On the decidability and complexity of problems for restricted hierarchical hybrid systems

We study variants of a recently introduced hybrid system model, called a Hierarchical Piecewise Constant Derivative (HPCD). These variants (loosely called Restricted HPCDs) form a class of natural models with similarities to many other well known hybrid system models in the literature such as Stopwatch Automata, Rectangular Automata and PCDs. We study the complexity of reachability and mortality problems for variants of RHPCDs and show a variety of results, depending upon the allowed powers. These models form a useful tool for the study of the complexity of such problems for hybrid systems, due to their connections with existing models. We show that the reachability problem and the mortality problem are co-NP-hard for bounded 3-dimensional RHPCDs (3-RHPCDs). Reachability is shown to be in PSPACE, even for n-dimensional RHPCDs. We show that for an unbounded 3-RHPCD, the reachability and mortality problems become undecidable. For a nondeterministic variant of 2-RHPCDs, the reachability problem is shown to be PSPACE-complete

On the decidability and complexity of problems for restricted hierarchical hybrid systems

We study variants of a recently introduced hybrid system model, called a Hierarchical Piecewise Constant Derivative (HPCD). These variants (loosely called Restricted HPCDs) form a class of natural models with similarities to many other well known hybrid system models in the literature such as Stopwatch Automata, Rectangular Automata and PCDs. We study the complexity of reachability and mortality problems for variants of RHPCDs and show a variety of results, depending upon the allowed powers. These models form a useful tool for the study of the complexity of such problems for hybrid systems, due to their connections with existing models. We show that the reachability problem and the mortality problem are co-NP-hard for bounded 3-dimensional RHPCDs (3-RHPCDs). Reachability is shown to be in PSPACE, even for n-dimensional RHPCDs. We show that for an unbounded 3-RHPCD, the reachability and mortality problems become undecidable. For a nondeterministic variant of 2-RHPCDs, the reachability problem is shown to be PSPACE-complete

Reachability problems for systems with linear dynamics

This thesis deals with reachability and freeness problems for systems with linear dynamics, including hybrid systems and matrix semigroups. Hybrid systems are a type of dynamical system that exhibit both continuous and discrete dynamic behaviour. Thus they are particularly useful in modelling practical real world systems which can both flow (continuous behaviour) and jump (discrete behaviour). Decision questions for matrix semigroups have attracted a great deal of attention in both the Mathematics and Theoretical Computer Science communities. They can also be used to model applications with only discrete components. For a computational model, the reachability problem asks whether we can reach a target point starting from an initial point, which is a natural question both in theoretical study and for real-world applications. By studying this problem and its variations, we shall prove in a formal mathematical sense that many problems are intractable or even unsolvable. Thus we know when such a problem appears in other areas like Biology, Physics or Chemistry, either the problem itself needs to be simplified, or it should by studied by approximation. In this thesis we concentrate on a specific hybrid system model, called an HPCD, and its variations. The objective of studying this model is twofold: to obtain the most expressive system for which reachability is algorithmically solvable and to explore the simplest system for which it is impossible to solve. For the solvable sub-cases, we shall also study whether reachability is in some sense easy or hard by determining which complexity classes the problem belongs to, such as P, NP(-hard) and PSPACE(-hard). Some undecidable results for matrix semigroups are also shown, which both strengthen our knowledge of the structure of matrix semigroups, and lead to some undecidability results for other models

Reachability problems for PAMs

Piecewise affine maps (PAMs) are frequently used as a reference model to show the openness of the reachability questions in other systems. The reachability problem for one-dimentional PAM is still open even if we define it with only two intervals. As the main contribution of this paper we introduce new techniques for solving reachability problems based on p-adic norms and weights as well as showing decidability for two classes of maps. Then we show the connections between topological properties for PAM's orbits, reachability problems and representation of numbers in a rational base system. Finally we show a particular instance where the uniform distribution of the original orbit may not remain uniform or even dense after making regular shifts and taking a fractional part in that sequence.Comment: 16 page

Mortality and Edge-to-Edge Reachability are Decidable on Surfaces

© 2022 Copyright held by the owner/author(s). Publication rights licensed to ACM. This is an open access paper distributed under the Creative Commons Attribution License, to view a copy of the license, see: https://creativecommons.org/licenses/by/4.0/The mortality problem for a given dynamical system S consists of determining whether every trajectory of S eventually halts. In this work, we show that this problem is decidable for the class of piecewise constant derivative systems on two-dimensional manifolds, also called surfaces (). Two closely related open problems are point-to-point and edge-to-edge reachability for . Building on our technique to establish decidability of mortality for , we show that the edge-to-edge reachability problem for these systems is also decidable. In this way we solve the edge-to-edge reachability case of an open problem due to Asarin, Mysore, Pnueli and Schneider [4]. This implies that the interval-to-interval version of the classical open problem of reachability for regular piecewise affine maps (PAMs) is also decidable. In other words, point-to-point reachability for regular PAMs can be effectively approximated with arbitrarily precision

Reachability Problems for One-Dimensional Piecewise Affine Maps

Piecewise affine maps (PAMs) are frequently used as a reference model to discuss the frontier between known and open questions about the decidability for reachability questions. In particular, the reachability problem for one-dimensional PAM is still an open problem, even if restricted to only two intervals. As the main contribution of this paper we introduce new techniques for solving reachability problems based on p-adic norms and weights as well as showing decidability for two classes of maps. Then we show the connections between topological properties for PAM’s orbits, reachability problems and representation of numbers in a rational base system. Finally we construct an example where the distribution properties of well studied sequences can be significantly disrupted by taking fractional parts after regular shifts. The study of such sequences could help with understanding similar sequences generated in PAMs or in well known Mahler’s 3/2 problem

Realizability of embedded controllers: from hybrid models to correct implementations

Un controller embedded \ue8 un dispositivo (ovvero, un'opportuna combinazione di componenti hardware e software) che, immerso in un ambiente dinamico, deve reagire alle variazioni ambientali in tempo reale. I controller embedded sono largamente adottati in molti contesti della vita moderna, dall'automotive all'avionica, dall'elettronica di consumo alle attrezzature mediche. La correttezza di tali controller \ue8 indubbiamente cruciale. Per la progettazione e per la verifica di un controller embedded, spesso sorge la necessit\ue0 di modellare un intero sistema che includa sia il controller, sia il suo ambiente circostante. La natura di tale sistema \ue8 ibrido. Esso, infatti, \ue8 ottenuto integrando processi ad eventi discreti (i.e., il controller) e processi a tempo continuo (i.e., l'ambiente). Sistemi di questo tipo sono chiamati cyber-physical (CPS) o sistemi ibridi. Le dinamiche di tali sistemi non possono essere rappresentati efficacemente utilizzando o solo un modello (i.e., rappresentazione) discreto o solo un modello continuo. Diversi tipi di modelli possono sono stati proposti per descrivere i sistemi ibridi. Questi si concentrano su obiettivi diversi: modelli dettagliati sono eccellenti per la simulazione del sistema, ma non sono adatti per la sua verifica; modelli meno dettagliati sono eccellenti per la verifica, ma non sono convenienti per i successivi passi di raffinamento richiesti per la progettazione del sistema, e cos\uec via. Tra tutti questi modelli, gli Automi Ibridi (HA) [8, 77] rappresentano il formalismo pi\uf9 efficace per la simulazione e la verifica di sistemi ibridi. In particolare, un automa ibrido rappresenta i processi ad eventi discreti per mezzo di macchine a stati finiti (FSM), mentre i processi a tempo continuo sono rappresentati mediante variabili "continue" la cui dinamica \ue8 specificata da equazioni differenziali ordinarie (ODE) o loro generalizzazioni (e.g., inclusioni differenziali). Sfortunatamente, a causa della loro particolare semantica, esistono diverse difficolt\ue0 nel raffinare un modello basato su automi ibridi in un modello realizzabile e, di conseguenza, esistono difficolt\ue0 nell'automatizzare il flusso di progettazione di sistemi ibridi a partire da automi ibridi. Gli automi ibridi, infatti, sono considerati dispositivi "perfetti e istantanei". Essi adottano una nozione di tempo e di variabili basata su insiemi "densi" (i.e., l'insieme dei numeri reali). Pertanto, gli automi ibridi possono valutare lo stato (i.e., i valori delle variabili) del sistema in ogni istante, ovvero in ogni infinitesimo di tempo, e con la massima precisione. Inoltre, sono in grado di eseguire computazioni o reagire ad eventi di sincronizzazione in modo istantaneo, andando a cambiare la modalit\ue0 di funzionamento del sistema senza alcun ritardo. Questi aspetti sono convenienti a livello di modellazione, ma nessun dispositivo hardware/software potrebbe implementare correttamente tali comportamenti, indipendentemente dalle sue prestazioni. In altre parole, il controller modellato potrebbe non essere implementabile, ovvero, esso potrebbe non essere realizzabile affatto. Questa tesi affronta questo problema proponendo una metodologia completa e gli strumenti necessari per derivare da modelli basati su automi ibridi, modelli realizzabili e le corrispondenti implementazioni corrette. In un modello realizzabile, il controller analizza lo stato del sistema ad istanti temporali discreti, tipicamente fissati dalla frequenza di clock del processore installato sul dispositivo che implementa il controller. Lo stato del sistema \ue8 dato dai valori delle variabili rilevati dai sensori. Questi valori vengono digitalizzati con precisione finita e propagati al controller che li elabora per decidere se cambiare la modalit\ue0 di funzionamento del sistema. In tal caso, il controller genera segnali che, una volta trasmessi agli attuatori, determineranno il cambiamento della modalit\ue0 di funzionamento del sistema. \uc8 necessario tener presente che i sensori e gli attuatori introducono ritardi che seppur limitati, non possono essere trascurati.An embedded controller is a reactive device (e.g., a suitable combination of hardware and software components) that is embedded in a dynamical environment and has to react to environment changes in real time. Embedded controllers are widely adopted in many contexts of modern life, from automotive to avionics, from consumer electronics to medical equipment. Noticeably, the correctness of such controllers is crucial. When designing and verifying an embedded controller, often the need arises to model the controller and also its surrounding environment. The nature of the obtained system is hybrid because of the inclusion of both discrete-event (i.e., controller) and continuous-time (i.e., environment) processes whose dynamics cannot be characterized faithfully using either a discrete or continuous model only. Systems of this kind are named cyber-physical (CPS) or hybrid systems. Different types of models may be used to describe hybrid systems and they focus on different objectives: detailed models are excellent for simulation but not suitable for verification, high-level models are excellent for verification but not convenient for refinement, and so forth. Among all these models, hybrid automata (HA) [8, 77] have been proposed as a powerful formalism for the design, simulation and verification of hybrid systems. In particular, a hybrid automaton represents discrete-event processes by means of finite state machines (FSM), whereas continuous-time processes are represented by using real-numbered variables whose dynamics is specified by (ordinary) differential equation (ODE) or their generalizations (e.g., differential inclusions). Unfortunately, when the high-level model of the hybrid system is a hybrid automaton, several difficulties should be solved in order to automate the refinement phase in the design flow, because of the classical semantics of hybrid automata. In fact, hybrid automata can be considered perfect and instantaneous devices. They adopt a notion of time and evaluation of continuous variables based on dense sets of values (usually R, i.e., Reals). Thus, they can sample the state (i.e., value assignments on variables) of the hybrid system at any instant in such a dense set R 650. Further, they are capable of instantaneously evaluating guard constraints or reacting to incoming events by performing changes in the operating mode of the hybrid system without any delay. While these aspects are convenient at the modeling level, any model of an embedded controller that relies for its correctness on such precision and instantaneity cannot be implemented by any hardware/software device, no matter how fast it is. In other words, the controller is un-realizable, i.e., un-implementable. This thesis proposes a complete methodology and a framework that allows to derive from hybrid automata proved correct in the hybrid domain, correct realizable models of embedded controllers and the related discrete implementations. In a realizable model, the controller samples the state of the environment at periodic discrete time instants which, typically, are fixed by the clock frequency of the processor implementing the controller. The state of the environment consists of the current values of the relevant variables as observed by the sensors. These values are digitized with finite precision and reported to the controller that may decide to switch the operating mode of the environment. In such a case, the controller generates suitable output signals that, once transmitted to the actuators, will effect the desired change in the operating mode. It is worth noting that the sensors will report the current values of the variables and the actuators will effect changes in the rates of evolution of the variables with bounded delays

Bayesian Learning of Coupled Biogeochemical-Physical Models

Predictive dynamical models for marine ecosystems are used for a variety of needs. Due to sparse measurements and limited understanding of the myriad of ocean processes, there is however significant uncertainty. There is model uncertainty in the parameter values, functional forms with diverse parameterizations, level of complexity needed, and thus in the state fields. We develop a Bayesian model learning methodology that allows interpolation in the space of candidate models and discovery of new models from noisy, sparse, and indirect observations, all while estimating state fields and parameter values, as well as the joint PDFs of all learned quantities. We address the challenges of high-dimensional and multidisciplinary dynamics governed by PDEs by using state augmentation and the computationally efficient GMM-DO filter. Our innovations include stochastic formulation and complexity parameters to unify candidate models into a single general model as well as stochastic expansion parameters within piecewise function approximations to generate dense candidate model spaces. These innovations allow handling many compatible and embedded candidate models, possibly none of which are accurate, and learning elusive unknown functional forms. Our new methodology is generalizable, interpretable, and extrapolates out of the space of models to discover new ones. We perform a series of twin experiments based on flows past a ridge coupled with three-to-five component ecosystem models, including flows with chaotic advection. The probabilities of known, uncertain, and unknown model formulations, and of state fields and parameters, are updated jointly using Bayes' law. Non-Gaussian statistics, ambiguity, and biases are captured. The parameter values and model formulations that best explain the data are identified. When observations are sufficiently informative, model complexity and functions are discovered.Comment: 45 pages; 18 figures; 2 table

Studying the effects of adding spatiality to a process algebra model

We use NetLogo to create simulations of two models of disease transmission originally expressed in WSCCS. This allows us to introduce spatiality into the models and explore the consequences of having different contact structures among the agents. In previous work, mean field equations were derived from the WSCCS models, giving a description of the aggregate behaviour of the overall population of agents. These results turned out to differ from results obtained by another team using cellular automata models, which differ from process algebra by being inherently spatial. By using NetLogo we are able to explore whether spatiality, and resulting differences in the contact structures in the two kinds of models, are the reason for this different results. Our tentative conclusions, based at this point on informal observations of simulation results, are that space does indeed make a big difference. If space is ignored and individuals are allowed to mix randomly, then the simulations yield results that closely match the mean field equations, and consequently also match the associated global transmission terms (explained below). At the opposite extreme, if individuals can only contact their immediate neighbours, the simulation results are very different from the mean field equations (and also do not match the global transmission terms). These results are not surprising, and are consistent with other cellular automata-based approaches. We found that it was easy and convenient to implement and simulate the WSCCS models within NetLogo, and we recommend this approach to anyone wishing to explore the effects of introducing spatiality into a process algebra model