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"Mathematical Logic"
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Souslin quasi-orders and bi-embeddability of uncountable structures
We provide analogues of the results from Friedman and Motto Ros (2011) and Camerlo, Marcone, and Motto Ros (2013) (which correspond
to the case
eBook
A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth
In this paper we introduce a general framework for the study of limits of relational structures and graphs in particular, which is
based on a combination of model theory and (functional) analysis. We show how the various approaches to graph limits fit to this
framework and that they naturally appear as “tractable cases” of a general theory. As an outcome of this, we provide extensions of known
results. We believe that this puts these into a broader context. The second part of the paper is devoted to the study of sparse
structures. First, we consider limits of structures with bounded diameter connected components and we prove that in this case the
convergence can be “almost” studied component-wise. We also propose the structure of limit objects for convergent sequences of sparse
structures. Eventually, we consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded
tree-depth, motivated by their role as “elementary bricks” these graphs play in decompositions of sparse graphs, and give an explicit
construction of a limit object in this case. This limit object is a graph built on a standard probability space with the property that
every first-order definable set of tuples is measurable. This is an example of the general concept of
eBook
Minimal Weak Truth Table Degrees and Computably Enumerable Turing Degrees
Two of the central concepts for the study of degree structures in computability theory are computably enumerable degrees and minimal degrees. For strong notions of reducibility, such as m-deducibility or truth table reducibility, it is possible for computably enumerable degrees to be minimal. For weaker notions of reducibility, such as weak truth table reducibility or Turing reducibility, it is not possible to combine these properties in a single degree. We consider how minimal weak truth table degrees interact with computably enumerable Turing degrees and obtain three main results. First, there are sets with minimal weak truth table degree which bound noncomputable computably enumerable sets under Turing reducibility. Second, no set with computable enumerable Turing degree can have minimal weak truth table degree. Third, no \\Delta^0_2 set which Turing bounds a promptly simple set can have minimal weak truth table degree.
eBook
The KenKen Method Puzzles for Beginners : 150 puzzles and solutions to make you smarter
\"KenKen is the world's fastest growing puzzle since sudoku. It was developed in a Japanese Classroom in 2004 by renowned educator, Tetsuya Miyamoto, and was later introduced internationally as both an adult and educational math and logic puzzle in 2008. Originally intended to improve reasoning, creativity, concentration, and perseverance, this simple yet sophisticated puzzle has since expanded beyond the classroom into a global sensation. KenKen can be found daily in The New York Times and is now available in over 200 publications worldwide. Unlimited KenKen puzzles can be accessed on www.kenkenpuzzle.com and can also be played on mobile devices via the KenKen Classic app. It is supported by the National Council of Teachers of Mathematics (NCTM), the world's largest math education organization, and is used by over 30,000 educators worldwide\"-- Provided by publisher.
Book
Fuzzy logic and mathematics : a historical perspective
The term “fuzzy logic” (FL) is a generic one, which stands for a broad variety of logical systems. Their common ground is the rejection of the most fundamental principle of classical logic—the principle of bivalence—according to which each declarative sentence has exactly two possible truth values—true and false. Each logical system subsumed under FL allows for additional, intermediary truth values, which are interpreted as degrees of truth. These systems are distinguished from one another by the set of truth degrees employed, its algebraic structure, truth functions chosen for logical connectives, and other properties. The book examines from the historical perspective two areas of research on fuzzy logic known as fuzzy logic in the narrow sense (FLN) and fuzzy logic in the broad sense (FLB), which have distinct research agendas. The agenda of FLN is the development of propositional, predicate, and other fuzzy logic calculi. The agenda of FLB is to emulate commonsense human reasoning in natural language and other unique capabilities of human beings. In addition to FL, the book also examines mathematics based on FL. One chapter in the book is devoted to overviewing successful applications of FL and the associated mathematics in various areas of human affairs. The principal aim of the book is to assess the significance of FL and especially its significance for mathematics. For this purpose, the notions of paradigms and paradigm shifts in science, mathematics, and other areas are introduced and employed as useful metaphors.
eBook
Logical Dynamics of Information and Interaction
This book develops a view of logic as a theory of information-driven agency and intelligent interaction between many agents - with conversation, argumentation and games as guiding examples. It provides one uniform account of dynamic logics for acts of inference, observation, questions and communication, that can handle both update of knowledge and revision of beliefs. It then extends the dynamic style of analysis to include changing preferences and goals, temporal processes, group action and strategic interaction in games. Throughout, the book develops a mathematical theory unifying all these systems, and positioning them at the interface of logic, philosophy, computer science and game theory. A series of further chapters explores repercussions of the 'dynamic stance' for these areas, as well as cognitive science.
eBook