Open are the double doors of the horizon; unlocked are its bolts.

Summary: This chapter is devoted to the study of the space complexity of computations, while focusing on two rather extreme cases. The first case is that of algorithms having logarithmic space complexity. We view such algorithms as utilizing the naturally minimal amount of temporary storage, where the term “minimal” is used here in an intuitive (but somewhat inaccurate) sense, and note that logarithmic space complexity seems a more stringent requirement than polynomial time. The second case is that of algorithms having polynomial space complexity, which seems a strictly more liberal restriction than polynomial time complexity. Indeed, algorithms utilizing polynomial space can perform almost all the computational tasks considered in this book (e.g., the class PSP ACε contains almost all complexity classes considered in this book). […]