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In arXiv:2208.12944 it is shown that an ordinal $\sup_{N<\omega}\psi_{\Omega_{1}}(\varepsilon_{\Omega_{\mathbb{S}+N}+1})$ is an upper bound for the proof-theoretic ordinal of a set theory ${\sf KP}\ell^{r}+(M\prec_{\Sigma_{1}}V)$. In this…
We look at equivalence relations on the set of models of a theory -- MERs, for short -- such that the class of equivalent pairs is itself an elementary class, in a language appropriate for pairs of models. We provide many examples of…
Let $m,n\ge 2$ be integers. Denote by $M_n$ the set of $n\times n$ complex matrices. Let $\|\cdot\|_{(p,k)}$ be the $(p,k)$ norm on $M_{mn}$ with $1\leq k\leq mn$ and $2<p<\infty$. We show that a linear map $\phi:M_{mn}\rightarrow M_{mn}$…
We propose a generalization of first-order logic originating in a neglected work by C.C. Chang: a natural and generic correspondence language for any types of structures which can be recast as Set-coalgebras. We discuss axiomatization and…
We present three syntactic forcing models for coherent logic. These are based on sites whose underlying category only depends on the signature of the coherent theory, and they do not presuppose that the logic has equality. As an application…
In this work, we reveal a rich combinatorial structure underlying exact minimax optimal algorithms for classical nonexpansive fixed-point problems. This viewpoint unifies all extremal optimal methods and provides a systematic and practical…
Automatic structures are first-order structures whose universe and relations can be represented as regular languages. It follows from the standard closure properties of regular languages that the first-order theory of an automatic structure…
Counterfactual explanations (CEs) offer a human-understandable way to explain decisions by identifying specific changes to the input parameters of a base or present model that would lead to a desired change in the outcome. For optimization…
Tractability results for the model checking problem of logics yield powerful algorithmic meta theorems of the form: Every computational problem expressible in a logic $L$ can be solved efficiently on every class $\mathscr{C}$ of structures…
It is well-known that the first order Peano axioms PA have a continuum of non-isomorphic countable models. The question, how close to being isomorphic such countable models can be, seems to be less investigated. A measure of closeness to…
We study succinctness as a measure of the expressive power of transformers. Succinctness -- how compactly a formalism can describe a language relative to other formalisms -- is a classical notion in logic and automata theory. We prove that…
We prove that if the linear-time and polynomial-time hierarchies coincide, then every model of $\Pi_1(\mathbb{N}) + \neg \Omega_1$ has a proper end-extension to a model of $\Pi_1(\mathbb{N})$, and so $\Pi_1(\mathbb{N}) + \neg \Omega_1…
Recent work on word ordering has argued that syntactic structure is important, or even required, for effectively recovering the order of a sentence. We find that, in fact, an n-gram language model with a simple heuristic gives strong…
An A-infinity bialgebra of type (m,n) is a Hopf algebra H equipped with a "compatible" operation \omega : H^{\otimes m} \to H^{\otimes n} of positive degree. We determine the structure relations for A-infinity bialgebras of type (m,n) and…
By reformulating a learning process of a set system L as a game between Teacher (presenter of data) and Learner (updater of the abstract independent set), we define the order type dim L of L to be the order type of the game tree. The theory…
Marker and Steinhorn shown that given two models $M\prec N$ of an o-minimal theory, if all 1-types over $M$ realized in $N$ are definable, then all types over $M$ realized in $N$ are definable. In this article we characterize pairs of…
We show that for every countable recursively saturated model $M$ of Peano Arithmetic and every subset $A \subseteq M$, there exists a full satisfaction class $S_A \subset M^2$ such that $A$ is definable in $(M,S_A)$ without parametres. It…
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by…
By operations on models we show how to relate completeness with respect to permissive-nominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissive-nominal…
The celebrated Trakhtenbrot's theorem states that the set of finitely valid sentences of first-order logic is not computably enumerable. In this note we will extend this theorem by proving that the finite satisfiability problem of any…