## Automata Theory and its ApplicationsThe theory of finite automata on finite stings, infinite strings, and trees has had a dis tinguished history. First, automata were introduced to represent idealized switching circuits augmented by unit delays. This was the period of Shannon, McCullouch and Pitts, and Howard Aiken, ending about 1950. Then in the 1950s there was the work of Kleene on representable events, of Myhill and Nerode on finite coset congruence relations on strings, of Rabin and Scott on power set automata. In the 1960s, there was the work of Btichi on automata on infinite strings and the second order theory of one successor, then Rabin's 1968 result on automata on infinite trees and the second order theory of two successors. The latter was a mystery until the introduction of forgetful determinacy games by Gurevich and Harrington in 1982. Each of these developments has successful and prospective applications in computer science. They should all be part of every computer scientist's toolbox. Suppose that we take a computer scientist's point of view. One can think of finite automata as the mathematical representation of programs that run us ing fixed finite resources. Then Btichi's SIS can be thought of as a theory of programs which run forever (like operating systems or banking systems) and are deterministic. Finally, Rabin's S2S is a theory of programs which run forever and are nondeterministic. Indeed many questions of verification can be decided in the decidable theories of these automata. |

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### Contents

II | vii |

III | xi |

IV | xii |

V | 3 |

VI | 6 |

VII | 9 |

VIII | 11 |

IX | 16 |

LXXX | 201 |

LXXXI | 202 |

LXXXIII | 204 |

LXXXIV | 207 |

LXXXV | 210 |

LXXXVII | 212 |

LXXXVIII | 214 |

LXXXIX | 216 |

X | 19 |

XI | 22 |

XII | 24 |

XIII | 27 |

XIV | 28 |

XV | 31 |

XVI | 32 |

XVII | 34 |

XVIII | 38 |

XX | 42 |

XXI | 48 |

XXII | 50 |

XXIV | 54 |

XXV | 58 |

XXVI | 59 |

XXVII | 62 |

XXVIII | 64 |

XXIX | 68 |

XXXI | 71 |

XXXII | 73 |

XXXIII | 77 |

XXXIV | 85 |

XXXVI | 87 |

XXXVII | 89 |

XXXVIII | 93 |

XL | 95 |

XLI | 97 |

XLIII | 98 |

XLIV | 101 |

XLV | 103 |

XLVI | 104 |

XLVII | 109 |

XLVIII | 113 |

L | 114 |

LI | 115 |

LII | 117 |

LIII | 119 |

LIV | 120 |

LV | 131 |

LVI | 135 |

LVIII | 139 |

LIX | 142 |

LXI | 143 |

LXII | 146 |

LXIII | 152 |

LXIV | 154 |

LXV | 155 |

LXVI | 159 |

LXVII | 162 |

LXVIII | 167 |

LXX | 171 |

LXXI | 176 |

LXXII | 179 |

LXXIV | 181 |

LXXV | 183 |

LXXVI | 186 |

LXXVII | 190 |

LXXVIII | 193 |

LXXIX | 194 |

XC | 217 |

XCI | 218 |

XCII | 223 |

XCIII | 228 |

XCIV | 231 |

XCV | 235 |

XCVI | 241 |

XCVII | 242 |

XCVIII | 251 |

XCIX | 254 |

C | 255 |

CI | 257 |

CII | 262 |

CIII | 264 |

CIV | 266 |

CV | 268 |

CVI | 273 |

CVII | 279 |

CVIII | 282 |

CIX | 284 |

CX | 287 |

CXI | 292 |

CXII | 293 |

CXIII | 296 |

CXIV | 298 |

CXV | 300 |

CXVI | 310 |

CXVIII | 311 |

CXIX | 319 |

CXX | 321 |

CXXI | 322 |

CXXII | 325 |

CXXIII | 328 |

CXXIV | 330 |

CXXV | 331 |

CXXVI | 332 |

CXXVII | 333 |

CXXVIII | 335 |

CXXIX | 338 |

CXXX | 341 |

CXXXI | 342 |

CXXXII | 346 |

CXXXIV | 350 |

CXXXV | 353 |

CXXXVII | 355 |

CXXXVIII | 357 |

CXXXIX | 361 |

CXLI | 364 |

CXLII | 366 |

CXLIII | 370 |

CXLIV | 374 |

CXLVI | 377 |

CXLVII | 380 |

CXLVIII | 382 |

CXLIX | 384 |

CL | 389 |

CLI | 395 |

415 | |

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### Common terms and phrases

ad-language algorithm alphabet assume automaton A accepts automaton A1 belongs binary binary relation Blue wins Boolean algebra Büchi automata clopen sets computation congruence relation Consider the set construct corollary countable decidable definition denote elements equivalence classes equivalence relation example Exercise exists Figure finite automata following properties forgetful strategy formal function game automaton game g game T(A GNFA graph Hence infinite string intersection isomorphic lemma is proved linearly ordered set monadic logic monadic second order move Müller automaton natural numbers nodes nondeterministic Note notion open game open set operation pair partially ordered set path Pathfinder placemarker Rabin automata regular expression satisfies second order logic second order theory sequence sequential Rabin set variables stage strategy f structure Suppose symbol transforms transition table unary algebras unary operation update network winning strategy wins the game words X-tree X-valued tree