Introduction to active sets and unification

Abstract

An active set is a unifying space being able to act as a "bridge" for transferring information, ideas and results between distinct types of uncertainties and different types of applications. An active set is a set of agents who independently deliver true or false values for a given proposition. It is not a simple vector of logic values for different propositions, the results are a vector but the set is not. The difference between an ordinary set and active set is that the ordinary set has passive elements with values of the attributes defined by an external agent, in the active set any element is an agent that internally defines the value of a given attribute for a passive element. So agents in many cases are in a logic conflict and this generates semantic uncertainty on the evaluation. Criteria and agents are the two variables by which we give different logic values to the same attribute or proposition. In modal logic, given a proposition, we can evaluate the proposition only when we know the worlds where the proposition is located. In epistemic logic any world is an agent that knows that the proposition is true or false. The active set is a set of agents as in epistemic logic but the difference with modal logic is that all the agents (worlds) are not separate but are joined in the evaluation of the given proposition. In active set for one agent and one criteria we have one logic value but for many agents and criteria the evaluation is not true or false but is a matrix of true and false. This matrix is not only a logic evaluation as in the modal logic but give us the conflicting structure of the active set evaluation. The agent multi dimensional space to evaluate active set also includes the Hilbert multidimensional space where is possible to simulate quantum logic gate. New logic operation are possible as fuzzy gate operations and more complex operations as conflicting solving, consensus operations, syntactic inconsistency, semantic inconsistency and knowledge integration. In the space of the agents evaluations morphotronic geometric operations are a new frontier to model new types of computers. In this paper we show the connection between classical, fuzzy, evidence theory and active sets. © 2012 Springer-Verlag.