Behavioural distances generally offer more fine-grained means of comparing quantitative systems than two-valued behavioural equivalences. They often relate to quantitative modalities, which generate quantitative modal logics that characterize a given behavioural distance in terms of the induced logical distance. We develop a unified framework for behavioural distances and logics induced by a special type of modalities that lift two-valued predicates to quantitative predicates. A typical example is the probability operator, which maps a two-valued predicate $A$ to a quantitative predicate on probability distributions assigning to each distribution the respective probability of $A$. Correspondingly, the prototypical example of our framework is $ε$-bisimulation distance of Markov chains, which has recently been shown to coincide with the behavioural distance induced by the popular Lévy-Prokhorov distance on distributions. Other examples include behavioural distance on metric transition systems and Hausdorff behavioural distance on fuzzy transition systems. Our main generic results concern the polynomial-time extraction of distinguishing formulae in two characteristic modal logics: A t
Infectious disease transmission in human populations has a complex two-way interaction with changes in host behaviour. It is increasingly recognised that incorporating adaptive behavioural change into epidemic models is important for improving understanding of infectious disease dynamics and developing policy-relevant modelling tools. An important aspect of behavioural dynamics is social contagion, where people tend to adopt behaviours exhibited by others around them. In a simple behavioural contagion model, the behaviour uptake rate increases linearly with the number of contacts who have adopted a given behaviour. Here, we explore an epidemic model with complex behavioural contagion, where the behaviour uptake rate is a nonlinear function of the number of behaving contacts. We identify key bifurcation parameters of the model, which include the basic reproduction number $R_0$, the strength of the behavioural effect on disease transmission, and the speed of behaviour uptake relative to behaviour abandonment. We show that, in some regions of parameter space, the model has multiple disease-free equilibria. In this situation, the occurrence of an epidemic in a population with an initia
Behavioural distances provide a quantitative approach to comparing the states of transition systems, moving beyond traditional Boolean notions of equivalence. In this paper, we develop a sound and complete axiomatisation of behavioural distance for nondeterministic processes using Milner's charts, a model that generalises finite-state automata by incorporating variable outputs. Charts provide a compelling setting for studying behavioural distances because they shift the focus from language equivalence to bisimilarity. Their axiomatic study lays the groundwork for quantitative analysis of more expressive models, such as weighted transition systems. To formalise this approach, we adopt string diagrams as our syntax of choice. String diagrams closely mirror the graphical structure of charts, while providing a rigorous formalism that supports inductive reasoning and compositional semantics. Unlike traditional algebraic syntaxes, which require additional mechanisms such as binders and substitution, string diagrams offer a variable-free representation where recursion naturally decomposes into simpler components. This makes them well-suited for reasoning about behavioural distances and al
Behavioural conformances -- e.g. behavioural equivalences, distances, preorders -- on a wide range of system types (non-deterministic, probabilistic, weighted etc.) can be dealt with uniformly in the paradigm of universal coalgebra. One of the most commonly used constructions for defining behavioural distances on coalgebras arises as a generalization of the well-known Wasserstein metric. In this construction, couplings of probability distributions are replaced with couplings of more general objects, depending on the functor describing the system type. In many cases, however, the set of couplings of two functor elements is empty, which causes such elements to have infinite distance even in situations where this is not desirable. We propose an approach to defining behavioural distances and preorders based on a more liberal notion of coupling where the coupled elements are matched laxly rather than on-the-nose. We thereby substantially broaden the range of behavioural conformances expressible in terms of couplings, covering, e.g., refinement of modal transition systems and behavioural distance on metric labelled Markov chains.
Coinduction is a widely used technique for establishing behavioural equivalence of programs in higher-order languages. In recent years, the rise of languages with quantitative (e.g.~probabilistic) features has led to extensions of coinductive methods to more refined types of behavioural conformances, most notably notions of behavioural distance. To guarantee soundness of coinductive reasoning, one needs to show that the behavioural conformance at hand forms a program congruence, i.e. it is suitably compatible with the operations of the language. This is usually achieved by a complex proof technique known as \emph{Howe's method}, which needs to be carefully adapted to both the specific language and the targeted notion of behavioural conformance. We develop a uniform categorical approach to Howe's method that features two orthogonal dimensions of abstraction: (1) the underlying higher-order language is modelled by an \emph{abstract higher-order specification} (AHOS), a novel and very general categorical account of operational semantics, and (2) notions of behavioural conformance (such as relations or metrics) are modelled via fibrations over the base category of an AHOS. Our main res
The 1mm roundworm C. elegans is a model organism used in many sub-areas of biology to investigate different types of biological processes. In order to complement the n-vivo analysis with computer-based investigations, several methods have been proposed to simulate the worm behaviour. These methods extract discrete behavioural units from the flow of the worm movements using different types of tracking techniques. Nevertheless, these techniques require a clear view of the entire worm body, which is not always achievable. For example, this happens in high density worm conditions, which are particularly informative to understand the influence of the social context on the single worm behaviour. In this paper, we illustrate and evaluate a method to extract behavioural units from recordings of C. elegans movements which do not necessarily require a clear view of the entire worm body. Moreover, the behavioural units are defined by an unsupervised automatic pipeline which frees the process from predefined assumptions that inevitably bias the behavioural analysis. The behavioural units resulting from the automatic method are interpreted by comparing them with hand-designed behavioural units.
This PhD thesis discusses how European law could improve privacy protection in the area of behavioural targeting. Behavioural targeting, also referred to as online profiling, involves monitoring people's online behaviour, and using the collected information to show people individually targeted advertisements. To protect privacy in the area of behavioural targeting, the EU lawmaker mainly relies on the consent requirement for the use of tracking technologies in the e-Privacy Directive, and on general data protection law. With informed consent requirements, the law aims to empower people to make choices in their best interests. But behavioural studies cast doubt on the effectiveness of the empowerment approach as a privacy protection measure. Many people click "I agree" to any statement that is presented to them. Therefore, to mitigate privacy problems such as chilling effects, this study argues for a combined approach of protecting and empowering the individual. Compared to the current approach, the lawmaker should focus more on protecting people. The PhD thesis is a legal study, but it also incorporates insights from other disciplines, such as computer science, behavioural economic
Behavioural distances provide a robust alternative to notions of equivalence such as bisimilarity in the context of probabilistic transition systems. They can be defined as least fixed points, whose universal property allows us to exhibit upper bounds on the distance between states, showing them to be at most some distance apart. In this paper, we instead consider the problem of bounding distances from below, showing states to be at least some distance apart. Contrary to upper bounds, it is possible to reason about lower bounds inductively. We exploit this by giving an inductive derivation system for lower bounds on an existing definition of behavioural distance for labelled Markov chains. This is inspired by recent work on apartness as an inductive counterpart to bisimilarity. Proofs in our system will be shown to closely match the behavioural distance by soundness and (approximate) completeness results. We further provide a constructive correspondence between our derivation system and formulas in a modal logic with quantitative semantics. This logic was used in recent work of Rady and van Breugel to construct evidence for lower bounds on behavioural distances. Our constructions p
The paper introduces a novel behavioural translation style space (BTSS) that describes possible behavioural translation patterns. The suggested BTSS is organized as a hierarchical structure that entails various embedded processing layers. We posit that observable translation behaviour - i.e. eye and finger movements - is fundamental when executing the physical act of translation but it is caused and shaped by higher-order cognitive processes and affective translation states. We analyse records of keystrokes and gaze data as indicators of the hidden mental processing structure and organize the behavioural patterns as a multi-layered embedded BTSS. We develop a perspective in which the BTSS serves as the basis for a computational translation agent to simulate the temporal dynamics of affect, behavioural routines and cognition during human translation production.
Behavioural analytics provides insights into individual and crowd behaviour, enabling analysis of what previously happened and predictions for how people may be likely to act in the future. In defence and security, this analysis allows organisations to achieve tactical and strategic advantage through influence campaigns, a key counterpart to physical activities. Before action can be taken, online and real-world behaviour must be analysed to determine the level of threat. Huge data volumes mean that automated processes are required to attain an accurate understanding of risk. We describe the mathematical basis of technologies to analyse quotes in multiple languages. These include a Bayesian network to understand behavioural factors, state estimation algorithms for time series analysis, and machine learning algorithms for classification. We present results from studies of quotes in English, French, and Arabic, from anti-violence campaigners, politicians, extremists, and terrorists. The algorithms correctly identify extreme statements; and analysis at individual, group, and population levels detects both trends over time and sharp changes attributed to major geopolitical events. Group
In behavioural testing, system functionalities underrepresented in the standard evaluation setting (with a held-out test set) are validated through controlled input-output pairs. Optimising performance on the behavioural tests during training (behavioural learning) would improve coverage of phenomena not sufficiently represented in the i.i.d. data and could lead to seemingly more robust models. However, there is the risk that the model narrowly captures spurious correlations from the behavioural test suite, leading to overestimation and misrepresentation of model performance -- one of the original pitfalls of traditional evaluation. In this work, we introduce BeLUGA, an analysis method for evaluating behavioural learning considering generalisation across dimensions of different granularity levels. We optimise behaviour-specific loss functions and evaluate models on several partitions of the behavioural test suite controlled to leave out specific phenomena. An aggregate score measures generalisation to unseen functionalities (or overfitting). We use BeLUGA to examine three representative NLP tasks (sentiment analysis, paraphrase identification and reading comprehension) and compare
Behavioural types provide a promising way to achieve lightweight, language-integrated verification for communication-centric software. However, a large barrier to the adoption of behavioural types is that the current state of the art expects software to be written using the same tools and typing discipline throughout a system, and has little support for components over which a developer has no control. This position paper describes the outcomes of a working group discussion at Dagstuhl Seminar 24051 (Next-Generation Protocols for Heterogeneous Systems). We propose a methodology for integrating multiple behaviourally-typed components, written in different languages. Our proposed approach involves an extensible protocol description language, a session IR that can describe data transformations and boundary monitoring and which can be compiled into program-specific session proxies, and finally a session middleware to aid session establishment. We hope that this position paper will stimulate discussion on one of the most pressing challenges facing the widespread adoption of behavioural typing.
Behavioural metrics provide a quantitative refinement of classical two-valued behavioural equivalences on systems with quantitative data, such as metric or probabilistic transition systems. In analogy to the linear-time/branching-time spectrum of two-valued behavioural equivalences on transition systems, behavioural metrics vary in granularity, and are often characterized by fragments of suitable modal logics. In the latter respect, the quantitative case is, however, more involved than the two-valued one; in fact, we show that probabilistic metric trace distance cannot be characterized by any compositionally defined modal logic with unary modalities. We go on to provide a unifying treatment of spectra of behavioural metrics in the emerging framework of graded monads, working in coalgebraic generality, that is, parametrically in the system type. In the ensuing development of quantitative graded semantics, we introduce algebraic presentations of graded monads on the category of metric spaces. Moreover, we provide a general criterion for a given real-valued modal logic to characterize a given behavioural distance. As a case study, we apply this criterion to obtain a new characteristic
Bisimulation is a concept that captures behavioural equivalence of states in a variety of types of transition systems. It has been widely studied in discrete-time settings where a key notion is the bisimulation metric which quantifies "how similar two states are". In [ 11], we generalized the concept of bisimulation metric in order to metrize the behaviour of continuous-time Markov processes. Similarly to the discrete-time case, we constructed a pseudometric following two iterative approaches - through a functional and through a real-valued logic, and showed that the outcomes coincide: the pseudometric obtained from the logic is a specific fixpoint of the functional which yields our first pseudometric. However, different from the discrete-time setting, in which the process has a step-by-step dynamics, the behavioural pseudometric we constructed applies to Markov processes that evolve continuously through time, such as diffusions and jump diffusions. While our treatment of the pseudometric in [11] relied on the time-indexed Markov kernels, in [ 8 , 9, 10 ], we showed the importance of trajectories in the consideration of behavioural equivalences for true continuous-time Markov proce
Bisimilarity is a central notion for coalgebras. In recent work, Geuvers and Jacobs suggest to focus on apartness, which they define by dualising coalgebraic bisimulations. This yields the possibility of finite proofs of distinguishability for a wide variety of state-based systems. We propose behavioural apartness, defined by dualising behavioural equivalence rather than bisimulations. A motivating example is the subdistribution functor, where the proof system based on bisimilarity requires an infinite quantification over couplings, whereas behavioural apartness instantiates to a finite rule. In addition, we provide optimised proof rules for behavioural apartness and show their use in several examples.
This chapter examines the policy implications of behavioural sciences insights for the regulation of privacy on the Internet, by focusing in particular on behavioural targeting. This marketing technique involves tracking people's online behaviour to use the collected information to show people individually targeted advertisements. Enforcing data protection law may not be enough to protect privacy in this area. I argue that, if society is better off when certain behavioural targeting practices do not happen, policymakers should consider banning them.
Federated Learning (FL), a privacy-aware approach in distributed deep learning environments, enables many clients to collaboratively train a model without sharing sensitive data, thereby reducing privacy risks. However, enabling human trust and control over FL systems requires understanding the evolving behaviour of clients, whether beneficial or detrimental for the training, which still represents a key challenge in the current literature. To address this challenge, we introduce Federated Behavioural Planes (FBPs), a novel method to analyse, visualise, and explain the dynamics of FL systems, showing how clients behave under two different lenses: predictive performance (error behavioural space) and decision-making processes (counterfactual behavioural space). Our experiments demonstrate that FBPs provide informative trajectories describing the evolving states of clients and their contributions to the global model, thereby enabling the identification of clusters of clients with similar behaviours. Leveraging the patterns identified by FBPs, we propose a robust aggregation technique named Federated Behavioural Shields to detect malicious or noisy client models, thereby enhancing secu
This paper describes methods for training autonomous agents to play the game "Doom 2" through Imitation Learning (IL) using only pixel data as input. We also explore how Reinforcement Learning (RL) compares to IL for humanness by comparing camera movement and trajectory data. Through behavioural cloning, we examine the ability of individual models to learn varying behavioural traits. We attempt to mimic the behaviour of real players with different play styles, and find we can train agents that behave aggressively, passively, or simply more human-like than traditional AIs. We propose these methods of introducing more depth and human-like behaviour to agents in video games. The trained IL agents perform on par with the average players in our dataset, whilst outperforming the worst players. While performance was not as strong as common RL approaches, it provides much stronger human-like behavioural traits to the agent.
Trajectory sampling in the Frenet(road-aligned) frame, is one of the most popular methods for motion planning of autonomous vehicles. It operates by sampling a set of behavioural inputs, such as lane offset and forward speed, before solving a trajectory optimization problem conditioned on the sampled inputs. The sampling is handcrafted based on simple heuristics, does not adapt to driving scenarios, and is oblivious to the capabilities of downstream trajectory planners. In this paper, we propose an end-to-end learning of behavioural input distribution from expert demonstrations or in a self-supervised manner. Our core novelty lies in embedding a custom differentiable trajectory optimizer as a layer in neural networks, allowing us to update behavioural inputs by considering the optimizer's feedback. Moreover, our end-to-end approach also ensures that the learned behavioural inputs aid the convergence of the optimizer. We improve the state-of-the-art in the following aspects. First, we show that learned behavioural inputs substantially decrease collision rate while improving driving efficiency over handcrafted approaches. Second, our approach outperforms model predictive control meth
Reinforcement learning (RL) agents under partial observability often condition actions on internally accumulated information such as memory or inferred latent context. We formalise such information-conditioned interaction patterns as behavioural dependency: variation in action selection with respect to internal information under fixed observations. This induces a probe-relative notion of $ε$-behavioural equivalence and a within-policy behavioural distance that quantifies probe sensitivity. We establish three structural results. First, the set of policies exhibiting non-trivial behavioural dependency is not closed under convex aggregation. Second, behavioural distance contracts under convex combination. Third, we prove a sufficient local condition under which gradient ascent on a skewed mixture objective decreases behavioural distance when a dominant-mode gradient aligns with the direction of steepest contraction. Minimal bandit and partially observable gridworld experiments provide controlled witnesses of these mechanisms. In the examined settings, behavioural distance decreases under convex aggregation and under continued optimisation with skewed latent priors, and in these experi