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Published work

31 published item(s)

preprint2026arXiv

Guiding Neuro-Symbolic Scenario Generation with Spatio-Temporal Logic

The rapid advancement of autonomous driving (AD) technologies has outpaced the development of robust safety evaluation methods. Conventional testing relies on exposing AD systems to vast numbers of real-world traffic scenes -- a brute-force approach that is prohibitively expensive and statistically ineffective at capturing the rare, safety-critical edge cases essential for validating real-world robustness. To address this fundamental limitation, we introduce STRELGen, a scalable framework for the targeted generation of safety-critical driving scenarios. STRELGen synergistically combines a multi-agent trajectory-generation diffusion model (DM) with Spatio-Temporal Logic (STREL) specifications that encode complex safety and realism properties through a highly interpretable formalism. Crucially, monitoring satisfaction levels of these specifications is differentiable, enabling gradient-based search. At inference time, we optimize directly over the DM latent space to maximize STREL formula satisfaction. The result is efficient generation of highly plausible yet safety-critical multi-agent scenarios that lie within the learned data distribution. STRELGen thus provides a flexible, interpretable, and powerful tool for stress-testing autonomous driving systems, moving beyond the limitations of brute-force data collection.

preprint2022arXiv

Graph Neural Networks for Propositional Model Counting

Graph Neural Networks (GNNs) have been recently leveraged to solve several logical reasoning tasks. Nevertheless, counting problems such as propositional model counting (#SAT) are still mostly approached with traditional solvers. Here we tackle this gap by presenting an architecture based on the GNN framework for belief propagation (BP) of Kuch et al., extended with self-attentive GNN and trained to approximately solve the #SAT problem. We ran a thorough experimental investigation, showing that our model, trained on a small set of random Boolean formulae, is able to scale effectively to much larger problem sizes, with comparable or better performances of state of the art approximate solvers. Moreover, we show that it can be efficiently fine-tuned to provide good generalization results on different formulae distributions, such as those coming from SAT-encoded combinatorial problems.

preprint2022arXiv

Learning Model Checking and the Kernel Trick for Signal Temporal Logic on Stochastic Processes

We introduce a similarity function on formulae of signal temporal logic (STL). It comes in the form of a kernel function, well known in machine learning as a conceptually and computationally efficient tool. The corresponding kernel trick allows us to circumvent the complicated process of feature extraction, i.e. the (typically manual) effort to identify the decisive properties of formulae so that learning can be applied. We demonstrate this consequence and its advantages on the task of predicting (quantitative) satisfaction of STL formulae on stochastic processes: Using our kernel and the kernel trick, we learn (i) computationally efficiently (ii) a practically precise predictor of satisfaction, (iii) avoiding the difficult task of finding a way to explicitly turn formulae into vectors of numbers in a sensible way. We back the high precision we have achieved in the experiments by a theoretically sound PAC guarantee, ensuring our procedure efficiently delivers a close-to-optimal predictor.

preprint2022arXiv

Resilience of Bayesian Layer-Wise Explanations under Adversarial Attacks

We consider the problem of the stability of saliency-based explanations of Neural Network predictions under adversarial attacks in a classification task. Saliency interpretations of deterministic Neural Networks are remarkably brittle even when the attacks fail, i.e. for attacks that do not change the classification label. We empirically show that interpretations provided by Bayesian Neural Networks are considerably more stable under adversarial perturbations of the inputs and even under direct attacks to the explanations. By leveraging recent results, we also provide a theoretical explanation of this result in terms of the geometry of the data manifold. Additionally, we discuss the stability of the interpretations of high level representations of the inputs in the internal layers of a Network. Our results demonstrate that Bayesian methods, in addition to being more robust to adversarial attacks, have the potential to provide more stable and interpretable assessments of Neural Network predictions.

preprint2020arXiv

Bounding Mean First Passage Times in Population Continuous-Time Markov Chains

We consider the problem of bounding mean first passage times for a class of continuous-time Markov chains that captures stochastic interactions between groups of identical agents. The quantitative analysis of such probabilistic population models is notoriously difficult since typically neither state-based numerical approaches nor methods based on stochastic sampling give efficient and accurate results. Here, we propose a technique that extends recently developed methods using semi-definite programming to determine bounds on mean first passage times. We further apply the technique to hybrid models and demonstrate its accuracy and efficiency for some examples from biology.

preprint2020arXiv

Density-embedding layers: a general framework for adaptive receptive fields

The effectiveness and performance of artificial neural networks, particularly for visual tasks, depends in crucial ways on the receptive field of neurons. The receptive field itself depends on the interplay between several architectural aspects, including sparsity, pooling, and activation functions. In recent literature there are several ad hoc proposals trying to make receptive fields more flexible and adaptive to data. For instance, different parameterizations of convolutional and pooling layers have been proposed to increase their adaptivity. In this paper, we propose the novel theoretical framework of density-embedded layers, generalizing the transformation represented by a neuron. Specifically, the affine transformation applied on the input is replaced by a scalar product of the input, suitably represented as a piecewise constant function, with a density function associated with the neuron. This density is shown to describe directly the receptive field of the neuron. Crucially, by suitably representing such a density as a linear combination of a parametric family of functions, we can efficiently train the densities by means of any automatic differentiation system, making it adaptable to the problem at hand, and computationally efficient to evaluate. This framework captures and generalizes recent methods, allowing a fine tuning of the receptive field. In the paper, we define some novel layers and we experimentally validate them on the classic MNIST dataset.

preprint2020arXiv

Robustness of Bayesian Neural Networks to Gradient-Based Attacks

Vulnerability to adversarial attacks is one of the principal hurdles to the adoption of deep learning in safety-critical applications. Despite significant efforts, both practical and theoretical, the problem remains open. In this paper, we analyse the geometry of adversarial attacks in the large-data, overparametrized limit for Bayesian Neural Networks (BNNs). We show that, in the limit, vulnerability to gradient-based attacks arises as a result of degeneracy in the data distribution, i.e., when the data lies on a lower-dimensional submanifold of the ambient space. As a direct consequence, we demonstrate that in the limit BNN posteriors are robust to gradient-based adversarial attacks. Experimental results on the MNIST and Fashion MNIST datasets with BNNs trained with Hamiltonian Monte Carlo and Variational Inference support this line of argument, showing that BNNs can display both high accuracy and robustness to gradient based adversarial attacks.

preprint2019arXiv

Rejection-Based Simulation of Non-Markovian Agents on Complex Networks

Stochastic models in which agents interact with their neighborhood according to a network topology are a powerful modeling framework to study the emergence of complex dynamic patterns in real-world systems. Stochastic simulations are often the preferred - sometimes the only feasible - way to investigate such systems. Previous research focused primarily on Markovian models where the random time until an interaction happens follows an exponential distribution. In this work, we study a general framework to model systems where each agent is in one of several states. Agents can change their state at random, influenced by their complete neighborhood, while the time to the next event can follow an arbitrary probability distribution. Classically, these simulations are hindered by high computational costs of updating the rates of interconnected agents and sampling the random residence times from arbitrary distributions. We propose a rejection-based, event-driven simulation algorithm to overcome these limitations. Our method over-approximates the instantaneous rates corresponding to inter-event times while rejection events counterbalance these over-approximations. We demonstrate the effectiveness of our approach on models of epidemic and information spreading.

preprint2016arXiv

Location Aggregation of Spatial Population CTMC Models

In this paper we focus on spatial Markov population models, describing the stochastic evolution of populations of agents, explicitly modelling their spatial distribution, representing space as a discrete, finite graph. More specifically, we present a heuristic approach to aggregating spatial locations, which is designed to preserve the dynamical behaviour of the model whilst reducing the computational cost of analysis. Our approach combines stochastic approximation ideas (moment closure, linear noise), with computational statistics (spectral clustering) to obtain an efficient aggregation, which is experimentally shown to be reasonably accurate on two case studies: an instance of epidemic spreading and a London bike sharing scenario.

preprint2016arXiv

Matching models across abstraction levels with Gaussian Processes

Biological systems are often modelled at different levels of abstraction depending on the particular aims/resources of a study. Such different models often provide qualitatively concordant predictions over specific parametrisations, but it is generally unclear whether model predictions are quantitatively in agreement, and whether such agreement holds for different parametrisations. Here we present a generally applicable statistical machine learning methodology to automatically reconcile the predictions of different models across abstraction levels. Our approach is based on defining a correction map, a random function which modifies the output of a model in order to match the statistics of the output of a different model of the same system. We use two biological examples to give a proof-of-principle demonstration of the methodology, and discuss its advantages and potential further applications.

preprint2016arXiv

Policy learning for time-bounded reachability in Continuous-Time Markov Decision Processes via doubly-stochastic gradient ascent

Continuous-time Markov decision processes are an important class of models in a wide range of applications, ranging from cyber-physical systems to synthetic biology. A central problem is how to devise a policy to control the system in order to maximise the probability of satisfying a set of temporal logic specifications. Here we present a novel approach based on statistical model checking and an unbiased estimation of a functional gradient in the space of possible policies. The statistical approach has several advantages over conventional approaches based on uniformisation, as it can also be applied when the model is replaced by a black box, and does not suffer from state-space explosion. The use of a stochastic gradient to guide our search considerably improves the efficiency of learning policies. We demonstrate the method on a proof-of-principle non-linear population model, showing strong performance in a non-trivial task.

preprint2015arXiv

CARMA: Collective Adaptive Resource-sharing Markovian Agents

In this paper we present CARMA, a language recently defined to support specification and analysis of collective adaptive systems. CARMA is a stochastic process algebra equipped with linguistic constructs specifically developed for modelling and programming systems that can operate in open-ended and unpredictable environments. This class of systems is typically composed of a huge number of interacting agents that dynamically adjust and combine their behaviour to achieve specific goals. A CARMA model, termed a collective, consists of a set of components, each of which exhibits a set of attributes. To model dynamic aggregations, which are sometimes referred to as ensembles, CARMA provides communication primitives that are based on predicates over the exhibited attributes. These predicates are used to select the participants in a communication. Two communication mechanisms are provided in the CARMA language: multicast-based and unicast-based. In this paper, we first introduce the basic principles of CARMA and then we show how our language can be used to support specification with a simple but illustrative example of a socio-technical collective adaptive system.

preprint2015arXiv

Distribution approximations for the chemical master equation: comparison of the method of moments and the system size expansion

The stochastic nature of chemical reactions involving randomly fluctuating population sizes has lead to a growing research interest in discrete-state stochastic models and their analysis. A widely-used approach is the description of the temporal evolution of the system in terms of a chemical master equation (CME). In this paper we study two approaches for approximating the underlying probability distributions of the CME. The first approach is based on an integration of the statistical moments and the reconstruction of the distribution based on the maximum entropy principle. The second approach relies on an analytical approximation of the probability distribution of the CME using the system size expansion, considering higher-order terms than the linear noise approximation. We consider gene expression networks with unimodal and multimodal protein distributions to compare the accuracy of the two approaches. We find that both methods provide accurate approximations to the distributions of the CME while having different benefits and limitations in applications.

preprint2015arXiv

Efficient Checking of Individual Rewards Properties in Markov Population Models

In recent years fluid approaches to the analysis of Markov populations models have been demonstrated to have great pragmatic value. Initially developed to estimate the behaviour of the system in terms of the expected values of population counts, the fluid approach has subsequently been extended to more sophisticated interrogations of models through its embedding within model checking procedures. In this paper we extend recent work on checking CSL properties of individual agents within a Markovian population model, to consider the checking of properties which incorporate rewards.

preprint2015arXiv

Fluid Model Checking of Timed Properties

We address the problem of verifying timed properties of Markovian models of large populations of interacting agents, modelled as finite state automata. In particular, we focus on time-bounded properties of (random) individual agents specified by Deterministic Timed Automata (DTA) endowed with a single clock. Exploiting ideas from fluid approximation, we estimate the satisfaction probability of the DTA properties by reducing it to the computation of the transient probability of a subclass of Time-Inhomogeneous Markov Renewal Processes with exponentially and deterministically-timed transitions, and a small state space. For this subclass of models, we show how to derive a set of Delay Differential Equations (DDE), whose numerical solution provides a fast and accurate estimate of the satisfaction probability. In the paper, we also prove the asymptotic convergence of the approach, and exemplify the method on a simple epidemic spreading model. Finally, we also show how to construct a system of DDEs to efficiently approximate the average number of agents that satisfy the DTA specification.

preprint2014arXiv

Mean-Field approximation and Quasi-Equilibrium reduction of Markov Population Models

Markov Population Model is a commonly used framework to describe stochastic systems. Their exact analysis is unfeasible in most cases because of the state space explosion. Approximations are usually sought, often with the goal of reducing the number of variables. Among them, the mean field limit and the quasi-equilibrium approximations stand out. We view them as techniques that are rooted in independent basic principles. At the basis of the mean field limit is the law of large numbers. The principle of the quasi-equilibrium reduction is the separation of temporal scales. It is common practice to apply both limits to an MPM yielding a fully reduced model. Although the two limits should be viewed as completely independent options, they are applied almost invariably in a fixed sequence: MF limit first, QE-reduction second. We present a framework that makes explicit the distinction of the two reductions, and allows an arbitrary order of their application. By inverting the sequence, we show that the double limit does not commute in general: the mean field limit of a time-scale reduced model is not the same as the time-scale reduced limit of a mean field model. An example is provided to demonstrate this phenomenon. Sufficient conditions for the two operations to be freely exchangeable are also provided.

preprint2014arXiv

Proceedings Twelfth International Workshop on Quantitative Aspects of Programming Languages and Systems

This volume contains the proceedings of the Twelfth Workshop on Quantitative Aspects of Programming Languages and Systems (QAPL 2014), held in Grenoble, France, on 12 and 13 April, 2014. QAPL 2014 was a satellite event of the European Joint Conferences on Theory and Practice of Software (ETAPS). The central theme of the workshop is that of quantitative aspects of computation. These aspects are related to the use of physical quantities (storage space, time, bandwidth, etc.) as well as mathematical quantities (e.g. probability and measures for reliability, security and trust), and play an important (sometimes essential) role in characterising the behaviour and determining the properties of systems. Such quantities are central to the definition of both the model of systems (architecture, language design, semantics) and the methodologies and tools for the analysis and verification of the systems properties. The aim of this workshop is to discuss the explicit use of quantitative information such as time and probabilities either directly in the model or as a tool for the analysis of systems.

preprint2014arXiv

Smoothed Model Checking for Uncertain Continuous Time Markov Chains

We consider the problem of computing the satisfaction probability of a formula for stochastic models with parametric uncertainty. We show that this satisfaction probability is a smooth function of the model parameters. This enables us to devise a novel Bayesian statistical algorithm which performs statistical model checking simultaneously for all values of the model parameters from observations of truth values of the formula over individual runs of the model at isolated parameter values. This is achieved by exploiting the smoothness of the satisfaction function: by modelling explicitly correlations through a prior distribution over a space of smooth functions (a Gaussian Process), we can condition on observations at individual parameter values to construct an analytical approximation of the function itself. Extensive experiments on non-trivial case studies show that the approach is accurate and several orders of magnitude faster than naive parameter exploration with standard statistical model checking methods.

preprint2014arXiv

Stochastic HYPE: Flow-based modelling of stochastic hybrid systems

Stochastic HYPE is a novel process algebra that models stochastic, instantaneous and continuous behaviour. It develops the flow-based approach of the hybrid process algebra HYPE by replacing non-urgent events with events with exponentially-distributed durations and also introduces random resets. The random resets allow for general stochasticity, and in particular allow for the use of event durations drawn from distributions other than the exponential distribution. To account for stochasticity, the semantics of stochastic HYPE target piecewise deterministic Markov processes (PDMPs), via intermediate transition-driven stochastic hybrid automata (TDSHA) in contrast to the hybrid automata used as semantic target for HYPE. Stochastic HYPE models have a specific structure where the controller of a system is separate from the continuous aspect of this system providing separation of concerns and supporting reasoning. A novel equivalence is defined which captures when two models have the same stochastic behaviour (as in stochastic bisimulation), instantaneous behaviour (as in classical bisimulation) and continuous behaviour. These techniques are illustrated via an assembly line example.

preprint2013arXiv

A temporal logic approach to modular design of synthetic biological circuits

We present a new approach for the design of a synthetic biological circuit whose behaviour is specified in terms of signal temporal logic (STL) formulae. We first show how to characterise with STL formulae the input/output behaviour of biological modules miming the classical logical gates (AND, NOT, OR). Hence, we provide the regions of the parameter space for which these specifications are satisfied. Given a STL specification of the target circuit to be designed and the networks of its constituent components, we propose a methodology to constrain the behaviour of each module, then identifying the subset of the parameter space in which those constraints are satisfied, providing also a measure of the robustness for the target circuit design. This approach, which leverages recent results on the quantitative semantics of Signal Temporal Logic, is illustrated by synthesising a biological implementation of an half-adder.

preprint2013arXiv

Fluid Model Checking

In this paper we investigate a potential use of fluid approximation techniques in the context of stochastic model checking of CSL formulae. We focus on properties describing the behaviour of a single agent in a (large) population of agents, exploiting a limit result known also as fast simulation. In particular, we will approximate the behaviour of a single agent with a time-inhomogeneous CTMC which depends on the environment and on the other agents only through the solution of the fluid differential equation. We will prove the asymptotic correctness of our approach in terms of satisfiability of CSL formulae and of reachability probabilities. We will also present a procedure to model check time-inhomogeneous CTMC against CSL formulae.

preprint2013arXiv

Hybrid Behaviour of Markov Population Models

We investigate the behaviour of population models written in Stochastic Concurrent Constraint Programming (sCCP), a stochastic extension of Concurrent Constraint Programming. In particular, we focus on models from which we can define a semantics of sCCP both in terms of Continuous Time Markov Chains (CTMC) and in terms of Stochastic Hybrid Systems, in which some populations are approximated continuously, while others are kept discrete. We will prove the correctness of the hybrid semantics from the point of view of the limiting behaviour of a sequence of models for increasing population size. More specifically, we prove that, under suitable regularity conditions, the sequence of CTMC constructed from sCCP programs for increasing population size converges to the hybrid system constructed by means of the hybrid semantics. We investigate in particular what happens for sCCP models in which some transitions are guarded by boolean predicates or in the presence of instantaneous transitions.

preprint2013arXiv

Learning Temporal Logical Properties Discriminating ECG models of Cardiac Arrhytmias

We present a novel approach to learn the formulae characterising the emergent behaviour of a dynamical system from system observations. At a high level, the approach starts by devising a statistical dynamical model of the system which optimally fits the observations. We then propose general optimisation strategies for selecting high support formulae (under the learnt model of the system) either within a discrete set of formulae of bounded complexity, or a parametric family of formulae. We illustrate and apply the methodology on an in-depth case study of characterising cardiac malfunction from electro-cardiogram data, where our approach enables us to quantitatively determine the diagnostic power of a formula in discriminating between different cardiac conditions.

preprint2013arXiv

On the Robustness of Temporal Properties for Stochastic Models

Stochastic models such as Continuous-Time Markov Chains (CTMC) and Stochastic Hybrid Automata (SHA) are powerful formalisms to model and to reason about the dynamics of biological systems, due to their ability to capture the stochasticity inherent in biological processes. A classical question in formal modelling with clear relevance to biological modelling is the model checking problem. i.e. calculate the probability that a behaviour, expressed for instance in terms of a certain temporal logic formula, may occur in a given stochastic process. However, one may not only be interested in the notion of satisfiability, but also in the capacity of a system to mantain a particular emergent behaviour unaffected by the perturbations, caused e.g. from extrinsic noise, or by possible small changes in the model parameters. To address this issue, researchers from the verification community have recently proposed several notions of robustness for temporal logic providing suitable definitions of distance between a trajectory of a (deterministic) dynamical system and the boundaries of the set of trajectories satisfying the property of interest. The contributions of this paper are twofold. First, we extend the notion of robustness to stochastic systems, showing that this naturally leads to a distribution of robustness scores. By discussing two examples, we show how to approximate the distribution of the robustness score and its key indicators: the average robustness and the conditional average robustness. Secondly, we show how to combine these indicators with the satisfaction probability to address the system design problem, where the goal is to optimize some control parameters of a stochastic model in order to best maximize robustness of the desired specifications.

preprint2013arXiv

Proceedings 11th International Workshop on Quantitative Aspects of Programming Languages and Systems

Quantitative aspects of computation are important and sometimes essential in characterising the behavior and determining the properties of systems. They are related to the use of physical quantities (storage space, time, bandwidth, etc.) as well as mathematical quantities (e.g. probability and measures for reliability, security and trust). Such quantities play a central role in defining both the model of systems (architecture, language design, semantics) and the methodologies and tools for the analysis and verification of system properties. The aim of this workshop is to discuss the explicit use of quantitative information such as time and probabilities either directly in the model or as a tool for the analysis of systems.

preprint2013arXiv

Proceedings Third International Workshop on Hybrid Autonomous Systems

The interest on autonomous systems is increasing both in industry and academia. Such systems must operate with limited human intervention in a changing environment and must be able to compensate for significant system failures without external intervention. The most appropriate models of autonomous systems can be found in the class of hybrid systems (which study continuous-state dynamic processes via discrete-state controllers) that interact with their environment. This workshop brings together researchers interested in all aspects of autonomy and resilience of hybrid systems.

preprint2012arXiv

Hybrid performance modelling of opportunistic networks

We demonstrate the modelling of opportunistic networks using the process algebra stochastic HYPE. Network traffic is modelled as continuous flows, contact between nodes in the network is modelled stochastically, and instantaneous decisions are modelled as discrete events. Our model describes a network of stationary video sensors with a mobile ferry which collects data from the sensors and delivers it to the base station. We consider different mobility models and different buffer sizes for the ferries. This case study illustrates the flexibility and expressive power of stochastic HYPE. We also discuss the software that enables us to describe stochastic HYPE models and simulate them.

preprint2012arXiv

Proceedings First International Workshop on Hybrid Systems and Biology

This volume contains the proceedings of the First International Workshop on Hybrid Systems and Biology (HSB 2012), that will be held in Newcastle upon Tyne, UK, on the 3rd September, 2012. HSB 2012 is a satellite event of the 23rd International Conference on Concurrency Theory (CONCUR 2012). This workshop aims at collecting scientists working in the area of hybrid modeling applied to systems biology, in order to discuss about current achieved goals, current challenges and future possible developments.

preprint2011arXiv

HYPE with stochastic events

The process algebra HYPE was recently proposed as a fine-grained modelling approach for capturing the behaviour of hybrid systems. In the original proposal, each flow or influence affecting a variable is modelled separately and the overall behaviour of the system then emerges as the composition of these flows. The discrete behaviour of the system is captured by instantaneous actions which might be urgent, taking effect as soon as some activation condition is satisfied, or non-urgent meaning that they can tolerate some (unknown) delay before happening. In this paper we refine the notion of non-urgent actions, to make such actions governed by a probability distribution. As a consequence of this we now give HYPE a semantics in terms of Transition-Driven Stochastic Hybrid Automata, which are a subset of a general class of stochastic processes termed Piecewise Deterministic Markov Processes.

preprint2011arXiv

Programmable models of growth and mutation of cancer-cell populations

In this paper we propose a systematic approach to construct mathematical models describing populations of cancer-cells at different stages of disease development. The methodology we propose is based on stochastic Concurrent Constraint Programming, a flexible stochastic modelling language. The methodology is tested on (and partially motivated by) the study of prostate cancer. In particular, we prove how our method is suitable to systematically reconstruct different mathematical models of prostate cancer growth - together with interactions with different kinds of hormone therapy - at different levels of refinement.