Logic Of Subset Spaces Research Articles

The Epistemology of Nondeterminism

This paper proposes new semantics for propositional dynamic logic (PDL), replacing the standard relational semantics. Under these new semantics, program execution is represented as fundamentally deterministic (i.e., functional), while nondeterminism emerges as an epistemic relationship between the agent and the system: intuitively, the nondeterministic outcomes of a given process are precisely those that cannot be ruled out in advance. We formalize these notions using topology and the framework of dynamic topological logic (DTL) (Kremer and Mints in Ann Pure Appl Logic 131:133–158, 2005). We show that DTL can be used to interpret the language of PDL in a manner that captures the intuition above, and moreover that continuous functions in this setting correspond exactly to deterministic processes. We also prove that certain axiomatizations of PDL remain sound and complete with respect to the corresponding classes of dynamic topological models. Finally, we extend the framework to incorporate knowledge using the machinery of subset space logic (Dabrowski et al. in Ann Pure Appl Logic 78:73–110, 1996), and show that the topological interpretation of public announcements as given in Bjorndahl (in: van Ditmarsch and Sandu (eds) Jaakko Hintikka on knowledge and game theoretical semantics, outstanding contributions to Logic, vol 12, Springer, 2018) coincides exactly with a natural interpretation of test programs.

EXPSPACE-Completeness of the Logics K4 × S5 and S4 × S5 and the Logic of Subset Spaces

It is known that the satisfiability problems of the product logics K4 × S5 and S4 × S5 are NEXPTIME-hard and that the satisfiability problem of the logic SSL of subset spaces is PSPACE-hard. Furthermore, it is known that the satisfiability problems of these logics are in N2EXPTIME. We improve the lower and the upper bounds for the complexity of these problems by showing that all three problems are in ESPACE and are EXPSPACE-complete under logspace reduction.

Topological facets of the logic of subset spaces (with emphasis on canonical models)

Abstract Among other things, this article makes a small contribution to the field of bi-topological modal logic, as it is examined herein to what extent Moss and Parikh's logic of subset spaces, LSS, is topologically relevant. For that purpose, several spatial characteristics are identified and proved to correspond with the axioms of this logic in the sense of topological definability first. It turns out that a certain bi-modal cover property plays a crucial part in doing so. Then, the question is raised whether these properties are valid on the canonical topo-model for LSS. Our investigation into this finally results in a topological characterization of that model, and it leads us to studying additional schemata in the same way, namely bi-modal commutation relations. As one of the questions opening our exposition asks for the nature of the topologies induced by the two LSS-modalities, we present a corresponding characterization in the concluding part of the article.

Temporal aspects of the modal logic of subset spaces

Starting out from the ‘topological’ modal logic of Moss and Parikh (Ann. Pure Appl. Logic 78 (1996) 73–110) we present a logic of knowledge and time in which subset trees appear as the intended semantical structures. Our main results concern completeness and decidability of a corresponding system as well as the complexity of the satisfiability problem for certain subclasses of formulas.

The Complexity of Certain Modal Formulas on Binary Ramified Subset Trees

In the present note we determine the complexity of the satisfiability problem of a certain modal logic of subset spaces which was proposed in [6]. The corresponding semantics is defined w.r.t. binary ramified subset trees consisting of a set X and a distinguished set O of subsets of X such that the relation of reverse set inclusion makes O a binary ramified tree. It turns out that the satisfiability problem of the logic is complete in NP.

Topological Modal Logics Satisfying Finite Chain Conditions

We modify the semantics of topological modal logic, a language due to Moss and Parikh. This enables us to study the corresponding theory of further classes of subset spaces. In the paper we deal with spaces where every chain of opens fulfils a certain finiteness condition. We consider both a local finiteness condition relevant to points and a global one concerning the whole frame. Completeness of the appearing logical systems, which turn out to be generalizations of the well-known modal system G, can be obtained in the same manner as in the case of the general subset space logic. It is our main purpose to show that the systems differ with regard to the finite model property.