\v{s}\'ak and D. Meza-Alc\'antara, we show that the ideal $\text{Fin}\times\text{Fin}$ is not critical among Borel ideals for extendability to a $\bf{\Pi^0_3}$ ideal. Finally, we apply our results in studies of Hindman spaces and in the context of analytic P-ideals.

Let $T$ be a polynomially bounded o-minimal theory extending the theory of real closed ordered fields. Let $K$ be a model of $T$ equipped with a $T$-convex valuation ring and a $T$-derivation. If this derivation is continuous with respect to the valuation topology, then we call $K$ a $T$-convex $T$-differential field. We show that every $T$-convex $T$-differential field has an immediate strict $T$-convex $T$-differential field extension which is spherically complete. In some important cases, the assumption of polynomial boundedness can be relaxed to power boundedness., Comment: 26 pages

In the Zermelo--Fraenkel set theory with the Axiom of Choice a forcing notion is "$\kappa$-distributive" if and only if it is "$\kappa$-sequential". We show that without the Axiom of Choice this equivalence fails, even if we include a weak form of the Axiom of Choice, the Principle of Dependent Choice for $\kappa$. Still, the equivalence may still hold along with very strong failures of the Axiom of Choice, assuming the consistency of large cardinal axioms. We also prove that while a $\kappa$-distributive forcing notion may violate Dependent Choice, it must preserve the Axiom of Choice for families of size $\kappa$. On the other hand, a $\kappa$-sequential can violate the Axiom of Choice for countable families. We also provide a condition of "quasiproperness" which is sufficient for the preservation of Dependent Choice, and is also necessary if the forcing notion is sequential., Comment: 12 pages; accepted version

We analyze the axiomatic strength of the following theorem due to Rival and Sands [28] in the style of reverse mathematics. Every infinite partial order P of finite width contains an infinite chain C such that every element of P is either comparable with no element of C or with infinitely many elements of C. Our main results are the following. The Rival–Sands theorem for infinite partial orders of arbitrary finite width is equivalent to $\mathsf {I}\Sigma ^0_{2} + \mathsf {ADS}$ over $\mathsf {RCA}_0$ . For each fixed $k \geq 3$ , the Rival–Sands theorem for infinite partial orders of width $\leq \!k$ is equivalent to $\mathsf {ADS}$ over $\mathsf {RCA}_0$ . The Rival–Sands theorem for infinite partial orders that are decomposable into the union of two chains is equivalent to $\mathsf {SADS}$ over $\mathsf {RCA}_0$ . Here $\mathsf {RCA}_0$ denotes the recursive comprehension axiomatic system, $\mathsf {I}\Sigma ^0_{2}$ denotes the $\Sigma ^0_2$ induction scheme, $\mathsf {ADS}$ denotes the ascending/descending sequence principle, and $\mathsf {SADS}$ denotes the stable ascending/descending sequence principle. To the best of our knowledge, these versions of the Rival–Sands theorem for partial orders are the first examples of theorems from the general mathematics literature whose strength is exactly characterized by $\mathsf {I}\Sigma ^0_{2} + \mathsf {ADS}$ , by $\mathsf {ADS}$ , and by $\mathsf {SADS}$ . Furthermore, we give a new purely combinatorial result by extending the Rival–Sands theorem to infinite partial orders that do not have infinite antichains, and we show that this extension is equivalent to arithmetical comprehension over $\mathsf {RCA}_0$ .

In this paper we will show that for every cut I of any countable nonstandard model $\mathcal {M}$ of $\mathrm {I}\Sigma _{1}$ , each I-small $\Sigma _{1}$ -elementary submodel of $\mathcal {M}$ is of the form of the set of fixed points of some proper initial self-embedding of $\mathcal {M}$ iff I is a strong cut of $\mathcal {M}$ . Especially, this feature will provide us with some equivalent conditions with the strongness of the standard cut in a given countable model $\mathcal {M}$ of $ \mathrm {I}\Sigma _{1} $ . In addition, we will find some criteria for extendability of initial self-embeddings of countable nonstandard models of $ \mathrm {I}\Sigma _{1} $ to larger models.

We force the existence of a non-trivial κ \kappa -complete ultrafilter over κ \kappa which fails to satisfy the Galvin property. This answers a question asked by Benhamou and Gitik [Ann. Pure Appl. Logic 173 (2022), Paper No. 103107].

One of the main open problems regarding decidability of the existential theory of rings is the analogue of Hilbert's Tenth Problem (HTP) for the ring of entire holomorphic functions in one variable. In the direction of a negative solution, we prove unsolvability of HTP for rings of exponential polynomials. This provides the first known case of HTP for a ring of entire holomorphic functions in one variable strictly containing the polynomials. The technique of proof consists of an interaction between Arithmetic, Analysis, Logic, and Functional Transcendence.

In this paper, we focus on the variety DHMSH of dually hemimorphic semi-Heyting algebras from a logical point of view. Firstly, we present a Hilbert-style axiomatization of a new logic called Dually hemimorphic semi-Heyting logic (DHMSH, for short), as an expansion of semi-intuitionistic logic SI (also called SH) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety DHMSH. It is deduced that the logic DHMSH is algebraizable in the sense of Blok and Pigozzi, with the variety DHMSH as its equivalent algebraic semantics and that the lattice of axiomatic extensions of DHMSH is dually isomorphic to the lattice of subvarieties of DHMSH. A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of DHMSH in which the Deduction Theorem holds. Thirdly, we present several new logics, extending the logic DHMSH, corresponding to several important subvarieties of the variety DHMSH. These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Lukasiewiczlogic. Surprisingly, many of these logics turn out to be connexive logics, a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan-Goedel logics and dually pseudocomplemented Goedel logics, Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems., Comment: 53 pages, 5 figures

Let $R$ be a ring equipped with a proper norm. We show that under suitable conditions on $R$, there is a natural basis under continuous linear injection for the set of Polish $R$-modules which are not countably generated. When $R$ is a division ring, this basis can be taken to be a singleton., Comment: 13 pages; final version for journal

In this paper, we determine the descriptive complexity of subsets of the Polish space of marked groups defined by various group theoretic properties. In particular, using Grigorchuk groups, we establish that the sets of solvable groups, groups of exponential growth and groups with decidable word problem are $\mathbf{\Sigma}^0_2$-complete and that the sets of periodic groups and groups of intermediate growth are $\mathbf{\Pi}^0_2$-complete. We also provide bounds for the descriptive complexity of simplicity, amenability, residually finiteness, Hopficity and co-Hopficity. This paper is intended to serve as a compilation of results on this theme., Comment: Some minor mistakes in the earlier version are fixed. Also some improvements and expansions are made

We present some general results implying nonfinite axiomatisability of many additively idempotent semirings with finitely based semigroup reducts. The smallest is a $3$-element commutative example, which we show also has \texttt{NP}-hard membership for its variety. As well as being the only nonfinite axiomatisable ai-semiring on $3$-elements, we are able to show that its nonfinite basis property infects many related semirings, including the natural ai-semiring structure on the semigroup $B_2^1$. We also extend previous group-theory based examples significantly, by showing that any finite additively idempotent semiring with a nonabelian nilpotent subgroup is not finitely axiomatisable for its identities.

In this paper, we introduce a logic based on team semantics, called $\mathbf {FOT} $ , whose expressive power is elementary, i.e., coincides with first-order logic both on the level of sentences and (possibly open) formulas, and we also show that a sublogic of $\mathbf {FOT} $ , called $\mathbf {FOT}^{\downarrow } $ , captures exactly downward closed elementary (or first-order) team properties. We axiomatize completely the logic $\mathbf {FOT} $ , and also extend the known partial axiomatization of dependence logic to dependence logic enriched with the logical constants in $\mathbf {FOT}^{\downarrow } $ .

Let $\Gamma$ be a countable group and let $G$ be the Schreier graph of the free part of the Bernoulli shift of $\Gamma$ (with respect to some finite subset $F \subseteq \Gamma$). We show that the Borel fractional chromatic number of $G$ is equal to $1$ over the measurable independence number of $G$. As a consequence, we asymptotically determine the Borel fractional chromatic number of $G$ when $\Gamma$ is the free group, answering a question of Meehan., Comment: 8 pp

An old theorem of Ad\'amek constructs initial algebras for sufficiently cocontinuous endofunctors via transfinite iteration over ordinals in classical set theory. We prove a new version that works in constructive logic, using "inflationary" iteration over a notion of size that abstracts from limit ordinals just their transitive, directed and well-founded properties. Borrowing from Taylor's constructive treatment of ordinals, we show that sizes exist with upper bounds for any given signature of indexes. From this it follows that there is a rich class of endofunctors to which the new theorem applies, provided one admits a weak form of choice (WISC) due to Streicher, Moerdijk, van den Berg and Palmgren, and which is known to hold in the internal constructive logic of many kinds of topos., Comment: In Proceedings ACT 2021, arXiv:2211.01102

Motivated by the Poisson Dixmier-Moeglin equivalence problem, a systematic study of commutative unitary rings equipped with a {\em biderivation}, namely a binary operation that is a derivation in each argument, is here begun, with an eye toward the geometry of the corresponding {\em $B$-varieties}. Foundational results about extending biderivations to localisations, algebraic extensions and transcendental extensions are established. Resolving a deficiency in Poisson algebraic geometry, a theory of base extension is achieved, and it is shown that dominant $B$-morphisms admit generic $B$-fibres. A bidifferential version of the Dixmier-Moeglin equivalence problem is articulated.

We explore the dual version of Gottschalk's conjecture recently introduced by Capobianco, Kari, and Taati, and the notion of dual surjunctivity in general. We show that dual surjunctive groups satisfy Kaplansky's direct finiteness conjecture for all fields of positive characteristic. By quantifying the notions of injectivity and post-surjectivity for cellular automata, we show that the image of the full topological shift under an injective cellular automaton is a subshift of finite type in a quantitative way. Moreover we show that dual surjunctive groups are closed under ultraproducts, under elementary equivalence, and under certain semidirect products (using the ideas of Arzhantseva and Gal for the latter); they form a closed subset in the space of marked groups, fully residually dual surjunctive groups are dual surjunctive, etc. We also consider dual surjunctive systems for more general dynamical systems, namely for certain expansive algebraic actions, employing results of Chung and Li., v2:Some improvements in Section 5 and in the proof of Theorem 2.3

We give a model-theoretic treatment of the fundamental results of Kechris-Pestov-Todorčević theory in the more general context of automorphism groups of not necessarily countable structures. One of the main points is a description of the universal ambit as a certain space of types in an expanded language. Using this, we recover results of Kechris et al. (Funct Anal 15:106–189, 2005), Moore (Fund Math 220:263–280, 2013), Ngyuen Van Thé (Fund Math 222: 19–47, 2013), in the context of automorphism groups of not necessarily countable structures, as well as Zucker (Trans Am Math Soc 368, 6715–6740, 2016).

We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is proof-theoretical naturalness: the system consists of all and only the inference rules generated by the single, simple, linear scheme of the recently introduced subatomic logic. Thanks to this regularity, cuts are eliminated via a natural construction. The second reason is that the system generates efficient proofs. Indeed, we show that a certain class of tautologies due to Statman, which cannot have better than exponential cut-free proofs in the sequent calculus, have polynomial cut-free proofs in our system. We achieve this by using the same construction that we use for cut elimination. In summary, by expanding the language of propositional logic, we make its proof theory more regular and generate more proofs, some of which are very efficient. That design is made possible by considering atoms as superpositions of their truth values, which are connected by self-dual, non-commutative connectives. A proof can then be projected via each atom into two proofs, one for each truth value, without a need for cuts. Those projections are semantically natural and are at the heart of the constructions in this paper. To accommodate self-dual non-commutativity, we compose proofs in deep inference., Comment: To appear on ACM Transactions on Computational Logic

The paper introduces a broad family of metrics applicable to finite and countably infinite strings, or, by extension, to formal structures serving as semantics for countable languages. The main focus is on applications to sets of pointed Kripke models, a semantics for modal logics. For the resulting metric spaces, the paper classifies topological properties including which metrics are topologically equivalent, providing sufficient conditions for compactness, characterizing clopen sets and isolated points, and characterizing the metrical topologies by a concept of logical convergence. We then apply the approach to maps from dynamic epistemic logic, showing that product updates with action models yield continuous maps, hence allowing for an interpretation of the iterated updates as discrete time dynamical systems.

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 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 homeomorphisms is Borel bireducible to the universal orbit equivalence relation.

A recent strand of research in structural proof theory aims at exploring the notion of analytic calculi (i.e. those calculi that support general and modular proof-strategies for cut elimination), and at identifying classes of logics that can be captured in terms of these calculi. In this context, Wansing introduced the notion of proper display calculi as one possible design framework for proof calculi in which the analiticity desiderata are realized in a particularly transparent way. Recently, the theory of properly displayable logics (i.e. those logics that can be equivalently presented with some proper display calculus) has been developed in connection with generalized Sahlqvist theory (aka unified correspondence). Specifically, properly displayable logics have been syntactically characterized as those axiomatized by analytic inductive axioms, which can be equivalently and algorithmically transformed into analytic structural rules so that the resulting proper display calculi enjoy a set of basic properties: soundness, completeness, conservativity, cut elimination and subformula property. In this context, the proof that the given calculus is complete w.r.t. the original logic is usually carried out syntactically, i.e. by showing that a (cut free) derivation exists of each given axiom of the logic in the basic system to which the analytic structural rules algorithmically generated from the given axiom have been added. However, so far this proof strategy for syntactic completeness has been implemented on a case-by-case base, and not in general. In this paper, we address this gap by proving syntactic completeness for properly displayable logics in any normal (distributive) lattice expansion signature. Specifically, we show that for every analytic inductive axiom a cut free derivation can be effectively generated which has a specific shape, referred to as pre-normal form., arXiv admin note: text overlap with arXiv:1604.08822 by other authors

Let At denote the set of infinite sequences of effective dimension t. We determine both how close and how far an infinite sequence of dimension s can be from one of dimension t, measured using the Besicovitch pseudometric. We also identify classes of sequences for which these infima and suprema are realized as minima and maxima. When t < s, we find d(X,At) is minimized when X is a Bernoulli p-random, where H(p)=s, and maximized when X belongs to a class of infinite sequences that we call s-codewords. When s < t, the situation is reversed., 28 pages, 3 figures

We show that the set of prime numbers has exponential alternating complexity, proving a conjecture by Fijalkow. We further show that the set of squarefree integers has essentially maximal possible alternating complexity.

In [11] Sklinos proved that any uncountable free group is not $\aleph_1$-homogenenous. This was later generalized by Belegradek in [1] to torsion-free residually finite relatively free groups, leaving open whether the assumption of residual finiteness was necessary. In this paper we use methods arising from the classical analysis of relatively free groups in infinitary logic to answer Belegradek's question in the negative. Our methods are general and they also applications in varieties with torsion, for example we show that if $V$ contains a non-solvable group, then any uncountable $V$-free group is not $\aleph_1$-homogenenous.

We identify natural conditions for a countable group acting on a countable tree which imply that the orbit equivalence relation of the induced action on the Gromov boundary is Borel hyperfinite. Examples of this condition include acylindrical actions. We also identify a natural weakening of the aforementioned conditions that implies measure hyperfinitenss of the boundary action. We then document examples of group actions on trees whose boundary action is not hyperfinite.

We define LE-ALC, a generalization of the description logic ALC based on the propositional logic of general (i.e.~not necessarily distributive) lattices, and semantically interpreted on relational structures based on formal contexts from Formal Concept Analysis (FCA). The description logic LE-ALC allows us to formally describe databases with objects, features, and formal concepts, represented according to FCA as Galois-stable sets of objects and features. We describe ABoxes and TBoxes in LE-ALC, provide a tableaux algorithm for checking the consistency of LE-ALC knowledge bases with acyclic TBoxes, and show its termination, soundness and completeness. Interestingly, consistency checking for LE-ALC with acyclic TBoxes is in PTIME, while the complexity of the consistency checking of classical ALC with acyclic TBoxes is PSPACE-complete.

Based on earlier work of the third author, we construct a Chang-type model with supercompact measures extending a derived model of a given hod mouse with a regular cardinal $\delta$ that is both a limit of Woodin cardinals and a limit of ${\Theta$. This complements Woodin's generalized Chang model, which satisfies $\mathsf{AD}^+ + \mathsf{AD}_{\mathbb{R}}+\omega_1$ is supercompact, assuming a proper class of Woodin cardinals that are limits of Woodin cardinals., Comment: 22 pages

The absolute Galois group Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$ of the field $\mathbb{Q}$ of rational numbers can be presented as a highly computable object, under the notion of type-2 Turing computation. We formalize such a presentation and use it to address several effectiveness questions about Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$: the difficulty of computing Skolem functions for this group, the arithmetical complexity of various definable subsets of the group, and the extent to which countable subgroups defined by complexity (such as the group of all computable automorphisms of the algebraic closure $\overline{\mathbb{Q}}$) may be elementary subgroups of the overall group.

This paper studies nested sequents for quantified modal logics. In particular, it considers extensions of the propositional modal logics definable by the axioms D, T, B, 4, and 5 with varying, increasing, decreasing, and constant domains. Each calculus is proved to have good structural properties: weakening and contraction are height-preserving admissible and cut is (syntactically) admissible. Each calculus is shown to be equivalent to the corresponding axiomatic system and, thus, to be sound and complete. Finally, it is argued that the calculi are internal -- i.e., each sequent has a formula interpretation -- whenever the existence predicate is expressible in the language., accepted to TABLEAUX 2023

This paper focuses on the characterization of $\aleph_0$-categorical theories of trees in the following sense: for any $\aleph_0$-cateorical theory $T$ of trees there is a tree plan $\Gamma$ such that $T=Th(\Gamma(\omega))$ where $\Gamma(\omega)$ is the generic model of a Fra\"iss\'e class $\mathbf{K}(\Gamma)$ obtained from the tree plan $\Gamma$. Also, it is shown that $\mathbf{K}(\Gamma)$ forms an asymptotic class, and its model-theoretic properties have been studied. Moreover, it is demonstrated that asymptotic classes of finite trees yield $\aleph_0$-categorical ultraproducts, and a characterization of supersimple, finite rank trees, using the notion of tree plan, is provided.