This chapter is devoted to the introduction of annotations, procedures, recursion and repetitions, all concepts highly relevant to programming practice and programming methodology. In 2.1 we introduce Hoare triples as a specification method. Hoare triples are used in 2.2 for correctness proofs by annotation. In 2.3 and 2.4 we treat procedures in a programming language like Pascal. The specification and invocation rules are discussed in Section 2.3. The correctness of recursive procedures is treated in Section 2.4. The methods presented here are not new but deserve to be promoted.

In Section 2.5 we present and prove an abstract version of the rule for total correctness of recursive procedures. In 2.6 we introduce homomorphisms, functions from commands to predicate transformers that satisfy the standard laws of wp and wlp. Homomorphisms are used in 2.7 to give Hoare's Induction Rule for conditional correctness of recursive procedures, and a related rule for the necessity of preconditions. Finally, in Section 2.8, the results on recursive procedures are specialized to the repetition.

With respect to recursive procedures, this chapter is not ‘well-founded’. We only postulate some properties and proof rules, but the definition of the semantics of recursion (i.e., of the functions wp and wlp) and the proof of the postulates are postponed to Chapter 4.

Specification with Hoare triples

Weakest preconditions provide the easiest way to present predicate–transformation semantics. The formalism of Hoare triples, however, is completely equivalent and more convenient for program derivations and proofs of program correctness.

INTRODUCTION

Recent years have seen considerable work on two approaches to belief revision: the so-called foundations and coherence approaches. The foundations approach supposes that a rational agent derives its beliefs from justifications or reasons for these beliefs: in particular, that the agent holds some belief if and only if it possesses a satisfactory reason for that belief. According to the foundations approach, beliefs change as the agent adopts or abandons reasons. The coherence approach, in contrast, maintains that pedigrees do not matter for rational beliefs, but that the agent instead holds some belief just as long as it logically coheres with the agent's other beliefs. More specifically, the coherence approach supposes that revisions conform to minimal change principles and conserve as many beliefs as possible as specific beliefs are added or removed. The artificial intelligence notion of reason maintenance system (Doyle, 1979) (also called “truth maintenance system”) has been viewed as exemplifying the foundations approach, as it explicitly computes sets of beliefs from sets of recorded reasons. The so-called AGM theory of Alchourrón, Gärdenfors and Makinson (1985; 1988) exemplifies the coherence approach with its formal postulates characterizing conservative belief revision.

Although philosophical work on the coherence approach influenced at least some of the work on the foundations approach (e.g., (Doyle, 1979) draws inspiration from (Quine, 1953; Quine and Ullian, 1978)), Harman (1986) and Gärdenfors (1990) view the two approaches as antithetical. Gärdenfors has presented perhaps the most direct argument for preferring the coherence approach to the foundations approach.

INTRODUCTION

Since the beginning of artificial intelligence research on action, researchers have been concerned with reasoning about actions with preconditions and postconditions. Through the work of Moore (1980), Pratt's (1980) dynamic semantics soon established itself in artificial intelligence as the appropriate semantics for action. Mysteriously, however, actions with preconditions and postconditions were not given a proper treatment within the modal framework of dynamic logic. This paper offers such an analysis. Things are complicated by the need to deal at the same time with the notion of competence, or an actor's ability. Below, a logic of actions with preconditions and postconditions is given a sound and complete syntactic characterization, in a logical formalism in which it is possible to express actor competence, and the utility of this formalism is demonstrated in the generation and evaluation of plans.

The notion of actions with pre- and postconditions arose in artificial intelligence in the field of planning. In formulating a plan to reach some particular goal, there are a number of things which a planning agent must take into account. First, he will have to decide which actions can and may be undertaken in order to reach the goal. The physical, legal, financial and other constraints under which an actor must act will be lumped together below, since we will be interested in what is common to them all, namely that they restrict available options.

In this last chapter we shall survey a number of instances where infinite electrical networks are useful models of physical phenomena, or serve as analogs in some other mathematical disciplines, or are realizations of certain abstract entities. We shall simply describe those applications without presenting a detailed exposition. To do the latter would carry us too far afield into quite a variety of subjects. However, we do provide references to the literature wherein the described applications can be examined more closely.

Several examples are presented in Sections 8.1 and 8.2 that demonstrate how the theory of infinite electrical networks is helpful for finding numerical solutions of some partial differential equations when the phenomenon being studied extends over an infinite region. The basic analytical tool is an operator version of Norton's representation, which is appropriate for an infinite grid that is being observed along a boundary. In effect, the infinite grid is replaced by a set of terminating resistors and possibly equivalent sources connected to the boundary nodes. In this way, the infinite domain of the original problem can be reduced to a finite one – at least so far as one of the spatial dimensions is concerned. This can save computer time and memory-storage requirements.

In Section 8.3 we describe two classical problems in the theory of random walks on infinite graphs and state how infinite-electrical-network theory solves those problems. Indeed, resistive networks are analogs for random walks.

In 1971 Harley Flanders [51] opened a door by showing how a unique, finite-power, voltage-current regime could be shown to exist in an infinite resistive network whose graph need not have a regular pattern. To be sure, infinite networks had been examined at least intermittently from the earliest days of circuit theory, but those prior works were restricted to simple networks of various sorts, such as ladders and grids. For example, infinite uniform ladder networks were analyzed in [31], [73], and [139], works that appeared 70 to 80 years ago. Early examinations of uniform grids and the discrete harmonic operators they generate can be found in [35], [44], [45], [47], [52], [54], [65], [84], [126], [135], [143].

Flanders' theorem, an exposition of which starts this chapter, is restricted to locally finite networks with a finite number of sources. Another tacit assumption in his theory is that only open-circuits appear at the infinite extremities of the network. The removal of these restrictions, other extensions, and a variety of ramifications [158], [159], [163], [177] comprise the rest of this chapter. However, the assumptions that the network consists only of linear resistors and independent sources and is in a finite-power regime are maintained throughout this chapter.

Actually, finite-power theories for nonlinear networks are now available, and they apply just as well to linear networks as special cases. One is due to Dolezal [40], [41], and the other to DeMichelle and Soardi [37].

We have already seen that the idea of “connections at infinity” can be encompassed within electrical network theory through the invention of 1-nodes. A consequence is the genesis of transfinitely connected graphs, that is, graphs having pairs of nodes that are connected through transfinite paths but not through finite ones. An example of this is provided by Figure 3.4; there is no finite path connecting a node of branch a to a node of branch α, but there are 1-paths that do so. The maximal, finitely connected subnetworks of that figure are the 0-sections of the 1-graph, and the 1-nodes described in Example 3.2-5 connect those infinitely many 0-sections into a 1-graph.

There is an incipient inductive process arising here. Just as 0-nodes connect branches together to produce a 0-graph, so too do 1-nodes connect 0-sections together to produce 1-graphs. The purpose of the present chapter is to pursue this induction. While doing so, we will discover an infinite hierarchy of transfinite graphs. It turns out that most of the electrical network theory discussed so far can be transferred to such graphs to obtain an infinite hierarchy of transfinite networks. Thus, an electrical parameter in one branch can affect the voltage-current pair of another branch, even when every path connecting the two branches must pass through an “infinity of infinite extremities.”

To think about this in another way, let each 0-section of Figure 3.4 be replaced by replicates of the entire 1-graph of that figure.

The preceding chapters were focused on the mathematical foundations of infinite electrical networks, existence and uniqueness theorems being their principal result. Generality was a concomitant aim of those discussions. For the remaining chapters, we shift our attention to particular kinds of networks (namely, the infinite cascades and grids) that are more closely related to physical phenomena. Two examples of this significance were given in Section 1.7, and more will be discussed in Chapter 8. Our proofs will now be constructive, and consequently methods for finding voltage-current regimes will be encompassed. Moreover, various properties of voltage-current regimes will examined.

We must now be specific about any network we hope to analyze. In particular, its graph and element values need to be stipulated everywhere. An easy way of doing this is to impose some regularity upon the network. Most of this chapter (Sections 6.1 to 6.8) is devoted to the simplest of such regularities, the periodic two-times chainlike structures. They are 0-networks appearing in two forms. One form will be called a one-ended grounded cascade and is illustrated in Figure 6.1; it has an infinite node as one of its spines, and a one-ended 0-path as the other spine. The other form will be called a one-ended ungrounded cascade and is shown in Figure 6.2; in this case, both spines are one-ended 0-paths. The third possibility of both spines being infinite nodes is a degenerate case and will not be discussed.

The theory of electrical networks became fully launched, it seems fair to say, when Gustav Kirchhoff published his voltage and current laws in 1847 [72]. Since then, a massive literature on electrical networks has accumulated, but almost all of it is devoted to finite networks. Infinite networks received scant attention, and what they did receive was devoted primarily to ladders, grids, and other infinite networks having periodic graphs and uniform element values. Only during the past two decades has a general theory for infinite electrical networks with unrestricted graphs and variable element values been developing. The simpler case of purely resistive networks possesses the larger body of results. Nonetheless, much has also been achieved with regard to RLC networks. Enough now exists in the research literature to warrant a book that gathers the salient features of the subject into a coherent exposition.

As might well be expected, the jump in complexity from finite electrical networks to infinite ones is comparable to the jump in complexity from finite-dimensional spaces to infinite-dimensional spaces. Many of the questions we conventionally ask and answer about finite networks are unanswerable for infinite networks – at least at the present time.

As was indicated in Example 1.6-3, the total power dissipated in the resistances by a voltage-current regime, satisfying Ohm's law, Kirchhoff's current law at finite nodes, and Kirchhoff's voltage law around finite loops, need not be finite. Moreover, these laws need not by themselves determine the regime uniquely. However, if voltage-current pairs are assigned to certain branches, the infinite-power regime may become uniquely determined. The latter result requires in addition the “nonbalancing” of various subnetworks, as is explained in the next section. In which branches the voltage-current pairs can be arbitrarily chosen and how the nonbalancing criterion can be specified are the issues resolved in this chapter. The discussion is based on a graph-theoretic decomposition of the countably infinite network into a chainlike structure, which was first discovered by Halin for locally finite graphs [63]. That result has been extended to graphs having infinite nodes [166]. The chainlike structure implies a partitioning of the network into a sequence of finite subnetworks, which can be analyzed recursively to determine the voltage-current pair for every branch. We call this a limb analysis.

As was mentioned before, in most of this book we restrict our attention to resistive networks. However, a limb analysis can just as readily embrace complex-valued voltages, currents, and branch parameters. In short, a limb analysis can be used for a phasor representation of an AC regime or for the complex representation of a Laplace-transformed transient regime in a linear RLC network [166].

Perhaps the most important infinite electrical networks with respect to physical phenomena – putting aside the finite networks – are the infinite grids. This is because finite-difference approximations of various partial differential equations have realizations as electrical networks whose nodes are located at the sample points of the approximation. Those sample points are distributed in accordance with increments in each of the coordinates, hence the gridlike structure. Moreover, if the phenomenon exists throughout an infinite domain, it is natural (but, to be sure, not always necessary) to choose an infinity of sample points. In this way, one is led to infinite electrical grids as models for the so-called “exterior problems” of certain partial differential equations. Two cases of this were presented in Section 1.7, and more will be discussed in the next chapter. The grids we examine are of two general types: the grounded grids, wherein a resistor connects each node to a common ground node, and the ungrounded grids, wherein those grounding resistors are entirely absent. Grounded grids are readily analyzed, but ungrounded grids are more problematic because of a singularity in a certain function that characterizes the network.

This chapter is devoted to rectangular grids, the natural finite difference model for Cartesian coordinates, but the analysis can be extended to other coordinate systems such as cylindrical and spherical ones [180], [183], [184], [188].

The adjective “nonlinear” will be used inclusively by taking “linear” to be a special case of “nonlinear.” As promised, we present in this chapter two different theories for nonlinear infinite networks. The first one is due to Dolezal and is very general in scope – except that it is restricted to 0-networks. It is an infinite-dimensional extension of the fundamental theory for scalar, finite, linear networks [67], [115], [127]. In particular, it examines nonlinear operator networks, whose voltages and currents are members of a Hilbert space ℋ; in fact, infinite networks whose parameters can be nonlinear, multivalued mappings restricted perhaps to subsets of ℋ are encompassed. As a result, virtually all the different kinds of parameters encountered in circuit theory – resistors, inductors, capacitors, gyrators, transformers, diodes, transistors, and so forth – are allowed. However, there is a price to be paid for such generality: Its existence and uniqueness theorems are more conceptual than applicable, because their hypotheses may not be verifiable for particular infinite networks. (In the absence of coupling between branches, the theory is easy enough to apply; see Corollary 4.1-7 below.) Nonetheless, with regard to the kinds of parameters encompassed, this is the most powerful theory of infinite networks presently available. Dolezal has given a thorough exposition of it in his two books [40], [41]. However, since no book on infinite electrical networks would be complete without some coverage of Doleza's work, we shall present a simplified version of his theory.

The purposes of this initial chapter are to present some basic definitions about infinite electrical networks, to show by examples that their behaviors can be quite different from that of finite networks, and to indicate how they approximately represent various partial differential equations in infinite domains. Finally, we explain how the transient responses of linear RLC networks can be derived from the theory of purely resistive networks; this is of interest because most of the results of this book are established in the context of resistive networks.

Notations and Terminology

Let us start by reviewing some symbols and phraseology so as to dispel possible ambiguities in our subsequent discussions. We follow customary usage; hence, this section may be skipped and referred to only if the need arises. Also, an Index of Symbols is appended for the more commonly occurring notations in this book; it cites the pages on which they are defined.

Let X be a set. X is called denumerably infinite or just denumerable if its members can be placed in a one-to-one correspondence with all the natural numbers: 0, 1, 2,. … X is called countable if it is either finite or denumerable. In this book the set of branches of any network will always be countable.

The notation {x ∈ X: P(x)}, or simply {x: P(x)} if X is understood, denotes the set of all x ∈ X for which the proposition P(x) concerning x is true.