A ubiquitous feature of human experience is using one thing to represent another. Abstraction, generalization, categorization, approximation, and reformulation all describe this process, from different (and often imprecisely defined) perspectives. This volume seeks to understand this space of concepts historically and across disciplines, distinguish them from one another, develop a formal theory of abstraction, and show its relevance to artificial intelligence (AI) and complex systems.

The first five chapters are a broad survey of this set of concepts in domains as diverse as philosophy, natural language, mathematics, computer science (CS), art, cognitive science, and vision. After a brief introduction in chapter 1, chapter 2 reviews how the concepts emerge in these domains. Anticipating the book’s particular focus on AI, chapter 3 reviews its relevance to planning, constraint satisfaction, knowledge representation, and agent-based modeling. Next, chapter 4 reviews previous definitions, mostly from formal logic and CS. Chapter 5 summarizes various characteristics of abstraction proposed by different researchers and seeks to distinguish it from generalization and categorization on the one hand, and approximation and reformulation on the other. In the authors’ view, abstraction (along with approximation and reformulation) is an intensional property (pertaining to descriptions), while generalization and categorization are extensional (pertaining to instances).

To distinguish abstraction from approximation and reformulation, the authors present a formal model, KRA (knowledge, reformulation, and abstraction), described in chapters 6 through 8. The heart of the theory is the notion of a query to be answered, which requires both observations from the system under study and some theory. The observations from the world are formalized in terms of a description frame describing the things that can be observed in the system. The set of all possible descriptions is the system’s configuration space, of which a subspace (let’s call it the compatible subspace) will be compatible with a given set of observations. The query and theory, along with the description frame, data structure definitions, and the language in which the theory and the query are expressed, are captured in the query environment. Abstraction, approximation, and reformulation are all achieved by applying operators to transform one description frame and its associated query environment into another. Specifically: in abstraction, the compatible subspace of the transformed configuration space is a strict superset of the compatible subspace of the original configuration space; in reformulation, the two compatible subspaces are identical; and in approximation, neither compatible subspace is a subset of the other.

Chapter 9 applies the KRA theory to machine learning. Chapter 10 applies it to complex systems. Chapter 11 illustrates its use in system design, cartography, and hidden Markov models. The last two chapters offer discussion and summary. Eight appendices and a website offer supporting details, including a collection of abstraction operators. The 592 references are encyclopedic, ranging from Plato in the fourth century BC to 2012. The theory is rather cumbersome and the discussion is sometimes difficult to follow, but the book brings together an amazing range of concepts within a coherent framework that will make it an essential part of ongoing discussions on this powerful set of concepts.