Related papers: Dividing Lines between Positive Theories
We present several known formalizations of theorems from computational complexity in bounded arithmetic and formalize the PCP theorem in the theory PV1 (no formalization of this theorem was known). This includes a formalization of the…
First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem, parameterized by solution size. On the other hand, FO cannot express the very simple algorithmic question of…
We present an adequacy theorem for a concurrent extension of probabilistic GCL. The underlying denotational semantics is based on the so-called mixed powerdomains, which combine non-determinism with probabilistic behaviour. The theorem…
We prove several results on the behavior of Kim-independence upon changing the base in NSOP$_{1}$ theories. As a consequence, we prove that Kim-independence satisfies transitivity and that this characterizes NSOP$_{1}$. Moreover, we…
We study descriptive complexity of counting complexity classes in the range from #P to #$\cdot$NP. A corollary of Fagin's characterization of NP by existential second-order logic is that #P can be logically described as the class of…
The compactness theorem for a logic states, roughly, that the satisfiability of a set of well-formed formulas can be determined from the satisfiability of its finite subsets, and vice versa. Usually, proofs of this theorem depend on the…
We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng. It turns out that these theorems…
We provide examples of a large class of one dimensional higher order field theories with kink solutions which asymptotically have a power-law tail either at one end or at both ends. We provide analytic solutions for the kinks in a few cases…
Lindstr\"om theorems characterize logics in terms of model-theoretic conditions such as Compactness and the L\"owenheim-Skolem property. Most existing characterizations of this kind concern extensions of first-order logic. But on the other…
A study of the set N_p of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p>0 was initiated by Shalev and continued by the present author. The main goal…
In this paper, we propose a new logic for expressing and reasoning about probabilistic hyperproperties. Hyperproperties characterize the relation between different independent executions of a system. Probabilistic hyperproperties express…
We prove that, in order to establish that a theory is NSOP$_{1}$, it suffices to show that no formula in a single free variable has SOP$_{1}$.
We prove that the expressive power of first-order logic with team semantics plus contradictory negation does not rise beyond that of first-order logic (with respect to sentences), and that the totality atoms of arity k +1 are not definable…
We consider the following property of a first order theory T with a distinguished unary predicate P: every model of the theory of P occurs as the P-part of some model of T. We call this property the Gaifman property. Gaifman conjectured…
In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of "generic stability" in arbitrary theories. Among other things, we show that the standard definition of generic…
In the paper, we investigate the following fundamental question. For a set $\mathcal{K}$ in $\mathbb{L}^0(\mathbb{P})$, when does there exist an equivalent probability measure $\mathbb{Q}$ such that $\mathcal{K}$ is uniformly integrable in…
Let $T$ be the theory of dense cyclically ordered sets with at least two elements. We determine the classifying space of $\mathsf{Mod}(T)$ to be homotopically equivalent to $\mathbb{CP}^\infty$. In particular,…
We study the logical content of several maximality principles related to the finite intersection principle ($F\IP$) in set theory. Classically, these are all equivalent to the axiom of choice, but in the context of reverse mathematics their…
We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that…
We develop a framework, in the style of Adler, for interpreting the notion of "witnessing" that has appeared (usually as a variant of Kim's Lemma) in different areas of neostability theory as a binary relation between abstract independence…