Model checking with the standard Kripke models used in (Dynamic) Epistemic Logic leads to scalability issues. Hence alternative representations have been developed, in particular symbolic structures based on Binary Decision Diagrams (BDDs) and succinct models based on mental programs. While symbolic structures have been shown to perform well in practice, their theoretical complexity was not known so far. On the other hand, for succinct models model checking is known to be PSPACE-complete, but no implementations are available. We close this gap and directly relate the two representations. We show that model checking DEL on symbolic structures encoded with BDDs is also PSPACE-complete. In fact, already model checking Epistemic Logic without dynamics is PSPACE-complete on symbolic structures. We also provide direct translations between BDDs and mental programs. Both translations yield exponential outputs. For the translation from mental programs to BDDs we show that no small translation exists. For the other direction we conjecture the same.

Originating in psychology, $\textit{Theory of Mind}$ (ToM) has attracted significant attention across multiple research communities, especially logic, economics, and robotics. Most psychological work does not aim at formalizing those central concepts, namely $\textit{goals}$, $\textit{intentions}$, and $\textit{beliefs}$, to automate a ToM-based computational process, which, by contrast, has been extensively studied by logicians. In this paper, we offer a different perspective by proposing a computational framework viewed through the lens of game theory. On the one hand, the framework prescribes how to make boudedly rational decisions while maintaining a theory of mind about others (and recursively, each of the others holding a theory of mind about the rest); on the other hand, it employs statistical techniques and approximate solutions to retain computability of the inherent computational problem.

Random permutation set (RPS) is a recently proposed framework designed to represent order-structured uncertain information. Measuring the distance between permutation mass functions is a key research topic in RPS theory (RPST). This paper conducts an in-depth analysis of distances between RPSs from two different perspectives: random finite set (RFS) and transferable belief model (TBM). Adopting the layer-2 belief structure interpretation of RPS, we regard RPST as a refinement of TBM, where the order in the ordered focus set represents qualitative propensity. Starting from the permutation, we introduce a new definition of the cumulative Jaccard index to quantify the similarity between two permutations and further propose a distance measure method for RPSs based on the cumulative Jaccard index matrix. The metric and structural properties of the proposed distance measure are investigated, including the positive definiteness analysis of the cumulative Jaccard index matrix, and a correction scheme is provided. The proposed method has a natural top-weightiness property: inconsistencies between higher-ranked elements tend to result in greater distance values. Two parameters are provided to the decision-maker to adjust the weight and truncation depth. Several numerical examples are used to compare the proposed method with the existing method. The experimental results show that the proposed method not only overcomes the shortcomings of the existing method and is compatible with the Jousselme distance, but also has higher sensitivity and flexibility.

Abstract--Dempster-Shafer Theory (DST) provides a powerful framework for modeling uncertainty and has been widely applied to multi-attribute classification tasks. However, traditional DST - based attribute fusion-based classifiers suffer from oversimplified membership function modeling and limited exploitation of the belief structure brought by basic probability assignment (BPA), reducing their effectiveness in complex real-world scenarios. This paper presents an enhanced attribute fusion-based classifier that addresses these limitations through two key innovations. First, we adopt a selective modeling strategy that utilizes both single Gaussian and Gaussian Mixture Models (GMMs) for membership function construction, with model selection guided by cross-validation and a tailored evaluation metric. Second, we introduce a novel method to transform the possibility distribution into a BPA by combining simple BPAs derived from normalized possibility distributions, enabling a much richer and more flexible representation of uncertain information. Furthermore, we apply the belief structure-based BPA generation method to the evidential K-Nearest Neighbors (EKNN) classifier, enhancing its ability to incorporate uncertainty information into decision-making. Comprehensive experiments on benchmark datasets are conducted to evaluate the performance of the proposed attribute fusion-based classifier and the enhanced evidential K-Nearest Neighbors classifier in comparison with both evidential classifiers and conventional machine learning classifiers. The results demonstrate that the proposed classifier outperforms the best existing evidential classifier, achieving an average accuracy improvement of 4.86%, while maintaining low variance, thus confirming its superior effectiveness and robustness.

Theory of Mind is a fundamental cognitive ability that underpins human social behavior, enabling individuals to infer the beliefs, intentions, and knowledge of others. In this paper, we propose BeliefNest, an open-source simulator designed to support research on collaborative behavior in embodied agents endowed with Theory of Mind capabilities. Recent advances in embodied agents powered by large language models (LLMs) have shown promising progress. However, there is still no platform that can explicitly represent nested belief states and integrate them with action generation mechanisms. BeliefNest addresses this gap by providing a flexible simulation framework that incorporates both hierarchical belief structures and prompt generation support. BeliefNest offers the following features: Explicit representation of nested belief states, as studied in Theory of Mind, using hierarchical simulators (see Section 3) Support for prompt generation based on each belief state, enabling the design and evaluation of methods for agent control with LLMs (see Section 5) Integration with the Minecraft environment, which is widely used in LLM agent research [1-4], and support for open-domain tasks In this paper, we describe the design and functionality of BeliefNest and demonstrate its effectiveness through experiments on false-belief tasks.

We show how the AGM framework for belief change (expansion, revision, contraction) can be extended to deal with conditioning in the so-called Desirability-Indifference framework, based on abstract notions of accepting and rejecting options, as well as on abstract notions of events. This level of abstraction allows us to deal simultaneously with classical and quantum probability theory.

In a recent paper Miranda & Zaffalon (2020) some results about compatibility or consistency of coherent sets of gambles or lower previsisons have been derived and it was remarked that these results were in fact results of the theory of information or valuation algebras (Kohlas, 2003). This point of view, however, was not worked out by Miranda & Zaffalon (2020). In this paper this issue is taken up and it is shown that coherent sets of gambles, strictly desirable sets of gambles, coherent lower and upper previsions indeed form idempotent information algebras. Like in group theory, certain results concerning particular groups follow from general group theory, so many known results about desirable gambles, lower and linear previsions are indeed properties of an information algebra and follow from the corresponding general theory. Some of these results are discussed in this paper, but there are doubtless many other properties which can be derived from the theory of information algebra.

Its application involves a wide range of area including expert systems[3][4][5], information fusion[6], pattern classfication[7][8][9], risk evaluation [10,11] [12], image recognition [13], classification[14,15] and data mining [16] etc. The original DS theory requires deterministic belie degrees and belief structures. However, in practical situations, evidence coming from multiple sources may be influenced by unexpected extraneous factors. The lack of information, linguistic ambiguity or vagueness and cognitive bias all contribute to the uncertain evidence obtained in practical situations. For example, during risk assessment, expert may be unable to provide a precise assessment if he/she is not 100% sure.

Much work on social media opinion polarization focuses on identifying separate or orthogonal beliefs from media traces, thereby missing points of agreement among different communities. This paper develops a new class of Non-negative Matrix Factorization (NMF) algorithms that allow identification of both agreement and disagreement points when beliefs of different communities partially overlap. Specifically, we propose a novel Belief Structured Matrix Factorization algorithm (BSMF) to identify partially overlapping beliefs in polarized public social media. BSMF is totally unsupervised and considers three types of information: (i) who posted which opinion, (ii) keyword-level message similarity, and (iii) empirically observed social dependency graphs (e.g., retweet graphs), to improve belief separation. In the space of unsupervised belief separation algorithms, the emphasis was mostly given to the problem of identifying disjoint (e.g., conflicting) beliefs. The case when individuals with different beliefs agree on some subset of points was less explored. We observe that social beliefs overlap even in polarized scenarios. Our proposed unsupervised algorithm captures both the latent belief intersections and dissimilarities. We discuss the properties of the algorithm and conduct extensive experiments on both synthetic data and real-world datasets. The results show that our model outperforms all compared baselines by a great margin.

We introduce a semantics for epistemic logic exploiting a belief base abstraction. Differently from existing Kripke-style semantics for epistemic logic in which the notions of possible world and epistemic alternative are primitive, in the proposed semantics they are non-primitive but are defined from the concept of belief base. We show that this semantics allows us to define the universal epistemic model in a simpler and more compact way than existing inductive constructions of it. We provide (i) a number of semantic equivalence results for both the basic epistemic language with "individual belief" operators and its extension by the notion of "only believing", and (ii) a lower bound complexity result for epistemic logic model checking relative to the universal epistemic model.