Categorisation of Computable Data-Base Queries

We present an alternative approach to that of Chandra and Harel [CHAN80] and Abiteboul and Vianu [ABIT90] in considering computable database queries, which are mappings from sets of records to sets of records. In particular, we view a computable database query as being realised via a Turing-computable mapping from strings to strings and an encoding, which encodes the input set of records into an appropriate string. An encoding of a set of records consists of two components: an ordering function, which orders the records in a set as well as the values of each record in the set, and an isomorphism, which maps the values in the records of the set to strings. An important class of encodings, called free encodings, whose isomorphism has the same semantics as the identity mapping on record values, is also defined. Our analysis of computable database queries elucidates the notion of a computable database query by dealing with the problem of how a database language can be implemented on a standard Turing machine that does not cater directly for mappings from sets of records to sets of records. We carry out our analysis by categorising computable database queries into subclasses and by establishing the relationships that exist amongst these subclasses. We also investigate an equivalence relation on computable database queries; two computable database queries are related if they are realised via the same Turing-computable mapping, say δ. We prove the following interesting result regarding the cardinality of the equivalence class of a computable query with respect to the said equivalence relation: either δ does not realise any computable query, or δ realises exactly one computable query, or δ realises a countably infinite set of computable queries. Our final result shows that, by adding membership queries to the class of encodingindependent computable queries, the closure of the resulting extended class under composition of mappings is the set of all isomorphism-independent computable queries.