The decision problem for the probabilities of higher-order properties

The probability of a property on the class of all finite relational structures is the limit as <italic>n</italic> → ∞ of the fraction of structures with <italic>n</italic> elements satisfying the property, provided the limit exists. It is known that <italic>0-1 laws</italic> 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 <italic>iterative logic</italic> 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 <italic>strict</italic> &Sgr;<supscrpt>1</supscrpt><subscrpt>1</subscrpt>. This class of properties is obtained by restricting appropriately the first-order part of existential second-order sentences. Every strict &Sgr;<supscrpt>1</supscrpt><subscrpt>1</subscrpt> property is NP-computable and there are strict &Sgr;<supscrpt>1</supscrpt><subscrpt>1</subscrpt> properties that are NP-complete, such as 3-colorability. We show that the 0-1 law holds for strict &Sgr;<supscrpt>1</supscrpt><subscrpt>1</subscrpt> 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.