Relations, residuals, regular interiors, and relative regular equivalence

Given a relation α (a binary sociogram) and an a priori equivalence relation π, both on the same set of individuals, it is interesting to look for the largest equivalence πo that is contained in and is regular with respect to α. The equivalence relation πo is called the regular interior of π with respect to α. The computation of πo involves the left and right residuals, a concept that generalized group inverses to the algebra of relations. A polynomial-time procedure is presented (Theorem 11) and illustrated with examples. In particular, the regular interior gives meet in the lattice of regular equivalences: the regular meet of regular equivalences is the regular interior of their intersection. Finally, the concept of relative regular equivalence is defined and compared with regular equivalence.

Using set operations to deal with the frame problem

This paper presents a simple approach to the so-called frame problem based on some ordinary set operations, which does not require non-monotonic reasoning. Following the notion of the situation calculus, we shall represent a state of the world as a set of fluents, where a fluent is simply a Boolean-valued property whose truth-value is dependent on the time. High-level causal laws are characterised in terms of relationships between actions and the involved world states. An effect completion axiom is imposed on each causal law, which guarantees that all the fluents that can be affected by the performance of the corresponding action are always totally governed. It is shown that, compared with other techniques, such a set operation based approach provides a simpler and more effective treatment to the frame problem.

A framework for state-based time-series analysis and prediction

Time-series analysis and prediction play an important role in state-based systems that involve dealing with varying situations in terms of states of the world evolving with time. Generally speaking, the world in the discourse persists in a given state until something occurs to it into another state. This paper introduces a framework for prediction and analysis based on time-series of states. It takes a time theory that addresses both points and intervals as primitive time elements as the temporal basis. A state of the world under consideration is defined as a set of time-varying propositions with Boolean truth-values that are dependent on time, including properties, facts, actions, events and processes, etc. A time-series of states is then formalized as a list of states that are temporally ordered one after another. The framework supports explicit expression of both absolute and relative temporal knowledge. A formal schema for expressing general time-series of states to be incomplete in various ways, while the concept of complete time-series of states is also formally defined. As applications of the formalism in time-series analysis and prediction, we present two illustrating examples.