Recursive Relations

By definition, recursion is the process of reducing a problem into a smaller version of itself (Malik, 2006).  Recursion can be further classified as either direct recursion that calls itself or indirect recursion that is dependent on another component (Malik, 2006).  Recursion is a powerful programming tool however it can incur a steep cost as it uses considerable amounts of memory and in the worst case can overflow the stack (Malik, 2006; Schalkoff, 1990).  With this basis, recursion requires careful design to ensure it leads to a base case and eventually resolves (Malik, 2006).

Relational recursion is based on recursive joins and predicate logic (Codd, 1990).  In a RDBMS setting, relational recursion can be succinctly described using Datalog notation (Garcia-Molina, Ullman & Widom, 2009. p. 222).  Datalog is a logical query language based on predicate logic and is syntactically and semantically similar to Prolog (Garcia-Molina, Ullman & Widom, 2009; Wilson & Clark, 2001).  Similar to Prolog, Datalog facilitates the representation of relational if-then rules that can query a database’s relations.  In this framework, the database may be viewed as a knowledge base and the relation’s tuples comprise the knowledge base’s asserted facts.

In the Datalog format, relations form the head and body of a Datalog query separated by an arrow (e.g. head ç body).  A rule is defined to be recursive if a predicate symbol appears in the head and also appears in the body as this will require a recursive join (Codd, 1990; Van Le, 1993).  It is common to call the head and body’s relations clauses and the head may be further distinguished as the goal (Van Le, 1993; Wilson & Clark, 2001).  From an interpretation perspective, if the Datalog body’s sub-goals can be met (i.e. tuple-based facts exist), a Boolean value is returned and the Datalog’s head or goal follows.  This type of resolution is particularly relevant to problem spaces that can be represented as directed dependency graphs.  In this instance, the system can determine if a solution is valid or exists by determining if a path of arcs exist from the sub-goals (i.e. body) to the goal (i.e. head).   This framework is consistent with SQL relational and predicate logic principles as the programmer encodes the goal or the desired result rather than how to achieve the result (Wilson & Clark, 2001).

SQL 99 includes the provision to support recursion as presented by Codd through the RECURSION and WITH keywords (Codd, 1990; Garcia-Molina, Ullman & Widom, 2009).  SQL requires recursive statements be preceded by the keyword RECURSIVE.  The WITH statement allows the programmer to define temporary relations and allows these temporary relations to be used in subsequent queries.  This allows the programmer to break up a problem space into smaller components in accord with the recursive definitional basis above.  SQL allows these temporary relations to be mutually recursive where they are defined in terms of other relations in addition to their own relation.  With this basis SQL supports both direct and indirect recursion however if an SQL query is mutually recursive, SQL limits the body to include just a single mutually recursive sub-goal.

An important facet of recursive relational logic programming is its support for anonymous variables (Wilson & Clark, 2001).  An anonymous variable is allowed to range over all possible attribute instance values of a relation’s column.  Functionally this allows an anonymous variable to be iteratively set to all values in a column and tested or compared in additional sub-goals.  If all of the body’s sub-goals are met resulting in a Boolean value of true, the anonymous variable’s value is instantiated and this value becomes part of the solution query or relation (i.e. the head clause).

Summary

SQL’s added support for recursive programming in 1999 represents an important step forward in accord with Codd’s relational theory (Codd, 1990).   The ability to program Datalog equivalent SQL queries demonstrates that SQL is truly a fourth generation language.  Furthermore, support for recursion demonstrates that today’s RDBMS are moving towards knowledge-base functionality.  This is remarkable step forward citing that knowledge-bases are central to Expert System (ES) development and serve as the critical underpinnings of semantic interoperability, knowledge representation and transfer (Graham et al., 2005; Jackson, 1999; Nagarajan, Verma, Sheth & Miller, 2007; Sheth, Gomadam & Lathem, 2007).

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