This book is a collection of chapters authored primarily by Jeffrey Zheng, who is also the editor. The preface lays out a broad roadmap for expanding vector 0-1 logical systems, highlighting the areas of variant logic, variant measurement, and variant mapping. These are all based on the construction of a phase space generalizing the space of truth functions with operations of permutation and complementation. This is ostensibly analogous to invariant theory, which studies the action of a group on various spaces.

The book first covers theoretical foundations, followed by various application areas where the theory has been applied. The chapters span a broad set of application areas, from measuring the edge accuracy of digital images, to cryptography, to analyzing electrocardiogram (ECG) sequences.

The writing style is difficult to follow. The material is quite dense and requires prior knowledge of areas such as coding theory to make sense. It is also quite difficult to understand the motivation of various definitions and constructions that seems to be implicit in the chapters. For example, even in the first chapter’s introduction to the structures underlying the theory under development, it is difficult to understand the motivation for introducing the permutation and complementary operations. Page 11 provides an example tabulation (analogous to a truth table) of a particular Boolean function and its effect under a complementary operation, but the motivation for this construction--that is, how it may be applied--and for the choice of parameters in the table is missing. I also spent quite some time trying to figure out if there was something special about the number 178 used in the example, or for that matter the complementary parameter 11001100.

Some of the chapters are tantalizing, that is, I was able to follow the individual steps of construction and reasoning, but it was very difficult to grasp the bigger picture. For example, the chapter on triangular numbers organizes binary sequences of a given length using the number of ones and the number of bit flips in the sequence. The chapter shows how this is related to binomial coefficients and the triangular deconstruction of binomial coefficients. This leads to novel sequences that have apparently not been studied, that is, not listed in the On-Line Encyclopedia of Integer Sequences (OEIS). However, it is not at all clear how this insight can be applied. It almost seems like the insights presented were serendipitous discoveries.

Another notable chapter is “Recursive Measures of Edge Accuracy on Digital Images,” which ostensibly proposes a methodology to evaluate different edge detection schemes. However, the chapter is very sparse in terms of details. Because the chapter is not self-contained, I was unable to understand how the proposed technique could be applied in practice, or reused in different contexts.

Other chapters in the book are puzzling. For example, “Meta Model on Concept Cell” addresses knowledge representation theory. This informal chapter introduces classifications of knowledge representations; however, it is not clear how it relates to the preceding chapters.

The overall impression left by the collection is that it presents an interesting approach to understanding diverse processes by constructing a mathematical phase space based on binary strings. However, it may be difficult for a broader audience to understand and absorb, as a significant amount of implicit knowledge is assumed.