Review of The Foundations of Computability Theory (Second Edition) by Borut Robič

Computability theory forms the foundation for much of theoretical computer science. Many of our great unsolved questions stem from the need to understand what problems can even be solved. The greatest question of computer science, P vs. NP, even sidesteps this entirely, asking instead how efficiently we can find solutions for the problems that we know are solvable. For many students both at the undergraduate and graduate level, a first exposure to computability theory follows a standard sequence on data structures and algorithms and students often marvel at the first results they see on undecidability - how could we possibly prove that we can never solve a problem? This book, in contrast with other books that are often used as first exposures to computability, finite automata, Turing machines, and the like, focuses very specifically on the notion of what is computable and how computability theory, as a science unto itself, fits into the grander scheme. The book is appropriate for advanced undergraduates and beginning graduate students in computer science or mathematics who are interested in theoretical computer science. Robič sidesteps the standard theoretical computer science progression - understanding finite automata and pushdown automata before moving into Turing machines - by setting the stage with Hilbert's program and mathematical prerequisites before introducing the Turing machine absent the usual prerequisites, and then introducing advanced topics often absent in introductory texts. Most chapters are relatively short and contain problem sets, making it appropriate for both a classroom text or for self-study.