Even with the recent theoretical advancements that dramatically reduced the complexity of Integer Programming (IP), heuristics remain the dominant problem-solvers for this difficult category. This study seeks to establish the groundwork for Integer Evolution Strategies (IESs), a class of randomized search heuristics inherently designed for continuous spaces. IESs already excel in treating IP in practice, but accomplish it via discretization and by applying sophisticated patches to their continuous operators, while persistently using the $\ell_2$-norm as their operation pillar. We lay foundations for discrete search, by adopting the $\ell_1$-norm, accounting for the suitable step-size, and questioning alternative measures to quantify correlations over the integer lattice. We focus on mutation distributions for unbounded integer decision variables. We briefly discuss a couple of candidate discrete probabilities induced by the uniform and binomial distributions, which we show to possess less appealing theoretical properties, and then narrow down to the Truncated Normal (TN) and Double Geometric (DG) distributions. We explore their theoretical properties, including entropy functions, and propose a procedure to generate scalable correlated mutation distributions. Our investigations are accompanied by extensive numerical simulations, which consistently support the claim that the DG distribution is better suited for unbounded integer search. We link our theoretical perspective to empirical evidence indicating that an IES with correlated DG mutations outperformed other strategies over non-separable quadratic IP. We conclude that while the replacement of the default TN distribution by the DG is theoretically justified and practically beneficial, the truly crucial change lies in adopting the $\ell_1$-norm over the $\ell_2$-norm.

On a drizzly recent Sunday night, the Art Deco lobby of 30 Rockefeller Plaza was clogged with celebrities attending "Saturday Night Live" 's fiftieth-anniversary show. Taran Killam, who played Donald Trump on the show in 2015, strolled in wearing a double-breasted tux and stopped cold when he noticed a curly-haired man in a green suit standing beside a rig that looked like a ten-foot-tall Swiss Army knife. "Oh, here it is," Killam said. The man was Cole Walliser, a forty-three-year-old director, and the gizmo was the Glambot, a high-speed camera mounted on a giant robotic arm. Walliser oversees the device during the E! network's red-carpet coverage, and the resulting dramatic slo-mo celebrity action clips are posted to TikTok and Instagram.

Standpoint EL is a multi-modal extension of the popular description logic EL that allows for the integrated representation of domain knowledge relative to diverse standpoints or perspectives. Advantageously, its satisfiability problem has recently been shown to be in PTime, making it a promising framework for large-scale knowledge integration. In this paper, we show that we can further push the expressivity of this formalism, arriving at an extended logic, called Standpoint EL+, which allows for axiom negation, role chain axioms, self-loops, and other features, while maintaining tractability. This is achieved by designing a satisfiability-checking deduction calculus, which at the same time addresses the need for practical algorithms. We demonstrate the feasibility of our calculus by presenting a prototypical Datalog implementation of its deduction rules.

The tractability of the lightweight description logic EL has allowed for the construction of large and widely used ontologies that support semantic interoperability. However, comprehensive domains with a broad user base are often at odds with strong axiomatisations otherwise useful for inferencing, since these are usually context-dependent and subject to diverging perspectives. In this paper we introduce Standpoint EL, a multi-modal extension of EL that allows for the integrated representation of domain knowledge relative to diverse, possibly conflicting standpoints (or contexts), which can be hierarchically organised and put in relation to each other. We establish that Standpoint EL still exhibits EL's favourable PTime standard reasoning, whereas introducing additional features like empty standpoints, rigid roles, and nominals makes standard reasoning tasks intractable.

Throughout her career, Maya Rudolph has excelled in a number of roles and is a very talented and flexible performer. She is a major player in the entertainment world with a variety of talents that include comedy, singing, and acting. Have you seen her perform as a regular cast member on "Saturday Night Live"? With her impersonations of Donatella Versace, Beyoncé, and even Oprah Winfrey, Rudolph has made a lasting impression. She was a regular cast member on the show from 2000 until 2007, and during that period, she received an Emmy nomination for her performance.

Recently, the field of knowledge representation is drawing a lot of inspiration from database theory. In particular, in the area of description logics and ontology languages, interest has shifted from satisfiability checking to query answering, with various query notions adopted from databases, like (unions of) conjunctive queries or different kinds of path queries. Likewise, the finite model semantics is being established as a viable and interesting alternative to the traditional semantics based on unrestricted models. In this paper, we investigate diverse database-inspired reasoning problems for very expressive description logics (all featuring the worrisome trias of inverses, counting, and nominals) which have in common that role paths of unbounded length can be described (in the knowledge base or of the query), leading to a certain non-locality of the reasoning problem. We show that for all the cases considered, undecidability can be established by very similar means. Most notably, we show undecidability of finite entailment of unions of conjunctive queries for a fragment of SHOIQ (the logic underlying the OWL DL ontology language), and undecidability of finite entailment of conjunctive queries for a fragment of SROIQ (the logical basis of the more recent and popular OWL 2 DL standard).

Existential rules (also known as Datalog /- or tuple-generating dependencies) have been intensively studied in recent years as a prominent formalism in knowledge representation and database systems. We consider them here as a querying formalism, extending classical Datalog, the language of deductive databases. It is well known that the classes of databases recognized by (Boolean) existential rule queries are closed under homomorphisms. Also, due to the existence of a semi-decision procedure (the chase), these database classes are recursively enumerable. We show that, conversely, every homomorphism-closed recursively enumerable query can be expressed as an existential rule query, thus arriving at a precise characterization of existential rules by model-theoretic and computational properties. Although the result is very intuitive, the proof turns out to be non-trivial. This result can be seen as a very expressive counterpart of the prominent Lyndon-Los-Tarski-Theorem characterizing the homomorphism-closed fragment of first-order logic. Notably, our result does not presume the existence of any additional built-in structure on the queried data, such as a linear order on the domain, which is a typical requirement for other characterizations in the spirit of descriptive complexity.

Formal Concept Analysis (FCA) is a prominent field of applied mathematics using object-attribute relationships to define formal concepts -- groups of objects with common attributes -- which can be ordered into conceptual hierarchies, so-called concept lattices. We consider the problem of satisfiability of membership constraints, i.e., to determine if a formal concept exists whose object and attribute set include certain elements and exclude others. We analyze the computational complexity of this problem in general and for restricted forms of membership constraints. We perform the same analysis for generalizations of FCA to incidence structures of arity three (objects, attributes and conditions) and higher. We present a generic answer set programming (ASP) encoding of the membership constraint satisfaction problem, which allows for deploying available highly optimized ASP tools for its solution. Finally, we discuss the importance of membership constraints in the context of navigational approaches to data analysis.

Abstract dialectical frameworks (ADFs) are a recently introduced powerful generalization of Dung’s popular abstract argumentation frameworks (AFs). Inspired by similar work for AFs, we introduce a decomposition scheme for ADFs, which proceeds along the ADF’s strongly connected components. We find that, for several semantics, the decomposition-based version coincides with the original semantics, whereas for others, it gives rise to a new semantics. These new semantics allow us to deal with pertinent problems such as odd-length negative cycles in a more general setting, that for instance also encompasses logic programs. We perform an exhaustive analysis of the computational complexity of these new, so-called naive-based semantics. The results are quite interesting, for some of them involve little-known classes of the so-called Boolean hierarchy (another hierarchy in between classes of the polynomial hierarchy). Furthermore, in credulous and sceptical entailment, the complexity can be different depending on whether we check for truth or falsity of a specific statement.

In 2017 I decided to find out what would happen if I trained a neural net on 240 Christmas carols (collected by The Times of London and reader/neural net hobbyist Erik Svensson). Run, run Rudolph, run, run Rudolph, run, run Rudolph, run, run Rudolph, run, run Rudolph, run, run Rudolph, run, run Rudolph, run, run Rudolf the new born King. You can kind of understand where the confusion came from. But that was 2017, when I was training char-rnn from scratch on my laptop. Now in 2019 I have access to the much more powerful GPT-2, trained by OpenAI on 40GB of text from the internet.