The Generation-Recognition Asymmetry: Six Dimensions of a Fundamental Divide in Formal Language Theory

TL;DR

This paper explores the fundamental asymmetries between generation and recognition in formal language theory across six dimensions, challenging common assumptions and connecting them to recent NLP developments.

Contribution

It identifies six key dimensions of the generation-recognition divide, introduces two new dimensions, and relates temporal asymmetry to surprisal, offering a comprehensive unified perspective.

Findings

Generation under constraints can be NP-hard.

Parsing is always constrained, generation need not be.

Bidirectional systems have existed for fifty years but are underused.

Abstract

Every formal grammar defines a language and can in principle be used in three ways: to generate strings (production), to recognize them (parsing), or -- given only examples -- to infer the grammar itself (grammar induction). Generation and recognition are extensionally equivalent -- they characterize the same set -- but operationally asymmetric in multiple independent ways. Inference is a qualitatively harder problem: it does not have access to a known grammar. Despite the centrality of this triad to compiler design, natural language processing, and formal language theory, no survey has treated it as a unified, multidimensional phenomenon. We identify six dimensions along which generation and recognition diverge: computational complexity, ambiguity, directionality, information availability, grammar inference, and temporality. We show that the common characterization "generation is easy, parsing is hard" is misleading: unconstrained generation is trivial, but generation under constraints can be NP-hard. The real asymmetry is that parsing is always constrained (the input is given) while generation need not be. Two of these dimensions -- directionality and temporality -- have not previously been identified as dimensions of the generation-recognition asymmetry. We connect the temporal dimension to the surprisal framework of Hale (2001) and Levy (2008), arguing that surprisal formalizes the temporal asymmetry between a generator (surprisal = 0) and a parser that predicts under uncertainty (surprisal > 0). We review bidirectional systems in NLP and observe that bidirectionality has been available for fifty years yet has not transferred to most domain-specific applications. We conclude with a discussion of large language models, which architecturally unify generation and recognition while operationally preserving the asymmetry.