Introducing a general framework for many-valued temporal and spatial logic.
Date:
Reasoning with temporal and spatial information is crucial in many real-world ap- plications from different fields. In order to do so, many modal temporal and spatial logics have been proposed throughout the years, leveraging modal operators to treat the relations between points in time (Linear Temporal Logic [1]) and space (Compass Logic [2, 3]), or even time intervals (Halpern and Shoham’s Interval Temporal Logic [4]) or areas in an image (Lutz and Wolter’s Logic of Topological Relations [5]). At the same time, these applications live in scenarios characterized by uncertainty and vagueness in the data. This problem is usually addressed using fuzzy logics, such as G ̈odel logic (G) [6], Lukasiewicz logic [7], and product logic (Π) [8].
In terms of temporal logic, only a few attempts have been made to study point- based temporal languages in the fuzzy case, with early contributions [9, 10, 11] that limited to the fuzzification of the propositional side of formulas, while [12] provides a generalization of Linear Temporal Logic allowing to express uncertainty in both atomic propositions and temporal relations. In the case of interval temporal logic, [13, 14, 15] introduced and studied a many-valued extension of Halpern and Shoham’s Interval Temporal Logic over a Heyting algebra [16], based on a work by Fitting [17]. On the other hand, there seem to be no attempts for a many-valued version of spatial logics. We aim to introduce a possible many-valued extension for well-known temporal and spatial logics based on the family of finite FLew-algebras [18], allowing for a unified treatment of both fuzzy logics and Heyting algebras, leveraging the notion of a many- valued linear order and allowing for the definition of a many-valued semantics of modal frames. This framework is also implemented as part of an open-source framework for representing, reasoning, and learning from structured and unstructured data, namely Sole.jl [19], serving as the foundation for reasoning and learning tools leveraging many- valued temporal and spatial logics.
References
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