A Second Course in Formal Languages and Automata Theory Jeffrey Shallit Index More Information

Cambridge University Press 978-0-521-86572-2 - A Second Course in Formal Languages and Automata Theory Jeffrey Shallit Index More information

Index

2DFA, 66 Berstel, J., 48, 106 2DPDA, 213 Biedl, T., 135 3-SAT, 21 Van Biesbrouck, M., 139 Birget, J.-C., 107 abelian cube, 47 blank symbol, B, 14 abelian square, 47, 137 Blum, N., 106 Ackerman, M., xi Boolean closure, 221 Adian, S. I., 43, 47, 48 Boolean grammar, 223 Aho, A. V., 173, 201, 223 Boolean matrices, 73, 197 Alces, A., 29 Boonyavatana, R., 139 alfalfa, 30, 34, 37 border, 35, 104 Allouche, J.-P., 48, 105 bordered, 222 alphabet, 1 bordered word, 35 unary, 1 Borges, M., xi always-halting TM, 209 Borwein, P., 48 ambiguous, 10 Boyd, D., 223 NFA, 102 Brady, A. H., 200 Angluin, D., 105 branch point, 113 angst Brandenburg, F.-J., 139 existential, 54 Brauer, W., 106 antisymmetric, 92 Brzozowski, J., 27, 106 assignment Bucher, W., 138 satisfying, 20 Buntrock, J., 200 automaton Burnside problem for groups, 42 pushdown, 11 Buss, J., xi, 223 synchronizing, 105 busy beaver problem, 183, 184, 200 two-way, 66 Axelrod, R. M., 105 calliope, 3 Cantor, D. C., 201 Bader, C., 138 Cantor set, 102 balanced word, 137 cardinality, 1 Bar-Hillel, Y., 201 census generating function, 133 Barkhouse, C., xi Cerny’s conjecture, 105 base alphabet, 4 Chaitin, G. J., 200 Berman, P., 200 characteristic sequence, 28

233

234 Index

chess, 41 cycle operation, 59 German rule in, 42 cyclic shift, 34, 60 laws of, 41 CYK algorithm, 141, 142, 144 Chomsky hierarchy, 212 Chomsky, N., 27, 201, 212, 223 Damon, T., xi Chomsky normal form, 10 Davis, M., 27 clickomania, 135 DCFL’s, 126 closed, 7 closure under complement, 128 closure properties, 7 decision problem co-CFL, 132 solvable, 17 Cocke, J., 141 deeded, 47 cold, 3 deified, 3 commutative image, 122 derivation, 10 comparable, 92 derivation tree, 10 complement deterministic context-free languages, of a set, 1 126 complete item, 145 deterministic finite automaton, 4 complex, 178 definition, 4 complexity class, 19 deterministic PDA, 126 compression, 178 DFA, 4 optimal, 179 DiRamio, M., xi computing model directed graph, 73 robust, 66 Dirichlet’s theorem, 95 concatenation division-free sequence, 93 of languages, 3 drawer, 3 of strings, 3 Drobot, V., 173 configuration, 12, 14 Du, D.-Z., 139 conflict, 156 conjugate, 34, 44, 60, 137 Earley, J., 173 conjunctive grammar, 222 Earley’s parsing method, 144 conjunctive language, 222 Eggan, L. C., 105 conjunctive normal form, 20 Ehrig, H., 138 context-free grammar, 8, 9 Ellul, K., xi definition of, 9 empty stack pure, 133 acceptance by in PDA, 12 context-free language empty string, 2 definition of, 10 Engels, G., 138 context-free languages enlightenment, 2 Boolean closure of, 221 entanglement, 34 closed under inverse morphism, 109 -production, 10 closed under substitution, 108 -transition, 6 context-sensitive grammar, 202 equivalence classes, 77 context-sensitive language, 202 equivalence relation, 77 Cook, S. A., 26, 223 index, 77 Coppersmith, D., 106 Myhill-Nerode, 78 Crochemore, M., 47 refinement of, 77 CSG, 202 right invariant, 78 CSL, 202 Euwe, M., 42, 48 cube, 37 exponent abelian, 47 of a group, 43 cubefree, 37 extended regular expression, 23

Index 235

factor, 27 Hadley, R., xi factoring, 162 Haines, L. H., 107 Fermat’s last theorem, 36 Hall, M., 43 Fernandes, S., xi halting state, 14 Fibonacci number, 132 Hanan, J., 222 final state, 4 handle, 164 acceptance by in PDA, 12 Harju, T., 48, 106 Fine, N. J., 48 Harrison, M. A., 107, 223 finite index, 77 Hartmanis, J., 105 finite-state transducer, 61, 102 Hashiguchi, K., 105 Fischer, P. C., 139 high information content, 178 Floyd, R. W., 201 homomorphism, 29, 54 FORTRAN, 133 iterated, 29 fractional power, 34 Hopcroft, J. E., 26, 106, 173, 201, 223 Franz, J., xi Friedman property, 100 Ibarra, O. H., 105, 223 full configuration immediate left recursion, 161 of a 2DPDA, 214 Immerman, N., 210, 223 function incidence matrix, 73 partial, 14 incompressibility method, 180, 182 Gabarro, J., 138 incompressible, 179 Garey, M. R., 27 index Gaspar, C., xi of an equivalence relation, 77 Gazdar, G., 138 finite, 77 general sequential machine, indistinguishable, 82 106 inductive counting, 210 German rule, 42 infinite sequence, 28 Ginsburg, S., 106, 138, 139, 201 infinite string, 28 Glaister, I., 107 infinite word, 28 Glover, M., xi Ingalls, C., 48 Golbeck, R., xi inherently ambiguous, 114, 115 Goldstine, J., 138, 139 initial state, 5 Golod, D., xi input grammar, 8 accepted, 4 Boolean, 223 rejected, 4 conjunctive, 222 integer context-free, 8 random, 181 context-sensitive, 202 interchange lemma, 118–121 length-increasing, 203 invalid computation, 190 Type 0, 200 invariance theorem, 178 unrestricted, 174–176 inverse morphism, 57 Gray, J. N., 223 inverse substitution, 97, 135 Greibach, S. A., 139 alternate definition, 98, 135 Gries, D., 106 Istead, J., xi group item, 163 exponent of, 43 complete, 145 infinite periodic, 46 periodic, 43 Jiang, T., 105, 107 torsion, 43 Johnson, D. S., 27 Gupta, N., 47 Jones, N. D., 223

236 Index

Kalbfleisch, R., xi Lin, S., 200 Kao, J.-Y., xi Lindenmayer, A., 222 Karhumaki,¨ J., 48, 106 linear, 122 Karp reduction, 20 linear bounded automaton, 206 Kasami, T., 141 linear grammar, 130 Keanie, A., xi linear languages, 130 Kfoury, A. J., 47 pumping lemma for, 131 Kleene closure, 4 Linster, B. G., 105 Klein, E., 138 Litt, A., xi Knuth, D. E., 173 Liu, L. Y., 138 Knuth DFA, 164 log(L), 76 Knuth NFA-, 165 Lorentz, R. J., 200 Ko, K.-I., 139 Lothaire, M., 48, 107 Kolmogorov complexity, 177, 178 LR(0) grammar uncomputable, 179 definition of, 167 Koo, A., xi Lucier, B., xi Kozen, D., 105 Lutz, B., xi Kreowski, H.-J., 138 Lyndon, R. C., 30, 48 Krieger, D., xi Lyndon–Schutzenberger¨ theorems, 30, 31 Kuroda, S. Y., 223 MacDonald, I., xi Landry, D., xi Main, M. G., 138 language, 2 Martin, A., xi conjunctive, 222 Martin, G., xi context-free, 10 Martin, J. C., 26, 139 inherently ambiguous, 115 Martinez, A., xi prefix, 3, 54, 96 Marxen, H., 200, 201 recursive, 15 Maslov, A. N., 107 recursively enumerable, 15 matrices regular, 4 Boolean, 73, 197 suffix, 3 matrix multiplication, 75 languages, 3 Maurer, H. A., 138 product of, 3 McCulloch, W. S., 27 Laplace, P. S., 177 McNaughton, R., 105, 106 Lawson, M. V., 106 Mealy, G. H., 106 LBA, 206 Mealy machine, 49 deterministic, 223 Mehta, G., xi leading zeros Miltchman, O., xi removing, 54, 62 minimal Lee, J., xi in partially ordered set, 94 left quotient, 106 minimal word, 133 leftmost derivation, 10 mirror invariant, 45 length-increasing grammar, 203 Mobius¨ function, 46 letters, 1 Montanari, U., 138 Levi’s lemma, 30 Moore, E. F., 106 Lewis, H. R., 26 Moore machine, 49 lexicographic order, 3 moose, 3 lexicographically median, 100 antlers, 29 Li, M., 200 morphism, 29, 54 Ligocki, S., 201 convention when defining, 29 Ligocki, T. J., 201 iterated, 29

Index 237

nonerasing, 104 outshout, 35 overlap-free, 43 overlap-free, 37, 39 Morse, M., 38 morphism, 43 Motwani, R., 26 Moura, A., 138 P (polynomial time complexity class), 19 multigrade problem, 40 Palacios, A., xi murmur, 104 palindrome, 3 Myhill, J., 106 palindromic closure, 104 Myhill–Nerode equivalence relation, Panini, 27 78 Papadimitriou, C. H., 26, 27 Myhill–Nerode theorem, 77 Papert, S., 106 Parikh map, 122 n! trick, 114 Parikh, R. J., 139 Nazari, S., xi parse tree, 10 Nerode, A., 106 partial configuration Nevraumont, A. F., 200 of a 2DPDA, 218 NFA, 6 partial function, 14 ambiguous, 102 partial order, 92 NFA-, 6 pattern, 104, 199 Ng, S., xi pattern matching, 199 Nguyen, L., xi PCF grammar, 133 Nichols, M., xi PCP, 186 Nivat’s theorem PDA for transducers, 63 acceptance by empty stack, 12 nonconstructive, 54 acceptance by final state, 12 nondeterministic finite automaton, 6 deterministic, 126 nondeterministic state complexity, 90 perfect shuffle, 3, 120 nonerasing morphism, 104 period, 29 nonpalindromes, 9 periodic nontrivial purely, 29 prefix, 2 ultimately, 29 suffix, 2 Perles, M., 201 Nørgard,˚ P., 42 permutations, 97 normal form for transducers, 64 Perrin, D., 47, 48 Novikov, P. S., 43 photograph, 35 Nowotka, D., 48 π, 43 Nozaki, A., 106 Pilling, D. L., 139 NP-complete, 20 Pin, J.-E., 47, 48, 105 NP (nondeterministic polynomial time), Pitts, W., 27 20 positive closure, 4, 103 NPSPACE, 21 Post correspondence problem, 186, 201 Post, E., 201 ODDPAL, 9 power, 33 Ogden, W., 138, 139 power set, 1 Ogden’s lemma, 112 prefix, 2 example of, 114 nontrivial, 2 Okhotin, A., 222, 223 proper, 2 ω-language, 47 prefix language, 3, 54, 96 order prefix-free encoding, 181 lexicographic, 3 preperiod, 29 radix, 3 Price, J. K., 138

238 Index

prime number theorem, 180 removing trailing zeros, 54 primes repaper, 3 in unary, 7 reversal, 3 PRIMES2 (primes in base 2), 2, 8, 17 reward, 3 irregularity of, 7 Rice, H. G., 138 primitive Rideout, J., xi, 200 root, 33 right invariant, 78 string, 33 right quotient, 106 primitive strings, 222 right sentential form, 164 enumeration of, 46 rightmost derivation, 10 primitive word, 138 Robinson, R., xi proper prefix, 2 Robson, J. M., 105 proper suffix, 2 robustness, 66 Prouhet, E,´ 40 Rose, G. F., 201 Prusinkiewicz, P., 222 Rose, K., xi PSPACE, 21 Rosenkrantz, D. J., 173 Pullum, G., 138 Ross, R. J., 138, 139 pure context-free grammar, 133 rotator, 3 purely periodic, 29 Rozenberg, G., 138 pushdown Rubinstein, A., 105 automaton, 11 pushdown store, 11 Sag, I., 138 Salomaa, K., 107 quotient Sanov, I. N., 43 of languages, 52 Santean, N., xi SAT, 20 Rabin, M. O., 26, 27 satisfying assignment, 20 radar, 3 Savitch’s theorem, 196 radix order, 3 Schutzenberger,¨ M. P., 30, 48, 139, 201 Rado, T., 200 Scott, D., 26, 27 Rampersad, N., xi Seiferas, J. I., 105, 106 random, 178, 179 semilinear, 122 random integer, 181 sentence, 1 rat, 3 sentential form, 9 rational relation, 63 sequence Ravikumar, B., 107 infinite, 28 reappear, 47 sequential transducer, 106 Reardon, J., xi Sethi, R., 173 recurrent, 45 Shallit, J., 48, 105, 107 recursive, 15 Shamir, E., 201 recursively enumerable, 15 Shepherdson, J. C., 106 redivider, 3 Shieber, S. M., 138 refinement, 77–79, 82, 84 shift-reduce parsing, 168 reflexive, 92 shoe, 3 regular expressions, 4 shuffle, 57, 97, 132 regular language, 4 perfect, 3, 120 regularity-preserving transformation, 105 Shyr, H. J., 48 relation, 77 Slutzki, G., 139 equivalence, 77 Socha, J., xi rational, 63 Solomonoff, R., 200 removing leading zeros, 54, 62 solvable, 17

Index 239

Spanier, E. H., 106, 138, 139 Tikuisis, A., xi sparse, 101 time complexity, 19 square, 37 TM, 13 abelian, 47 trailing zeros squarefree, 37, 39, 45 removing, 54 squarefree-preserving, 121, 136 transducer stack, 11 finite-state, 61, 102 star height, 99 normal form for, 64 start state, 5 sequential, 106 start symbol, 9 subsequential, 106 state complexity transduction, 63 nondeterministic, 90 transition diagram, 5 Stearns, R. E., 105, 173 transition function, 4 Stebila, D., xi extended, 5 Strachey, C., 27 transitive, 92 string, 1 triple construction, 116 infinite, 28 Tromp, J., 200 length of, 2 Turing, A. M., 26, 27 primitive, 33, 222 Turing machine, 13, 14 right-infinite, 28 always-halting, 209 strings, 2 computing a function, 16 subsequence, 2, 27, 92, 93, 100, 134 Turing reduction, 18 subsequence ordering, 92 two-way finite automaton, 66 subsequential transducer, 106 Type 0 grammar, 200, 212 substitution, 55 Type 1 grammar, 212 inverse, 97, 135 Type 2 grammar, 212 by regular languages, 55 Type 3 grammar, 212 subword, 2, 27 subword ordering, 92 Ullman, J. D., 26, 106, 173, 201, subword-doubling closure, 99 223 subword-doubling map, 99 ultimately periodic, 29 suffix, 2 unambiguous, 10 nontrivial, 2 unary alphabet, 1 proper, 2 unary context-free language, 111 suffix language, 3 unbalanced word, 137 Sun, W., xi unbordered, 134 supersequence, 93 unbordered word, 35 symbols, 1 uncomputable, 27 synchronizing automaton, 105 undecidable, 27 Szelepcsenyi,´ R., 210, 223 uneven word, 45 unit productions, 10 Tarry–Escott problem, 40, 48 universality problem, 193, 197 terminator, 218 unrestricted grammar, 174–176 Thierrin, G., 48 unsolvable, 27 Thomas, W., 47 Urbanek, F., 173 Thornton, R. J., 27 useless symbol, 24 Thue, A., 38, 48 Thue–Morse morphism, 30, 44 Valiant, L., 140, 173 Thue–Morse sequence, 38, 40–42, 45, 48 valid music and, 42 for viable prefix, 164 Thue–Morse word, 137 valid computation, 190

240 Index

viable preﬁx, 164 unbordered, 35, 134 Vitanyi,´ P., 200 uneven, 45 Wotschke, D., 138 Wallace, J., xi Wright, E. M., 48 Wang, X., xi Weiner, P., 138 Xu, Z., xi Wilf, H. S., 48 Winklmann, K., 138, 139 yield, 10 Winograd, S., 106 Young, C., xi Wolfram, S., 105 Younger, D., 141 Wood, D., 106 Younger, D. H., 173 Woods, D. R., 48 Yu, S., 106, 107 word, 1 balanced, 137 Zhang, G.-Q., 106 bordered, 35, 104 Zhang, N., xi inﬁnite, 28 Zhao, L, 200 primitive, 138 Zhuang, Q., 107 unbalanced, 137 Zimny, B., xi