China
Africa
Explore chapters and articles related to this topic
Digital Fuzzy Operations Using Multi-Valued Fredkin Gates
Published in Hafiz Md. Hasan Babu, VLSI Circuits and Embedded Systems, 2023
Fuzzy set theory and the corresponding logic is quite transitioning from the traditional set theory and the concept of uncertainty. When A is a fuzzy set and x is a relevant object, the proposition “x is a member of A″ is not necessarily true or false, it may be true only to some degree. It is most common to express the degrees of membership by numbers in the closed interval [0, 1]. In this chapter, digital fuzzy set is considered where the membership-value space is discretized. The standard set operations and the concept of fuzzy relations are defined based on these digital fuzzy sets and their realizations. In this chapter, the composition of fuzzy relations and a systolic array structure are described to compute it. Collections of fuzzy if-then rules or fuzzy algorithms are mathematically equivalent to fuzzy relations and the problem of inference of (evaluating them with specific values) is mathematically equivalent to composition.
A
Published in Splinter Robert, Illustrated Encyclopedia of Applied and Engineering Physics, 2017
[computational, general] A virtual concept proposed under special relativistic conditions describing the apparent correlation between multiple inertial systems outlining a frame of reference. Absolute simultaneity entails that the observations made by observers of two simultaneous event observed in one reference frame midway in space between the two events would also be experienced by observers in another inertial system as simultaneous. This, in fact, is not necessarily true, primarily because of the specific definition of the boundary conditions of each reference frame in space and time. The discrepancy results from the relativistic concept of time dilation, where time is relative to the observer’s inertial frame of reference and consequently distance is subjected to the boundary conditions of the reference frame of the observer (see Figure A.8).
Introduction to Systems of ODEs
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
Many real-world problems can be modeled by differential equations containing more than one dependent variable to be considered. Mathematical and computer simulations of such problems can give rise to systems of ordinary differential equations. Another reason to study these systems is the fact that all higher order differential equations can be written as an equivalent system of first order equations. The inverse statement is not necessarily true. For this reason, most computer programs are written to approximate the solutions for a first order system of ordinary differential equations.
Fuzzy intensional semantics
Published in Journal of Applied Non-Classical Logics, 2018
One of the simplest enhancements of intensional frames, known as the Kripke semantics, is the addition of a binary accessibility relation between possible worlds. In Kripke models, a formula is necessarily true in a possible world iff it is true in all worlds accessible from this world, and possibly true iff true in some accessible world. The notions of necessity and possibility from Definition 2.3 are thus a special case when the accessibility relation is total. Modifying the properties of the accessibility relation results in different kinds of necessity and possibility modalities (for a detailed overview of classical modal logics see, e.g. Chagrov & Zakharyaschev, 1997).
Reasoning about manipulation in multi-agent systems
Published in Journal of Applied Non-Classical Logics, 2022
Christopher Leturc, Grégory Bonnet
It is commonly accepted to consider the effects of action as an equivalence relationship, such as in STIT. Thus, for any agent , is a reflexive, transitive and Euclidean relationship.5 Indeed, when an agent implements one or more actions, if he has carried them out, it means that this is the case and that the consequence φ is true. The reflexivity, denoted , expresses the fact that once the actions leading to a certain φ consequence have been performed by an agent i, then that consequence φ is necessarily true in the current world. This constraint gives the axiom T. From this constraint, we immediately deduce that this relation is also serial. This translates that if an agent i ensures that one or several actions lead to a certain consequence φ, it is not the case that this action or series of actions can lead to the opposite in the current world. So we deduce in such a system the property D. The relation is also transitive since when the actions of agent i lead to φ, these actions also lead to the fact that these actions are done properly, i.e.: Finally, if an agent i does not perform actions that lead to some consequences φ, then agent i indirectly performs actions that lead to not realise the actions that lead to φ. Thus the relation is Euclidean, i.e.: Let us notice that we do not consider positive and negative introspection with knowledge because since represents the effects of actions, some effects may not be known by the agent i as side-effects.
Axiomatic and dual systems for constructive necessity, a formally verified equivalence
Published in Journal of Applied Non-Classical Logics, 2019
Lourdes del Carmen González-Huesca, Favio E. Miranda-Perea, P. Selene Linares-Arévalo
The above definition is an extension of the one given by von Plato (2014, Definition 3.6) (it is also implicitly given by the rules of the system HK of Hakli & Negri, 2012) and generates a deductive system by means of the following inductive definition: We name the system . It stems from system HK of Hakli and Negri (2012) with the following differences: since we do not consider negation, the axiom of stability is removed. Further, system is adapted to handle constructive necessity incorporating the modal axioms and . Observe that the (MP) rule is given in a so-called multiplicative style (see Sørensen & Urzyczyn, 2006, Chapter 5), since it considers two different collections of hypotheses from which A and are derived. This formulation is inherited from system HK, but the context concatenation is swapped in the conclusion: there it is given as instead of our . The reason to this change is related to our proof approach and to the fact that here the contexts are lists, whereas in Hakli and Negri (2012) they are multisets, or lists where the order plays no role as stated by von Plato (2014, p. 51). In that work, the modus ponens is stated exactly as here but later used in a lax manner since sometimes the contexts are swapped in the conclusion without further notice. This practice is a harmless mathematical procedure whose direct mechanisation is forbidden by the proof assistant. Regarding the box operator, the necessitation rule (Nec) reflects the fact that if A is a theorem then we can infer that A is necessarily true, that is, we can infer from any collection of assumptions. Recall that, the inductive definition of the judgment generates a formal notion of derivability through a shallow embedding.