 # Introduction to Active Sets

05/16/2013
Speaker(s) : Germano Resconi (Department of Mathematics and Physics, Catholic University, Italie)
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. An active set 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. Agents in the active set with a special criteria gives the logic value for the same attribute. So agents in many cases are in a logic conflict and this generate semantic uncertainty on the logic evaluation. Criteria and agents are the two variables by which we give different logic values to the same attribute or proposition. Active sets is beyond the modal logic. In fact given a proposition in modal logic we can evaluate the proposition only when we know the worlds where the proposition is locate. When we evaluate one proposition in one world we cannot evaluate the same proposition in another world. Now in epistemic logic any world is an agent that know that the proposition is true or false. Now the active set is a set of agents as in the 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 and 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. Matrix agent is the vector subspace of the true false agent multi dimension space. Operations among active set include operations in the traditional set , fuzzy sets and rough set as special cases. The agent multi dimensional space to evaluate active set include also 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 the new frontier to model new types of computers , new type of model for wireless communications as cognitive radio. In conclusion Active set open new possibility and new models for the logic.
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