Optimization and quantization in gradient symbol systems: A framework for integrating the continuous and the discrete in cognition

Paul Smolensky, Matthew Goldrick*, Donald Mathis

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

70 Scopus citations


Mental representations have continuous as well as discrete, combinatorial properties. For example, while predominantly discrete, phonological representations also vary continuously; this is reflected by gradient effects in instrumental studies of speech production. Can an integrated theoretical framework address both aspects of structure? The framework we introduce here, Gradient Symbol Processing, characterizes the emergence of grammatical macrostructure from the Parallel Distributed Processing microstructure (McClelland, Rumelhart, & The PDP Research Group, 1986) of language processing. The mental representations that emerge, Distributed Symbol Systems, have both combinatorial and gradient structure. They are processed through Subsymbolic Optimization-Quantization, in which an optimization process favoring representations that satisfy well-formedness constraints operates in parallel with a distributed quantization process favoring discrete symbolic structures. We apply a particular instantiation of this framework, λ-Diffusion Theory, to phonological production. Simulations of the resulting model suggest that Gradient Symbol Processing offers a way to unify accounts of grammatical competence with both discrete and continuous patterns in language performance.

Original languageEnglish (US)
Pages (from-to)1102-1138
Number of pages37
JournalCognitive Science
Issue number6
StatePublished - Aug 2014


  • Combinatorial structure
  • Distributed representation
  • Harmonic grammar
  • Optimization
  • Selection
  • Speech errors

ASJC Scopus subject areas

  • Experimental and Cognitive Psychology
  • Cognitive Neuroscience
  • Artificial Intelligence


Dive into the research topics of 'Optimization and quantization in gradient symbol systems: A framework for integrating the continuous and the discrete in cognition'. Together they form a unique fingerprint.

Cite this