The previous three chapters have introduced some basic elements of the theory of coalgebras, focusing on coalgebraic system descriptions, homomorphisms, behaviour, finality and bisimilarity. So far, only relatively simple coalgebras have been used, for inductively defined classes of polynomial functors, on the category Sets of sets and functions. This chapter will go beyond these polynomial functors and will consider other examples. But more important, it will follow a different, more systematic approach, relying not on the way functors are constructed but on the properties they satisfy – and work from there. Inevitably, this chapter will technically be more challenging, requiring more categorical maturity from the reader.

The chapter starts with a concrete description of two new functors, namely the multiset and distribution functors, written as M and D, respectively. As we shall see, on the one hand, from an abstract point of view, they are much like powerset P, but on the other hand they capture different kinds of computation: D is used for probabilistic computation and M for resourcesensitive computation.

Subsequently, Sections 4.2–4.5 will take a systematic look at relation lifting – used in the previous chapter to define bisimulation relations. Relation lifting will be described as a certain logical operation, which will be developed on the basis of a moderate amount of categorical logic, in terms of so-called factorisation systems. This will give rise to the notion of ‘logical bisimulation’ in Section 4.5. It is compared with several alternative formulations. For weak pullback-preserving functors on Sets these different formulations coincide. With this theory in place Section 4.6 concentrates on the existence of final coalgebras. Recall that earlier we skipped the proof of Theorem 2.3.9, claiming the existence of final coalgebras for finite Kripke polynomial functors. Here we present general existence results, for ‘bounded’ endofunctors on Sets. Finally, Section 4.7 contains another characterisation of simple polynomial functors in terms of size and preservation properties. It also contains a characterisation of more general ‘analytical’ functors, which includes for instance the multiset functor M.

Multiset and Distribution Functors

A set is a collection of elements. Such an element, if it occurs in the set, occurs only once. This sounds completely trivial. But one can imagine situations in which multiple occurrences of the same element can be relevant.

Mathematics is about the formal structures underlying counting, measuring, transforming etc. It has developed fundamental notions like number systems, groups, vector spaces (see e.g. [345]) and has studied their properties. In more recent decades also ‘dynamical’ features have become a subject of research. The emergence of computers has contributed to this development. Typically, dynamics involves a ‘state of affairs’, which can possibly be observed and modified. For instance, the contents of a tape of a Turing machine contribute to its state. Such a machine may thus have many possible states and can move from one state to another. Also, the combined contents of all memory cells of a computer can be understood as the computer's state. A user can observe part of this state via the screen (or via the printer) and modify this state by typing commands. In reaction, the computer can display certain behaviour. Describing the behaviour of such a computer system is a non-trivial matter. However, formal descriptions of such complicated systems are needed if we wish to reason formally about their behaviour. Such reasoning is required for the correctness or security of these systems. It involves a specification describing the required behaviour, together with a correctness proof demonstrating that a given implementation satisfies the specification.

Both mathematicians and computer scientists have introduced various formal structures to capture the essence of state-based dynamics, such as automata (in various forms), transition systems, Petri nets and event systems. The area of coalgebra has emerged within theoretical computer science with a unifying claim. It aims to be the mathematics of computational dynamics. It combines notions and ideas from the mathematical theory of dynamical systems and from the theory of state-based computation. The area of coalgebra is still in its infancy but promises a perspective on uniting, say, the theory of differential equations with automata and process theory and with biological and quantum computing, by providing an appropriate semantical basis with associated logic. The theory of coalgebras may be seen as one of the original contributions stemming from the area of theoretical computer science.

To begin the study of data structures, I demonstrate the usefulness of even quite simple structures by working through a detailed motivating example. We shall afterward come back to the basics and build up our body of knowledge incrementally.

The algorithm presented in this introduction is due to R. S. Boyer and J S. Moore and solves the string matching problem in a surprisingly efficient way. The techniques, although sophisticated, do not require any advanced mathematical tools for their understanding. It is precisely because of this simplicity that the algorithm is a good example of the usefulness of data structures, even the simplest ones. In fact, all that is needed to make the algorithm work are two small arrays storing integers.

There are two sorts of algorithms that, when first encountered, inspire both perplexity and admiration. The first is an algorithm so complicated that one can hardly imagine how its inventors came up with the idea, triggering a reaction of the kind, “How could they think of that?” The other possibility is just the opposite – some flash of ingeniousity that gives an utterly simple solution, leaving us with the question, “How didn't I think of that?” The Boyer–Moore algorithm is of this second kind.

We encounter on a daily basis instances of the string matching problem, defined generically as follows: given a text T = T[1]T[2] · · · T[n] of length n characters and a string S = S[1]S[2] · · · S[m] of length m, find the (first, or all) location(s) of S in T, if one appears there at all. In the example of Figure 1.1, the string S = TRYME is indeed found in T, starting at position 22.

To solve the problem, we imagine that the string is aligned underneath the text, starting with both text and string left justified. One can then compare corresponding characters, until a mismatch is found, which enables us to move the string forward to a new potential matching position. We call this the naive approach. It should be emphasized that our discourse of moving the pattern along an imaginary sliding path is just for facilitating understanding.