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What Was Wrong with the Chomsky Hierarchy?

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What Was Wrong with the Chomsky Hierarchy? What Was Wrong with the Chomsky Hierarchy? Makoto Kanazawa Hosei University Chomsky Hierarchy of Formal Languages recursively enumerable context- sensitive context- free regular Enduring impact of Chomsky’s work on theoretical computer science. 15 pages devoted to the Chomsky hierarchy. Hopcroft and Ullman 1979 No mention of the Chomsky hierarchy. 450 INDEX The same with the latest edition of Carmichael, R. D., 444 CNF-formula, 302 Cartesian product, 6, 46 Co-Turing-recognizableHopcroft, language, 209 Motwani, and Ullman (2006). CD-ROM, 349 Cobham, Alan, 444 Certificate, 293 Coefficient, 183 CFG, see Context-free grammar Coin-flip step, 396 CFL, see Context-free language Complement operation, 4 Chaitin, Gregory J., 264 Completed rule, 140 Chandra, Ashok, 444 Complexity class Characteristic sequence, 206 ASPACE(f(n)),410 Checkers, game of, 348 ATIME(t(n)),410 Chernoff bound, 398 BPP,397 Chess, game of, 348 coNL,354 Chinese remainder theorem, 401 coNP,297 Chomsky normal form, 108–111, 158, EXPSPACE,368 198, 291 EXPTIME,336 Chomsky, Noam, 444 IP,417 Church, Alonzo, 3, 183, 255 L,349 Church–Turing thesis, 183–184, 281 NC,430 CIRCUIT-SAT,386 NL,349 Circuit-satisfiability problem, 386 NP,292–298 CIRCUIT-VALUE,432 NPSPACE,336 Circular definition, 65 NSPACE(f(n)),332 Clause, 302 NTIME(f(n)),295 Clique, 28, 296 P,284–291,297–298 CLIQUE,296 PH,414 Sipser 2013Closed under, 45 PSPACE,336 Closure under complementation RP,403 context-free languages, non-, 154 SPACE(f(n)),332 deterministic context-free TIME(f(n)),279 languages, 133 ZPP,439 P,322 Complexity theory, 2 regular languages, 85 Composite number, 293, 399 Closure under concatenation Compositeness witness, 401 context-free languages, 158 COMPOSITES,293 NP,322 Compressible string, 267 P,322 Computability theory, 3 regular languages, 47, 60 decidability and undecidability, Closure under intersection 193–210 context-free languages, non-, 154 recursion theorem, 245–252 regular languages, 46 reducibility, 215–239N: finite alphabet of nonterminal Semi-Thue System/StringClosure under Rewriting star Turing machines, 165–182 context-free languages, 158 Computable function, 234 System NP,327 Computation history symbols P,327 context-free languages, 225–226 regular languages, 62 defined, 220 Closure under union linear bounded automata,Σ: finite alphabet of terminal symbols context-free languages, 158 221–225 Production α → βNP,322 α, β ∈ (N ∪ Σ)* Post Correspondence Problem, P,322 227–233 regular languages, 45, 59 reducibility, 220–233 One-step rewriting γαδ ⇒ γβδ Language L(G) = { w ∈ Σ* ∣ S ⇒* w } Type 0 unrestricted Type 1 |α| ≤ |β| Type 2 α ∈ N Type 3 α ∈ N, β ∈ Σ N LBA: linear-bounded automata = rewriting machine systems models NSPACE(n) Turing machines PDA: pushdown automata type 0 recursively Turing machines enumerable FA: finite automata context- LBA type 1 sensitive type 2 context- free PDA type 3 regular FA recursively Context-sensitive languages are enumerable type 0 computationally very complex. Some context-sensitive languages are type 1 context- sensitive PSPACE-complete. PSPACE-complete context- type 2 free regular type 3 • Context-sensitive languages are computationally very complex. • There is a huge gap between context-free and context- sensitive. • Many intermediate classes of languages have been considered in formal language theory. • In particular, “mildly context-sensitive” grammars have been investigated by computational linguists. Indexed Grammars Hopcroft and Ullman 1979 recursively What type of rewriting system is an enumerable type 0 indexed grammar?? type 1 context- sensitive ?? indexed context- type 2 free regular type 3 Type 0 Production α → β One-step rewriting γαδ ⇒ γβδ Indexed Production T → ABC One-step rewriting Tfg ⇒ AfgBfgCfg An indexed grammar is not an instance of a type 0 grammar. recursively Indexed languages are also sort of enumerable type 0 complex. type 1 context- sensitive indexed NP-complete context- type 2 free regular type 3 68 STUART M. SHffiBER 68 STUART M. SHffiBER 2. Robert Berwick and Amy Weinberg. The Grammatical Basis of Linguistic Performance: Language Use and Acquisition. MIT Press, Cambridge, Massachusetts, 1984. APPLICABILITYGERALD GAZDAR OF INDEXED GRAMMARS 71 70 GERALD GAZDAR2. Robert Berwick and Amy Weinberg. The Grammatical Basis of Linguistic Performance: Gazdar 1988 3. JoanLanguage Bresnan, Use editor. and Acquisition. The Mental MIT Representation Press, Cambridge, of Grammatical Massachusetts, Relations. 1984. MIT Press, GERALD GAZDAR UNIVERSITY OF SUSSEX push items onto, pop3. JoanCambridge, items Bresnan, from,Massachusetts, editor. andThe Mental 1982.copy Representation the stack. of What Grammatical we Relations.end up MIT Press, of orthographic conventions: nonterminal symbols are indicated by upper 4. LucaCambridge, Cardelli Massachusetts, and Peter 1982.Wegner. Understanding types, data abstraction and UNIVERSITY OF SUSSEX with now is no longer4. polymorphism.Luca equivalent Cardelli Draft andto thepaper,Peter CF-PSGs 1985.Wegner. Understanding but is significantly types, data more abstraction and case letters (A, B, C),SCHOOL terminal OF SOCIALsymbols SCIENCES by lower case letters (a, b, e), 5. Noampolymorphism. Chomsky. Aspects Draft paper, of the 1985.Theory of Syntax. MIT Press, Cambridge, Massachusetts, possibly empty stringsSCHOOL of terminals OF SOCIAL and nonterminals SCIENCES by W, W , W , etc., powerful, namely 5.the Noam1965. indexed Chomsky. Aspectsgrammars of the Theory (Aho, of Syntax. 1968). MIT Press,This Cambridge, class Massachusetts,of BRIGHTON 1 2 grammars has been6. Noamalluded1965. Chomsky. to aLectures number on Government of times and in Binding.the recent Foris Publications,linguistic Dordrecht, indices by lower case italic lettersBRIGHTON k), and stacks of indices by square Holland, 1982. 6. Noam Chomsky. Lectures on Government and Binding. Foris Publications, Dordrecht, brackets and periods ([], [ .. ], [i, .. ]) where [i, .. ] is a stack whose topmost literature: by Klein7. Steven Holland,(1981) Fortune, 1982. in Daniel connection Leivant, and Michaelwith O'Donnell.nested Thecomparative Expressiveness of Simple and Second Order Type Structures. Research Report RC 8542, IBM Thomas J. Watson 7. Steven Fortune, Daniel Leivant, and Michael O'Donnell. The Expressiveness of Simple index is i, [] is an empty, and [.. ] is a possibly empty stack of indices. As for constructions, by DahlResearch (1982) Center, in connectionYorktown Heights, with New topicalised York, 1980. pronouns, by and Second Order Type Structures. Research Report RC 8542, IBM Thomas J. Watson the rules themselves, Aho (1968)APPLICABILITY uses one notation, and Hopcroft and Engdahl (1982) and8. Gerald ResearchGazdar Gazdar. Center, (1982)Phrase Yorktown Structurein Heights,connection Grammar, New York, pages with1980. 131-186. Scandinavian D. Reidel, Dordrecht, APPLICABILITY 8. GeraldHolland, Gazdar. 1982.] Phrase Structure Grammar, pages 131-186. D. Reidel, Dordrecht, Ullman (1979) use another.OF INDEXED The notation GRAMMARS used below is essentially just a unbounded dependencies,9. Gerald Gazdar, by Huybregts Ewan Klein, Geoffrey(1984) K. ilDdPullum, Pulman and Ivan andA. Sag. Ritchie Generalized Phrase Holland, 1982.] OF INDEXED GRAMMARS (1984) in connection9. GeraldStructure with Gazdar, Grammar. Dutch, Ewan Blackwell Klein, by Geoffrey MarshPublishing, K. Pullum, andOxford, Parteeand England, Ivan A. and(1984) Sag. Harvard Generalized in University Phrase redundant, and intendedlyTO NATURAL more perspicuous, LANGUAGES variant of that employed by connection with variablePress,Structure binding, Cambridge, Grammar. and Massachusetts, Blackwell doubtless Publishing, 1985. elsewhere Oxford, England, as well. and Harvard University Hopcroft and Ullman. TO NATURAL LANGUAGES 10. RonaldPress, Kaplan.Cambridge, Personal Massachusetts, communication, 1985. 1985. Indexed grammars11.10. MartinRonald fall Kaplan.Kay.in Unificationbetween Personal communication,Grammar. CF-PSGs Technical 1985. and Report, context-sensitive Xerox Palo Alto Research In the standard formulations, an indexed grammar can contain rules of Center, Palo Alto, California, 1983. APPLICABILITY OF INDEXED GRAMMARS 73 grammars in their generative11. Martin Kay. capacity. Unification Every Grammar. context-free Technical Report, language Xerox Palo(CPL) Alto Research three72 differentGERALD sorts: GAZDAR 12. FernandoCenter, Palo C. N. Alto, Pereira California, and Stuart 1983. M. Shieber. The semantics of grammar formalisms seen as computer languages. In of the Tenthn International Conference on is an indexed language12. Fernando (IL), C.but N. notPereira conversely. and Stuart M. ThusShieber. anbne The semanticsis an of grammarIL, for formalisms stack to all nonterminal .daughters 1. in all rule* types, and (b) restricts (4) 51] Computationalseen as computer Linguistics, languages. StanfoId In University,of Stanford, the Tenth California, International 2-7 JulyConference 1984. on (1) 1. A[.. ] -> W[ .. ] INTRODUCI10N example, but it is not13. Fernando a CPL. C. And N. Pereira every and ILDavid is H.a D.context-sensitive Warren. Parsing as deduction. language, In 1. 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