A Note on Join and Auto-Intersection of n-ary Rational Relations

A finite-state machine with n tapes describes a rational (or regular) relation on n strings. It is more expressive than a relational database table with n columns, which can only describe a finite relation. We describe some basic operations on n-ary rational relations and propose notation for them. (For generality we give the semiring-weighted case in which each tuple has a weight.) Unfortunately, the join operation is problematic: if two rational relations are joined on more than one tape, it can lead to non-rational relations with undecidable properties. We recast join in terms of “auto-intersection” and illustrate some cases in which difficulties arise. We close with the hope that partial or restricted algorithms may be found that are still powerful enough to have practical use.