DOCSLIB.ORG

Data Structure Choices for On-The-Fly Model Checking of Real-Time Systems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Data Structure Choices for On-The-Fly Model Checking of Real-Time Systems Data Structure Choices for On-the-Fly Model Checking of Real-Time Systems Peter Fontana Rance Cleaveland Department of Computer Science Department of Computer Science University of Maryland, College Park University of Maryland, College Park Email: [email protected] Email: [email protected] Abstract—This paper studies the performance of sparse- checking based on proof search: a formula correspond- matrix-based data structures to represent clock zones (con- ing to the assertion that a timed automaton satisfies a vex sets of clock values) in an on-the-fly predicate equation mu-calculus property can be checked in a goal-driven system model checker for timed automata. We analyze the fashion to determine its truth. Zhang and Cleaveland [6] impact of replacing the dense difference bound matrix demonstrated the efficiency of this approach vis a` vis (DBM) with both the linked-list CRDZone and array-list CRDArray data structure. From analysis on the paired- other real-time model-checking approaches. example-by-example differences in time performance, we In this paper we consider the special model check- infer the DBM is either competitive with or slightly faster ing case of timed automata and timed modal equation than the CRDZone, and both perform faster than the systems representing safety properties (also studied in CRDArray. Using similar analysis on space performance, [6]), for which there are still many opportunities for we infer the CRDZone takes the least space, and the DBM performance improvements. One component of such a takes less space than the CRDArray. model checker that has a noticeable influence on perfor- mance is the data structure for the sets of clock values. I. INTRODUCTION When analyzing safety properties, each desired set of Automatic verification of real-time systems is under- clock values forms a convex set of clock values, or clock taken using notations for verifiable programs and check- zone (see Definition 3). The conventional way to store able specifications (see [1]–[6]). One common program a clock zone is as a difference bound matrix (DBM) notation is timed automata [7]. There are specification (see Definition 4) [15], which stores the constraints as notations such as timed computation tree logic (TCTL) a matrix. This approach is used by UPPAAL [16] and [1], [8] and timed extensions of a modal mu-calculus, described in [17]. To potentially save space and time, including one in [3] and another given in [5]. Specifica- instead of representing the set of constraints as a matrix, tions in a timed modal mu-calculus can be written as lists one can represent the set as an ordered linked path of of equations, known as timed modal equation systems constraints where any clock difference not on the path [5], [6], [9]. For information on the untimed modal-mu has the implicit constraint < 1. If we generalize this calculi, see [10]–[12], and see [10], [11] for information to allow for a union of zones to be represented by a on modal equation systems. directed graph of constraints (representing a tree of paths One approach to model checking timed automata as opposed to a single path), we get a clock restriction with timed modal mu-calculus specifications is to use diagram (CRD) [18]. If we compress the nodes to have predicate equation systems (PES), which were invented them represent upper and lower bound constraints as well independently by Groote and Willemse (as parameter- as explicitly encoding both valid and invalid paths, we ized boolean equation systems) [13] and by Zhang and get a clock difference diagram (CDD) [2]. These two Cleaveland [6], [9]. Predicate equation systems provide structures are extensions of binary decision diagrams a general framework for program models including para- (BDDs) (see [19] for information). metric timed automata [6] and Presburger systems [14]. To improve performance, we take the above idea of They also admit a natural on-the-fly approach to model a linked implementation and incorporate the sparseness of the implementation of CRDs while simplifying (or Research supported by NSF Grant CCF-0820072. shrinking) the structure to only support a single clock zone (CRDs and CDDs in general can encode unions of • I : L −! Φ(CX) gives a clock constraint for each clock zones). This simplified structure is a sparse sorted location l. I(l) is called the invariant of l. linked-list implementation of a DBM, the CRDZone (see • E ⊆ L×Σ×Φ(CX)×2CX ×L is the set of edges. Definition 5). We also implement an array-list version In an edge e = (l; a; φ, Y; l0) from l to l0 with action of the CRDZone, the CRDArray (see Definition 6). A a, φ 2 Φ(CX) is the guard of e, and Y represents CRDZone may be seen as a sparse sorted linked-list the set of clocks to reset to 0. implementation of a DBM, and the CRDArray a sparse array-list implementation of the CRDZone. We examine Some sources [6], [23] and our PES checker allow the time and space performance of all three clock zone clock assignments (x1 := x2) in addition to clock resets implementations: the matrix DBM, linked-list CRDZone on edges, other sources [17] allow constraints on clock and array-list CRDArray. differences and other sources [1] allow states to be The contributions of this paper are: labelled with atomic propositions that each state satisfies. Timed automata use clock valuations ν 2 V (V = • We run experiments comparing time and space CX −! ≥0 is the set of all clock valuations), which performance of a model checker (on safety (reach- R at any moment stores a non-negative real value for each ability) properties) with the DBM, CRDZone and clock x 2 CX. The semantics of a timed automaton are CRDArray data structure implementations. described as an infinite-state machine, where each state • We formalize and extend the analysis style per- is a location-valuation pair (l; ν). Transitions represent formed in the model checking experiments of [2], either time advances or edge executions (performing an [6], [9], [18], [20], [21] by utilizing paired data action). For a formal definition of the semantics of a (each implementation checked the same examples) timed automaton, see [7]. and applying descriptive statistics on the paired example-by-example differences on time and space Example 1 (Example of a timed automaton). Consider consumption. See Section VI for details on the the timedReal-Time automaton in Model Figure 1, which Checking models a train statistics and Section VI-B for the analysis. in the generalized railroad crossing (GRC) protocol. After analyzing the experimental results, for time per- exit, x > 1 formance we infer the DBM is either competitive with or 1 slightly faster than the CRDZone and both perform faster 0: far 1: near 2: in than the CRDArray for the examples in this experiment. x < 4 x < 15 approach, x1 := 0 1 in, x1 = 4, 1 In terms of space, we infer the CRDZone takes up the x1 := 0 least space, and the DBM and takes less space than the TCTL (Invalid): AF [near ∨in] CRDArray for the examples in this experiment. Fig. 1. Timed automaton TA<∞1, a model of a train in the generalized railroad crossing (GRC) protocol. TCTL (Valid): AG<∞[near ! AF!TP+TDU[far]] II. PROGRAM MODEL AND SPECIFICATIONS There are three locations—0: far (initial location), A. Timed Automata 1: near and 2: in, with one clock x1. There are the actions 2 A timed automaton encodes the behavior of a real-time approach, in and exit in Σ. Here, location 1 has the system [7], [22]. invariant x1 ≤ 4 while 0 has no invariant. The edge (1: near; in; x1 = 4; fx1g;2: in) has the guard x1 = 4 Definition 1 (Clock constraint φ 2 Φ(CX)). Given a set and resets x1 to 0. of clocks CX, a clock constraint φ is constructed with the following grammar, where xi is a clock and c 2 Z: B. Modal Equation Systems (MES) We use a modal equation system (MES) to represent φ ::= x < c j x ≤ c j x > c j x ≥ c j φ ^ φ i i i i real-time temporal properties that timed automata can Φ(CX) is the set of all possible clock constraints. possess. A MES is an ordered list of equations with variables on the left hand side and basic timed temporal Definition 2 (Timed automaton). A timed automaton logical formulae on the right. Each equation involves a TA = (L; L0; Σ; CX; I; E) is a tuple where: variable X, a basic formula φ and a greatest fixpoint (ν) • L is a finite set of locations with the initial set of or a least fixpoint (µ), and the equation is labeled with ν µ locations L0 ⊆ L. the fixpoint (X = φ or X = φ). For a formal definition • Σ is the set of actions and CX is the set of clocks. of MES syntax and semantics, see [6], [9]. Example 2 (Continuation of Example 1). Again consider While converting to a canonical form takes a consid- the timed automaton in Figure 1. The MES erable amount of time, it is needed to simplify and stan- ν dardize the algorithms for the zone operations includ- X1 = far ^ 8([ − ](X1)) (1) ing time successor (succ(z)) computations and subset represents the safety property “the train is always in checks. For time successor, having the zone in canonical form allows the time elapse operation to simply set all state 0: far”, read as “the variable X1 is the greatest fixpoint of being in state 0: far and for all time advances single-clock upper bound constraints to < 1.
Load more

Recommended publications
  • A Boolean Approach to Unbounded, Fully Symbolic Model Checking of Timed Automata
    A Boolean Approach to Unbounded, Fully Symbolic Model Checking of Timed Automata Sanjit A. Seshia Randal E. Bryant March 2003 CMU-CS-03-117 School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 A shorter version of this paper will appear at CAV '03. Abstract We present a new approach to unbounded, fully symbolic model checking of timed automata that is based on an efficient translation of quantified separation logic to quantified Boolean logic. Our technique preserves the interpretation of clocks over the reals and can check any property expressed in the timed µ calculus. The core operations of eliminating quantifiers over real variables and deciding separation logic are respectively translated to eliminating quantifiers on Boolean variables and checking Boolean satisfiability (SAT). We can thus leverage well-known techniques for Boolean formulas, including Binary Decision Diagrams (BDDs) and recent advances in SAT and SAT-based quantifier elimination. We present preliminary empirical results for a BDD-based implementation of our method. This research was supported in part by a National Defense Science and Engineering Graduate Fellowship and by ARO grant DAAD19-01-1-0485. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwith- standing any copyright annotation thereon. The views and conclusions contained in this document are those of the authors, and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Department of Defense or the U.S. Government. Keywords: Timed automata, Model checking, Boolean logic, Separation logic, Quantifier elimination 1 Introduction Timed automata [2] have proved to be a useful formalism for modeling real-time systems.
    [Show full text]
  • Automata on Timed Structures 1
    Automata on Timed Structures 1 Samy Jaziri June 14, 2019 1This thesis have been supported by ERC project EQualIS Contents I Timed Systems Modeling 7 1 Basic Notions on Transition Systems 8 1.1 Partial Functions . .9 1.2 Labeled Transition System . .9 1.3 Bisimulation on Labeled Transition Systems . 11 2 Timed Structures 14 2.1 Timed Domains . 15 2.2 Updates and Timed Structures . 19 2.3 About Guards and Guarded Timed Structures . 20 3 Automata on Timed Structures 25 3.1 Definition and Representation . 26 3.2 Bisimulation for Automata on Timed Structures . 31 4 Controls 36 4.1 General Definition . 37 4.2 Timed Controls . 39 4.3 Graph and Bisimulation of Timed Controlled Automata . 42 II Determinization of Timed Systems 47 5 Timed Control Determinization 48 5.1 Determinization of Timed Controlled Automata on Timed Systems . 50 5.2 Application to Timed Automata . 56 6 Timed Structure Equivalences 59 6.1 From Product to Maps . 60 6.2 From Maps to Bounded Markings . 67 7 Application to Classes of Timed Automata 74 7.1 Application to Timed Automata . 75 7.2 About Other Determinizable Classes . 76 8 Full Control Determinization 80 8.1 Full Controlled ATS are always Strongly Determinizable . 81 8.2 Application to Input-Determined Models . 85 1 III Diagnosis of Timed Automata 90 9 Diagnosis for Timed Automata 91 9.1 General Context . 92 9.2 Diagnosis with Automata On Timed Structures . 93 10 Timed Set Algebra 97 10.1 Timed Sets . 98 10.2 Regular Timed Interval . 104 10.3 Regular Timed Markings .
    [Show full text]
  • Fully Symbolic Model Checking of Timed Systems Using Di Erence
    p URL httpwwwelsevier nl lo ca te ent cs vol ume htm l pages Fully Symb olic Mo del Checking of Timed Systems using Dierence Decision Diagrams Jesp er Mller Jakob Lichtenberg Henrik R Andersen and Henrik Hulgaard The IT University in Copenhagen Glentevej DK Copenhagen NV Denmark Email jmjlhrahenrikitudk Abstract Current approaches for analyzing timed systems are based on an explicit enumera tion of the discrete states and thus these techniques are only capable of analyzing systems with a handful of timers and a few thousand states We address this lim itation by describing how to analyze a timed system fully symb olically ie by representing sets of discrete states and their asso ciated timing information implic itly We demonstrate the eciency of the symb olic technique by computing the set of reachable states for a nontrivial timed system and compare the results with the stateoftheart to ols Kronos and Uppaal With an implementation based on dif ference decision diagrams the runtimes are several orders of magnitudes b etter The key op eration in obtaining these results is the abilitytoadvance time symb olically Weshow how to do this eciently by essentially quantifying out aspecialvariable z which is used to represent the constant zero The symb olic manipulations given in this pap er are sucienttoverify TCTLformulae fully symb olically Intro duction Mo del checking is to day used extensively for formal verication of nite state systems such as digital circuits and emb edded software The success of the technique is primarily due to
    [Show full text]
  • Clocks, Dbms and States in Timed Systems
    Clocks, DBMs and States in Timed Systems Johan Bengtsson A Dissertation submitted for the Degree of Doctor of Philosophy Department of Information Technology Uppsala University June 2002 Dissertation for the Degree of Doctor of Philosophy in Computer science with specialization in Real Time Systems presented at Uppsala University in 2002. ABSTRACT Bengtsson, J. 2002: Clocks, DBMs and States in Timed Systems. Acta Universitatis Upsaliensis Uppsala Dissertations from the Faculty of Science and Technology 39. 143 pp. Uppsala. ISBN 91-554-5350-3 Today, computers are used to control various technical systems in our society. In many cases, time plays a crucial role in the operation of computers embedded in such systems. This thesis is about techniques and tools for the analysis of timing behaviours of computer systems. Its main contributions are in the development and implementation of UPPAAL, a tool designed to automate the analysis process of systems modelled as timed automata. As the first contribution, we present a software package for timing constraints represented as Difference Bound Matrices. We describe in details, all data-structures and operations for DBMs needed in state-space exploration of timed automata, as well as techniques for efficient imple- mentation. In particular, we have developed two normalisation algorithms to guarantee termina- tion of reachability analysis for timed automata containing constraints on clock differences, that transform DBMs according to not only maximal constants of clocks as in algorithms published in the literature, but also difference constraints appearing in the automata. The second contribu- tion of this thesis is a collection of low level optimisations on the internal data-structures and algorithms of UPPAAL to minimise memory and time consumption.
    [Show full text]
  • Context-Dependent Reachability Analysis in Hybrid System Have Been Explored
    The present work was submitted to the LuFG Theory of Hybrid Systems MASTER OF SCIENCE THESIS CONTEXT-DEPENDENT REACHABILITY ANALYSIS FOR HYBRID AUTOMATA Justin Winkens Examiners: Prof. Dr. Erika Ábrahám Prof. Dr. Thomas Noll Additional Advisor: Stefan Schupp M.Sc. Aachen, May 22, 2018 Abstract This thesis is focused on the analysis of hybrid automata, which incorporate discrete and continuous behavior of a given system into one model. A common approach to analyze these automata for safety is so called flowpipe construction that iteratively constructs a set of over-approximative geometric objects that contain all reachable states of the automaton. Implementations of this approach can be found in toolboxes like Flow*, HyPro/HyDRA or SpaceEx. Unfortunately, the computational cost of flowpipe construction is exponential in the number of variables/dimensions. This thesis aims at improving the HyPro/HyDRA toolbox by implement- ing a context-based approach to flowpipe analysis using variable set separation, in which the state space of the underlying automaton is decomposed into syn- tactically independent sets of variables allowing for separate analysis of the now lower dimensional subspaces. The different subspaces can then be analyzed for their properties enabling us to perform a context-based analysis where we will be able to dynamically choose data structures and algorithms based on the context defined by the automaton’s current location and the state’s subspaces. In the process HyPro is augmented with a new representation called differ- ence bound matrix that is especially suited for the analysis of timed automata. Experimental results in common benchmarks show a speedup of up to 11 000% for decomposed analysis.
    [Show full text]
  • Controller Program Synthesis for Industrial Machines
    R S V E I T I A N S U S S A I R S A V I E N CONTROLLER PROGRAM SYNTHESIS FOR INDUSTRIAL MACHINES by Hans-Jorg¨ Peter A diploma thesis in the DEPARTMENT 6.2 - COMPUTER SCIENCE ADVISORS Prof. Bernd Finkbeiner, PhD Prof. Dr.-Ing. habil. Hartmut Janocha SAARLAND UNIVERSITY, GERMANY November 2005 Erklarung¨ Hiermit versichere ich an Eides statt, dass ich die vorliegende Arbeit selbst¨andig und unter ausschließlicher Verwendung der angegebenen Hilfsmittel angefertigt habe. Saarbr¨ucken, im November 2005 i This work is dedicated to my grandparents Hannelore and Nikolaus Peter ii Abstract In this thesis, a new synthesis algorithm for industrial controller programs is presented. Verification and synthesis are the two basic approaches to guarantee that a system is correct. While verification requires the programmer to provide both the specification and the implementation, synthesis automatically transforms the specification into an implemen- tation that is correct by construction. The presented approach includes a new specification language that is geared towards usability in an industrial set-up. Specifications consist of two parts: a generic description of the machine components that is reused for different programs, and a description of the production goals that is specific to each program. The behaviour of the machine compo- nents is described by timed automata, while the production goals are captured by safety and bounded liveness properties. The advantage of this approach is that the description of the goals, and thus of the behaviour of the overall system, is decoupled from the technical details of the machine components. This results in a high degree of re-usability, adaptivity, and maintainability.
    [Show full text]
  • Symbolic Implementation of Model-Checking Probabilistic Timed Automata
    SYMBOLIC IMPLEMENTATION OF MODEL-CHECKING PROBABILISTIC TIMED AUTOMATA by FUZHI WANG A thesis submitted to The University of Birmingham for the degree of DOCTOR OF PHILOSOPHY School of Computer Science The University of Birmingham September 2006 Abstract In this thesis, we present symbolic implementation techniques for model checking prob- abilistic timed automata as models for systems, for example, communication networks and randomised distributed algorithms. Given a system model as probabilistic timed au- tomata and a specification, such as, “a leader will be elected within 5 time units with probability 0.999” and “the message can be successfully delivered within 3 time units with probability 0.985”, a probabilistic real-time model checker can automatically verify if the model satisfies the specification. Motivated by the success of symbolic approaches to both non-probabilistic real-time model checking and untimed probabilistic model checking, we present a symbolic implementation of model checking for probabilistic timed automata, which is based on the data structure called Multi-terminal Binary Decision Diagrams (MTBDDs). Our MTBDD-based symbolic implementation is generic with respect to the support of the real-time information. Two data structures representing the real-time information are supported, Difference Bound Matrices (DBMs) and Difference Decision Diagrams (DDDs). We develop explicit and symbolic implementations of model check- ing for probabilistic timed automata, focusing on probabilistic reachability properties. The explicit implementation concerns both forward and backward generation of the zone graph, where the latter results in non-convex zones in the case of calculating the minimum probability. The symbolic implementation is of the forward exploration and is built on the PRISM software.
    [Show full text]
  • Symbolic Model Checking of Timed Guarded Commands Using
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector THE JOURNAL OF LOGIC AND ALGEBRAIC PROGRAMMING The Journal of Logic and Algebraic Programming 52–53 (2002) 53–77 www.elsevier.com/locate/jlap Symbolic model checking of timed guarded commands using difference decision diagramsୋ Jesper Møller∗, Henrik Hulgaard, Henrik Reif Andersen Department of Innovation, The IT University of Copenhagen, Denmark Abstract We describe a novel methodology for analyzing timed systems symbolically. Given a formula representing a set of states, we describe how to determine a new formula that represents the set of states reachable by taking a discrete transition or by advancing time. The symbolic representations are given as formulae expressed in a simple first-order logic over difference constraints of the form x − y d which can be combined with Boolean operators and existentially quantified. We also show how to symbolically determine the set of states that can reach a given set of states (i.e., a backward step), thus making it possible to verify timed CTL-formulae symbolically. The main contribution is a way of advancing time symbolically essentially by quantifying out a special variable z which is used to represent the current zero point in time. We also describe a data structure called DDDs for representing difference constraint formulae, and we demonstrate the efficiency of the symbolic technique by analyzing two scheduling protocols using a DDD-based model checker. © 2002 Elsevier Science Inc. All rights reserved. Keywords: Real-time systems; Timed automata; Timed guarded commands; Symbolic model check- ing; Timed CTL; Difference constraint systems; Difference decision diagrams; Quantifier elimination 1.
    [Show full text]
  • What Else Is Decidable About Integer Arrays? Peter Habermehl, Radu Iosif, Tomáš Vojnar
    What Else Is Decidable about Integer Arrays? Peter Habermehl, Radu Iosif, Tomáš Vojnar To cite this version: Peter Habermehl, Radu Iosif, Tomáš Vojnar. What Else Is Decidable about Integer Arrays?. Foun- dations of Software Science and Computational Structures, 11th International Conference, FOSSACS 2008, Mar 2008, Budapest, Hungary. 10.1007/978-3-540-78499-9_33. hal-01418914 HAL Id: hal-01418914 https://hal.archives-ouvertes.fr/hal-01418914 Submitted on 17 Dec 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. What else is decidable about integer arrays? Peter Habermehl1, Radu Iosif2, and Tom´aˇsVojnar3 1 LSV, ENS Cachan, CNRS, INRIA; 61 av. du Pr´esident Wilson, F-94230 Cachan, France, e-mail:[email protected] 2 VERIMAG,CNRS, 2 av. de Vignate, F-38610 Gi`eres, France, e-mail:[email protected] 3 FIT BUT, Boˇzetˇechova 2, CZ-61266, Brno, Czech Republic, e-mail: [email protected] Abstract. We introduce a new decidable logic for reasoning about infinite arrays of integers. The logic is in the ∃∗∀∗ first-order fragment and allows (1) Presburger constraints on existentially quantified variables, (2) difference constraints as well as periodicity constraints on universally quantified indices, and (3) difference constraints on values.
    [Show full text]
  • Arxiv:2108.00893V2 [Cs.LG] 23 Aug 2021 Ning for Autonomous Systems
    Static analysis of ReLU neural networks with tropical polyhedra Eric Goubault?1, S´ebastienPalumby1, Sriram Sankaranarayanan2, Sylvie Putot1, and Louis Rustenholz1 1 LIX, Ecole Polytechnique, CNRS and Institut Polytechnique de Paris, 91128 Palaiseau, France [email protected] 2 Engineering Center Computer Science, University of Colorado at Boulder, USA [email protected] Abstract. This paper studies the problem of range analysis for feedfor- ward neural networks, which is a basic primitive for applications such as robustness of neural networks, compliance to specifications and reacha- bility analysis of neural-network feedback systems. Our approach focuses on ReLU (rectified linear unit) feedforward neural nets that present spe- cific difficulties: approaches that exploit derivatives do not apply in gen- eral, the number of patterns of neuron activations can be quite large even for small networks, and convex approximations are generally too coarse. In this paper, we employ set-based methods and abstract inter- pretation that have been very successful in coping with similar difficulties in classical program verification. We present an approach that abstracts ReLU feedforward neural networks using tropical polyhedra. We show that tropical polyhedra can efficiently abstract ReLU activation func- tion, while being able to control the loss of precision due to linear com- putations. We show how the connection between ReLU networks and tropical rational functions can provide approaches for range analysis of ReLU neural networks. We report on a preliminary evaluation of our approach using a prototype implementation. 1 Introduction and related work Neural networks are now widely used in numerous applications including speech recognition, natural language processing, image segmentation, control and plan- arXiv:2108.00893v2 [cs.LG] 23 Aug 2021 ning for autonomous systems.
    [Show full text]
  • Incremental Closure for Systems of Two Variables Per Inequality
    City Research Online City, University of London Institutional Repository Citation: Howe, J. M. ORCID: 0000-0001-8013-6941, King, A. and Simon, A. (2019). Incremental closure for systems of two variables per inequality. Theoretical Computer Science, 768, pp. 1-42. doi: 10.1016/j.tcs.2018.12.001 This is the published version of the paper. This version of the publication may differ from the final published version. Permanent repository link: https://openaccess.city.ac.uk/id/eprint/21241/ Link to published version: http://dx.doi.org/10.1016/j.tcs.2018.12.001 Copyright: City Research Online aims to make research outputs of City, University of London available to a wider audience. Copyright and Moral Rights remain with the author(s) and/or copyright holders. URLs from City Research Online may be freely distributed and linked to. Reuse: Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. City Research Online: http://openaccess.city.ac.uk/ [email protected] Theoretical Computer Science 768 (2019) 1–42 Contents lists available at ScienceDirect Theoretical Computer Science www.elsevier.com/locate/tcs Incremental Closure for Systems of Two Variables Per Inequality ∗ ∗ Jacob M. Howe a, , Andy King b, , Axel Simon c a Department of Computer Science, City, University of London, EC1V 0HB, UK b School of Computing, University of Kent, Canterbury, CT2 7NF, UK c Google, Mountain View, California, CA 94043, USA a r t i c l e i n f o a b s t r a c t Article history: Subclasses of linear inequalities where each inequality has at most two variables are Received 1 December 2017 popular in abstract interpretation and model checking, because they strike a balance Received in revised form 14 October 2018 between what can be described and what can be efficiently computed.
    [Show full text]
  • Tropical Abstractions of Max-Plus Linear Systems
    Tropical Abstractions of Max-Plus Linear Systems B Muhammad Syifa’ul Mufid1( ), Dieky Adzkiya2, and Alessandro Abate1 1 Department of Computer Science, University of Oxford, Oxford, UK {muhammad.syifaul.mufid,alessandro.abate}@cs.ox.ac.uk 2 Department of Mathematics, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia [email protected] Abstract. This paper describes the development of finite abstractions of Max-Plus-Linear (MPL) systems using tropical operations. The idea of tropical abstraction is inspired by the fact that an MPL system is a discrete-event model updating its state with operations in the tropical algebra. The abstract model is a finite-state transition system: we show that the abstract states can be generated by operations on the tropical algebra, and that the generation of transitions can be established by tropical multiplications of matrices. The complexity of the algorithms based on tropical algebra is discussed and their performance is tested on a numerical benchmark against an existing alternative abstraction approach. Keywords: MPL system · Tropical algebra · Definite form Difference-bound matrix · Abstraction · Reachability 1 Introduction Tropical mathematics has been a rapidly growing subject since it was firstly introduced [15]. It has branches in mathematical fields such as tropical geometry [11] and tropical algebra [15]. The latter denotes an algebraic structure that uses max or min for addition and + for multiplication, respectively - hence, it is well known as max-plus or min-plus algebra. In this paper, we use the former operation to define the tropical algebra. A class of discrete-event system (DES) based on tropical algebra is the Max- Plus-Linear (MPL) one [5].
    [Show full text]