Reactive Valuations

TL;DR

This thesis explores reactive valuations in sequential logic, analyzing their properties, axiomatizations, and semantics, and establishing soundness, completeness, independence, and {}-completeness for specific classes.

Contribution

It provides a detailed study of reactive valuations, including axiomatization, semantics, and proofs of soundness, completeness, independence, and -completeness for certain classes.

Findings

Proved soundness and completeness of axiomatizations.

Established independence of axioms, showing no redundancies.

Demonstrated -completeness for two classes of reactive valuations.

Abstract

In sequential logic there is an order in which the atomic propositions in an expression are evaluated. This order allows the same atomic proposition to have different values depending on which atomic propositions have already been evaluated. In the sequential propositional logic discussed in this thesis, such valuations are called "reactive" valuations, in contrast to "static" valuations as are common in e.g. ordinary propositional logic. There are many classes of these reactive valuations e.g., we can define a class of reactive valuations such that the value for each atomic proposition remains the same until another atomic proposition is evaluated. This Master of Logic thesis consists of a study of some of the properties of this logic. We take a closer look at some of the classes of reactive valuations. We particularly focus on the relation between the axiomatization and the semantics. Consequently, the main part of this thesis focuses on proving soundness and completeness. Furthermore, we show that the axioms in the provided axiomatizations are independent i.e., there are no redundant axioms present. Finally, we show {\omega}-completeness for two classes of reactive valuations.