Related papers: Acyclic Comprehension is equal to Stratified Compr…

When quantitative models are used to support decision-making on complex and important topics, understanding a model's ``reasoning'' can increase trust in its predictions, expose hidden biases, or reduce vulnerability to adversarial attacks.…

This paper will develop a single framework for unifying, simplifying and extending our prior results about axiom systems that retain a partial knowledge of their own consistency, via an axiomatic declaration of self-consistency. Its perhaps…

In 1993 Stanley showed that if a simplicial complex is acyclic over some field, then its face poset can be decomposed into disjoint rank $1$ boolean intervals whose minimal faces together form a subcomplex. Stanley further conjectured that…

We call $n$ a cyclic number if every group of order $n$ is cyclic. It is implicit in work of Dickson, and explicit in work of Szele, that $n$ is cyclic precisely when $\gcd(n,\phi(n))=1$. With $C(x)$ denoting the count of cyclic $n\le x$,…

There is no single definition of complexity (Edmonds 1999; Gershenson 2008; Mitchell 2009; De Domenico, et al., 2019), as it acquires different meanings in different contexts. A general notion is the amount of information required to…

Even though every mathematician knows intuitively what it means to "simplify" a mathematical expression, there is still no universally accepted rigorous mathematical definition of "simplify". In this paper, we shall give a simple and…

Reasoning has long been understood as a pathway between stages of understanding. Proper reasoning leads to understanding of a given subject. This reasoning was conceptualized as a process of understanding in a particular way, i.e.,…

For a fixed prime $p$, we consider a filtration of the commuting complex of elements of order $p$ in the symmetric group $\mathfrak{S}_n$. The filtration is obtained by imposing successively relaxed bounds on the number of disjoint…

A higher associativity was introduced by Jim Stasheff in [Sta63] with higher coherence conditions and now becomes one of the most important structures on spaces and algebras. He also claims that the condition on unit can be weakened, using…

We construct a pairing, which we call factorization homology, between framed manifolds and higher categories. The essential geometric notion is that of a vari-framing of a stratified manifold, which is a framing on each stratum together…

The paper demonstrates that falsifiability is fundamental to learning. We prove the following theorem for statistical learning and sequential prediction: If a theory is falsifiable then it is learnable -- i.e. admits a strategy that…

We prove that computing a single pair of vertices that are mapped onto each other by an isomorphism $\phi$ between two isomorphic graphs is as hard as computing $\phi$ itself. This result optimally improves upon a result of G\'{a}l et al.…

The classical view of epistemic logic is that an agent knows all the logical consequences of their knowledge base. This assumption of logical omniscience is often unrealistic and makes reasoning computationally intractable. One approach to…

We consider the problem of translating between irreducible closed sets and implicational bases in closure systems. To date, the complexity status of this problem is widely open, and it is further known to generalize the notorious hypergraph…

Automated decision-making systems are becoming increasingly ubiquitous, which creates an immediate need for their interpretability and explainability. However, it remains unclear whether users know what insights an explanation offers and,…

In \cite{sp25}, continuous information frames were introduced that capture exactly all continuous domains. They are obtained from the information frames considered in \cite{sp21} by omitting the conservativity requirement. Information…

The stratified resampling mechanism is one of the resampling schemes commonly used in the resampling steps of particle filters. In the present paper, we prove a central limit theorem for this mechanism under the assumption that the initial…

This short article contains the construction of a construction that generalizes the concept of the derivative of a function of one variable, using the theory of filters. The paper presents a new concept, demonstrates that it really…

Complexity of patterns is a key information for human brain to differ objects of about the same size and shape. Like other innate human senses, the complexity perception cannot be easily quantified. We propose a transparent and universal…

This paper studies abelian categories that can be decomposed into smaller abelian categories via iterated recollements - such a decomposition we call a stratification. Examples include the categories of (equivariant) perverse sheaves and…