In Chap. 4, we discuss observability of Boolean control networks (BCNs). Intuitively, a BCN is observable if one can use an input sequence and the corresponding output sequence to determine the initial state. Once the initial state has been obtained, all subsequent states can be determined by input sequences since a BCN is deterministic.

A finite-state automaton (FSA) (see Sect. 2.3) is obtained from a standard nondeterministic finite automaton (NFA, see Sect. 1.3) by removing all accepting states and also replacing the unique initial state by a set of initial states. Such a modification can be used to better describe discrete-event systems (DESs) (Wonham and Cai 2019; Cassandras and Lafortune 2010). DESs usually refer to as event-driven processes that cannot be described by differential equations, where the latter usually represent time-driven processes. The FSA-based DESs are usually simple and many properties of FSAs are decidable, so FSAs are rather welcome in engineering-oriented studies. However, despite being simple, FSAs can also perform many good properties that are of both theoretical and practical importance. During the past three decades, plenty of properties and their verification or synthesis techniques on DESs in the framework of FSAs have been proposed and developed, e.g., controllability and observability (The notion of observability in the supervisory control framework is totally different from that studied in this book (see Chaps. 4, 5, 7).) (Ramadge and Wonham 1987; Ramadge 1986; Lin and Wonham 1988), diagnosability (Lin 1994; Sampath 1995), detectability (Shu et al. 2007; Shu and Lin 2011; Zhang 2017), opacity (Saboori and Hadjicostis 2014, 2012, 2011, 2013; Lin 2011), etc., where controllability and observability are defined on formal languages, diagnosability is defined on events, but the others are defined on states (except for that the result of Lin (2011) is defined on formal languages).

In Chap. 9, we studied the verification and complexity problem of the notions of strong detectability and weak detectability for finite-state automata. In this chapter, we characterize these notions for labeled Petri nets (see Sect. 2.4).

In the past few years, important applications of nondeterministic finite-transition systems (NFTSs) in formal verification and synthesis of (infinite-state) continuous (or hybrid) control systems have been witnessed (Tabuada 2009; Belta et al. 2017; Kloetzer and Belta 2008; Reissig 2011; Zamani 2014; Girard and Pappas 2007). In this methodology, the requirements or specifications for the control systems are described using temporal logics or automata Baier and Katoen (2008). Then, one constructs finite and often nondeterministic abstractions (NFTSs) of the control systems with the property that discrete (or symbolic) controllers designed on the abstractions by using automata-theoretic algorithms from computer science can be refined into controllers on the original control systems to make the requirements be met.

In this chapter, we introduce basic concepts and properties of discrete-time and discrete-space dynamical systems which will be discussed in this book, including Boolean control networks, nondeterministic finite-transition systems, finite automata, labeled Petri nets, and cellular automata.

Verifying observability and reconstructibility of Boolean control networks (BCNs) is NP-hard in the number of nodes. A BCN is observable (reconstructible) if one can use an input sequence and the corresponding output sequence to determine the initial (current) state. In this article, we study when a node aggregation approach can be used to overcome the computational complexity in verifying these properties. We first define a class of node aggregations with subnetworks being BCNs. For acyclic node aggregations in this class, all corresponding subnetworks being observable (reconstructible) implies that the whole BCN is observable (reconstructible), although the converse is not true. In general, for cyclic node aggregations, the whole BCN being observable (reconstructible) does not imply that all subnetworks are observable (reconstructible), or vice versa. We design an algorithm to search for all acyclic node aggregations in this class, and show that finding acyclic node aggregations with small subnetworks can significantly reduce the computational complexity in verifying observability (reconstructibility). We also define a second class of node aggregations with subnetworks being finite-transition systems (more general than BCNs), which compensates for the drawback of the first class when the BCN has only a small number of output nodes. Finally, we use a BCN T-cell receptor kinetics model from the literature with 37 state nodes and 3 input nodes to illustrate the efficiency of the results derived from the two node aggregation methods. For this model, we derive the unique minimal set of 16 state nodes needed to be directly measured to make the overall BCN observable. We also compute 5 of the 16 state nodes needed to be directly measured to make the model reconstructible.

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, (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.) 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).

As stated before, initially a Boolean control network (BCN) (see Σ1 in Fig. 3.1) was in a state, then as inputs were fed into the BCN one by one, state transitions occurred successively, yielding a sequence of outputs. What may interest us is: Could the above process be reversed? That is, whether there exists another BCN (see Σ2 in Fig. 3.1) that reverses the inputs and outputs of Σ1. In this chapter, we prove a series of fundamental results on this problem, and apply these results to the mammalian cell cycle.

In Chaps. 4 and 5, we investigated how to verify different notions of observability and detectability for Boolean control networks (BCNs), and also studied how to determine the initial state (current state) of a BCN according to a particular notion of observability (detectability). In addition, we proved that the problems of verifying these notions are all NP-hard in the number of nodes. Hence, these problems are generally intractable. Actually, in general, for a BCN with more than 30 nodes, one cannot obtain whether it is observable or detectable in a reasonable amount of time by using a personal computer (PC). Hence BCNs with more than 30 nodes can be regarded as large-scale.