Complexity Theory

The Complexity Theory Companion is an accessible, algorithmically oriented, research-centered, up-to-date guide to some of the most interesting techniques of complexity theory.

Modern cryptology employs mathematically rigorous concepts and methods from complexity theory. Conversely, current research in complexity theory often is motivated by questions and problems arising in cryptology. This book takes account of this trend, and therefore its subject is what may be dubbed "cryptocomplexity,'' some sort of symbiosis of these two areas.

Parameterized complexity theory is a recent branch of computational complexity theory that provides a framework for a refined analysis of hard algorithmic problems. The central notion of the theory, fixed-parameter tractability, has led to the development of various new algorithmic techniques and a whole new theory of intractability.

This advanced textbook presents a broad and up-to-date view of the computational complexity theory of Boolean circuits. It combines the algorithmic and the computability-based approach, and includes extensive discussion of the literature to facilitate further study.

This book gives a broad overview of core topics of finite model theory: expressive power, descriptive complexity, and zero-one laws, together with selected applications to database theory and artificial intelligence, especially, constraint databases and constraint satisfaction problems.

This book delivers stimulating input for a broad range of researchers, from geographers and ecologists to psychologists interested in spatial perception and physicists researching in complex systems.

This book examines key concepts and analytical approaches in complexity theory as it applies to landscape ecology, including complex networks, connectivity, criticality, feedback, and self-organisation. It then reviews the ways that these ideas have led to new insights into the nature of ecosystems and the role of processes in landscapes.

Computation theory is a discipline that strives to use mathematical tools and concepts in order to expose the nature of the activity that we call “computation” and to explain a broad range of observed computational phenomena. Why is it harder to perform some computations than others? Are the differences in difficulty that we observe inherent, or are they artifacts of the way we try to perform the computations? Even more basically: how does one reason about such questions?

A complete treatment of fundamentals and recent advances in complexity theory Complexity theory studies the inherent difficulties of solving algorithmic problems by digital computers. This comprehensive work discusses the major topics in complexity theory, including fundamental topics as well as recent breakthroughs not previously available in book form. Theory of Computational Complexity offers a thorough presentation of the fundamentals of complexity theory, including NP-completeness theory, the polynomial-time hierarchy, relativization, and the application to cryptography.

Intuitively, a sequence such as 101010101010101010… does not seem random, whereas 101101011101010100…, obtained using coin tosses, does. How can we reconcile this intuition with the fact that both are statistically equally likely? What does it mean to say that an individual mathematical object such as a real number is random, or to say that one real is more random than another? And what is the relationship between randomness and computational power. The theory of algorithmic randomness uses tools from computability theory and algorithmic information theory to address questions such as these.