It may not be possible to implement a specification. The requirements may be inconsistent, the specification may be meaningless or ill-defined, or, surprisingly, the function specified may not be computable. The concept of an incomputable function comes from computability theory, a branch of mathematical logic with particular relevance to computer science.

Chapter 12 introduces some of the key ideas of computability theory, both as motivation for subsequent study and to provide the background necessary to appreciate the significance of an incomputability claim. In particular, we will prove that some functions are not computable by any C program, explain why such problems are deemed to be algorithmically unsolvable, and list a number of unsolvable problems of practical importance involving context-free grammars.

Additional Reading

Good presentations of elementary computability theory may be found in [RS86, Sip97, Jon97, GH98, LP98, HMU01].