Pooling multiple beliefs and opinions is often crucial in multi-agent systems. Specifically, we consider pooling in the context of multi-hypothesis social learning problems, where agents gather evidence and pool information to determine the true hypothesis. Whilst past research has largely focused on designing probability pooling operators to satisfy certain desirable properties, there has been limited exploration of learning operators directly from data. The conventional method, relying on well-motivated axioms and heuristics for pooling, constrains the solution space for operators and choosing between different operators often requires domain specific knowledge. We introduce a machine learning approach to opinion pooling that derives operators via end-to-end reinforcement learning incorporating DeepSets neural networks. We provide empirical results showing that our approach outperforms existing baselines across a range of social learning contexts and exhibits robust generalisation to the number of agents pooled in a single operation. Furthermore, we provide an analysis of the learned neural network representations, offering insights into optimal opinion pooling strategies.

Threshold models in which an individual's response to a particular state of the world depends on whether an associated measured value exceeds a given threshold are common in a variety of social learning and collective decision-making scenarios in both natural and artificial systems. If thresholds are heterogeneous across a population of agents, then graded population level responses can emerge in a context in which individual responses are discrete and limited. In this article, I propose a threshold-based model for social learning of shared quality categories. This is then combined with the voting model of fuzzy categories to allow individuals to learn membership functions from their peers, which can then be used for decision-making, including ranking a set of available options. I use agent-based simulation experiments to investigate variants of this model and compare them to an individual learning benchmark when applied to the ranking problem. These results show that a threshold-based approach combined with category-based voting across a social network provides an effective social mechanism for ranking that exploits emergent vagueness.

In explainable artificial intelligence, there is increasing interest in understanding the behaviour of autonomous agents to build trust and validate performance. Modern agent architectures, such as those trained by deep reinforcement learning, are currently so lacking in interpretable structure as to effectively be black boxes, but insights may still be gained from an external, behaviourist perspective. Inspired by conceptual spaces theory, we suggest that a versatile first step towards general understanding is to discretise the state space into convex regions, jointly capturing similarities over the agent's action, value function and temporal dynamics within a dataset of observations. We create such a representation using a novel variant of the CART decision tree algorithm, and demonstrate how it facilitates practical understanding of black box agents through prediction, visualisation and rule-based explanation.

The evidence available to a multi-agent system can take at least two distinct forms. There can be direct evidence from the environment resulting, for example, from sensor measurements or from running tests or experiments. In addition, agents also gain evidence from other individuals in the population with whom they are interacting. We, therefore, envisage an agent's beliefs as a probability distribution over a set of hypotheses of interest, which are updated either on the basis of direct evidence using Bayesian updating, or by taking account of the probabilities of other agents using opinion pooling. This paper investigates the relationship between these two processes in a multi-agent setting. We consider a possible Bayesian interpretation of probability pooling and then explore properties for pooling operators governing the extent to which direct evidence is diluted, preserved or amplified by the pooling process. We then use simulation experiments to show that pooling operators can provide a mechanism by which a limited amount of direct evidence can be efficiently propagated through a population of agents so that an appropriate consensus is reached. In particular, we explore the convergence properties of a parameterised family of operators with a range of evidence propagation strengths.

This paper investigates the problem of measuring consensus in multi-perspective Multi-Criteria Group Decision Making (MCGDM) problems, in which participants have individual views on the relative importance of different evaluation criteria. A novel dual consensus measure for multi-perspective MCGDM problems is introduced. The proposed measure determines the level of agreement between participants' opinions based on: (i) the global performance or satisfaction of alternatives, (ii) their partial performances of alternatives under each criterion, and (iii) the similarity between the perspectives of participants regarding criteria weights. Preliminary experiments are conducted for an example multi-perspective MCGDM scenario. The degree to which global and partial performance information are jointly taken into account - together with the actual pairwise distances between the opinions of participants - are shown to directly affect the overall measurement of consensus in the group. An application example is introduced in a MCGDM problem on selecting the safest logistic route to transport hazardous materials.

Autonomous robots must operate in complex and changing environments subject to requirements on their behaviour. Verifying absolute satisfaction (true or false) of these requirements is challenging. Instead, we analyse requirements that admit flexible degrees of satisfaction. We analyse vague requirements using fuzzy logic, and probabilistic requirements using model checking. The resulting analysis method provides a partial ordering of system designs, identifying trade-offs between different requirements in terms of the degrees to which they are satisfied. A case study involving a home care robot interacting with a human is used to demonstrate the approach.

In this paper, we investigate the best-of-n distributed decision problem in robot swarms. In this context, we compare the weighted voter model [25] with a three-valued model that incorporates an intermediate belief state meaning either `uncertain' or `indifferent'. We focus particularly on the trade-off between speed of convergence to a shared belief, and robustness to the presence of unreliable individuals in the population. By means of both simulation and embodied experiments in real robot swarms of 400 Kilobots, we show that the three-valued model is much more robust than the weighted voter model, but with decreased speed of convergence.

We introduce a hierarchical framework for conjunctive concept combination based on conceptual spaces and random set theory. The model has the flexibility to account for composition of concepts at various levels of complexity. We show that the conjunctive model includes linear combination as a special case, and that the more general model can account for non-compositional behaviours such as overextension, non-commutativity, preservation of necessity and impossibility of attributes and to some extent, attribute loss or emergence. We investigate two further aspects of human concept use, the conjunction fallacy and the ‘guppy effect’.

The idea of a Cloned Controller to approximate optimised control algorithms in a real-time environment is introduced. A Cloned Controller is demonstrated using Linguistic Decision Trees (LDTs) to clone a Model Predictive Controller (MPC) based on Mixed Integer Linear Programming (MILP) for Unmanned Aerial Vehicle (UAV) path planning through a hostile environment. Modifications to the LDT algorithm are proposed to account for attributes with circular domains, such as bearings, and discontinuous output functions. The cloned controller is shown to produce near optimal paths whilst significantly reducing the decision period. Further investigation shows that the cloned controller generalises to the multi-obstacle case although this can lead to situations far outside of the training dataset and consequently result in decisions with a high level of uncertainty. A modification to the algorithm to improve the performance in regions of high uncertainty is proposed and shown to further enhance generalisation. The resulting controller combines the high performance of MPC–MILP with the rapid response of an LDT while providing a degree of transparency/interpretability of the decision making.

This paper is a review of the label semantics framework as an epistemic approach to modeling granular information represented by linguistic labels and label expressions. The focus of label semantics is on the decision-making process that a rational communicating agent must undertake in order to establish which available labels can be appropriately used to describe their perceptual information in such a way as they are consistent with the linguistic conventions of the population. As such, it provides an approach to characterizing the relationship between labels and the underlying perceptual domain which, we propose, lies at the heart of what is meant by information granules. Furthermore, it is then shown that there is an intuitive relationship between label semantics and prototype theory, which provides a clear link with Zadeh's original conception of information granularity. For information propagation, linguistic mappings are introduced, which provide a mechanism to infer labeling information about a decision variable from the available labeling information about a set of input variables. Finally, a decision-making process is outlined whereby from linguistic descriptions of input variables, we can infer a linguistic description of the decision variable and, where required, select a single expression describing that variable or a single estimated value.

The seven papers in this special section provide an overview of the main ideas and research directions for computing with words.

Linguistic Decision Trees based on label semantics have been used as a classifier or predictor in many areas. A linguistic decision tree presents information propagation from input attributes to a goal variable based on transparent linguistic rules. The relationship between input attributes and the goal variable is often highly nonlinear. Cerebellar Model Articulation Controller (CMAC) belongs to the family of feed-forward networks with a single linear trainable layer. A CMAC has the feature of fast learning, and is suitable for modeling any non-linear relationship. Combining label semantics and an original CMAC, a linguistic CMAC based on Mass Assignment on labels is proposed to map the relationship between the attributes and the goal variable. The proposed LCMAC model is functionally equivalent to a linguistic decision tree, and takes the advantage of fast local training of the original CMAC and the advantage of transparency of a linguistic decision tree.

We investigate the propagation of label informa tion for multi-attribute decision maldng problems downwards a linguistic attribute hierarchy, which represents the complex and often imprecise functional relationships between low level attributes or measurements and high-level decision or classification variables. The downward propagation algorithm identifies the branches in linguistic decision trees for which the probability of a high-level goal exceeds a given threshold. The sensitivity of the method to this threshold is then reduced by integrating with respect to a probability distribution on high threshold values.

This paper focuses on the application of LID3 (linguistic decision tree induction algorithm) to real-time flood forecasting. Specifically the prediction of the river level at locations along the River Severn, Britainpsilas largest river. Modelling river dynamics implies modelling a system that changes over time. It is therefore inappropriate to use a static model to model river levels, that are driven by an underlying dynamic system. Hence, an updateable version of LID3 is proposed. There are two main features of ULID3 (updateable LID3). The first being error-based updating, which weights new instances depending on the treepsilas current ability to describe each new example. The ability to update probability distributions at each node enables the tree to adapt and capture the new dynamic concept more effectively. The second feature is the ability to extend both the input and output domains, given new examples. This is necessary when the data available for updating, exceeds the current domain set by the training data. An algorithm is presented to update the new probability distributions throughout the tree, without the need for storing the complete set of examples at each node.

We propose label semantics as an integrated representation framework for probabilistic uncertainty and fuzziness in multiple-attribute decision making problems. Linguistic attribute hierarchies are then introduced as a means of modelling the complex and often imprecise functional relationships between low-level attributes or measurements and high-level decision or classification variables. Within this framework we introduce linguistic decision trees as a tool for information aggregation in multi-attribute decision problems and describe the process of information propagation through a hierarchy of linked decision trees. In addition, we consider the ranking of different alternatives or examples based on their linguistic descriptions of a high-level utility variable. Finally, we discuss how linguistic decision trees embedded in attribute hierarchies can be learnt from data.

In this paper, we introduce a new comparison relation on fuzzy numbers based on their alpha-cut representation and comparison probabilities of interval values. Basically, this comparison process combines a widely accepted interpretation of fuzzy sets together with the uncertain characteristics inherent in the representation of fuzzy numbers. The proposed comparison relation is then applied to the issue of ranking fuzzy numbers using fuzzy targets in terms of target-based evaluations. Some numerical examples are used to illuminate the proposed ranking technique as well as to compare with previous methods. More interestingly, according to the interpretation of the new comparison relation on fuzzy numbers, we provide a fuzzy target-based decision model as a solution to the problem of decision making under uncertainty, with which an interesting link between the decision maker's different attitudes about target and different risk attitudes in terms of utility functions can be established. Moreover, an application of the proposed comparison relation to the fuzzy target-based decision model for the problem of fuzzy decision making with uncertainty is provided. Numerical examples are also given for illustration.

A fuzzy Bayesian algorithm is introduced, allowing for the incorporation of both uncertainty and fuzziness into data derived models. This is applied to predicting the sea-level near the Thames Estuary at Sheerness, from tidal gauge measurements down the east coast, astronomical tidal prediction, and meteorological data. We show that this approach can result in accurate, low-dimensional models with low computational costs and relatively fast execution times.

We propose label semantics as an integrated representation framework for probabilistic uncertainty and fuzziness in multiple-attribute decision making problems. Linguistic attribute hierarchies are then introduced as a means of modelling the complex and often imprecise functional relationships between low-level attributes or measurements and high-level decision or classification variables.

This paper focuses on the application of LID3 (linguistic decision tree induction algorithm) to the classification of weather radar images. In radar analysis a phenomenon known as bright band occurs. This essentially is an amplification in reflectivity due to melted snow and leads to overestimation of precipitation. It is therefore beneficial to detect this bright band region and apply the appropriate corrections. This paper uses LID3 in order to identify the bright band region pixel by pixel in real time. This is not possible with the current differencing methods currently used for bright band detection. LID3 also allows us to infer a set of linguistic rules to further our understanding of the relationship between radar measurements and the classification of bright band. A new idea called conditional labeling is proposed, which attempts to ensure a more efficiently partitioned space, omitting relatively sparse branches caused by attribute dependencies.

In the sequel, we shall be concerned with the automated induction of prototypes to represent a database in a way that combines transparency and accuracy. More precisely, our aim is to automatically summarize the information from a set of data using prototypes and simultaneously decide on the number of prototypes needed to the represent the data adequately. We propose to use fuzzy prototypes, which correspond to groupings of similar objects represented by tuples of fuzzy sets over attribute universes. In the case of numerical attributes, the universes will be discretized using linguistic variables and the attribute values will be described by fuzzy sets on words. To learn the prototypes, we shall present an algorithm inspired by hierarchical clustering techniques, which makes use of new measures of prototype similarity and of prototype addition. Also, a pruning algorithm will be incorporated which uses a new heuristic measure of prototype quality to fuse overly specific prototypes. The potential of the resulting method will be illustrated by its application to a number of classification problems and comparing its performance with that of previous approaches in the literature.

With the wealth of information available in the world today the challenge of how to extract information from several data sources in an intuitive and transparent manner has emerged in machine learning. This article describes a method for learning models from a database using linguistic descriptions on fuzzy sets and fusion methods in a data-mining framework. It focuses on AND/OR combination functions and shows how these can be optimized for knowledge extraction and classification of data. © 2003 Wiley Periodicals, Inc.

A new approach to normalizing fuzzy sets is introduced where it is assumed that the normalization method is compatible with a given t-norm. In this context it is proved that the most usual ways to normalize fuzzy subsets correspond to the most common t-norms. For a given fuzzy subset μ, the corresponding normalized fuzzy subset [Formula: see text] can be viewed as the distribution of μ conditioned on the (degree of) existence of its elements with maximal membership. From this view point we investigate the less specific normal fuzzy subset of X among the most similar fuzzy subsets to μ and the normal fuzzy subset generating the same fuzzy T-preorder as μ.

A framework for modelling with words is introduced based on label semantics. It is shown how within this framework that linguistic prototypes, defined as vectors of mass assignments on sets of labels, can be used to evaluate linguistic queries. This provides a flexible knowledge representation framework for data mining and knowledge discovery as well as an environment well suited to information fusion and modelling with words in general.

The problem of normalisation of non-normalised fuzzy sets/possibility distributions is considered. A definition of a valid normalised possibility distribution is given at the mass assignment level and a characterisation of this definition at the distribution level is proposed. A number of possible normalisations are considered these being related to certain epistemic principles. Finally an interval approach to reasoning with partially inconsistent knowledge is introduced.

A c-fuzzy means algorithm is described. The algorithm learns fuzzy prototypes to represent data sets and is based on ideas taken from mass assignment theory. The potential of this approach for both unsupervised and supervised learning is illustrated by its application to a number of benchmark and model problems.

The mass assignment ID3 (MA-ID3) algorithm for generating linguistic decision trees is introduced together with the mass assignment semantics for linguistic variables. The potential of this algorithm for learning control rules is illustrated by means of the Van de Pol system. A data set of control paths is generated using an existing on-line controller. This is then used to generate a set of quantified linguistic control rules. The effectiveness and robustness of this rule-base is then demonstrated.

A form of prototypes defined as tuples of marginal probability distributions is introduced. Addition and subtraction operations on such prototypes are then described. A brief introduction to the mass assignment theory of the probability of fuzzy events is given, and it is shown how fuzzy sets can serve as conceptual descriptions of probability distributions. Hence fuzzy descriptions of prototypes can be derived and these can be used for inference as well as enabling rule based representations of a set of prototypes to be formed. A prototype induction algorithm, based on these ideas together with the addition and subtraction operations, is described. The potential of this approach is then illustrated by its application to a number of model and real world machine learning problems. ©1999 John Wiley & Sons, Inc.

Notions of generalised fuzzy conditional and equivalence rules relative to a combination function are introduced and a framework for reasoning with such rules is described. The applicability of this framework to machine learning and knowledge discovery problems is demonstrated. Methods for the automatic generation of two particular types of generalised rule are proposed. The two rule forms considered are rules with weighted AND/OR combination functions, as suggested by Zimmerman and Zysno, and evidential logic equivalence rules as defined by Baldwin. The process of generating rule bases is divided into the problem of generating fuzzy sets from data and that of finding combination functions to optimise the performance of the system given these fuzzy sets. For the former problem a mass assignment based approach is adopted and for the latter semantic discrimination analysis is used in conjunction with customised optimisation algorithms. The potential of rule bases of both forms is illustrated by their application to a number of model and real world machine learning problems.

A mass assignment based ID3 algorithm for learning probabilistic fuzzy decision trees is introduced. Fuzzy partitions are used to discretize continuous feature universes and to reduce complexity when universes are discrete but with large cardinalities. Furthermore, the fuzzy partitioning of classification universes facilitates the use of these decision trees in function approximation problems. Generally the incorporation of fuzzy sets into this paradigm overcomes many of the problems associated with the application of decision trees to real-world problems. The probabilities required for the trees are calculated according to mass assignment theory applied to fuzzy labels. The latter concept is introduced to overcome computational complexity problems associated with higher dimensional mass assignment evaluations on databases. ©1997 John Wiley & Sons, Inc.