A Theory and Calculus for Reasoning about Sequential Behavior

TL;DR

This paper develops a formal theory and calculus for reasoning about sequential behavior using a generalized implicant concept, providing new proof methods for logical/temporal dependencies in sequences.

Contribution

It introduces a generalized implicant notion for sequences of Boolean expressions and establishes a necessary and sufficient condition for implicants of regular sets, enabling new proof techniques.

Findings

Defines a generalized implicant for sequences of Boolean expressions.

Provides a necessary and sufficient condition for implicants of regular sets.

Introduces sequential resolution and normalization proof methods.

Abstract

Basic results in combinatorial mathematics provide the foundation for a theory and calculus for reasoning about sequential behavior. A key concept of the theory is a generalization of Boolean implicant which deals with statements of the form: A sequence of Boolean expressions alpha is an implicant of a set of sequences of Boolean expressions A This notion of a generalized implicant takes on special significance when each of the sequences in the set A describes a disallowed pattern of behavior. That is because a disallowed sequence of Boolean expressions represents a logical/temporal dependency, and because the implicants of a set of disallowed Boolean sequences A are themselves disallowed and represent precisely those dependencies that follow as a logical consequence from the dependencies represented by A. The main result of the theory is a necessary and sufficient condition for a sequence of Boolean expressions to be an implicant of a regular set of sequences of Boolean expressions. This result is the foundation for two new proof methods. Sequential resolution is a generalization of Boolean resolution which allows new logical/temporal dependencies to be inferred from existing dependencies. Normalization starts with a model (system) and a set of logical/temporal dependencies and determines which of those dependencies are satisfied by the model.