Many type systems include the concept of subtyping, allowing a value of one type (the subtype) to be used as if it is a value of another type (the supertype). The aim is to allow greater flexibility in programming, while maintaining safety. In this chapter, we see how subtyping can be included in our system of session types. We build on the language in Chapter 4, using replicated input, rather than recursive process definitions, to express repetitive behaviour.

This chapter describes the evolution of computing systems, from data processing to an emphasis on communication, and motivates a corresponding evolution of the concept of typing. Data types codify the structure of data, and go back to the early days of programming languages. This book is about session types, which codify the structure of communication – they are type-theoretic specifications of communication protocols. The chapter summarises the assumptions about communication that are necessary for the theory of session types, and describes the behavioural safety properties that are guaranteed by checking session types.

The earlier chapters present session type systems declaratively, focusing on how typing judgements describe the way in which processes use channels. In order to apply session types to programming languages, it is essential to be able to implement an efficient typechecking algorithm which answers the question: given a candidate typing judgement, is it derivable? The declarative typing rules, however, are not immediately suitable for implementation. In this chapter we explain the problem and how to overcome it.

This chapter develops a theory of infinite session types in order to describe communication protocols that allow unbounded behaviour. The theory is based on the technical machinery of recursive types, coalgebras and coinduction, which the chapter introduces at an elementary level. Recursive process definitions are introduced so that unbounded behaviour can be implemented. The type safety results of Chapter 2 are extended to the new setting.

Digital behavior does not occur in isolation within space, but is instead ever-present, vying for a user’s time against other alternative behaviors. The determinants of behavior choice are diverse, yet in behavioral sciences, this “behavioral competition” is operationalized by alterations in the value of the contingent reinforcer to latent behaviors. In this competitive environment, where the user has limited time to enact certain behaviors, they must choose by seeking a balance that maximizes their satisfaction (law of diminishing marginal utility & utility maximization model). This shifting process is known as behavioral contrast, which reflects a variation in some behavioral component due to the change in the value of reinforcers associated with any of the present behaviors. In the design of digital behaviors, understanding this process is fundamental, as it directs the designer towards potential enhancements of the digital service (through improving the reinforcers) to better its positioning against competitors.

Corecursive algebras are algebras that admit unique solutions of recursive equation systems. We study these and a generalization: completely iterative algebras. The terminal coalgebra turns out to be the initial corecursive algebra as well as the initial completely iterative algebra. Dually, the initial algebra is the initial (parametrically) recursive coalgebra. These results explain the title of the chapter. We apply recursive coalgebras in order to obtain a new proof of the Initial Algebra Theorem from Chapter 6.

Well-founded coalgebras generalize well-foundedness for graphs, and they capture the induction principle for well-founded orders on an abstract level. Taylor’s General Recursion Theorem shows that, under hypotheses, every well-founded coalgebra is parametrically recursive. We give a new proof of this result, and we show that it holds for all set functors, and for all endofunctors preserving monomorphisms on a complete and well-powered category with smooth monomorphisms. The converse of the theorem holds for set functors preserving inverse images. We provide an iterative construction of the well-founded part of a given coalgebra: It is carried by the least fixed point of Jacobs’ next-time operator.

This chapter presents a number of sufficient conditions to guarantee that an endofunctor has an initial algebra or a terminal coalgebra. We generalize Kawahara and Mori’s notion of a bounded set functor and prove that for a cocomplete and co-well-powered category with a terminal object, every endofunctor bounded by a generating set has a terminal coalgebra. We use this to show that every accessible endofunctor on a locally presentable category has an initial algebra and a terminal coalgebra. We introduce pre-accessible functors and prove that on a cocomplete and co-well-powered category, the initial-algebra chain of a pre-accessible functor converges, and so the initial algebra exists. If the base category is locally presentable and the functor preserves monomorphisms, then the terminal coalgebra exists.

This chapter presents initial algebras and terminal coalgebras obtained by the most common method. For the initial algebra, this is by iteration starting from the initial object through the natural numbers. For the terminal coalgebra, this is the dual: the iteration begins with the terminal object. The chapter is mainly concerned with examples drawn from sets, posets, complete partial orders, and metric spaces.

This chapter highlights connections of the book’s topics to structures used in all areas of mathematics. Cantor famously proved that no set can be mapped onto its power set. We present some analogous results for metric spaces and posets. On the category of topological spaces, we consider endofunctors built from the Vietoris endofunctor using products, coproducts, composition, and constant functors restricted for Hausdorff spaces. Every such functor has an initial algebra and a terminal coalgebra. Similar results hold for the Hausdorff functor on (complete) metric spaces. Extending a result of Freyd, we exhibit structures on the unit interval [0, 1] making it a terminal coalgebra of an endofunctor on bipointed metric spaces. The positive irrationals and other subsets of the real line are described as terminal coalgebras or corecursive algebras for some set functors, calling on results from the theory of continued fractions.

Discussing ethics within the highly competitive technology sector can be complex. Although companies outwardly express their commitment to ethical practices, internally, the topic is often regarded as uncomfortable due to its implications for corporate finances. In digital behavior design, ethical consideration begins with defining the various drives and reinforcers that will guide the design process. As a sub-field of Human Factors and Engineering Psychology, digital behavior design can draw from their established Codes of Ethics and Codes of Conduct. However, beyond incorporating the ethical principles that govern Human Factors and psychology, it is also imperative to acknowledge unique ethical principles that highlight the particularities of their application. In seeking these specific ethical norms, it is important to identify the essential desirable value inherent in the professional practice of digital behavior designers. Primarily, the genuine value provided by digital behavior design is utilitarian in nature; that is, it fulfills user needs through satisfaction. From this continuous satisfaction may arise a dependency on these digital services for happiness, leading to problematic online behaviors. Therefore, poor design or the unethical use of discriminative stimuli (nudges) and reinforcers can be highly hazardous for populations with certain psychosocial and neural vulnerabilities. This chapter introduces certain standards to guide ethical and responsible conduct for designers when creating digital services. It also proposes a solution in the form of an algorithm that could be implemented in digital services to detect and support compulsive behaviors.

This chapter is an indispensable part of the framework for the design of digital behaviors, as it lays the foundation upon which user behavior when engaging with digital services can be anticipated. The core principle to grasp is that the probability of a user repeating the use of a digital or virtual service is primarily linked to two variables: satiation of the drive and the emotion generated. The satiation of the drive is elucidated by the drive reduction theory, whereas the proposal of how emotion or excitement selects behavior is explained by Frederick Sheffield’s induced emotion theory. The key point here is the attainment of the reinforcer, which is the element that induces changes both in the reduction of the drive and in the induction of emotion. In relation to this, it has also been suggested to differentiate among drive, motivation, and reinforcer. Drive is an internal state of physiological or psychological origin, acting as a vector of behavior, but it does not energize the behavior per se. This drive triggers a biochemical cascade within the organism that intensifies over time, and its non-satiation progressively energizes behavior, resulting in a subjective experience defined as an emotion, which initiates goal-directed behavior. Finally, a list of different types of drives (survival, sexual, social, and meta-cognitive) has also been offered, indicating the potential needs of users when turning to a digital or virtual service.

The theme of this chapter is the relation between the initial algebra for a set functor and the terminal coalgebra, assuming that both exist and that the endofunctor is non-trivial. We introduced a notion called pre-continuity. Pre-continuous set functors generalize finitary and continuous set functors. For such functors, the initial algebra and the terminal coalgebra have the same Cauchy completion and the same ideal completion: the $\omega$-iteration of the terminal-coalgebra chain. It follows that for a non-trivial continuous set functor, the terminal coalgebra is the Cauchy completion of the initial algebra.

The rational fixed point of an endofunctor is a fixed point which is in general different from its initial algebra and its terminal coalgebra. It collects precisely the behaviours of all ‘finite’ coalgebras of a given endofunctor. For sets, they are those with finitely many states. Examples of rational fixed points include regular languages, eventually periodic and rational streams, etc. To study the rational fixed point in categories beyond sets, we discuss locally finitely presentable categories, and we do so in some detail. We characterize the rational fixed point as an initial iterative algebra. The chapter goes into details on many examples, such as rational fixed points in nominal sets. It discusses the rational fixed point and several other fixed points as well, and it summarizes much of what is known about them.

Chapter 6 may be regarded as an extension of Chapter 5, aiming to discuss several important aspects concerning secondary or psychological drives, given that digital services widely consumed by the general population tend to cater to the satiation of one or more of these drives. These drives or needs have emerged within a cultured society that reaches into the human psyche, taking on a significant role in regulating interactions among different members of society. With the advent of social networks, and the continuous interactions we engage in through them, certain psychological drives such as status or control over identity are becoming a powerful mobilizing force for goal-directed behavior. Hence, it has been deemed timely to delve deeper into some specific drives in this regard.