This chapter presents the main results of the theory of S-systems and T-systems. It is divided into two sections of similar structure, one for each of these two classes. The main results of each section are a Liveness Theorem, which characterizes liveness, a Boundedness Theorem, which characterizes b-boundedness of live systems, and a Reachability Theorem, which characterizes the set of reachable markings of live systems. Additionally, both sections contain a Shortest Sequence Theorem, which states that every reachable marking can be reached by an occurrence sequence whose length is bounded by a small polynomial in the number of transitions of the net (linear in the case of S-systems and quadratic for T-systems).

S-systems

Recall from the Introduction that S-systems are systems whose transitions have exactly one input place and one output place.

Definition 3.1S-nets, S-systems

A net is an S-net if |•t| = 1 = |t•| for every transition t.

A system (N, M0) is an S-system if N is an S-net.

The fundamental property of S-systems is that all reachable markings contain exactly the same number of tokens. In other words, the total number of tokens of the system remains invariant under the occurrence of transitions.

Proposition 3.2Fundamental property of S-systems

Let (N, M0) be an S-system. If M is a reachable marking, then M0(S) = M(S), where S is the set of places of N.

A home marking of a system is a marking which is reachable from every reachable marking; in other words, a marking to which the system may always return. The identification of home markings is an interesting issue in system analysis. A concurrent interactive system performs some initial behaviour and then settles in its ultimate cyclic (repetitive) mode of operation. A typical example of such a design is an operating system which, at boot time, carries out a set of initializations and then cyclically waits for, and produces, a variety of input/output operations. The states that belong to the ultimate cyclic behavioural component determine the central function of this type of system. The markings modelling such states are the home markings.

In Section 8.1 we show that live and bounded free-choice systems have home markings. In Section 8.2 we prove a stronger result: the home markings are the reachable markings which mark all the proper traps of the net.

Existence of home markings

Definition 8.1Home marking

Let (N, M0) be a system. A marking M of the net N is a, home marking of (N, M0) if it is reachable from every marking of [M0〉.

We say that (N, M0) has a home marking if some reachable marking is a home marking.

Using the results of Chapter 3, we can easily prove the following proposition.

Proposition 8.2Home markings of live S- and T-systems

Every reachable marking of a live S-system or a live T-system is a home marking.

This book developed out of lecture courses in Uncertain Reasoning given at the University of Manchester in 1989 and 1991 as part of the Master of Science degree in Mathematical Logic and is aimed at readers with some mathematical background who wish to understand the mathematical foundations of the subject. Thus the emphasis is on providing mathematical formulations, analyses and justifications of what I see as some of the major questions and assumptions underlying present day theories of Uncertain Reasoning whilst avoiding, as far as possible, lengthy philosophical discussions on the one hand and precise computer algorithms on the other. Much of the material presented appears already, in some form or other, in published papers, so that my main contribution is the assembling and presenting of it within a unified framework.

My hope is that by doing so I might encourage more ‘mathematicians’ to take an active interest in the subject whilst at the same time offering to those currently working on practical applications in the field easy access to some of the mathematics underlying their assumptions. In short it is the sort of book which I wish had been available to me when I first entered the area.

The subject of Uncertain Reasoning (also referred to as Approximate Reasoning and Reasoning under Uncertainty) dates back to Plato, if not beyond, but has seen an exponential expansion in the last decade with the drive towards intelligent computers, especially so called expert systems.

Petri nets are one of the most popular formal models of concurrent systems, used by both theoreticians and practitioners. The latest compilation of the scientific literature related to Petri nets, dating from 1991, contains 4099 entries, which belong to such different areas of research as databases, computer architecture, semantics of programming languages, artificial intelligence, software engineering and complexity theory. There are also several introductory texts to the theory and applications of Petri nets (see the bibliographic notes).

The problem of how to analyze Petri nets – i.e., given a Petri net and a property, how to decide if the Petri net satisfies it or not – has been intensely studied since the early seventies. The results of this research point out a very clear trade-off between expressive power and analyzability. Even though most interesting properties are decidable for arbitrary Petri nets, the decision algorithms are extremely inefficient. In this situation it is important to explore the analyzability border, i.e., to identify a class of Petri nets, as large as possible, for which strong theoretical results and efficient analysis algorithms exist.

It is now accepted that this border can be drawn very close to the class of free-choice Petri nets. Eike Best coined the term ‘free-choice hiatus’ in 1986 to express that, whereas there exists a rich and elegant theory for free-choice Petri nets, few of its results can be extended to larger classes. Since 1986, further developments have deepened this hiatus, and reinforced its relevance in Petri net theory.

This chapter introduces elementary definitions, concepts and results concerning arbitrary Petri nets. We start with a short section on mathematical notation. Section 2.2 is devoted to the definition and properties of nets, markings, the occurrence rule and incidence matrices. Section 2.3 defines net systems as nets with a distinguished initial marking. We give formal definitions of some behavioural properties of systems: liveness, deadlock-freedom, place-liveness, boundedness. Section 2.4 introduces S- and T-invariants, an analysis technique used throughout the book. The relationship between these invariants and the behavioural properties of Section 2.3 is discussed.

The chapter includes six simple but important results, which are very often used in later chapters. They are the Monotonicity, Marking Equation, Exchange, Boundedness, and Reproduction Lemma, and the Strong Connectedness Theorem. We encourage the reader to become familiar with them before moving to the next chapters.

Mathematical preliminaries

We use the standard definitions on sets, numbers, relations, sequences, vectors and matrices. The purpose of this section is to fix some additional notations.

Notation 2.1Sets, numbers, relations

Let X and Y be sets. We write X ⊆ Y if X is a subset of Y, including the case X = Y. X ⊂ Y denotes that X is a proper subset of Y, i.e., X ⊆ Y and X ≠ Y. X\Y denotes the set of elements of X that do not belong to Y. |X| denotes the cardinality of X.

Free-choice Petri nets have been around for more than twenty years, and are a successful branch of net theory. Nearly all the introductory texts on Petri nets devote some pages to them. This book is intended for those who wish to go further. It brings together the classical theorems of free-choice theory obtained by Commoner and Hack in the seventies, and a selection of new results, like the Rank Theorem, which were so far scattered among papers, reports and theses, some of them difficult to access.

Much of the recent research which found its way into the book was funded by the ESPRIT II BRA Action DEMON, and the ESPRIT III Working Group CALIBAN. The book is self-contained, in the sense that no previous knowledge of Petri nets is required. We assume that the reader is familiar with naïve set theory and with some elementary notions of graph theory (e.g. path, circuit, strong connectedness) and linear algebra (e.g. linear independence, rank of a matrix). One result of Chapter 4 requires some knowledge of the theory of NP-completeness.

The book can be the subject of an undergraduate course of one semester if the proofs of the most difficult theorems are omitted. If they are included, we suggest the course be restricted to Chapters 1 through 5, which contain most of the classical results on S- and T-systems and free-choice Petri nets. A postgraduate course could cover the whole book.

All chapters are accompanied by a list of exercises.

This part, which consists of two chapters, gives the basic material.

Chapter 1 is a brief discussion of the required background from (modal-free) propositional logic. There should be nothing in this chapter which you don't know already, although perhaps the style of presentation will be new to you. The variety of modal languages are described in Chapter 2. The polymodal versions are introduced right from the beginning and this brings a greater cohesion to the subject.

Introduction

The propositional modal language is an extension of the pure propositional language formed by adding a battery of new 1-ary connectives (known informally as box connectives). Originally there was just one new connective □, however for many purposes it is necessary to add several (possibly infinitely many) such connectives [i], one for each element i of an index set I. Thus there are many possible modal languages, one for each index set I. The syntax, semantics, and proof systems associated with modal languages are designed to subsume those of the proposition language, in fact, propositional logic can be regarded as the extreme version of modal logic where I = ∅.

The element i of I are called labels and I itself is called the signature of the modal language. Thus two languages are identical precisely when they have the same signature. (We are never going to consider how one language may be be translated into another, so we need not worry about comparison of signatures.)

Unlike the propositional connectives ¬, →, ∧, ∨, ⊤, and ⊥, the box connectives [i] do not have a fixed interpretation. For each formula φ (of the modal language) we may use [i] to obtain a new formula

[i]φ

This may be read in several ways, and different readings suggest different semantics and proof systems.