Strong Completeness and the Finite Model Property for Bi-Intuitionistic Stable Tense Logics

Bi-Intuitionistic Stable Tense Logics (BIST Logics) are tense logics with a Kripke semantics where worlds in a frame are equipped with a pre-order as well as with an accessibility relation which is ‘stable’ with respect to this pre-order. BIST logics are extensions of a logic, BiSKt, which arose in the semantic context of hypergraphs, since a special case of the pre-order can represent the incidence structure of a hypergraph. In this paper we provide, for the first time, a Hilbert-style axiomatisation of BISKt and prove the strong completeness of BiSKt. We go on to prove strong completeness of a class of BIST logics obtained by extending BiSKt by formulas of a certain form. Moreover we show that the finite model property and the decidability hold for a class of BIST logics

Axiomatizing Discrete Spatial Relations

Qualitative spatial relations are used in artificial intelligence to model commonsense notions such as regions of space overlapping, touching only at their boundaries, or being separate. In this paper we extend earlier work on qualitative relations in discrete space by pre- senting a bi-intuitionistic modal logic with universal modalities, called UBiSKt. This logic has a semantics in which formulae are interpreted as subgraphs. We show how a variety of qualitative spatial relations can be defined in UBiSKt. We make essential use of a sound and complete axiomatisation of the logic and an implementation of a tableau based theorem prover to establish novel properties of these spatial relations. We also explore the role of UBiSKt in expressing spatial relations at more than one level of detail. The features of the logic allow it to rep- resent how a subgraph at a detailed level is approximated at a coarser level