The *-composition -A Novel Generating Method of Fuzzy Implications: An Algebraic Study

Fuzzy implications are one of the two most important fuzzy logic connectives, the other being t-norms. They are a generalisation of the classical implication from two-valued logic to the multivalued setting. A binary operation I on [0; 1] is called a fuzzy implication if (i) I is decreasing in the first variable, (ii) I is increasing in the second variable, (iii) I(0; 0) = I(1; 1) = 1 and I(1; 0) = 0. The set of all fuzzy implications defined on [0; 1] is denoted by I. Fuzzy implications have many applications in fields like fuzzy control, approximate reasoning, decision making, multivalued logic, fuzzy image processing, etc. Their applicational value necessitates new ways of generating fuzzy implications that are fit for a specific task. The generating methods of fuzzy implications can be broadly categorised as in the following: (M1): From binary functions on [0; 1], typically other fuzzy logic connectives, viz., (S;N)-, R-, QL- implications, (M2): From unary functions on [0,1], typically monotonic functions, for instance, Yager’s f-, g- implications, or from fuzzy negations, (M3): From existing fuzzy implications

Elementary Canonical Formulae: A Survey on Syntactic, Algorithmic, and Modeltheoretic Aspects

In terms of validity in Kripke frames, a modal formula expresses a universal monadic second-order condition. Those modal formulae which are equivalent to first-order conditions are called elementary. Modal formulae which have a certain persistence property which implies their validity in all canonical frames of modal logics axiomatized with them, and therefore their completeness, are called canonical. This is a survey of a recent and ongoing study of the class of elementary and canonical modal formulae. We summarize main ideas and results, and outline further research perspectives

As Below So Above: Reconstructing the Neo-Babylonian Worldview

To add to our knowledge about a Near Eastern culture, this project examines through textual evidence how the early first millennium BCE Neo-Babylonians thought, reasoned, and wrote in order to partially reconstruct the shared, generally held worldview of the Neo-Babylonian people using the transdisciplinary approach of worldview analysis. Worldviews are what we use to think with, not what we think about. Underlying surficial cultural behaviors are deeper levels of cognition regarding how to reason, perceive the world, prioritize values, prescribe behavior, and explain all of life. Specifically, this work examines the language and logic reflected in the textual archive, believing that this is the foundational level of any worldview. I argue that one finds two related components: (1) that they were linguistically programmed to be attuned to the full context over particularities, verbal actions over agential subjects, the continuity of substances over discrete objects, the standard use of maleness over femaleness, and the affective power of spoken or written words, and (2) that they were logically programmed to prefer gradations over distinctions, functional properties over inherent attributes, radials and/or rhizomes over linearity, and relationships and/or comparisons over abstractions and algorithms. By addressing this underlying, implicit cognitive software the Neo-Babylonians used, one is better able to understand the society’s more observable and obvious religious, ethical, legal, political, and social features. This has the potential to present a more contextualized view of Neo-Babylonian civilization. Reconstructing the ancient Neo-Babylonian worldview allows scholars to compare and contrast the linguistic and logical features to other ancient nearby cultures in order to understand continuities and differences and what accounts for them. One can open a dialogue between these ancient societies at a deeper level. It demonstrates the uniqueness of Neo-Babylonia. And it provides a basis for understanding how Neo-Babylonians contributed to the roots of Western civilization and thought. Most current worldview analysis examines modern or postmodern worldviews. By examining an ancient worldview, one can begin to more clearly understand any common aspects which exist for all worldviews and any elements that exist in ancient ones which are missing from cataloging more modern worldviews. Thus, the cataloging of an ancient worldview helps to open new vistas within worldview studies. This study invites similar ones within ancient Near Eastern studies and within ancient studies in general

Searching the space of representations: reasoning through transformations for mathematical problem solving

The role of representation in reasoning has been long and widely regarded as crucial. It has remained one of the fundamental considerations in the design of information-processing systems and, in particular, for computer systems that reason. However, the process of change and choice of representation has struggled to achieve a status as a task for the systems themselves. Instead, it has mostly remained a responsibility for the human designers and programmers. Many mathematical problems have the characteristic of being easy to solve only after a unique choice of representation has been made. In this thesis we examine two classes of problems in discrete mathematics which follow this pattern, in the light of automated and interactive mechanical theorem provers. We present a general notion of structural transformation, which accounts for the changes of representation seen in such problems, and link this notion to the existing Transfer mechanism in the interactive theorem prover Isabelle/HOL. We present our mechanisation in Isabelle/HOL of some specific transformations identified as key in the solutions of the aforementioned mathematical problems. Furthermore, we present some tools that we developed to extend the functionalities of the Transfer mechanism, designed with the specific purpose of searching efficiently the space of representations using our set of transformations. We describe some experiments that we carried out using these tools, and analyse these results in terms of how close the tools lead us to a solution, and how desirable these solutions are. The thorough qualitative analysis we present in this thesis reveals some promise as well as some challenges for the far-reaching problem of representation in reasoning, and the automation of the processes of change and choice of representation

Computer science: the hardware software and heart of IT

1st edition, 201