Searcharxiv⌕ Search

arXiv subjects

Rupak Majumdar

Publications and source records attributed to Rupak Majumdar.

At least 91 records · Page 5Linked to original sources
arXiv2017

Parameter Optimization in Control Software using Statistical Fault Localization Techniques

Embedded controllers for cyber-physical systems are often parameterized by look-up maps representing discretizations of continuous functions on metric spaces. For example, a non-linear control action may be represented as a table of pre-computed values, and the output action of the controller for a given input is computed by using interpolation. For industrial-scale control systems, several man-hours of effort is spent in tuning the values within the look-up maps, and sub-optimal performance is often associated with inappropriate values in look-up maps. Suppose that during testing, the controller code is found to have sub-optimal performance. The parameter fault localization problem asks which parameter values in the code are potential causes of the sub-optimal behavior. We present a statistical parameter fault localization approach based on binary similarity coefficients and set spectra methods. Our approach extends previous work on software fault localization to a quantitative setting where the parameters encode continuous functions over a metric space and the program is reactive. We have implemented our approach in a simulation workflow for automotive control systems in Simulink. Given controller code with parameters (including look-up maps), our framework bootstraps the simulation workflow to return a ranked list of map entries which are deemed to have most impact on the performance. On a suite of industrial case studies with seeded errors, our tool was able to precisely identify the location of the errors.

eess.SY↗
arXiv2017

Compositional Abstractions of Interconnected Discrete-Time Stochastic Control Systems

This paper is concerned with a compositional approach for constructing abstractions of interconnected discrete-time stochastic control systems. The abstraction framework is based on new notions of so-called stochastic simulation functions, using which one can quantify the distance between original interconnected stochastic control systems and their abstractions in the probabilistic setting. Accordingly, one can leverage the proposed results to perform analysis and synthesis over abstract interconnected systems, and then carry the results over concrete ones. In the first part of the paper, we derive sufficient small-gain type conditions for the compositional quantification of the distance in probability between the interconnection of stochastic control subsystems and that of their abstractions. In the second part of the paper, we focus on the class of discrete-time linear stochastic control systems with independent noises in the abstract and concrete subsystems. For this class of systems, we propose a computational scheme to construct abstractions together with their corresponding stochastic simulation functions. We demonstrate the effectiveness of the proposed results by constructing an abstraction (totally 4 dimensions) of the interconnection of four discrete-time linear stochastic control subsystems (together 100 dimensions) in a compositional fashion.

eess.SY↗
arXiv2017

Compositional Construction of Finite State Abstractions for Stochastic Control Systems

Controller synthesis techniques for continuous systems with respect to temporal logic specifications typically use a finite-state symbolic abstraction of the system. Constructing this abstraction for the entire system is computationally expensive, and does not exploit natural decompositions of many systems into interacting components. We have recently introduced a new relation, called (approximate) disturbance bisimulation for compositional symbolic abstraction to help scale controller synthesis for temporal logic to larger systems. In this paper, we extend the results to stochastic control systems modeled by stochastic differential equations. Given any stochastic control system satisfying a stochastic version of the incremental input-to-state stability property and a positive error bound, we show how to construct a finite-state transition system (if there exists one) which is disturbance bisimilar to the given stochastic control system. Given a network of stochastic control systems, we give conditions on the simultaneous existence of disturbance bisimilar abstractions to every component allowing for compositional abstraction of the network system.

eess.SY↗
arXiv2017

Compositional Abstraction-Based Controller Synthesis for Continuous-Time Systems

Controller synthesis techniques for continuous systems with respect to temporal logic specifications typically use a finite-state symbolic abstraction of the system model. Constructing this abstraction for the entire system is computationally expensive, and does not exploit natural decompositions of many systems into interacting components. We describe a methodology for compositional symbolic abstraction to help scale controller synthesis for temporal logic to larger systems. We introduce a new relation, called (approximate) disturbance bisimulation, as the basis for compositional symbolic abstractions. Disturbance bisimulation strengthens the standard approximate alternating bisimulation relation used in control. It extends naturally to systems which are composed of weakly interconnected sub-components possibly connected in feedback, and models the coupling signals as disturbances. After proving this composability of disturbance bisimulation for metric systems we apply this result to the compositional abstraction of networks of input-to-state stable deterministic non-linear control systems. We give conditions that allow to construct finite-state abstractions compositionally for each component in such a network, so that the abstractions are simultaneously disturbance bisimilar to their continuous counterparts. Combining these two results, we show conditions under which one can compositionally abstract a network of non-linear control systems in a modular way while ensuring that the final composed abstraction is disturbance bisimilar to the original system. We discuss how we get a compositional abstraction-based controller synthesis methodology for networks of such systems against local temporal specifications as a by-product of our construction.

eess.SY↗
arXiv2017

The Robot Routing Problem for Collecting Aggregate Stochastic Rewards

We propose a new model for formalizing reward collection problems on graphs with dynamically generated rewards which may appear and disappear based on a stochastic model. The *robot routing problem* is modeled as a graph whose nodes are stochastic processes generating potential rewards over discrete time. The rewards are generated according to the stochastic process, but at each step, an existing reward disappears with a given probability. The edges in the graph encode the (unit-distance) paths between the rewards' locations. On visiting a node, the robot collects the accumulated reward at the node at that time, but traveling between the nodes takes time. The optimization question asks to compute an optimal (or epsilon-optimal) path that maximizes the expected collected rewards. We consider the finite and infinite-horizon robot routing problems. For finite-horizon, the goal is to maximize the total expected reward, while for infinite horizon we consider limit-average objectives. We study the computational and strategy complexity of these problems, establish NP-lower bounds and show that optimal strategies require memory in general. We also provide an algorithm for computing epsilon-optimal infinite paths for arbitrary epsilon > 0.

eess.SY↗
arXiv2017

Multilevel Monte Carlo Method for Statistical Model Checking of Hybrid Systems

We study statistical model checking of continuous-time stochastic hybrid systems. The challenge in applying statistical model checking to these systems is that one cannot simulate such systems exactly. We employ the multilevel Monte Carlo method (MLMC) and work on a sequence of discrete-time stochastic processes whose executions approximate and converge weakly to that of the original continuous-time stochastic hybrid system with respect to satisfaction of the property of interest. With focus on bounded-horizon reachability, we recast the model checking problem as the computation of the distribution of the exit time, which is in turn formulated as the expectation of an indicator function. This latter computation involves estimating discontinuous functionals, which reduces the bound on the convergence rate of the Monte Carlo algorithm. We propose a smoothing step with tunable precision and formally quantify the error of the MLMC approach in the mean-square sense, which is composed of smoothing error, bias, and variance. We formulate a general adaptive algorithm which balances these error terms. Finally, we describe an application of our technique to verify a model of thermostatically controlled loads.

eess.SY↗
arXiv2017

Shrinking Horizon Model Predictive Control with Signal Temporal Logic Constraints under Stochastic Disturbances

We present Shrinking Horizon Model Predictive Control (SHMPC) for discrete-time linear systems with Signal Temporal Logic (STL) specification constraints under stochastic disturbances. The control objective is to maximize an optimization function under the restriction that a given STL specification is satisfied with high probability against stochastic uncertainties. We formulate a general solution, which does not require precise knowledge of the probability distributions of the (possibly dependent) stochastic disturbances; only the bounded support intervals of the density functions and moment intervals are used. For the specific case of disturbances that are independent and normally distributed, we optimize the controllers further by utilizing knowledge of the disturbance probability distributions. We show that in both cases, the control law can be obtained by solving optimization problems with linear constraints at each step. We experimentally demonstrate effectiveness of this approach by synthesizing a controller for an HVAC system.

eess.SY↗
arXiv2016

Hitting Families of Schedules for Asynchronous Programs

We consider the following basic task in the testing of concurrent systems. The input to the task is a partial order of events, which models actions performed on or by the system and specifies ordering constraints between them. The task is to determine if some scheduling of these events can result in a bug. The number of schedules to be explored can, in general, be exponential. Empirically, many bugs in concurrent programs have been observed to have small bug depth; that is, these bugs are exposed by every schedule that orders $d$ specific events in a particular way, irrespective of how the other events are ordered, and $d$ is small compared to the total number of events. To find all bugs of depth $d$, one needs to only test a $d$-hitting family of schedules: we call a set of schedules a $d$-hitting family if for each set of $d$ events, and for each allowed ordering of these events, there is some schedule in the family that executes these events in this ordering. The size of a $d$-hitting family may be much smaller than the number of all possible schedules, and a natural question is whether one can find $d$-hitting families of schedules that have small size. In general, finding the size of optimal $d$-hitting families is hard, even for $d=2$. We show, however, that when the partial order is a tree, one can explicitly construct $d$-hitting families of schedules of small size. When the tree is balanced, our constructions are polylogarithmic in the number of events.

cs.DM↗
arXiv2016

Dynamic Hierarchical Reactive Controller Synthesis

In the formal approach to reactive controller synthesis, a symbolic controller for a possibly hybrid system is obtained by algorithmically computing a winning strategy in a two-player game. Such game-solving algorithms scale poorly as the size of the game graph increases. However, in many applications, the game graph has a natural hierarchical structure. In this paper, we propose a modeling formalism and a synthesis algorithm that exploits this hierarchical structure for more scalable synthesis. We define local games on hierarchical graphs as a modeling formalism which decomposes a large-scale reactive synthesis problem in two dimensions. First, the construction of a hierarchical game graph introduces abstraction layers, where each layer is again a two-player game graph. Second, every such layer is decomposed into multiple local game graphs, each corresponding to a node in the higher level game graph. While local games have the potential to reduce the state space for controller synthesis, they lead to more complex synthesis problems where strategies computed for one local game can impose additional requirements on lower-level local games. Our second contribution is a procedure to construct a dynamic controller for local game graphs over hierarchies. The controller computes assume-admissible winning strategies that satisfy local specifications in the presence of environment assumptions, and dynamically updates specifications and strategies due to interactions between games at different abstraction layers at each step of the play. We show that our synthesis procedure is sound: the controller constructs a play which satisfies all local specifications. We illustrate our results through an example controlling an autonomous robot in a known, multistory building.

eess.SY↗
arXiv2016

Computing Distances between Reach Flowpipes

We investigate quantifying the difference between two hybrid dynamical systems under noise and initial-state uncertainty. While the set of traces for these systems is infinite, it is possible to symbolically approximate trace sets using \emph{reachpipes} that compute upper and lower bounds on the evolution of the reachable sets with time. We estimate distances between corresponding sets of trajectories of two systems in terms of distances between the reachpipes. In case of two individual traces, the Skorokhod distance has been proposed as a robust and efficient notion of distance which captures both value and timing distortions. In this paper, we extend the computation of the Skorokhod distance to reachpipes, and provide algorithms to compute upper and lower bounds on the distance between two sets of traces. Our algorithms use new geometric insights that are used to compute the worst-case and best-case distances between two polyhedral sets evolving with time.

eess.SY↗
arXiv2015

Approximate Counting in SMT and Value Estimation for Probabilistic Programs

#SMT, or model counting for logical theories, is a well-known hard problem that generalizes such tasks as counting the number of satisfying assignments to a Boolean formula and computing the volume of a polytope. In the realm of satisfiability modulo theories (SMT) there is a growing need for model counting solvers, coming from several application domains (quantitative information flow, static analysis of probabilistic programs). In this paper, we show a reduction from an approximate version of #SMT to SMT. We focus on the theories of integer arithmetic and linear real arithmetic. We propose model counting algorithms that provide approximate solutions with formal bounds on the approximation error. They run in polynomial time and make a polynomial number of queries to the SMT solver for the underlying theory, exploiting "for free" the sophisticated heuristics implemented within modern SMT solvers. We have implemented the algorithms and used them to solve the value problem for a model of loop-free probabilistic programs with nondeterminism.

cs.LO↗
arXiv2015

Automatic Synthesis of Geometry Problems for an Intelligent Tutoring System

This paper presents an intelligent tutoring system, GeoTutor, for Euclidean Geometry that is automatically able to synthesize proof problems and their respective solutions given a geometric figure together with a set of properties true of it. GeoTutor can provide personalized practice problems that address student deficiencies in the subject matter.

cs.AI↗
arXiv2015

Reachability Analysis of Reversal-bounded Automata on Series-Parallel Graphs

Extensions to finite-state automata on strings, such as multi-head automata or multi-counter automata, have been successfully used to encode many infinite-state non-regular verification problems. In this paper, we consider a generalization of automata-theoretic infinite-state verification from strings to labeled series-parallel graphs. We define a model of non-deterministic, 2-way, concurrent automata working on series-parallel graphs and communicating through shared registers on the nodes of the graph. We consider the following verification problem: given a family of series-parallel graphs described by a context-free graph transformation system (GTS), and a concurrent automaton over series-parallel graphs, is some graph generated by the GTS accepted by the automaton? The general problem is undecidable already for (one-way) multi-head automata over strings. We show that a bounded version, where the automata make a fixed number of reversals along the graph and use a fixed number of shared registers is decidable, even though there is no bound on the sizes of series-parallel graphs generated by the GTS. Our decidability result is based on establishing that the number of context switches is bounded and on an encoding of the computation of bounded concurrent automata to reduce the emptiness problem to the emptiness problem for pushdown automata.

cs.FL↗
arXiv2015

Dynamic Bayesian Networks as Formal Abstractions of Structured Stochastic Processes

We study the problem of finite-horizon probabilistic invariance for discrete-time Markov processes over general (uncountable) state spaces. We compute discrete-time, finite-state Markov chains as formal abstractions of general Markov processes. Our abstraction differs from existing approaches in two ways. First, we exploit the structure of the underlying Markov process to compute the abstraction separately for each dimension. Second, we employ dynamic Bayesian networks (DBN) as compact representations of the abstraction. In contrast, existing approaches represent and store the (exponentially large) Markov chain explicitly, which leads to heavy memory requirements limiting the application to models of dimension less than half, according to our experiments. We show how to construct a DBN abstraction of a Markov process satisfying an independence assumption on the driving process noise. We compute a guaranteed bound on the error in the abstraction w.r.t.\ the probabilistic invariance property; the dimension-dependent abstraction makes the error bounds more precise than existing approaches. Additionally, we show how factor graphs and the sum-product algorithm for DBNs can be used to solve the finite-horizon probabilistic invariance problem. Together, DBN-based representations and algorithms can be significantly more efficient than explicit representations of Markov chains for abstracting and model checking structured Markov processes.

eess.SY↗
arXiv2015

Model Checking Parameterized Asynchronous Shared-Memory Systems

We characterize the complexity of liveness verification for parameterized systems consisting of a leader process and arbitrarily many anonymous and identical contributor processes. Processes communicate through a shared, bounded-value register. While each operation on the register is atomic, there is no synchronization primitive to execute a sequence of operations atomically. We analyze the case in which processes are modeled by finite-state machines or pushdown machines and the property is given by a Büchi automaton over the alphabet of read and write actions of the leader. We show that the problem is decidable, and has a surprisingly low complexity: it is NP-complete when all processes are finite-state machines, and is PSPACE-hard and in NEXPTIME when they are pushdown machines. This complexity is lower than for the non-parameterized case: liveness verification of finitely many finite-state machines is PSPACE-complete, and undecidable for two pushdown machines. For finite-state machines, our proofs characterize infinite behaviors using existential abstraction and semilinear constraints. For pushdown machines, we show how contributor computations of high stack height can be simulated by computations of many contributors, each with low stack height. Together, our results characterize the complexity of verification for parameterized systems under the assumptions of anonymity and asynchrony.

cs.DC↗
arXiv2015

Quantifying Conformance using the Skorokhod Metric (full version)

The conformance testing problem for dynamical systems asks, given two dynamical models (e.g., as Simulink diagrams), whether their behaviors are "close" to each other. In the semi-formal approach to conformance testing, the two systems are simulated on a large set of tests, and a metric, defined on pairs of real-valued, real-timed trajectories, is used to determine a lower bound on the distance. We show how the Skorkhod metric on continuous dynamical systems can be used as the foundation for conformance testing of complex dynamical models. The Skorokhod metric allows for both state value mismatches and timing distortions, and is thus well suited for checking conformance between idealized models of dynamical systems and their implementations. We demonstrate the robustness of the system conformance quantification by proving a \emph{transference theorem}: trajectories close under the Skorokhod metric satisfy "close" logical properties. Specifically, we show the result for the timed linear time logic \TLTL augmented with a rich class of temporal and spatial constraint predicates. We provide a window-based streaming algorithm to compute the Skorokhod metric, and use it as a basis for a conformance testing tool for Simulink. We experimentally demonstrate the effectiveness of our tool in finding discrepant behaviors on a set of control system benchmarks, including an industrial challenge problem.

eess.SY↗
arXiv2014

Computing the Skorokhod Distance between Polygonal Traces (Full Paper)

The \emph{Skorokhod distance} is a natural metric on traces of continuous and hybrid systems. For two traces, from $[0,T]$ to values in a metric space $O$, it measures the best match between the traces when allowed continuous bijective timing distortions. Formally, it computes the infimum, over all timing distortions, of the maximum of two components: the first component quantifies the {\em timing discrepancy} of the timing distortion, and the second quantifies the mismatch (in the metric space $O$) of the values under the timing distortion. Skorokhod distances appear in various fundamental hybrid systems analysis concerns: from definitions of hybrid systems semantics and notions of equivalence, to practical problems such as checking the closeness of models or the quality of simulations. Despite its popularity and extensive theoretical use, the \emph{computation} problem for the Skorokhod distance between two finite sampled-time hybrid traces has remained open. We address in this work the problem of computing the Skorokhod distance between two polygonal traces (these traces arise when sampled-time traces are completed by linear interpolation between sample points). We provide the first algorithm to compute the exact Skorokhod distance when trace values are in $\reals^n$ for the $L_1$, $L_2$, and $L_{\infty}$ norms. Our algorithm, based on a reduction to Frechet distances, is fully polynomial-time, and incorporates novel polynomial-time procedures for a set of geometric primitives in $\reals^n$ over the three norms.

eess.SY↗