Abstract Primal (propositional) logic PL is the {∧, →} fragment of intuitionistic logic, and primal (propositional) infon logic PIL is a conservative extension of PL with the quotation construct p said. Logic PIL was introduced by Gurevich and Neeman in 2009 in connection with the DKAL project. The derivation problem for PIL (and therefore for PL) is solvable in linear time, and yet PIL allows one to express many common access control scenarios. The most obvious limitations on the expressivity of logics PL and PIL are the failures of the transitivity rules pref x → z respectively where pref ranges over quotation prefixes p said q said . . .. Here we investigate the extension T of PL with an axiom x → x and the inference rule (trans0) as well as the extension qT of PIL with an axiom pref x → x and the inference rule (trans). • [Subformula property] T has the subformula property: if Γ y then there is a derivation of y from Γ comprising only subformulas of Γ ∪ {y}. qT has a similar locality property. • [Complexity] The derivation problems for T and qT are solvable in quadratic time. • [Soundness and completeness] We define Kripke models for qT (resp. T) and show that the semantics is sound and complete. • [Small models] T has the one-element-model property: if Γ y then there is a one-element counterexample. Similarly small (though not one-element) counterexamples exist for qT.

Abstract. Primal infon logic was introduced in 2009 in connection with access control. In addition to traditional logic constructs, it contains unary connectives p said indispensable in the intended access control applications. Propositional primal infon logic is decidable in linear time, yet suffices for many common access control scenarios. The most obvious limitation on its expressivity is the failure of the transitivity law for implication: x → y and y → z do not necessarily yield x → z. Here we introduce and investigate equiexpressive “transitive ” extensions TPIL and TPIL ∗ of propositional primal infon logic as well as their quote-free fragments TPIL0 and TPIL0 ∗ respectively. We prove the subformula property for TPIL0 ∗ and a similar property for TPIL ∗ ; we define Kripke models for the four logics and prove the corresponding soundness-and-completeness theorems; we show that, in all these logics, satisfiable formulas have small models; but our main result is a quadratic-time derivation algorithm for TPIL ∗. §1. Introduction. In a brick-and-mortar setting, some access control policies may be vague and even unwritten. The clerks ordinarily know them. When in doubt, they know whom to ask. In the cloud there are no clerks, and policies have to be managed automatically. The most challenging are federated scenarios where the policies of different