Communications and Control Engineering Kuize Zhang Lijun Zhang Lihua Xie Discrete-Time and Discrete-Space Dynamical Systems Communications and Control Engineering Series Editors Alberto Isidori, Roma, Italy Jan H van Schuppen, Amsterdam, The Netherlands Eduardo D Sontag, Boston, USA Miroslav Krstic, La Jolla, USA Communications and Control Engineering is a high-level academic monograph series publishing research in control and systems theory, control engineering and communications It has worldwide distribution to engineers, researchers, educators (several of the titles in this series find use as advanced textbooks although that is not their primary purpose), and libraries The series reflects the major technological and mathematical advances that have a great impact in the fields of communication and control The range of areas to which control and systems theory is applied is broadening rapidly with particular growth being noticeable in the fields of finance and biologically-inspired control Books in this series generally pull together many related research threads in more mature areas of the subject than the highly-specialised volumes of Lecture Notes in Control and Information Sciences This series’s mathematical and control-theoretic emphasis is complemented by Advances in Industrial Control which provides a much more applied, engineering-oriented outlook Indexed by SCOPUS and Engineering Index Publishing Ethics: Researchers should conduct their research from research proposal to publication in line with best practices and codes of conduct of relevant professional bodies and/or national and international regulatory bodies For more details on individual ethics matters please see: https://www.springer.com/gp/authors-editors/journal-author/journal-authorhelpdesk/publishing-ethics/14214 More information about this series at http://www.springer.com/series/61 Kuize Zhang Lijun Zhang Lihua Xie • • Discrete-Time and Discrete-Space Dynamical Systems 123 Kuize Zhang School of Electrical Engineering and Computer Science KTH Royal Institute of Technology Stockholm, Sweden Lijun Zhang School of Marine Science and Technology Northwestern Polytechnical University Xi’an, China Lihua Xie School of Electrical and Electronic Engineering Nanyang Technological University Singapore, Singapore ISSN 0178-5354 ISSN 2197-7119 (electronic) Communications and Control Engineering ISBN 978-3-030-25971-6 ISBN 978-3-030-25972-3 (eBook) https://doi.org/10.1007/978-3-030-25972-3 © Springer Nature Switzerland AG 2020 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface Discrete-time and discrete-space dynamical systems are widely used in various areas, e.g., decision-making or computation For instance, Boolean control networks were initially proposed to model genetic regulatory networks; finite automata and labeled Petri nets as models of discrete-event systems have been applied to many engineering fields such as manufacturing processes, production scheduling; nondeterministic finite-transition systems have been applied to model checking and automated synthesis of cyber-physical systems; cellular automata have been used in quantum physics, biological dynamics as well as computational mathematics, just to name a few Let us introduce what discrete-time and discrete-space dynamical systems mean in this book (transitioned from the well-known dynamical systems over Euclidean spaces) It seems widely accepted nowadays that the world consists of time elapsing “continuously” and space arranged “seamlessly” Evolution of a process within this setting can be described by nonlinear differential equations in a (locally) Euclidean space Such description is intuitive but explicit solutions are not easy to obtain, which makes it difficult to analyze their long-term behavior To bypass the obstacle of finding an explicit solution, one can discretize time to generate sequences of points (called trajectories) by iterations of maps Another difficulty of analyzing long-term behavior of a dynamical system over a continuous state space lies in its continuity, since it is almost impossible to separate the crucial locations from the redundant ones To overcome this dilemma, space is also discretized to guarantee every sequence has a convergent subsequence For example, this property holds in any finite metric space; in addition, every sequence has an increasing subsequence (and thus convergent (possibly to infinity)) in the countable metric space Nn (as a subspace of Rn ); this property even holds in some uncountable spaces, like the Cantor space, where every point can be regarded as a mapping from Zn to a common alphabet It is worth mentioning that none of the three kinds of spaces above is locally Euclidean; indeed, they have topological dimension (in the sense of Čech–Lebesgue covering dimension) while Euclidean spaces or general manifolds that are not the singleton have positive topological dimensions For dynamical systems over these zero-dimensional spaces, explicit solutions can be found without v vi Preface any restriction on the system maps and therefore the focus is on the spaces In this book, such dynamical systems are called discrete-time and discrete-space dynamical systems What makes it challenging to study these systems mainly lies in the unrestricted system maps Among diverse fundamental properties (controllability, observability, detectability, and stabilizability) in control theory, observability and detectability stand out: both of them deduce internal information out of external information Compared to controllability and stabilizability which focus directly on trajectories, observability and detectability are somewhat indirect; nevertheless, the latter provide basis for quantitative analysis of long-term behavior as well as prerequisites for (automated) controller synthesis Exploring long-term behavior is a long-lasting topic in dynamical systems In this book, we will study decidability and complexity of observability and detectability as well as their variants for different kinds of discrete-time and discrete-space dynamical systems, such as the aforementioned Boolean control networks, finite automata, and labeled Petri nets Various methods will be introduced to study different kinds of systems Fundamental characterization for observability and detectability will (1) provoke new related studies such as state estimation and automated synthesis; (2) help reveal relations and essential differences among different types of systems For instance, for finite automata, strong detectability is verifiable in polynomial time while weak detectability is PSPACE-complete; in contrast for labeled Petri nets, strong detectability is EXPSPACE-hard while weak detectability is even undecidable We will investigate another “indirect” property that is called “invertibility” or “reversibility” It means for control systems that an output sequence allows uniquely determining the corresponding input sequence, but it means for dynamical systems without control that every trajectory has a unique backward-in-time extension For Boolean control networks, a tool (equivalent to cellular automata) beyond them is demanding to characterize invertibility, which in turn reveals the importance of invertibility in unveiling some link between Boolean control networks and cellular automata Finally, let us point out several potential applications of fundamental properties of discrete-time and discrete-space dynamical systems (particularly of such systems with finitely many states) in formal verification and synthesis of hybrid (control) systems Since the verification problem for basic properties such as controllability and observability of (infinite-state) hybrid systems is mostly formidable and it is likely that many properties are undecidable By constructing a finite-state system as an approximation which (bi)simulates a given hybrid system while preserving some useful property, one can benefit in verifying this property over the finite approximating system instead Automated synthesis can be dealt with in an analogous way From this perspective, results presented in this book can also be related to the active field of formal verification and synthesis emerging in the last two decades Preface vii This book will first introduce (in Chap 1) basic mathematical preliminaries such as graph theory, the semitensor product of matrices, finite automata, and topology to support the study throughout the book, and also the differences between different types of zero-dimensional spaces and Euclidean spaces based on these preliminaries Second, it will discuss (in Chap 2) different types of discrete-time and discrete-space dynamical systems (Boolean control networks, finite automata, nondeterministic finite-transition systems, Petri nets, and cellular automata), highlighting their similarities and differences These essential differences show that there exists no unified method available to deal with these types of dynamical systems Then in the main parts (the remaining chapters), it will collect a series of recent fundamental results in control-theoretic and topological dynamical problems of discrete-time and discrete-space dynamical systems, e.g., invertibility, observability, detectability, reversibility, etc., by developing new techniques In addition, the book will also contain some practical applications of these problems in systems biology, etc In order to study different types of systems, various methods, e.g., a Chapter Detectability of FSAs Chapter 2.3 FSAs Chapter 10 Detectability of LPNs Chapter 2.4 LPNs Chapter 1.1 Graph theory Chapter 1.3 Finite automata Chapter 1.2 Semitensor product Chapter 1.4 Topology Chapter 2.2 NFTSs Chapter 2.1 BCNs Chapter 3.2 Invertibility of BCNs Chapter 2.5 CAs Chapter Observability of NFTSs Chapter Detectability of NFTSs Chapter Observability of BCNs Chapter Detectability of BCNs Chapter 3.3 Nonsingularity of BCNs Chapter Observability and detectability of large-scale BCNs Chapter 11 Generalized reversibility of CAs Fig 0.1 Reading flow of the book viii Preface semitensor product method, a graph-theoretic method, a finite-automaton method, a topological method, etc., will be adopted The book is aiming at bringing the reader new understanding of discrete-time and discrete-space dynamical systems While reading the book, the reader could refer to the reading flow shown in Fig 0.1 through arrow lines of the same type We did not introduce the mathematical tools used to handle labeled Petri nets in Chap 1, but introduced them in Chap 10 when labeled Petri nets were studied, because such tools quite depend on the labeled Petri nets themselves We are in debt to Dr Shaoshuai Mou at Purdue University, USA, Dr Rong Su at Nanyang Technological University, Singapore, Dr Karl Henrik Johansson at KTH Royal Institute of Technology, Sweden, Dr Ting Liu and Dr Daizhan Cheng both at Academy of Mathematics and Systems Science, Chinese Academy of Sciences, PR China, Dr Majid Zamani at University of Colorado Boulder, USA, and Dr Alessandro Giua at University of Cagliari, Italy, who have all coauthored with us a few papers, which have been included in this book We would like to thank Dr Chuang Xu at University of Copenhagen, Denmark, and Mr Ping Sun and Mr Zhenkun Wang at Harbin Engineering University, PR China, who helped a lot in proofreading the manuscripts We are also indebted to Mr Oliver Jackson for his patient support Stockholm, Sweden Xi’an, China Singapore Kuize Zhang Lijun Zhang Lihua Xie Contents Part I Introduction Preliminaries 1.1 Graph Theory 1.2 The Semitensor Product of Matrices 1.3 Finite Automata and Regular Languages 1.4 Topological Spaces and Discrete Spaces 1.4.1 Topological Spaces 1.4.2 Separability 1.4.3 Compactness 1.4.4 Connectedness 1.4.5 Metric Spaces 1.4.6 Topological Dimensions 1.4.7 Several Discrete Topological Spaces Notes References 10 17 17 18 18 19 20 21 24 31 31 Different Types of Discrete-Time and Discrete-Space Dynamical Systems 2.1 Boolean Control Networks 2.2 Nondeterministic Finite-Transition Systems 2.3 Finite-State Automata 2.4 Labeled Petri Nets 2.5 Cellular Automata Notes References 35 35 39 40 42 46 54 55 Invertibility and Nonsingularity of Boolean Control Networks 3.1 Notions of Invertibility and Nonsingularity 59 59 Part II Boolean Control Networks ix 206 10 Detectability of Labeled Petri Nets 10.4 Decidability of Weak Detectability In this section, we characterize weak detectability of labeled Petri nets Theorem 10.2 It is undecidable to verify if a labeled Petri net G is weakly detectable Proof We reduce the undecidable language inclusion problem to the non-weak detectability problem (see Fig 10.5 as a sketch) Given an -free labeled Petri net G and two copies of another -free labeled Petri net G , where G and G share the same alphabet, we effectively construct another labeled Petri net G by adding places, transitions, labels, and arcs (all with weight 1) into the three nets as follows: Add place p0 with exactly token; add places p1 , , p6 , p4 , , p6 ; Add three labels x, a, b; Add transitions tx1 , , tx6 all with label x, add transitions ta1 , ta2 , ta3 all with label a, and add transitions tb1 , , tb6 all with labels b; Add arcs p0 → tx1 → p1 → tx4 → p2 → ta1 → p2 → tb1 → p3 → tb2 → p3 , p0 → tx2 → p4 → tx5 → p5 → ta2 → p5 → tb3 → p6 → tb4 → p6 , p0 → tx3 → p4 → tx6 → p5 → ta3 → p5 → tb5 → p6 → tb6 → p6 ; For all transitions t of G add arcs t → p1 → t, for all places p of G add arcs p → ta1 , and for all transitions t of the first (resp second) copy of G add arcs t → p4 → t (resp t → p4 → t) For net G, initially only one of the transitions tx1 , tx2 , tx3 can fire If tx1 fires, then G can run After tx4 fires, G will stop and never run again If tx2 fires, then the first copy of G can run After tx5 fires, the first copy of G will stop and never run again The functionality of tx3 is similar to tx2 but induces the second copy of G to run Hence, one has Lω (G) ={xσ |σ ∈ Lω (G ) ∪ Lω (G )} n ω ∪ {xσ xa b |σ ∈ L(G ), n ∈ (G , σ )} ∪ {xσ xa n bω , xσ xa ω |σ ∈ L(G ), n ∈ N}, (10.9a) (10.9b) (10.9c) where (G , σ ) denotes { p∈P M( p)|M ∈ M(G , σ )}, i.e., the set of sums of numbers of tokens of all places of all reachable markings of G after finite label sequence σ has been observed, where P is the set of places of G Note that G is -free, then (G , σ ) is a finite set Note also that only one of G and the two copies of G can run Next we prove that L(G ) ⊂ L(G ) if and only if G is not weakly detectable Then the weak detectability of -free labeled Petri nets is undecidable by Proposition 10.1 Assume L(G ) ⊂ L(G ) Consider xσ in (10.9a) If σ ∈ Lω (G ), then after xσ has been observed, where σ σ , one has either p1 contains token (G runs), or p4 contains token (the first copy of G runs), or p4 contains token (the second copy of G runs) If σ ∈ Lω (G ) \ Lω (G ), then after xσ has been observed, where 10.4 Decidability of Weak Detectability 207 Fig 10.5 Sketch for the reduction in the undecidability proof of Theorem 10.2, where for each newly added transition, its label is just its subscript, reproduced from Masopust and Yin (2019) with permission @ 2019 Elsevier Ltd σ σ , one has either p4 contains token (the first copy of G runs) or p4 contains token (the second copy of G runs) Consider xσ xa n bω in (10.9b) or (10.9b) If σ ∈ L(G ), then after xσ xa n bm has been observed, where m ∈ Z+ , one has either p3 contains token (G runs), or p6 contains token (the first copy of G runs), or p6 contains token (the second copy of G runs) If σ ∈ L(G ) \ L(G ), then after xσ xa n bm has been observed, where m ∈ Z+ , one has either p6 contains token (the first copy of G runs) or p6 contains token (the second copy of G runs) Consider xσ xa ω in (10.9c) Note that here σ cannot belong to L(G ) since G is -free After xσ xa n has been observed, where n ∈ Z+ , one has either p5 contains token (the first copy of G runs) or p5 contains token (the second copy of G runs) Hence, G is not weakly detectable Assume L(G ) ⊂ L(G ) Arbitrarily choose σ ∈ L(G ) \ L(G ), and choose xσ xa k bω in (10.9a), where k = max{ (G , σ )} Then after xσ xa k bm has been observed, where m ∈ Z+ , one has p3 contains token, all places of G contain no token, the two copies of G are in their initial marking, and all other newly added places contain no token Hence, G is weakly detectable 208 10 Detectability of Labeled Petri Nets Notes The decidability of strong detectability for labeled Petri nets without the promptness assumption is an interesting open problem In Zhang and Giua (2018), the weak detectability of labeled Petri nets with inhibitor arcs was proved to be undecidable by reducing the undecidable language equivalence problem (Proposition 10.1) to negation of the weak detectability problem The reduction is as follows: given two labeled Petri nets G and G with the same alphabet, effectively compute a labeled Petri net G with inhibitor arcs It was shown that if L(G ) = L(G ) then G is not weakly detectable When L(G ) = L(G ), without loss of generality there exists σ ∈ L(G ) \ L(G ), an infinitely long label sequence σ generated by G having σ as a subsequence was found such that after observing a special prefix of σ containing σ , all places of G became empty, resulting in that G is weakly detectable It is not known whether such a reduction could be found for labeled Petri nets However, later on in Masopust and Yin (2019), when the language inclusion problem was chosen instead of the language equivalence problem, the above idea of clearing all places of G was implemented for labeled Petri nets (see the proof of Theorem 10.2) by a similar novel reduction, resulting in the stronger result (i.e., Theorem 10.2) Results on other interesting variants of notions of strong detectability and weak detectability for labeled Petri nets can be found in the manuscript (Zhang and Giua 2018) References Atig MF, Habermehl P (2009) On Yen’s path logic for Petri nets In: Bournez O, Potapov I (eds) Reachability problems Springer, Berlin, pp 51–63 Cassandras CG, Lafortune S (2010) Introduction to discrete event systems, 2nd edn Springer Publishing Company Dickson LE (1913) Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors Am J Math 35(4):413–422 Hack M (1976) Petri net languages Technical report Cambridge, MA, USA Li Z, Zhou M (2009) Deadlock resolution in automated manufacturing systems: a novel petri net approach, 1st edn Springer Publishing Company Lipton RJ (1976) The reachability problem requires exponential space Yale University Department of computer science Research report Department of computer science, Yale University Masopust T, Yin X (2019) Deciding detectability for labeled petri nets Automatica 104:238–241 Mazaré L (2004) Using unification for opacity properties Verimag Tech Rep Rackoff C (1978) The covering and boundedness problems for vector addition systems Theor Comput Sci 6(2):223–231 Reutenauer C (1990) The mathematics of petri nets Upper Saddle River, Prentice-Hall Inc, NJ, USA Saboori A, Hadjicostis CN (2013) Verification of initial-state opacity in security applications of discrete event systems Inf Sci 246:115–132 References 209 Seatzu C, Silva M, van Schuppen JH (eds) (2013) Control of discrete-event systems: automata and petri-net perspectives Lecture notes in control and information sciences, vol 433 Springer, London, p 478 Tong Y et al (2017) Decidability of opacity verification problems in labeled Petri net systems Automatica 80:48–53 Yen HC (1992) A unified approach for deciding the existence of certain petri net paths Inf Comput 96(1):119–137 Zhang K, Giua A (2018) On detectability of labeled Petri nets and finite automata https://arxiv.org/ abs/1802.07551 Zhang K, Giua A (2018) Weak (approximate) detectability of labeled petri net systems with inhibitor arcs IFAC-PapersOnLine 51(7):167–171 14th IFAC workshop on discrete event systems WODES Part VI Cellular Automata Chapter 11 Generalized Reversibility of Cellular Automata Reversibility is a fundamental property of microscopic physical systems, implied by the laws of quantum mechanics, which seems to be at odds with the Second Law of Thermodynamics (Schiff 2008; Toffoli and Margolus 1990) Nonreversibility always implies energy dissipation, in practice, in the form of heat Using reversible cellular automata (CAs) to simulate such systems has caused wide attention since the early days of the investigation of CAs (Toffoli and Margolus 1990; Kari 2005) On the other hand, if a CA is not reversible but reversible over an invariant closed subset, e.g., the limit set (Taaki 2007), it can also be used to describe physical systems locally In this chapter,1 we present a formal definition to represent this class of generalized reversible CAs, and investigate some of their topological properties We refer the reader to Zhang and Zhang (2015), Taaki (2007) for further reading Other variants of generalized reversibility can be found in Castillo-Ramirez and Gadouleau (2017) A CA is a dynamical system which consists of a regular network of finite-state automata (cells) that change their states synchronously depending on the states of their neighbors, according to a local update rule The update rule is quite simple, but CAs show complicated behavior (Schiff 2008; Wolfram 2002) 11.1 Cellular Automata Let Z and d be the set of integers and a given positive integer, respectively Zd denotes a d-dimensional cellular space, and elements of Zd are called cells A finite set S that has at least two elements denotes the state set A map c : Zd → S that specifies d the states of all cells is called a configuration (point) The symbol S Z denotes the d d set of all configurations, which is an uncountable set A function τv : S Z → S Z Theorems 11.1, 11.2, and 11.4 were reproduced from Zhang and Zhang (2015) with permission @ 2015 Old City Publishing Inc Theorems 11.3 and 11.5 were reproduced from Taaki (2007) with permission @ 2007 Old City Publishing Inc © Springer Nature Switzerland AG 2020 K Zhang et al., Discrete-Time and Discrete-Space Dynamical Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-25972-3_11 213 214 11 Generalized Reversibility of Cellular Automata is called a translation determined by vector v ∈ Zd if for all c ∈ S Z , all u ∈ Zd , d τv (c)(u) = c(u + v) A configuration c ∈ S Z is said to be uniform if all its cells are Zd in the same state A configuration c ∈ S is called spatially periodic if there exist linearly independent vectors v1 , , vd ∈ Zd such that τv1 (c) = c, …, τvd (c) = c It d can be seen that each open subset of S Z contains a spatially periodic configuration (this can be obtained by the topological base consisting of cylinders, see Sect 2.5), d then the set of all spatially periodic configurations is dense in S Z d Endowed with the following metric: for all configurations c, e ∈ S Z , d d(e, c) = if c = e, − min{|v||v∈Zd ,c(v)=e(v)} if c = e, where | · | denotes the Euclidean norm or the max-norm, which induce the same d topology, the whole configuration space S Z forms a compact, totally disconnected (that is, any two distinct points can be separated by two disjoint clopen sets) and perfect (that is, there is no isolated point) space (Proposition 1.27) A function G : d d S Z → S Z is called a CA function if G is continuous and commutes with every d d translation In this case, (S Z , G) is called a CA For any two CAs (S Z , G) and Zd (S , H ), the composition H ◦ G (or briefly H G) is also a CA function It is known d d that (Proposition 2.2) G : S Z → S Z is a CA function if and only if there is a finite d ordered set N = {n , , n m } ⊂ Z of m distinct vectors, called neighborhood, and d a local rule g : S m → S such that for all c ∈ S Z , all n ∈ Zd , G(c)(n) = g(c(n + n ), , c(n + n m )) In a CA G, every initial configuration evolves under iterations of the global function G A CA G is said to be injective (surjective), if its global function G is injective (surjective) It is said to be reversible if its global function G is bijective and the inverse function G −1 is also a CA function All injective CAs are reversible (Proposition 2.3) For a reversible CA, every configuration has one and only one predecessor Intuitively, in a reversible CA, each configuration can evolve both forward and backward uniquely at any time step In Amoroso and Patt (1972), an algorithm for determining whether a given one-dimensional CA is injective or surjective was given However, in Kari (1994), it was proved that it is undecidable whether a given two- or higher dimensional CA is injective or surjective Hence, designing a reversible CA with certain behavior remains far from trivial d n Zd For a CA (S Z , G), its limit set G := ∞ n=0 G (S ) (Definition 2.4) is nonempty, closed, contains a uniform configuration, and satisfies G( G ) = G (Proposition 2.4) That is to say, the restriction of each CA on its limit set is surjective If the limit set of a CA is finite, then it contains only one configuration (Proposition 2.4) Such CAs are called nilpotent The nilpotency of a CA is undecidable (Kari 1992; Culik et al 1989) The limit set G is called reached in finite time d d if G = G n (S Z ) for some n ∈ Z+ Later on, usually we simply denote S Z =: X 11.2 Drazin Inverses 215 11.2 Drazin Inverses The notion of Drazin inverse has applications in many areas, such as in special matrix theory, singular differential and difference equations, finite Markov chains, and graph theory (Meyer 1975; Bu et al 2011; Zhang and Bu 2012; Wang et al 2004) In Drazin (1958), the concept of Drazin inverse was first proposed Definition 11.1 (Drazin 1958) Consider an associative ring (or a semigroup) Given G, H , two elements of the ring (or semigroup), H is called a Drazin inverse of G, written H = G D , if Gk H G = Gk , (11.1) G H = H G, HGH = H for some k ∈ N The least nonnegative integer k for which these equations hold is called the index, written Ind(G), of G By definition, if G is invertible, G D = G −1 Note that if an element G in a semigroup S has a Drazin inverse H ∈ S, then H is a Drazin inverse of G also in every semigroup containing S, and in any subsemigroup of S that contains G and H It is well known (proved in Drazin 1958) that any element G of any associative ring (or semigroup) has at most one Drazin inverse, and if G does have a Drazin inverse, G D commutes with any element that commutes with G Since we will use these properties, next we write a short proof Assume that H1 and H2 are both Drazin inverses of G Denote k = ind(G), then one has H1 = H12 G = H1k+1 G k = H1k+1 G k+1 H2 = H1k G k H2 Furthermore, one has H1 = H1k G k H2 = H1k−1 G k H22 = · · · = H1 G k H2k = G k H2k+1 = H2 Let C be an element such that C G = GC, next we show C H = H C By definition, one has H C = H k+1 G k C = H k+1 C G k = H k+1 C G k+1 H = H k+1 G k+1 C H = H k G k C H Furthermore, H C = H k G k C H = H k−1 G k C H = · · · = G k C H k+1 = C G k H k+1 = C H 11.3 Generalized Reversibility In this section, we characterize generalized reversibility of CAs Theorem 11.1 Consider a CA (X, F) If there exists a function H : X → X such that H is a Drazin inverse of F, then (i) F is reversible over F , (ii) F is reached in finite time, (iii) (X, H ) is also a CA, and (iv) F = H = H (X ) Proof Denote the index of F by t By (11.1), we have F t (X ) = (F t+1 H )(X ) ⊂ F t+1 (X ) ⊂ F t (X ) 216 11 Generalized Reversibility of Cellular Automata That is, F = F t (X ) =: , which shows that is reached in finite time It is easy to verify that is a compact subspace of X by the continuity of F and the compactness of X For every point c ∈ , there exists a point e ∈ X such that F t (e) = c Then we have (H | F| )(c) = (H | F| F t )(e) = (H F F t )(e) = F t (e) = c, (F| H | )(c) = (F| H | F t )(e) = (F H F t )(e) = F t (e) = c Then F is reversible over , and H | = (F| )−1 Since F| is continuous, by the compactness of , H | is also continuous Furthermore, H = H F H = H (F H )t = H t+1 F t = (H | )t+1 F t is continuous That is, (X, H ) is also a CA Since H | is reversible, we have H (X ) = (H t+1 F t )(X ) = H t+1 ( ) = H (X ) = H ( ) = Hence H = = F , = H (X ) Next based on Theorem 11.1, we give an equivalent algebraic characterization for reversibility of a CA over its limit set Theorem 11.2 Consider a CA (X, F) The following three expressions are equivalent: F is reached in finite time, and F| F is injective There exists a CA (X, H ) such that H = F D and F = There exists a function H : X → X such that H = F D H = H (X ) Proof (1)⇒(2): Since F is reached in finite time, we set min{k|k ≥ 0, F = F k (X )} =: t, and have F( F ) = F(F t (X )) = F t+1 (X ) = F t (X ) = F Also F| F is injective and F| F is bijective Define H = (F| F )−(t+1) F t , then H = F D , since (F| F| (F| (F| F F )−(t+1) F t F = (F| (F| F F F ) −(t+1) F ((F| F ) )−(t+1) F| )−(t+1) F t = F(F| )−(t+1) F t F(F| t F (F| −(t+1) F ) F| Ft = )−(t+1) F t , )−(t+1) F t = t F F F F (F| F ) −(t+1) (11.2) F = (F| t F ) −(t+1) F, t F )F = F t t Then from Theorem 11.1, (X, H ) is a CA and H = (2)⇒(3): Obvious (3)⇒(1): This implication holds by Theorem 11.1 F = H (X ) Note that in both cases (2) and (3) of Theorem 11.2, H is unique Finally, based on Theorems 11.1 and 11.2, we give the definition of generalized inverse CA 11.3 Generalized Reversibility 217 Definition 11.2 Consider a CA (X, F) If F has a Drazin inverse H , then CA (X, H ) is called the generalized inverse CA of CA (X, F) Next we prove an implication relation between injectivity of F and F being reached in finite time Actually, this result is a generalization of one result in Culik et al (1989): for a nilpotent CA, the limit set can be reached in finite time (the limit set for such a CA is a singleton, and hence the CA is injective over its limit set) Theorem 11.3 If a CA (X, G) is injective (i.e., bijective) over its limit set G is reached in finite time Proof One sees (by G( X X) = ⊂ G −1 ( x, G, then Proposition 2.4) X) ⊂ G −2 ( G) ⊂ · · · ⊂ X If for some k, G −k ( X ) = G −(k+1) ( X ), then for all m > k we have G −m ( X ) = G −k ( X ) In this case, we claim that G −k ( X ) = X Suppose on the contrary that X \ G −k ( X ) =: Y = ∅, then none of configurations in Y will go into X However, since Y is open, there is c ∈ Y that is spatially periodic (since the set of all spatially periodically configurations is dense) Note that such c is eventually periodic, and hence will go into X , a contradiction Hence, either G m (X ) = X for some m ∈ N, or for every m ∈ N there is a configuration c−m ∈ X that enters X after exactly m steps Now consider a CA such that G| X is injective, we show that X is reached in finite time Suppose on the contrary that X is not reached in finite time Then for (− j) (−( j−1)) , each i ∈ N, there exist configurations ci(−i) , , ci(0) such that G(ci ) = ci (− j) (0) (0) ∈ / X for all < j ≤ i For all i ∈ N, since ci ∈ X , there exist ci ∈ X , but ci ei ∈ X such that G(ei ) = ci(0) (by Proposition 2.4), then ei = ci(−1) , and ei (0) = ci(−1) (0) without loss of generality (by using proper translations) Consider sequences {ei }i≥0 , {ci(0) }i≥0 , {ci(−1) }i≥1 , {ci(−2) }i≥2 , Since X is compact, there is a strictly increasing function μ : N → N such that all sequences (0) (−1) }i≥0 , {cμ(i) }i≥0 {eμ(i) }i≥0 , {cμ(i) converge Denote their limits by e, c(0) , and c(−1) , respectively Then by the continuity of G (preserving limits), we have G(e) = c(0) = G(c(−1) ) Furthermore, e ∈ X because X is closed (by Proposition 2.4) Apparently e = c(−1) Similarly, we can find a converging subsequence of {ci(−2) }i≥2 such that the subsequence converges to c(−2) and G(c(−2) ) = c(−1) Repeating this procedure, we can find a sequence , c(−2) , c(−1) such that G(c−( j+1) ) = c(− j) for all j ∈ Z+ Hence, c(−1) ∈ X (again by Proposition 2.4), and G| X is not injective, a contradiction 218 11 Generalized Reversibility of Cellular Automata Remark 11.1 Note that G being reached in finite time does not imply the injectivity of G| G , since there are surjective CAs that are not injective, e.g., elementary CA 102, i.e., the CA specified by {0, 1}Z , f, {−1, 0, 1} , where f (1, 1, 1) = 0, f (0, 1, 1) = 0, f (1, 1, 0) = 1, f (0, 1, 0) = 1, f (1, 0, 1) = 1, f (0, 0, 1) = 1, f (1, 0, 0) = 0, f (0, 0, 0) = 0, (01101110) is the binary representation of 102 For this CA, every configuration has exactly two predecessors Based on Theorems 11.3 and 11.2, we can give an equivalent algebraic characterization for reversibility of a CA over its limit set Theorem 11.4 Consider a CA (X, G) The following three expressions are equivalent: G is injective over G There exists a CA (X, H ) such that H = G D and G = There exists a function H : X → X such that H = G D H = H (X ) Proof (1) ⇒ (2): Assume that G| G is injective From Theorem 11.3, G is reached in finite time Then from Theorem 11.1, there is a function H : X → X such that H = G D , H is continuous, and G = H = H (X ) Since G commutes with any translation, then H also commutes with any translation Hence, (X, H ) is a cellular automaton (2) ⇒ (3): Obvious (3) ⇒ (1): This implication follows from Theorem 11.2 On decidability of existence of a generalized inverse CA of a CA, the following result holds, which implies that it is far from trivial to design a CA that possesses a generalized inverse CA Theorem 11.5 It is undecidable whether a d-dimensional CA G (d ≥ 1) has a generalized inverse CA, i.e., whether G is reversible over G Proof The original proof of the undecidability of nilpotency in Kari (1992) (for d = 1, which implies undecidability for d > 1) can be used to prove the undecidability of reversibility of G over G Consider the undecidable NW-deterministic tiling problem (proved in Kari 1992): given a partial function φ : S × S → S, determine where S is a finite set, whether 11.3 Generalized Reversibility 219 the subshift φ := {c ∈ S Z |φ(x(i, j), x(i + 1, j)) = x(i, j + 1)∀i, j ∈ Z} defined by φ is empty A partial function φ can be extended to a function ψ : (S ∪ {q}) × (S ∪ {q}) → S ∪ {q} by ψ(a, b) = φ(a, b) if φ(a, b) is defined, and ψ(a, b) = q otherwise Then, it can be seen that the one-dimensional CA G with local rule ψ is nilpotent if and only if φ is empty If G is nilpotent, then its limit set G is a singleton (by Proposition 2.4), is reached in finite time (Culik et al 1989), and consists of xq with xq (i) = q for all i ∈ Z, hence φ = ∅ If G is not nilpotent, then there is a configuration x in G that differs from x q Without loss of generality, we assume that x(0) = q, then by arranging x with its predecessors, we obtain an upper half place of Z2 such that all cells in positions (i, j) ∈ Z2 with ≤ i ≤ j are in states of S, i.e., it tiles a whole 1/8 plane Z2 We then define a configuration sequence {τ(k,2k) (x)}k∈N , by the compactness of S Z (by Proposition 1.27), one subsequence of {τ(k,2k) (x)}k∈N converges and hence the corresponding limit has all cells in states of S, i.e., φ = ∅ In addition, if the above CA G is nilpotent, then it is obviously reversible over its limit set Otherwise, configuration xq ∈ G has at least two different preimages in G , one of which is x q , another of which is c0 that has all cells in state q but the cell We can choose c0 (0) as follows: since φ = ∅, we find an arbitrary configuration x ∈ φ , and set c0 (0) = x (0, 0) Then one easily sees that there exist configurations , c−2 , c−1 ∈ S Z such that G(c−( j+1) ) = c− j for all j ∈ N Add in addition, for each j ∈ N, c− j has all cells in state q but the cells 0, , j Hence c0 ∈ G Notes Results on other interesting variants of notions of generalized reversibility of CAs can be found in Castillo-Ramirez and Gadouleau (2017) References Amoroso S, Patt YN (1972) Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures J Comput Syst Sci 6(5):448–464 Bu C, Zhang K, Zhao J (2011) Representations of the Drazin inverse on solution of a class singular differential equations Linear Multilinear Algebr 59(8):863–877 Castillo-Ramirez A, Gadouleau M (2017) Von Neumann regular cellular automata In: Dennunzio Alberto et al (eds) Cellular automata and discrete complex systems Springer International Publishing, Cham, pp 44–55 Culik K II, Pachl J, Yu S (1989) On the limit sets of cellular automata SIAM J Comput 18(4):831– 842 220 11 Generalized Reversibility of Cellular Automata Drazin MP (1958) Pseudo-inverses in associative rings and semigroups Am Math Mon 65(7):506– 514 Kari J (1992) The nilpotency problem of one-dimensional cellular automata SIAM J Comput 21(3):571–586 Kari J (1994) Reversibility and surjectivity problems of cellular automata J Comput Syst Sci 48(1):149–182 Kari J (2005) Theory of cellular automata: a survey Theor Comput Sci 334(1):3–33 Meyer CD Jr (1975) The role of the group generalized inverse in the theory of finite Markov chains SIAM Rev 17(3):443–464 Schiff JL (2008) Cellular automata: a discrete view of the world, 1st edn Wiley-Interscience Taaki S (2007) Cellular automata reversible over limit set J Cell Autom 2:167–177 Toffoli T, Margolus NH (1990) Invertible cellular automata: a review Phys D: Nonlinear Phenom 45(1):229–253 Wang G, Wei Y, Qiao S (2004) Generalized inverses: theory and computations Science Press, Beijing/New York Wolfram S (2002) A new kind of science Wolfram Media Zhang K, Bu C (2012) Group inverses of matrices over right Ore domains Appl Math Comput 218(12):6942–6953 Zhang K, Zhang L (2015) Generalized reversibility of topological dynamical systems and cellular automata J Cell Autom 10:425–434 Index A Aggregated graph, 122 Alphabet, 10 Associated graph, 119 B Ball, 30 Boolean control network, 35 Boolean matrix, Boolean network, 35 C Cantor set, 26 Cantor space, 29 ˇ Cech–Lebesgue covering dimension, 22 Cell, 46 Cellular automaton, 46 Clopen, 26 Compact, 19 Complexity, 16 Configuration, 26, 46 Controllability, 38 Cover, 18 Cycle, Cylinder, 26 D Decidability, 16 Dependency graph, 119 Detectability, 106, 107, 167, 182, 199, 200 Detectability graph, 107 Deterministic finite automaton, 10 Deterministic Turing machine, 49 Dimension, 47 Directed graph, Discrete-event system, 40 Discrete space, Drazin inverse, 215 E Edge, -free labeled Petri net, 44 -nondeterministic finite automaton, 14 Event, 10 F Finite-state automaton, 41 Fire, 43 Formal language, 10 Fullshift, 53 G Generalized inverse cellular automaton, 217 Genetic regulatory network, 35 H Homeomorphism, 29 Homing input sequence, 106, 112, 167 I Invertibility, 59, 65 K Khatri–Rao product, © Springer Nature Switzerland AG 2020 K Zhang et al., Discrete-Time and Discrete-Space Dynamical Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-25972-3 221 222 L Labeled Petri net, 44 Labeling function, 44 Lebesgue number, 24 Left shift, 46 Limit set, 52 Logical matrix, M Marking, 43 Maximal run, 40 Mealy machine, 39 Moore machine, 39 N Node aggregation, 121 Nondeterministic finite automaton, 10 Nondeterministic finite-transition system, 39 Nonsingularity, 60, 65 Nonsingularity graph, 74 O Observability, 88–90, 146, 147 Observability graph, 91 ω-language, 44 Order, 21 P Partition, 21 Path, Pattern, 30 Perfect, 19 Petri net, 43 Place/transition net, 43 Index Post-incidence function, 43 Power-reducing matrix, Pre-incidence function, 43 R Regular language, 10 Resulting subnetwork, 122 Reversibility, 52 Run, 40 S Self-loop, Semitensor product, 6, 37 Strong connectedness, Subshift, 52 Swap matrix, Symbolic space, 26 T Topological dimension, 21 Totally disconnected, 19 Totally periodic, 52 Transition, 39, 43 Transition relation, 39, 145, 166 Translation, 46 V Vertex, W Weak connectedness, Weighted directed graph, Word, 26 ... https://www.springer.com/gp/authors-editors/journal-author/journal-authorhelpdesk/publishing-ethics/14214 More information about this series at http://www.springer.com/series/61 Kuize Zhang Lijun Zhang Lihua Xie • • Discrete-Time and Discrete-Space Dynamical Systems 123 Kuize. .. between Euclidean spaces and discrete spaces step by step © Springer Nature Switzerland AG 2020 K Zhang et al., Discrete-Time and Discrete-Space Dynamical Systems, Communications and Control Engineering,... a series of recent fundamental results in control-theoretic and topological dynamical problems of discrete-time and discrete-space dynamical systems, e.g., invertibility, observability, detectability,