Expressive Power and Decidability for Memory Logics

Taking as inspiration the hybrid logic HL(↓), we introduce a new family of logics that we call memory logics. In this article we present in detail two interesting members of this family defining their formal syntax and semantics. We then introduce a proper notion of bisimulation and investigate their expressive power (in comparison with modal and hybrid logics). We will prove that in terms of expressive power, the memory logics we discuss in this paper are more expressive than orthodox modal logic, but less expressive than HL(↓). We also establish the undecidability of their satisfiability problems. 1 Memory Logics: Hybrid Logics with a Twist Hybrid languages have been extensively investigated in the past years. HL, the simplest hybrid language, is usually presented as the basic modal language K extended with special symbols (called nominals) to name individual states in a model. These new symbols are simply a new sort of atomic symbols {i, j, k, . . .} disjoint from the set of standard propositional variables. While they behave syntactically exactly as propositional variables do, their semantic interpretation differ: nominals denote elements in the model, instead of sets of elements. This simple addition already results in increased expressive power. For example the formula i ∧ 〈r〉i is true in a state w, only if w is a reflexive point named by the nominal i. As the basic modal language is invariant under unraveling, there is no equivalent modal formula [?]. But as we said above, HL is just the simplest hybrid language. Once nominals have been added to the language, other natural extensions arise. Having names for states at our disposal we can introduce, for each nominal i, an operator @i that allows us to jump to the point named by i obtaining the language HL(@). The formula @iφ (read ‘at i, φ’) moves the point of evaluation to the state named by i and evaluates φ there. Intuitively, the @i operators internalize the satisfaction relation ‘|=’ into the logical language: M, w |= ∗INRIA Nancy Grand Est, France †LSV, ENS Cachan, CNRS, INRIA, France ‡Departamento de Computación, FCEyN, UBA, Argentina