Related papers: Dimension and the Structure of Complexity Classes
The purpose of this article is to examine and limit the conditions in which the P complexity class could be equivalent to the NP complexity class. Proof is provided by demonstrating that as the number of clauses in a NP-complete problem…
This article introduces three invariance principles under which P is different from NP. In the second part a theorem of convergence is proven. This theorem states that for any language L there exists an infinite sequence of languages from…
High dimensional data can have a surprising property: pairs of data points may be easily separated from each other, or even from arbitrary subsets, with high probability using just simple linear classifiers. However, this is more of a rule…
We discuss the theory of certain partially ordered sets that capture the structure of commutation classes of words in monoids. As a first application, it follows readily that counting words in commutation classes is #P-complete. We then…
We argue that parameterized complexity is a useful tool with which to study global constraints. In particular, we show that many global constraints which are intractable to propagate completely have natural parameters which make them…
The dimension of a partially ordered set $P$ (poset for short) is the least positive integer $d$ such that $P$ is isomorphic to a subposet of $\mathbb{R}^d$ with the natural product order. Dimension is arguably the most widely studied…
Resource-bounded measure is a generalization of classical Lebesgue measure that is useful in computational complexity. The central parameter of resource-bounded measure is the {\it resource bound} $\Delta$, which is a class of functions.…
This paper presents the following results on sets that are complete for NP. 1. If there is a problem in NP that requires exponential time at almost all lengths, then every many-one NP-complete set is complete under length-increasing…
Many natural systems show emergent phenomena at different scales, leading to scaling regimes with signatures of chaos at large scales and an apparently random behavior at small scales. These features are usually investigated quantitatively…
Compositionality in language refers to how much the meaning of some phrase can be decomposed into the meaning of its constituents and the way these constituents are combined. Based on the premise that substitution by synonyms is…
This thesis presents an alternative to Cantor's theory of cardinality, insofar as that is understood as a theory of set size. The alternative is based on a general theory, ClassSize. ClassSize contains all sentences in the first order…
We consider formal verification of recursive programs with resource consumption. We introduce prefix replacement systems with non-negative integer counters which can be incremented and reset to zero as a formal model for such programs. In…
A recent study on structural properties of regular and context-free languages has greatly promoted our basic understandings of the complex behaviors of those languages. We continue the study to examine how regular languages behave when they…
Epistemic logic programs (ELPs) are a popular generalization of standard Answer Set Programming (ASP) providing means for reasoning over answer sets within the language. This richer formalism comes at the price of higher computational…
In recent years, the development of large pretrained language models, such as BERT and GPT, significantly improved information extraction systems on various tasks, including relation classification. State-of-the-art systems are highly…
A given subset $A$ of natural numbers is said to be complete if every element of $\N$ is the sum of distinct terms taken from $A$. This topic is strongly connected to the knapsack problem which is known to be NP complete. The main goal of…
This paper is an exposition, with some new applications, of our results on the growth of entropy of convolutions. We explain the main result on $\mathbb{R}$, and derive, via a linearization argument, an analogous result for the action of…
In the context of learning formal languages, data about an unknown target language L is given in terms of a set of (word,label) pairs, where a binary label indicates whether or not the given word belongs to L. A (polynomial-size)…
The curse of dimensionality in the realm of association rules is twofold. Firstly, we have the well known exponential increase in computational complexity with increasing item set size. Secondly, there is a \emph{related curse} concerned…
We consider the existence of robust strange nonchaotic attractors (SNA's) in a simple class of quasiperiodically forced systems. Rigorous results are presented demonstrating that the resulting attractors are strange in the sense that their…