On the complexity of decision using destinies in H-bounded structures

Abstract

In this paper, two classes of decidable structures are compared. The first class is the class of H-bounded structures, in which quantifiers can be bounded by some recursive function H. The second class is the class of structures with recursive destinies, where destinies are finite organized trees embodying all Fraı̈sséan k-equivalence types of the structure. We show that H-bounded structures have recursive destinies, and we compare two decision algorithms in H-bounded structures: Ferrante–Rackoff's algorithm, which uses the function H and elimination of quantifiers, and Nézondet's algorithm, which precomputes destinies and evaluates sentences on these finite trees. We also prove that the inclusion between the two classes is strict, by exhibiting a structure with recursive destinies which is not H-bounded. Finally, we characterize the class of H-bounded structures as the class of structures with strongly recursive destinies, i.e. for which there exists an algorithm constructing the destinies of any tuple of elements.