Adding a temporal dimension to a logic system

Abstract

We introduce a methodology whereby an arbitrary logic system L can be enriched with temporal features to create a new system T(L). The new system is constructed by combining L with a pure propositional temporal logic T (such as linear temporal logic with "Since" and "Until") in a special way. We refer to this method as "adding a temporal dimension to L" or ju st"temporalising L". We show that the logic system T(L) preserves several properties of the original temporal logic like soundness, completeness, decidability, conservativeness and separation over linear flows of time. We then focus on the temporalisation of first-order logic, and a comparison is made with other first-order approaches to the handling of time.

BDI-modelling of complex intracellular dynamics

“…Based on the perspective put forward in (Finger and Gabbay, 1992), such an executable continuous time temporal modelling approach was introduced: the CTBDI-model, a continuous time extension of the BDI-model. In the processing of the CTBDI-model, a temporal simulation process replaces the inference process, which is usually applied to process the (non-temporalised) logic-based BDI-model.…”

“…Our temporalised BDI-model follows (Finger and Gabbay, 1992). In short, this approach considers a logic system L, and defines a temporal dimension for L based on time-indexed trajectories T, where a state α t at some time point t is characterised by the (state) properties of L that hold in α t .…”

“…For the BDI-model this means that the state properties for beliefs, desires and intentions are used to characterise the states in a trajectory. According to (Finger and Gabbay, 1992), within the temporal dimension, relationships between different states over time can be defined in an appropriate manner. In our temporal approach, for these temporal relationships, the 'leads to' relation introduced in Section 3 is taken as a basis.…”

Importing Logics: Soundness and Completeness Preservation

“…only appears in the source of unary edges. We now present some examples of signatures for modal logic [1,7], linear-time temporal logic [4,15] and intuitionistic logic [14], useful throughout the paper. Example 2.1 Signature for linear-time temporal logic.…”

“…Having in mind different fields of application, several asymmetric ways of combining logics have been reported in the literature, including temporalization [4], modalization [3], globalization [10], probabilization [2] and quantization [9]. We proposed in [12] importing as a general way of asymmetric combination of logics and showed that it subsumes such asymmetric combination mechanisms.…”