Reactive Valuations

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.

💡 Research Summary

The thesis investigates a sequential propositional logic in which the truth value of atomic propositions can change depending on the order in which they are evaluated. This dynamic behaviour is captured by the notion of “reactive valuations”, a concept originally introduced by Bergstra and Ponse. Unlike classical (static) valuations, reactive valuations incorporate a history of evaluated atoms, allowing the same atom to assume different truth values in different contexts.

The author first formalises reactive valuations using a ternary conditional operator (written a ▹ b ◃ c, meaning “if b then a else c”) and a history‑updating function ∂ₐ(H), which maps a valuation H to the valuation after atom a has been evaluated. The value of a compound term under a valuation H is then defined recursively in terms of the values of its sub‑terms under the appropriate updated histories.

Four main classes of reactive valuations are examined:

1. Free reactive valuations (FRV) – the most general class, imposing no restrictions beyond the sequential nature of evaluation.
2. Contractive valuations (CRV) – require that an atom’s value remains unchanged as long as no other atom is evaluated in between, modelling a kind of “local stability”.
3. Static valuations (SV) – ignore the evaluation order entirely, coinciding with ordinary propositional logic.
4. Repetition‑free valuations – a variant that forbids repeated evaluation of the same atom within a single evaluation sequence.

For each class a dedicated equational axiomatization is presented. The basic system CP (for FRV) consists of five axioms (CP1–CP5) that capture the behaviour of the conditional operator together with the constants T and F. Additional axioms are added for the other classes, yielding systems denoted CPᵣₚ, CPᶜʳ and CPˢᵗ.

The thesis proves soundness (all derivable equations hold under the intended semantics) and completeness (every semantically valid equation is derivable) for each system. The completeness proofs rely on a term‑rewriting approach: the axioms are shown to generate a confluent and terminating rewrite system, ensuring that any two semantically equivalent terms can be reduced to a common normal form.

A substantial part of the work is devoted to establishing independence of the axioms. For each axiom the author constructs a model that satisfies all other axioms but falsifies the one under consideration. This demonstrates that none of the axioms is redundant, confirming that the presented axiom sets are minimal.

The notion of ω‑completeness is then introduced. An axiomatization is ω‑complete if, whenever an infinite family of equations { tₙ = uₙ | n ∈ ℕ } is semantically valid, the system can prove a single equation that captures the whole family. The author proves ω‑completeness for the systems CP and CPˢᵗ by defining an ω‑rule and showing that it is admissible in these systems. This result is particularly relevant for reasoning about infinite behaviours such as recursion or non‑terminating processes in programming languages.

In the concluding chapter the author reflects on the relevance of reactive valuations for modelling side‑effects, short‑circuit evaluation, sequential circuits, and even everyday commonsense reasoning where facts can be retracted or revised. Potential future work includes extending the framework to richer operators (e.g., parallel composition, probabilistic choice), integrating the theory into automated theorem provers, and applying the semantics to real programming languages to capture their operational behaviour more faithfully.

Overall, the thesis provides a thorough logical foundation for sequential, history‑dependent evaluation, delivering sound, complete, and ω‑complete axiomatisations for several important subclasses, and establishing the minimality of these axiom sets. It bridges a gap between classical propositional logic and the needs of modern computational systems that exhibit stateful, order‑sensitive behaviour.