Computational model theory: an overview

The computational complexity of a problem is the amount of resources, such as time or space, required by a machine that solves the problem. The descriptive complexity of problems is the complexity of describing problems in some logical formalism over finite structures. One of the exciting developments in complexity theory is the discovery of a very intimate connection between computational and descriptive complexity. It is this connection between complexity theory and finite-model theory that we term computational model theory. In this overview paper we offer one perspective on computational model theory. Two important observations underly our perspective: (1) while computational devices work on encodings of problems, logic is applied directly to the underlying mathematical structures, and this ‘mismatch’ complicates the relationship between logic and complexity significantly, and (2) first-order logic has severely limited expressive power on finite structures, and one way to increase the limited expressive power of first-order logic is by adding fixpoint constructs. These observations motivated the introduction of two abstract formalisms: that of finite-variable infinitary logic and that of relational machines. This paper is the story of how these formalisms are used in computational model theory and their interrelationship.