Reasoning with many-valued interval temporal logic
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Unlike classical logic, which is grounded in the Boolean two-valued algebra, many- valued logics involve a more complex algebraic framework, thereby supporting a richer set of truth values. Such generalization has led to the development of a sophisticated algebraic taxonomy, accommodating different types of underlying domains and the properties of algebraic operators, which in turn influences the interpretation of logical connectives [1]. Noteworthy examples include Gödel algebras (G) [2], MV-algebras [3] on which Lukasiewicz logic is based [4], product algebras (Π) [5], and Heyting algebras (H) on which intuitionistic logic is based [6].
Standard (i.e., based on algebras whose lattice reduct is the interval [0, 1] in R) fuzzy logics (which can be seen as a sub-class of many-valued logics) are crucial in sev- eral mathematics and computer science areas, particularly within artificial intelligence. These logics are predominantly employed in rule-based classifier learning [7], to enhance the flexibility and expressive power of classical systems, but also to refine decision-tree classifiers by supporting more granular decision-making processes [8]. In the context of subsymbolic machine learning, they find their use in fuzzy neural networks, aid- ing in managing the inherent uncertainty in data improving network adaptability and performance [9].
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 [10, 11, 12] that limited to the fuzzification of the propositional side of formulas, while [13] provide a generaliza- tion of LTL allowing to express uncertainty in both atomic propositions and temporal relations. While standard algebras are relatively common in practical applications, the practical necessity for an infinite set of truth values is often debatable; for example, in the context of machine learning, datasets contain only a finite number of distinct values, giving rise to a finite number of distinct situations. At the same time, the conventional reliance on chain algebras can restrict modeling capabilities, disallowing the possibility of reasoning about different and incomparable experts’ viewpoints. A more general approach to many-valued modal (and therefore temporal) logics is that of Fitting [14]. In the case of interval temporal logic, [15, 16, 17] introduced and studied a many-valued extension of Halpern and Shoham’s interval temporal logic (HS [18]) over a Heyting algebra, providing a tableau system for it in the case of finite algebras.
In this work, we explore a further generalization of HS based on FLew -algebras [1]. An FLew -algebra is defined over a bounded commutative integral residuated lattice and naturally generalizes several common frameworks, including G, MV, Π, and H. To uniform the terminology, we shall use the term Many-Valued Halpern and Shoham’s interval temporal logic (MVHS). We extend Fitting’s tableau system to deal with MVHS over some finite FLew -algebra, we provide a working implementation for it as a part of our open-source reasoning and learning framework, and we test it to assess its practical usefulness.
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