Multidimensional extension to MCFGs

Develop the analogue of exact relative prime factorization, semantic residual splitting, and under-saturation controllers for multidimensional syntactic congruences and bounded-fan-out multiple context-free grammars.

Background

The paper’s theory is formulated for one-dimensional string concatenation, where relative classes, exact products, prime factorizations, and residual controllers are defined using ordinary word factors and concatenation.

The authors identify tuple congruence and non-permuting linear regular functions as a natural compositional basis for extending these notions to multidimensional and bounded-fan-out MCFG settings. The appropriate definitions and structural results remain unresolved.