Discontinuous-constituent phrase structure grammar

Discontinuous-constituent Phrase Structure Grammar (DCPSG) (distinct from Discontinuous Phrase Structure Grammar/DPSG) is a formalism for describing discontinuous phrase structures in natural language, such as verb phrases in VSO languages. The formalism was introduced in the slightly more constrained form of Discontinuous-constituent Phrase Structure Grammar with Subscripts and Deletes (DCPSGsd) in Harman (1963). DCPSGs describe a superset of the context-free languages, by means of rewrite rules that permit a limited amount of wrapping, similar to that found in Head grammar.

Description
Rewrite rules of a DCPSG are identical to those of a CFG, with the addition of a meta-symbol, denoted here as an underscore. DCPSG rules therefore have the general form $$X \to \alpha$$ where $$\alpha$$ is a string of terminal symbols and/or non-terminal symbols and at most one underscore.

The rewrite semantics of DCPSG are identical as those of a CFG when the rule being used does not contain an underscore: given a rule $$X \to \alpha$$, an occurrence of $$X$$ may be rewritten as $$\alpha$$.

For rules with an underscore, the rewrite semantics are slightly different: given a rule $$X \to \alpha \_ \beta$$, an occurrence of $$X$$ can be rewritten as $$\alpha$$, with $$\beta$$ being inserted immediately after the next non-terminal that is introduced at the same time. Using strict left-most productions, $$\beta$$ is simply inserted immediately after the non-terminal that follows $$X$$ prior to the rewrite.

Example
We can characterize the gross sentence structure of a VSO language such as Irish with the following rules (substituting English words for Irish words, and using subscripts solely for demonstration of discontinuity):











A derivation for the sentence saw John Susan, where John is the subject, and Susan is the direct object forming a VP with saw is: