ABSTRACT. This work addresses the problem of synthesizing fuzzy temporal logic rules from a set of given positive and negative examples. The examples are provided in the form of execution traces of finite length. Fuzzy Time Linear Temporal Logic over finite traces (FTL-f) is chosen as the language for rule synthesis. FTL-f is capable of capturing fuzzy temporal modalities, like, 'soon after', 'almost always', 'gradually' etc, that make the learnt rules simpler and more understandable than classical LTL representations. The proposed approach reduces the learning task to a multi-valued partial maximum satisfiability (WPMaxSAT) problem. This work is useful for generating interpretable explanations of complex system behaviours

ABSTRACT. Mīmāṁsā one of the Indian Philosophies deals with the in- terpretation of the Vedas. The Brāhmaṅas, a division of the Vedas, in- clude precise instructions (Vidhi) for the execution of rituals. Interpret- ing these immediately can be confusing. In order to fully understand this, the interpretive processes from Mīmāṁsā are utilized. The procedures en- compass various components, such as linguistic proficiency, grammatical comprehension, the individual’s capacity to execute the ritual, and log- ical attributes. In addition, Mīmāṁsā incorporates a mention of diverse sequencing systems known as krama, which precisely outlines the spec- ified sequence in which rituals should be performed. This paper takes inspiration from these sequential techniques and proposes a framework for temporal reasoning from a Logical perspective. This approach is in- corporated into the existing MIRA (Mīmāṁsā Inspired Representation of Actions) work, resulting in activity sequencing. Subsequently, it is uti- lized in the Large Language Models to autonomously produce a sequence of instructions in real-life situations.

ABSTRACT. The variable inclusion companions of logics have lately been thoroughly studied by multiple authors. There are broadly two types of these companions: the left and the right variable inclusion companions. Another type of companions of logics induced by Hilbert-style presentations (Hilbert-style logics) were introduced in a recent paper. A sufficient condition for the restricted rules companion of a Hilbert-style logic to coincide with its left variable inclusion companion was proved there, while a necessary condition remained elusive. The present article has two parts. In the first part, we give a necessary and sufficient condition for the left variable inclusion and the restricted rules companions of a Hilbert-style logic to coincide. In the rest of the paper, we recognize that the variable inclusion restrictions used to define variable inclusion companions of a logic $\langle\mathcal{L},\vdash\rangle$ are relations from $\mathcal{P}(\mathcal{L})$ to $\mathcal{L}$. This leads to a more general idea of a relational companion of a logical structure, a framework that we borrow from the field of universal logic. We end by showing that even Hilbert-style logics and the restricted rules companions of these can be brought under the umbrella of the general notions of logical structures and their relational companions that are discussed here.

ABSTRACT. In this paper, we propose a new version of \emph{complete} logic---\.{L}ogic of \.{I}nexact \.{K}nowledge ($\mathsf{LIK}$)---the model of which can reflect Williamson (1994)'s arguments on inexact knowledge in the sense that it has the following \emph{eight} features: (1) This model based on \emph{additively-semiordered qualitative conditional probability relation} that is a qualitatively-probabilistic counterpart of a \emph{JND} which is a psychophysical counterpart of a \emph{margin for error} can reflect the essence of \emph{inexact knowledge}. (2) We can formalize a \emph{margin for error principle} in $\mathsf{LIK}$. (3) The \emph{width} of a margin for error depends on the \emph{boundedly-rational cognitive capacities}. (4) In $\mathsf{LIK}$, a \emph{direct indiscriminability relation} is a \emph{primitive} and \emph{non-transitive} relation. (5) The reason why we introduce qualitative not absolute but \emph{conditional} probability is to make it possible to express the \emph{direct indiscriminability between the two events on the condition that either of the two occurs}. (6) In $\mathsf{LIK}$, \emph{inexact knowledge} is \emph{defined} in terms of this direct indiscriminability relation. (7) $\mathsf{LIK}$ has so \emph{rich expressive power} as to formalize several inferences about inexact knowledge. (8) The \emph{KK principle} is not valid in $\mathsf{LIK}$.