The Expressive Power of Memory Logics

Abstract

We investigate the expressive power of memory logics. These are modal logics extended with the possibility to store (or remove) the current node of evaluation in (or from) a memory, and to perform membership tests on the current memory. From this perspective, the hybrid logic ℋℒ (↓), for example, can be thought of as a particular case of a memory logic where the memory is an indexed list of elements of the domain.This work focuses in the case where the memory is a set, and we can test whether the current node belongs to the set or not. We prove that, in terms of expressive power, the memory logics we discuss here lie between the basic modal logic ${\cal K}$ and ℋℒ (↓). We show that the satisfiability problem of most of the logics we cover is undecidable. The only logic with a decidable satisfiability problem is obtained by imposing strong constraints on which elements can be memorized.

Characterization, definability and separation via saturated models

“…The framework can also be used for logics whose model theory has not been fully developed so far (e.g., Memory Logics [2,3]). In all cases, we only need to check that the requirements of the framework are met.…”

“…A pointed L-model is a pair M,w where M is an L-model andw ∈ |M| n for some fixed n that we will call the dimension of the pointed model. 3 Given an L-formula ϕ, we use the notation M,w ϕ for "ϕ is true atw in M" and M,w ϕ for "ϕ is false atw in M". The definition of depends on L. We only impose that ∧, ∨, and ⊥ have the usual interpretations:…”

“…Memory logics, introduced in [1] and further investigated in, e.g., [2,4,3], allow modeling dynamic behavior through explicit memory operators that change the structure where evaluation takes place. Memory logics extends the syntax and semantics of BML with operators that store and retrieve elements of the domain into a memory -a subset of the domains of the model.…”

“…It is shown in [3] that if L is any of these memory logics, then L-bisimulation implies L-equivalence. That is, if M, w and N , v are two memory models, then M,…”

Narcissists Are Easy, Stepmothers Are Hard

“…Also, satisfiability with respect to classes of models that are typically computationally well-behaved (e.g., linear, transitive or equivalence relation frames) is undecidable (except in the uni-modal case) [18]. Weakened versions of ↓ were investigated and also turned out to be undecidable [2].…”