Delta-Languages for Sets and sub-PTIME Graphs Transformers

This paper discusses three successively extending versions of a set theoretic Δ-language, as a prototype for “nested” data bases query language. Corresponding finite set operations (data base queries) may be realized in NLOGSPACE under representation of sets by extensional well-founded (acyclic) graphs. (In a previous work for another version of Δ-language an exact correspondence to PTIME-computability was established.) Moreover, each of the mentioned versions of the language is faithfully characterized in terms of corresponding three classes of the graph transformers, the last one being just all transformers definable in the First Order Logic with Transitive Closure operator. For simplicity we are considering here the case of “pure” hereditarily-finite sets, i.e. sets without atoms involved. They are naturally linear ordered, however this order is problematic to formally define in our present case (unlike the case corresponding to PTIME). The related question whether the last class of transformers and corresponding class of queries over HF coincide with all NLOGSPACE-computable ones is left open in this paper.