An algebraic approach to Multiple Context-Free Grammars

Abstract. We define an algebraic structure, Paired Complete Idempotent Semirings (pcis), which are appropriate for defining a denotational semantics for multiple context-free grammars of dimension 2 (2-mcfg). We demonstrate that homomorphisms of this structure will induce wellbehaved morphisms of the grammar, and generalize the syntactic concept lattice from context-free grammars to the 2-mcfg case. We show that this lattice is the unique minimal structure that will interpret the grammar faithfully and that therefore 2-mcfgs without mergeable nonterminals will have nonterminals that correspond to elements of this structure.