It was developed in the late 1970s by David E. Johnson and Paul Postal, and formalized in 1980 in the eponymous book Arc Pair Grammar.
[1] In the early 1970s, some linguists, such as Edward Keenan, began to challenge this notion from the transformationalist perspective, noting for instance the formation of relative clauses in Malagasy[1] and English passivization (see chômeur).
Relational grammar (RG) itself was never formalized in one place; instead, Keenan, Johnson, and others began writing aspects of the framework in a series of dissertations around this time.
Dissatisfied with the results and lack of formalization in RG, David Johnson and Paul Postal attempted to lay down a version of it using mathematical logic.
[2] APG takes grammatical relations, the graph theory notion of an arc, and two operations (SPONSOR and ERASE) as primitives, with all other rules being derived (many of them mathematically, rather than empirically).
Sentences of a language, understood as structures of a certain type, follow the set of linguistic laws and language-specific statements.
[2] A pair network consists of four components: the 'relational-graph,' the 'logical-graph,' the 'surface-graph' (R-, L-, S-graphs), and the two operations Sponsor and Erase.
The R-graph is simply the set of all items in the pair network, i.e., the structure as a whole of all arcs, labels (R-signs), and operations between them.
Sponsor operations are used between levels in the R-graph to establish different linguistic states (that is, a particular set of grammatical relations).