Two-sorted Modal Logic for Formal and Rough Concepts

In this paper, we propose two-sorted modal logics for the representation and reasoning of concepts arising from rough set theory (RST) and formal concept analysis (FCA). These logics are interpreted in two-sorted bidirectional frames, which are essentially formal contexts with converse relations. On one hand, the logic $\textbf{KB}$ contains ordinary necessity and possibility modalities and can represent rough set-based concepts. On the other hand, the logic $\textbf{KF}$ has window modality that can represent formal concepts. We study the relationship between \textbf{KB} and \textbf{KF} by proving a correspondence theorem. It is then shown that, using the formulae with modal operators in \textbf{KB} and \textbf{KF}, we can capture formal concepts based on RST and FCA and their lattice structures

A Note on the Completeness of Many-Valued Coalgebraic Modal Logic

In this paper, we investigate the many-valued version of coalgebraic modal logic through predicate lifting approach. Coalgebras, understood as generic transition systems, can serve as semantic structures for various kinds of modal logics. A well-known result in coalgebraic modal logic is that its completeness can be determined at the one-step level. We generalize the result to the finitely many-valued case by using the canonical model construction method. We prove the result for coalgebraic modal logics based on three different many-valued algebraic structures, including the finitely-valued {\L}ukasiewicz algebra, the commutative integral Full-Lambek algebra (FL$_{ew}$-algebra) expanded with canonical constants and Baaz Delta, and the FL$_{ew}$-algebra expanded with valuation operations.Comment: 17 pages, preprint for journal submissio

Floating Point Arithmetic Protocols for Constructing Secure Data Analysis Application

AbstractA large variety of data mining and machine learning techniques are applied to a wide range of applications today. There- fore, there is a real need to develop technologies that allows data analysis while preserving the confidentiality of the data. Secure multi-party computation (SMC) protocols allows participants to cooperate on various computations while retaining the privacy of their own input data, which is an ideal solution to this issue. Although there is a number of frameworks developed in SMC to meet this challenge, but they are either tailored to perform only on specific tasks or provide very limited precision. In this paper, we have developed protocols for floating point arithmetic based on secure scalar product protocols, which is re- quired in many real world applications. Our protocols follow most of the IEEE-754 standard, supporting the four fundamental arithmetic operations, namely addition, subtraction, multiplication, and division. We will demonstrate the practicality of these protocols through performing various statistical calculations that is widely used in most data analysis tasks. Our experiments show the performance of our framework is both practical and promising

Information theoretical analysis of two-party secret computation

Abstract. Privacy protection has become one of the most important issues in the information era. Consequently, many protocols have been developed to achieve the goal of accomplishing a computational task cooperatively without revealing the participants' private data. Practical protocols, however, do not guarantee perfect privacy protection, as some degree of privacy leakage is allowed so that resources can be used efficiently, e.g., the number of random bits required and the computation time. A metric for measuring the degree of information leakage based on an information theoretical framework was proposed i

Possibilistic Residuated Implication Logics and Applications

In this paper, we will develop a class of logics for reasoning about qualitative and quantitative uncertainty. The semantics of the logics is uniformly based on possibility theory. Each logic in the class is parameterized by a t-norm operation on [0,1], and we express the degree of implication between the possibilities of two formulas explicitly by using residuated implication with respect to the t-norm. The logics are then shown to be applicable to possibilistic reasoning, approximate reasoning, and nonmonotonic reasoning. Key Words:Possibilistic reasoning, similarity-based reasoning, nonmonotonic reasoning. 1 Introduction Knowledge representation and reasoning is fundamental to knowledge based systems. Due to the imprecision and incompleteness of acquired knowledge, uncertain reasoning is a key issue in knowledge representation. To accommodate different types of incomplete knowledge, many uncertainty reasoning methods have been proposed and extensively studied. Most methods focus e..

On the possibility theory-based semantics for logics of preference

AbstractIn this paper, we study the semantics for the logics of preference based on possibility theory. Possibility distributions representing the preference between worlds are associated with the possible world models for dynamic logics. Then the preference between actions are determined by comparing some measures of their consequences. We define different logics of preference by considering the comparisons of possibility measures and guaranteed possibility measures. Some properties of the proposed logics are studied and their relationships with deontic logics are also considered

Application schemes of possibility theory-based defaults

In this paper, we analyze the formalism proposed by Yager[6] to represent default knowledge in the framework of possibility theory. Three different application schemes are examined. The results produced by these application schemes are compared with those by Reiter’s default logic. It is shown that for a restricted class of default theories, there is some kinds of correspondence between Yager’s formalism and Reiter’s default logic. It is also exemplified that there is much mismatch between them for general default theories. Finally, the fixed point mechanism is suggested as an auxiliary scheme to eliminate such mismatch.

On the relationship between evidential structures and data tables

In this paper, we would like to investigate the relationship between evidential structures (ES)—the basic qualitative structures of Dempster-Shafer theory, and the data table based knowledge representation systems(KRS) subject to rough set analysis. It is shown that an ES has a natural representation as a data table and from a given data table and two of its attributes, an ES can be extracted. We also show that some important operations on ES’s can be realized in relational algebra. The results are then generalized to the fuzzy case. Consequentially, we further clarify the connection between evidence theory and rough set theory. Keywords: Knowledge representation, rough set, Dempster-Shafer theory, evidential structures, data tables