Completeness Results for Memory Logics

Fervari

Hoffmann

2013

Logic Journal of IGPL

We investigate dynamic modal operators that can change the model during evaluation. We define the logic SL by extending the basic modal language with the ♦ modality, which is a diamond operator that in addition has the ability to invert pairs of related elements in the domain while traversing an edge of the accessibility relation. SL is very expressive: it fails to have the finite and the tree model property. We show that SL is equivalent to a fragment of first-order logic by providing a satisfiability preserving translation. In addition, we provide an equivalence preserving translation from SL to the hybrid logic H(:, ↓). We also define a suitable notion of bisimulation for SL and investigate its expressive power, showing that it lies strictly between the basic modal logic and H(:, ↓). We finally show that its model checking problem is PSpace-complete and its satisfiability problem is undecidable.

Section: Changing the Modelmentioning

confidence: 99%

Swap logic

Fervari

Hoffmann

2013

Logic Journal of IGPL

“…The study we carried out in this paper draws a more detailed picture of the properties of of memory logics. We have investigated these logics in a number of recent papers (Areces, 2007;Areces et al, 2009aAreces et al, , 2009bMera, 2009) in which we present complete axiomatizations, tableaux calculi, complexity analysis for model checking, and preliminary results on the Beth and the interpolation properties for different fragments of this family. But there is still work to be done.…”

Section: Undecidable Memory Logicsmentioning

confidence: 99%

“…The original idea was introduced in Areces (2007). Areces et al (2009a) investigate tableau algorithms and model checking for memory logics, while Areces et al (2009b) discuss axiomatic completeness results. In this article we extend results originally presented in Areces et al (2008) and provide full proofs.…”

mentioning

confidence: 99%

The Expressive Power of Memory Logics

The Review of Symbolic Logic

Figueira

et al. 2011

Self Cite

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.

“…That is, we check simulation in L and satisfaction in FO. 4 We are almost ready to prove the Characterization theorem. In the proof we will need the following results.…”

Section: Characterizationmentioning

confidence: 99%

“…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.…”

Section: Memory Logicsmentioning

confidence: 99%

Characterization, definability and separation via saturated models

Theoretical Computer Science

Carreiro

Figueira

2014

Self Cite

Three important results about the expressivity of a modal logic L are the Characterization Theorem (that identifies a modal logic L as a fragment of a better known logic), the Definability theorem (that provides conditions under which a class of L-models can be defined by a formula or a set of formulas of L), and the Separation Theorem (that provides conditions under which two disjoint classes of L-models can be separated by a class definable in L).We provide general conditions under which these results can be established for a given choice of model class and modal language whose expressivity is below first order logic. Besides some basic constraints that most modal logics easily satisfy, the fundamental condition that we require is that the class of ω-saturated models in question has the Hennessy-Milner property with respect to the notion of observational equivalence under consideration. Given that the Characterization, Definability and Separation theorems are among the cornerstones in the model theory of L, this property can be seen as a test that identifies the adequate notion of observational equivalence for a particular modal logic.