Primal Infon Logic with Conjunctions as Sets

Primal infon logic was proposed by Gurevich and Neeman as an efficient yet expressive logic for policy and trust management. It is a propositional multimodal subintuitionistic logic decidable in linear time. However in that logic the principle of the replacement of equivalents fails. For example, \(\left(x \land y\right) \to z\) does not entail \(\left(y \land x\right) \to z\), and similarly \(w \to \left(\left(x \land y\right)\land z\right)\) does not entail \(w \to \left(x \land \left(y \land z\right)\right)\). Imposing the full principle of the replacement of equivalents leads to an NP-hard logic according to a recent result of Beklemishev and Prokhorov. In this paper we suggest a way to regain the part of this principle restricted to conjunction: We introduce a version of propositional primal logic that treats conjunctions as sets, and show that the derivation problem for this logic can be decided in linear expected time and quadratic worst-case time.