Birkhoff style proof systems for hybrid-dynamic quantum logic

TL;DR

This work proposes proof rules for reasoning about quantum clauses, investigates soundness and compactness properties that correspond to this proof calculus, and proves a Birkhoff completeness result for the fragment of hybrid-dynamic quantum logic determined by quantum clauses.

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.

Figures (6)

• Figure 1: Entailment systems
• Figure 2: Closed propositional symbols
• Figure 3: Basic proof rules
• Figure 4: Necessity
• Figure 5: Implication

Theorems & Definitions (27)

• theorem 1
• lemma 1
• remark 1
• remark 2
• remark 3
• theorem 2: Local satisfaction condition
• corollary 1
• theorem 3
• corollary 2
• lemma 2: Quantum implication