On cascade products of answer set programs

Describing complex objects by elementary ones is a common strategy in mathematics and science in general. In their seminal 1965 paper, Kenneth Krohn and John Rhodes showed that every finite deterministic automaton can be represented (or "emulated") by a cascade product of very simple automata. This led to an elegant algebraic theory of automata based on finite semigroups (Krohn-Rhodes Theory). Surprisingly, by relating logic programs and automata, we can show in this paper that the Krohn-Rhodes Theory is applicable in Answer Set Programming (ASP). More precisely, we recast the concept of a cascade product to ASP, and prove that every program can be represented by a product of very simple programs, the reset and standard programs. Roughly, this implies that the reset and standard programs are the basic building blocks of ASP with respect to the cascade product. In a broader sense, this paper is a first step towards an algebraic theory of products and networks of nonmonotonic reasoning systems based on Krohn-Rhodes Theory, aiming at important open issues in ASP and AI in general.Comment: Appears in Theory and Practice of Logic Programmin

Logic-Based Analogical Reasoning and Learning

Analogy-making is at the core of human intelligence and creativity with applications to such diverse tasks as commonsense reasoning, learning, language acquisition, and story telling. This paper contributes to the foundations of artificial general intelligence by developing an abstract algebraic framework for logic-based analogical reasoning and learning in the setting of logic programming. The main idea is to define analogy in terms of modularity and to derive abstract forms of concrete programs from a `known' source domain which can then be instantiated in an `unknown' target domain to obtain analogous programs. To this end, we introduce algebraic operations for syntactic program composition and concatenation and illustrate, by giving numerous examples, that programs have nice decompositions. Moreover, we show how composition gives rise to a qualitative notion of syntactic program similarity. We then argue that reasoning and learning by analogy is the task of solving analogical proportions between logic programs. Interestingly, our work suggests a close relationship between modularity, generalization, and analogy which we believe should be explored further in the future. In a broader sense, this paper is a first step towards an algebraic and mainly syntactic theory of logic-based analogical reasoning and learning in knowledge representation and reasoning systems, with potential applications to fundamental AI-problems like commonsense reasoning and computational learning and creativity

Sequential decomposition of propositional logic programs

The sequential composition of propositional logic programs has been recently introduced. This paper studies the sequential {\em decomposition} of programs by studying Green's relations $\mathcal{L,R,J}$ -- well-known in semigroup theory -- between programs. In a broader sense, this paper is a further step towards an algebraic theory of logic programming.Comment: arXiv admin note: text overlap with arXiv:2109.05300, arXiv:2009.0577

Proportional algebras, homomorphisms, congruences, and functors

This paper introduces proportional algebras as algebras endowed with the 4-ary analogical proportion relation where the fundamental concepts of subalgebras, homomorphisms, congruences, and functors are constructed

Analogical Proportions

Analogy-making is at the core of human and artificial intelligence and creativity with applications to such diverse tasks as commonsense reasoning, learning, language acquisition, and story telling. This paper introduces from first principles an abstract algebraic framework of analogical proportions of the form `$a$ is to $b$ what $c$ is to $d$' in the general setting of universal algebra. This enables us to compare mathematical objects possibly across different domains in a uniform way which is crucial for AI-systems. It turns out that our notion of analogical proportions has appealing mathematical properties. Most importantly, it turns out that the property of being in analogical proportion is a {\em local} property. As we construct our model from first principles using only elementary concepts of universal algebra, and since our model questions some basic properties of analogical proportions presupposed in the literature, to convince the reader of the plausibility of our model we show that it can be naturally embedded into first-order logic via model-theoretic types and prove from that perspective that analogical proportions are compatible with structure-preserving mappings. This provides conceptual evidence for its applicability. In a broader sense, this paper is a first step towards a theory of analogical reasoning and learning systems with potential applications to fundamental AI-problems like commonsense reasoning and computational learning and creativity

Boolean proportions

Analogy-making is at the core of human and artificial intelligence and creativity with applications to such diverse tasks as proving mathematical theorems and building mathematical theories, commonsense reasoning, learning, language acquisition, and story telling. This paper studies analogical proportions between booleans of the form `$a$ is to $b$ what $c$ is to $d$' called boolean proportions. Technically, we instantiate the abstract algebraic framework of analogical proportions recently introduced by the author in the boolean domain consisting of the booleans 0 and 1 together with boolean functions. It turns out that our notion of boolean proportions has a simple logical characterization which entails appealing mathematical properties. In a broader sense, this paper is a further step towards a theory of analogical reasoning and learning systems with potential applications to fundamental AI-problems like commonsense reasoning and computational learning and creativity.Comment: arXiv admin note: text overlap with arXiv:2006.0285

Fixed Point Semantics for Stream Reasoning

Arbeit an der Bibliothek noch nicht eingelangt - Daten nicht geprüftAbweichender Titel nach Übersetzung der Verfasserin/des VerfassersMenschen sind im Alltag mit Datenströmen (engl. ``streams'') aus ihrer Umgebung konfrontiert und das Schließen (engl. ``reasoning'') aus solchen Datenströmen ist für die menschliche Intelligenz von zentraler Bedeutung. Moderne digitale Dienste, wie zum Beispiel Google und Facebook, generieren in kürzester Zeit zahlreiche Daten die es maschinell zu verarbeiten gilt. Im letzten Jahrzehnt hat sich innerhalb der Künstlichen Intelligenz eine Forschungsrichtung dafür als besonders relevant hervorgehoben---das sogenannte Stream Reasoning. Vor kurzem wurde der regel-basierte Formalismus LARS für das nicht-monotone daten-basierte Schließen unter Anwendung der Antwortmengensemantik entwickelt. Syntaktisch sind LARS-Programme nichts anderes als Logikprogramme mit Negation, die zusätzlich Operatoren zum Ausdrücken zeitlicher Zusammenhänge erlauben, wobei der Fenster-Operator (engl. ``window operator'') von besonderem Interesse ist---dieser erlaubt es, relevante Zeitpunkte auszuwählen. Da LARS fixe Zeitintervalle für die Evaluierung von Programmen voraussetzt, ist der Formalismus in der aktuellen Form nicht flexibel genug, um konstruktiv mit sich rasch verändernden Daten umzugehen. Außerdem hat sich gezeigt, dass die von LARS verwendete und auf FLP-Redukten basierende Erweiterung der Antwortmengensemantik zirkuläre Schlüsse zulässt, wie sie auch von anderen Erweiterungen der klassischen Antwortmengensemantik bereits bekannt sind. Diese Doktorarbeit behebt alle erwähnten Schwächen von LARS und leistet einen Beitrag zu den Grundlagen des Stream Reasonings indem sie eine operationale Fixpunktsemantik für eine flexible Variante von LARS entwickelt die korrekt und konstruktiv in dem Sinne ist, dass Antwortmengen durch iterierte Anwendung eines Fixpunktoperators erzeugt werden und dadurch frei von zirkulären Schlüssen sind.Reasoning over streams of input data is an essential part of human intelligence. During the last decade stream reasoning has emerged as a research area within the AI-community with many potential applications. In fact, the increased availability of streaming data via services like Google and Facebook has raised the need for reasoning engines coping with data that changes at high rate. Recently, the rule-based formalism LARS for non-monotonic stream reasoning under the answer set semantics has been introduced. Syntactically, LARS programs are logic programs with negation incorporating operators for temporal reasoning, most notably window operators for selecting relevant time points. Unfortunately, by preselecting fixed intervals for the semantic evaluation of programs, the rigid semantics of LARS programs is not flexible enough to constructively cope with rapidly changing data dependencies. Moreover, we show that defining the answer set semantics of LARS in terms of FLP reducts leads to undesirable circular justications similar to other ASP extensions. This thesis fixes all of the aforementioned shortcomings of LARS. More precisely, we contribute to the foundations of stream reasoning by providing an operational fixed point semantics for a fully flexible variant of LARS and we show that our semantics is sound and constructive in the sense that answer sets are derivable bottom-up and free of circular justications.3

The relationship between the loess stratigraphy in the Vojvodina region of northern Serbia and the Saalian and Rissian Stage glaciations – a review

The regional loess stratigraphy in the Vojvodina region, in the southeastern Carpathian Basin has often been successfully correlated to global palaeoclimate. The loess record in the Carpathian Basin is a quasi-continuous data set on the sedimentary, climatic, and environmental conditions during the last four glacial/interglacial cycles. In this study, we present a standardized loess stratigraphy dataset and illustrate how it correlates with the marine oxygen isotope and Chinese loess stratigraphical records of palaeoclimate. We argue that the loess stratigraphy in Vojvodina region is an important link in the integration of European terrestrial stratigraphical schemes and the marine oxygen isotope stratigraphical model. Despite a few problems with these correlations, the suggested general framework represents a significant stratigraphical approach towards better and chronologically consistent, European stratigraphical models. We highlight how the loess record can better understand terrestrial environmental change through multiple glacial cycles when other records, such as glacial records. In these cases, evidence of glaciations are often missing due to the inherently fragmentary nature of these records, a situation common to most other terrestrial records. The loess sequences of the Carpathian (Pannonian) Basin provide an unique quasi-continuous environmental record that enables direct links to be made between the loess sediment records and their sources - glacial erosion in the Alps and other southern European mountains. This reveals evidence of glaciations during every glacial cycle of the Saalian Stage complex, equivalent to MIS 10, 8, and 6. It is argued, therefore, that loess has the potential to provide a direct link between terrestrial glaciations and wider records of global climate change, which is an enigma for many other continental records, especially during the Saalian Stage complex and equivalent Rissian Stage glaciations (MIS 10-6). The Serbian loess records display a strong relationship with the intensity of European glaciations during different glacial cycles. Loess sedimentation rates are highest in the most entensive European glaciation of the Saalian complex (MIS 6) and much lower during the weaker ‘missing’ glaciations equivalent to MIS 8 and 10.In contrast loess exposures in the Vojvodina region indicate that dust deposition was minimal during the interglacials, as was characteristic for interglacial loess in China. However, in contrast to the Chinese loess which saw deserts form in some interglacials, during the interglacials in Europe soils formed on the loess.A key observation from the Vojvodina loess is that a gradual increase in interglacial aridity through the late Middle Pleistocene (the Saalian complex). The explanation for the progressively increasing aridity in the southeastern Carpathian Basin and the Balkan region at this time is still unclear. However, this trend - of gradually increasing interglacial aridity - is consistent with the idea of the Saalian complex as representing a 400 ka mega glacial cycle modulated by shorter classic 100 ka glacial cycles