Birkhoff style proof systems for hybrid-dynamic quantum logic

TL;DR

This paper introduces a novel proof system for hybrid-dynamic quantum logic using Birkhoff style methods, focusing on quantum clauses and their logical properties within a Hilbert space framework.

Contribution

It develops a new proof calculus for quantum logic based on hybrid and dynamic modal logic, including soundness, compactness, and a Birkhoff completeness result.

Findings

Proposed quantum clause notion analogous to Horn clauses.

Established soundness and compactness of the proof rules.

Proved Birkhoff completeness for a fragment of the logic.

Abstract

We explore a simple approach to quantum logic based on hybrid and dynamic modal logic, where the set of states is given by some Hilbert space. In this setting, a notion of quantum clause is proposed in a similar way the notion of Horn clause is advanced in first-order logic, that is, to give logical properties for use in logic programming and formal specification. We propose proof rules for reasoning about quantum clauses and we investigate soundness and compactness properties that correspond to this proof calculus. Then we prove a Birkhoff completeness result for the fragment of hybrid-dynamic quantum logic determined by quantum clauses.