Ω-Algebarski sistemi

The research work carried out in this thesis is aimed   at fuzzifying algebraic and relational structures in the framework of Ω-sets, where Ω is a complete lattice. Therefore we attempt to synthesis universal algebra and fuzzy set theory. Our  investigations of Ω-algebraic structures are based on Ω-valued equality, satisability of identities and cut techniques. We introduce Ω-algebras, Ω-valued congruences,  corresponding quotient  Ω-valued-algebras and  Ω-valued homomorphisms and we investigate connections among these notions. We prove that there is an Ω-valued homomorphism from an Ω-algebra to the corresponding quotient Ω-algebra. The kernel of an Ω-valued homomorphism is an Ω-valued congruence. When dealing with cut structures, we prove that an Ω-valued homomorphism determines classical homomorphisms among the corresponding quotient structures over cut  subalgebras. In addition, an  Ω-valued congruence determines a closure system of classical congruences on cut subalgebras. In addition, identities are preserved under Ω-valued homomorphisms. Therefore in the framework of Ω-sets we were able to introduce Ω-attice both as an ordered and algebraic structures. By this Ω-poset is defined as an Ω-set equipped with  Ω-valued order which is  antisymmetric with respect to the corresponding Ω-valued equality. Thus defining the notion of pseudo-infimum and pseudo-supremum we obtained the definition of Ω-lattice as an ordered structure. It is also defined that the an Ω-lattice as an algebra is a bi-groupoid equipped with an Ω-valued equality fulfilling some particular lattice Ω-theoretical formulas. Thus using axiom of choice we proved that the two approaches are equivalent. Then we also introduced the notion of complete Ω-lattice based on Ω-lattice. It was defined as a generalization of the classical complete lattice. We proved results that characterizes Ω-structures and many other interesting results. Also the connection between Ω-algebra and the notion of weak congruences is presented. We conclude with what we feel are most interesting areas for future work.Tema ovog rada je fazifikovanje algebarskih i relacijskih struktura u okviru omega- skupova, gdeje Ω kompletna mreza. U radu se bavimo sintezom oblasti univerzalne algebre i teorije rasplinutih (fazi) skupova. Naša istraživanja omega-algebarskih struktura bazirana su na omega-vrednosnoj jednakosti,zadovoljivosti identiteta i tehnici rada sa nivoima. U radu uvodimo omega-algebre,omega-vrednosne kongruencije, odgovarajuće omega-strukture, i omega-vrednosne homomorfizme i istražujemo veze izmedju ovih pojmova. Dokazujemo da postoji Ω -vrednosni homomorfizam iz Ω -algebre na odgovarajuću količničku Ω -algebru. Jezgro Ω -vrednosnog homomorfizma je Ω- vrednosna kongruencija. U vezi sa nivoima struktura, dokazujemo da Ω -vrednosni homomorfizam odredjuje klasične homomorfizme na odgovarajućim količničkim strukturama preko nivoa podalgebri. Osim toga, Ω-vrednosna kongruencija odredjuje sistem zatvaranja klasične kongruencije na nivo podalgebrama. Dalje, identiteti su očuvani u Ω- vrednosnim homomorfnim slikama.U nastavku smo u okviru Ω-skupova uveli Ω-mreže kao uredjene skupove i kao algebre i dokazali ekvivalenciju ovih pojmova. Ω-poset je definisan kao Ω -relacija koja je antisimetrična i tranzitivna u odnosu na odgovarajuću Ω-vrednosnu jednakost. Definisani su pojmovi pseudo-infimuma i pseudo-supremuma i tako smo dobili definiciju Ω-mreže kao uredjene strukture. Takodje je definisana Ω-mreža kao algebra, u ovim kontekstu nosač te strukture je bi-grupoid koji je saglasan sa Ω-vrednosnom jednakošću i ispunjava neke mrežno-teorijske formule. Koristeći aksiom izbora dokazali smo da su dva pristupa ekvivalentna. Dalje smo uveli i pojam potpune Ω-mreže kao uopštenje klasične potpune mreže. Dokazali smo još neke rezultate koji karakterišu Ω-strukture.Data je i veza izmedju Ω-algebre i pojma slabih kongruencija.Na kraju je dat prikaz pravaca daljih istrazivanja

Reasoning in combinations of theories

Verification problems are often expressed in a language which mixes several theories. A natural question to ask is whether one can use decision procedures for individual theories to construct a decision procedure for the union theory. In the cases where this is possible one has a powerful method at hand to handle complex theories effectively. The setup considered in this thesis is that of one base theory which is extended by one or more theories. The question is if and when a given ground satisfiability problem in the extended setting can be effectively reduced to an equi-satisfiable problem over the base theory. A case where this reductive approach is always possible is that of so-called local theory extensions. The theory of local extensions is developed and some applications concerning monotone functions are given. Then the theory of local theory extensions is generalized in order to deal with data structures that exhibit local behavior. It will be shown that a suitable fragment of both the theory of arrays and the theory of pointers is local in this broader sense. Finally, the case of more than one theory extension is discussed. In particular, a modularity result is given that under certain circumstances the locality of each of the extensions lifts to locality of the entire extension. The reductive approach outlined above has become particularly relevant in recent years due to the rise of powerful solvers for background theories common in verification tasks. These so-called SMT-solvers effectively handle theories such as real linear or integer arithmetic. As part of this thesis, a program called H-PILoT was implemented which carries out reductive reasoning for local theory extensions. H-PILoT found applications in mathematics, multiple-valued logics, data-structures and reasoning in complex systems.Verifikationsaufgaben kombinieren oft verschiedene Theorien. Eine naheliegende Frage ist, ob man Entscheidungsverfahren für die Einzeltheorien auf die gesamte Theorie übertragen kann. In den Fällen, wo das möglich ist, hat man eine mächtige Technik zur Hand, um mit komplexen Theorien effizient umgehen zu können. Der Ansatz, der in dieser Arbeit betrachtet wird, ist stets der einer Hintergrundtheorie, die durch eine oder mehrere Theorien erweitert wird. Die Frage ist dann, ob und wann sich eine gegebene Beweisanfrage bezüglich der Theorieerweiterung effektiv auf eine äquivalente Beweisanfrage bezüglich der Hintergrundtheorie reduzieren lässt. Ein Fall, in dem diese Reduzierung immer möglich ist, ist derjenige der lokalen Theorieerweiterungen. Die Theorie der lokalen Erweiterungen wird entwickelt, und es werden einige Anwendungen für monotone Funktionen gegeben. Danach wird die Theorie der lokalen Erweiterungen verallgemeinert, um mit Datenstrukturen umgehen zu können, die Lokalitätseigenschaften aufweisen. Es wird gezeigt, dass sowohl ein geeignetes Fragment der Theorie der Arrays wie auch der Theorie der Zeiger lokal im erweiterten Sinne sind. Schließlich wird der Fall mehrerer Theorieerweiterungen betrachtet. Insbesondere wird ein Modularitätsresultat gezeigt, das besagt, dass unter gewissen Umständen die Lokalität der einzelnen Erweiterungen hinreichend ist, um die Lokalität der gesamten Erweiterung zu gewährleisten. Die oben erwähnte Reduzierung von Beweisaufgaben ist in jüngster Zeit besonders relevant geworden, weil leistungsfähige Beweiser für gängige Hintergrundtheorien entwickelt worden sind. Diese sogenannten SMT-Beweiser behandeln Theorien wie z.B. lineare Arithmetik der ganzen oder der reellen Zahlen effektiv. Als Teil der vorgelegten Arbeit wurde ein Programm namens H-PILoT entwickelt, welches die Reduzierung von Beweisaufgaben für lokale Theorien durchführt. H-PILoT hat Anwendungen in der Mathematik, bei mehrwertigen Logiken, bei der Verifikation von Datenstrukturen und in der Verifikation komplexer Systeme gefunden

Constructing and Extending Description Logic Ontologies using Methods of Formal Concept Analysis

Description Logic (abbrv. DL) belongs to the field of knowledge representation and reasoning. DL researchers have developed a large family of logic-based languages, so-called description logics (abbrv. DLs). These logics allow their users to explicitly represent knowledge as ontologies, which are finite sets of (human- and machine-readable) axioms, and provide them with automated inference services to derive implicit knowledge. The landscape of decidability and computational complexity of common reasoning tasks for various description logics has been explored in large parts: there is always a trade-off between expressibility and reasoning costs. It is therefore not surprising that DLs are nowadays applied in a large variety of domains: agriculture, astronomy, biology, defense, education, energy management, geography, geoscience, medicine, oceanography, and oil and gas. Furthermore, the most notable success of DLs is that these constitute the logical underpinning of the Web Ontology Language (abbrv. OWL) in the Semantic Web. Formal Concept Analysis (abbrv. FCA) is a subfield of lattice theory that allows to analyze data-sets that can be represented as formal contexts. Put simply, such a formal context binds a set of objects to a set of attributes by specifying which objects have which attributes. There are two major techniques that can be applied in various ways for purposes of conceptual clustering, data mining, machine learning, knowledge management, knowledge visualization, etc. On the one hand, it is possible to describe the hierarchical structure of such a data-set in form of a formal concept lattice. On the other hand, the theory of implications (dependencies between attributes) valid in a given formal context can be axiomatized in a sound and complete manner by the so-called canonical base, which furthermore contains a minimal number of implications w.r.t. the properties of soundness and completeness. In spite of the different notions used in FCA and in DLs, there has been a very fruitful interaction between these two research areas. My thesis continues this line of research and, more specifically, I will describe how methods from FCA can be used to support the automatic construction and extension of DL ontologies from data

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum