Logical Grammars, Logical Theories

Residuated lattices form one of the theoretical backbones of the Lambek Calculus as the standard free models. They also appear in grammatical inference as the syntactic concept lattice, an algebraic structure canonically defined for every language L based on the lattice of all distributionally definable subsets of strings. Recent results show that it is possible to build representations, such as context-free grammars, based on these lattices, and that these representations will be efficiently learnable using distributional learning. In this paper we discuss the use of these syntactic concept lattices as models of Lambek grammars, and use the tools of algebraic logic to try to link the proof theoretic ideas of the Lambek calculus with the more algebraic approach taken in grammatical inference. We can then reconceive grammars of various types as equational theories of the syntactic concept lattice of the language. We then extend this naturally from models based on concatenation of strings, to ones based on concatenations of discontinuous strings, which takes us from context-free formalisms to mildly context sensitive formalisms (multiple context-free grammars) and Morrill’s displacement calculus.