A Formal Semantics of SQL Queries, Its Validation, and Applications

While formal semantics of theoretical languages underlying SQL have been provided in the past, they all made simplifying assumptions ranging from changes in the syntax to omitting bag semantics and nulls. This situation is reminiscent of what happens in the field of programming languages, where semantics of formal calculi underlying the main features of languages are abundant, but formal semantics of real languages that people use are few and far between. We consider the basic class of SQL queries --- essentially SELECT-FROM-WHERE queries with subqueries, set/bag operations, and nulls --- and define a formal semantics for it, without any departures from the real language. This fragment already requires decisions related to the data model and handling variable names that are normally disregarded by simplified semantics. To justify our choice of the semantics, we validate it experimentally on a large number of randomly generated queries and databases. We give two applications of the semantics. One is the first formal proof of the equivalence of basic SQL and relational algebra that extends to bag semantics and nulls. The other application looks at the three-valued logic employed by SQL, which is universally assumed to be necessary to handle nulls. We prove however that this is not so, as three-valued logic does not add expressive power: every SQL query in our fragment can be evaluated under the usual two-valued Boolean semantics of conditions.