Tools for Template Dependencies

Template dependencies (TD’s) are a class of data dependencies that include multivalued and join dependencies and embedded versions of these. A collection of techniques, examples and results about TD’s are presented. The principal results are: 1) Finite implication (implication over relations with a finite number of tuples) is distinct from unrestricted implication for TD’s. 2) There are, for TD’s over three or more attributes, infinite chains of increasingly weaker and increasingly stronger full TD’s. 3) However, there are weakest (nontrivial) and strongest full TD’s over any given set of attributes. 4) Over two attributes, there are only three distinct TD’s. 5) There is no weakest (not necessarily full) TD over any set of three or more attributes. 6) There is a finite relation that obeys every strictly partial TD but no full TD. 7) The conjunction of each finite set of full TD’s is equivalent to a single full TD. However, the conjunction of a finite set of (not necessarily full) TD’s is not necessarily equivalent to a single TD and the disjunction of a finite set of full TD’s is not necessarily equivalent to a single TD. 8) There is a finite set of TD’s with an infinite Armstrong relation but no finite Armstrong relation. 9) A necessary and sufficient condition for the existence of finite Armstrong relations for sets of TD’s can be formulated in terms of the implication structure of TD’s.