On Algorithms and Complexity for Sets with Cardinality Constraints

Typestate systems ensure many desirable properties of imperative programs, including initialization of object fields and correct use of stateful library interfaces. Abstract sets with cardinality constraints naturally generalize typestate properties: relationships between the typestates of objects can be expressed as subset and disjointness relations on sets, and elements of sets can be represented as sets of cardinality one. Motivated by these applications, this paper presents new algorithms and new complexity results for constraints on sets and their cardinalities. We study several classes of constraints and demonstrate a trade-off between their expressive power and their complexity. Our first result concerns a quantifier-free fragment of Boolean Algebra with Presburger Arithmetic. We give a nondeterministic polynomial-time algorithm for reducing the satisfiability of sets with symbolic cardinalities to constraints on constant cardinalities, and give a polynomial-space algorithm for the resulting problem. In a quest for more efficient fragments, we identify several subclasses of sets with cardinality constraints whose satisfiability is NP-hard. Finally, we identify a class of constraints that has polynomial-time satisfiability and entailment problems and can serve as a foundation for efficient program analysis.Comment: 20 pages. 12 figure

A FRAMEWORK FOR AUTOMATICALLY GENERATING QUESTIONS FOR TOPICS IN DISCRETE MATHEMATICS

Automated question generation is critical for realizing personalized learning. Also, learning research shows that answering questions is a more effective method than rereading the textbook multiple times. However, creating different types of questions is intellectually challenging and time-intensive. Therefore, it emphasizes a necessity for an automated way to generate questions and evaluate them. In this research after analyzing the existing approaches to automated question generation, we conclude that most of the current systems use natural language process techniques to extract questions from the text, therefore, other topics such as mathematics are lacking an automated question generation system that could help learners to assess their knowledge.In this research we present a novel framework that automatically generates unlimited numbers of questions for different topics in discrete mathematics. We created multiple algorithms for various questions in four main topics using Python. Our final product is presented as an application programming interface (API) using Flask library, which makes it easy to gain access and use this system in any future developments. Finally, we discuss the potential extensions that can be added to our framework as future contributions. The repository for this framework is freely available at https://github.com/SalarHoushvand/discrete-math-restfulAPI

Crosslingual Document Embedding as Reduced-Rank Ridge Regression

There has recently been much interest in extending vector-based word representations to multiple languages, such that words can be compared across languages. In this paper, we shift the focus from words to documents and introduce a method for embedding documents written in any language into a single, language-independent vector space. For training, our approach leverages a multilingual corpus where the same concept is covered in multiple languages (but not necessarily via exact translations), such as Wikipedia. Our method, Cr5 (Crosslingual reduced-rank ridge regression), starts by training a ridge-regression-based classifier that uses language-specific bag-of-word features in order to predict the concept that a given document is about. We show that, when constraining the learned weight matrix to be of low rank, it can be factored to obtain the desired mappings from language-specific bags-of-words to language-independent embeddings. As opposed to most prior methods, which use pretrained monolingual word vectors, postprocess them to make them crosslingual, and finally average word vectors to obtain document vectors, Cr5 is trained end-to-end and is thus natively crosslingual as well as document-level. Moreover, since our algorithm uses the singular value decomposition as its core operation, it is highly scalable. Experiments show that our method achieves state-of-the-art performance on a crosslingual document retrieval task. Finally, although not trained for embedding sentences and words, it also achieves competitive performance on crosslingual sentence and word retrieval tasks.Comment: In The Twelfth ACM International Conference on Web Search and Data Mining (WSDM '19

Development of fuzzy syllogistic algorithms and applications distributed reasoning approaches

Thesis (Master)--Izmir Institute of Technology, Computer Engineering, Izmir, 2010Includes bibliographical references (leaves: 44-45)Text in English; Abstract: Turkish and Englishx, 65 leavesA syllogism, also known as a rule of inference or logical appeals, is a formal logical scheme used to draw a conclusion from a set of premises. It is a form of deductive reasoning that conclusion inferred from the stated premises. The syllogistic system consists of systematically combined premises and conclusions to so called figures and moods. The syllogistic system is a theory for reasoning, developed by Aristotle, who is known as one of the most important contributors of the western thought and logic. Since Aristotle, philosophers and sociologists have successfully modelled human thought and reasoning with syllogistic structures. However, a major lack was that the mathematical properties of the whole syllogistic system could not be fully revealed by now. To be able to calculate any syllogistic property exactly, by using a single algorithm, could indeed facilitate modelling possibly any sort of consistent, inconsistent or approximate human reasoning. In this work generic fuzzifications of sample invalid syllogisms and formal proofs of their validity with set theoretic representations are presented. Furthermore, the study discuss the mapping of sample real-world statements onto those syllogisms and some relevant statistics about the results gained from the algorithm applied onto syllogisms. By using this syllogistic framework, it can be used in various fields that can uses syllogisms as inference mechanisms such as semantic web, object oriented programming and data mining reasoning processes

Graph Structures for Knowledge Representation and Reasoning

This open access book constitutes the thoroughly refereed post-conference proceedings of the 6th International Workshop on Graph Structures for Knowledge Representation and Reasoning, GKR 2020, held virtually in September 2020, associated with ECAI 2020, the 24th European Conference on Artificial Intelligence. The 7 revised full papers presented together with 2 invited contributions were reviewed and selected from 9 submissions. The contributions address various issues for knowledge representation and reasoning and the common graph-theoretic background, which allows to bridge the gap between the different communities

Class Notes in Computer Science (First Edition)

These notes cover discrete structures and the theory of computer science

Using Diagrams to Understand Geometry

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/73219/1/0824-7935.00062.pd

Logical Comparison of Cases

Comparison between cases is a core issue in case-based reasoning. In this paper, we discuss a logical comparison approach in terms of the case model formalism. By logically generalizing the formulas involved in case comparison, our approach identifies analogies, distinctions and relevances. An analogy is a property shared between cases. A distinction is a property of one case ruled out by the other case, and a relevance is a property of one case, and not the other, that is not ruled out by the other case. The comparison approach is applied to HYPO-style comparison (where distinctions and relevances are not separately characterized) and to the temporal dynamics of case-based reasoning using a model of real world cases.</p