More About Recursive Structures: Descriptive Complexity and Zero-One Laws

This paper continues our work on innnite, recursive structures. We investigate the descriptive complexity of several logics over recursive structures, including rst-order, second-order, and xpoint logic, exhibiting connections between expressibility of a property and its computational complexity. We then address 0{1 laws, proposing a version that applies to recursive structures, and using it to prove several non-expressibility results. 0 Introduction Innnite recursive structures, with recursive graphs as a special case, have been studied quite extensively in the past. Most interesting properties of recursive graphs have been shown to be undecidable, and many are actually outside the arithmetic hierarchy In HH2] we considered recursive structures to be generalizations of nite relational data bases, and investigated the class of computable queries over them, the motivation being borrowed from CH1]. A computable query is a (partial) recursive function that is also generic, i.e., it preserves isomorphisms. One of the results of HH2] is that quantiier-free rst-order logic is complete for recursive data bases, meaning that it expresses precisely the recursive generic queries. Part 1 of this paper deals with languages that can express non-recursive queries. If we return for a moment to the world of nite structures, classical complexity classes can often be characterized in terms of their descriptive complexity, i.e., the ability of certain logics to express properties in the class. The rst important result was that of Fagin F1], who showed that existential second order logic (that is, 1 1 formulas) expresses exactly the properties in NP. An analogous match holds between each level of the second-order hierarchy and the corresponding level of the polynomial-time hierarchy. In Part 1 of the paper we carry out an analogous investigation over recursive structures, analyzing the expressibility of several logics, such as rst-order logic, second-order logic and xpoint logic. We are able to exhibit many connections between descriptive and computational complexity, such as the following result, which is analogous to that of Fagin: the properties of recursive structures expressible by a 1 k formula, for k 2, are exactly the generic properties in the complexity class 1 k of the analytic hierarchy.