The decision problem for the probabilities of higher-order properties

• Authors: P. Kolaitis;M. Vardi
• Venue: STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
• Year: 1987

The probability of a property on the class of all finite relational structures is the limit as n → ∞ of the fraction of structures with n elements satisfying the property, provided the limit exists. It is known that 0-1 laws hold for any property expressible in first-order logic or in fixpoint logic, i.e. the probability of any such property exists and is either 0 or 1. It is also known that the associated decision problem for the probabilities is PSPACE-complete and EXPTIME-complete for first-order logic and fixpoint logic respectively. The 0-1 law fails, however, in general for second-order properties and the decision problem becomes unsolvable.We investigate here logics which on the one hand go beyond fixpoint in terms of expressive power and on the other possess the 0-1 law. We consider first iterative logic which is obtained from first order logic by adding while looping as a construct. We show that the 0-1 law holds for this logic and determine the complexity of the associated decision problem. After this we study a fragment of second order logic called strict &Sgr;11. This class of properties is obtained by restricting appropriately the first-order part of existential second-order sentences. Every strict &Sgr;11 property is NP-computable and there are strict &Sgr;11 properties that are NP-complete, such as 3-colorability. We show that the 0-1 law holds for strict &Sgr;11 properties and establish that the associated decision problem is NEXPTIME-complete. The proofs of the decidability and complexity results require certain combinatorial machinery, namely generalizations of Ramsey's Theorem.