Computational Limits on Team Identi cation of Languages

A team of learning machines is essentially a multiset of learning machines A team is said to successfully identify a concept just in case each member of some nonempty subset of the team identi es the concept Team identi cation of programs for computable functions from their graphs has been investigated by Smith Pitt showed that this notion is essentially equivalent to function identi cation by a single probabilistic machine The present paper introduces motivates and studies the more di cult subject of team identi cation of grammars for languages from positive data It is shown that an analog of Pitt s result about equivalence of team func tion identi cation and probabilistic function identi cation does not hold for language identi cation and the results in the present paper reveal a very complex structure for team language identi cation It is also shown that for certain cases probabilistic language identi cation is strictly more powerful than team language identi cation Proofs of many results in the present paper involve very sophisticated diagonalization arguments Two very general tools are presented that yield proofs of new results from simple arithmetic manipulation of the parameters of known ones Some preliminary results were rst reported at the th International Collo quium on Automata Languages and Programming Warwick University July During the early stages of this work Sanjay Jain was a liated with the Department of Computer Science University of Rochester and the Depart ment of Computer and Information Sciences University of Delaware He was supported in part by NSF grant CCR at the University of Rochester His present address Institute of Systems Science National University of Sin gapore Singapore Republic of Singapore Email sanjay iss nus sg At the same time Arun Sharma was a liated with the Department of Computer Science SUNY at Bu alo Department of Computer and Infor mation Sciences University of Delaware and the Department of Brain and Cognitive Sciences MIT He was supported by NSF grant CCR at SUNY Bu alo and University of Delaware and by a Siemens Corporation grant at MIT