42,108 research outputs found
A note on many valued quantum computational logics
The standard theory of quantum computation relies on the idea that the basic
information quantity is represented by a superposition of elements of the
canonical basis and the notion of probability naturally follows from the Born
rule. In this work we consider three valued quantum computational logics. More
specifically, we will focus on the Hilbert space C^3, we discuss extensions of
several gates to this space and, using the notion of effect probability, we
provide a characterization of its states.Comment: Pages 15, Soft Computing, 201
Implication functions in interval-valued fuzzy set theory
Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a particular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory
Interval valued (\in,\ivq)-fuzzy filters of pseudo -algebras
We introduce the concept of quasi-coincidence of a fuzzy interval value with
an interval valued fuzzy set. By using this new idea, we introduce the notions
of interval valued (\in,\ivq)-fuzzy filters of pseudo -algebras and
investigate some of their related properties. Some characterization theorems of
these generalized interval valued fuzzy filters are derived. The relationship
among these generalized interval valued fuzzy filters of pseudo -algebras
is considered. Finally, we consider the concept of implication-based interval
valued fuzzy implicative filters of pseudo -algebras, in particular, the
implication operators in Lukasiewicz system of continuous-valued logic are
discussed
Interval-valued algebras and fuzzy logics
In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of āp implies qā and āp and qā, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter
Quantum Reality and Measurement: A Quantum Logical Approach
The recently established universal uncertainty principle revealed that two
nowhere commuting observables can be measured simultaneously in some state,
whereas they have no joint probability distribution in any state. Thus, one
measuring apparatus can simultaneously measure two observables that have no
simultaneous reality. In order to reconcile this discrepancy, an approach based
on quantum logic is proposed to establish the relation between quantum reality
and measurement. We provide a language speaking of values of observables
independent of measurement based on quantum logic and we construct in this
language the state-dependent notions of joint determinateness, value identity,
and simultaneous measurability. This naturally provides a contextual
interpretation, in which we can safely claim such a statement that one
measuring apparatus measures one observable in one context and simultaneously
it measures another nowhere commuting observable in another incompatible
context.Comment: 16 pages, Latex. Presented at the Conference "Quantum Theory:
Reconsideration of Foundations, 5 (QTRF5)," Vaxjo, Sweden, 15 June 2009. To
appear in Foundations of Physics
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