## Boolean Functions and Computation ModelsThe foundations of computational complexity theory go back to Alan Thring in the 1930s who was concerned with the existence of automatic procedures deciding the validity of mathematical statements. The first example of such a problem was the undecidability of the Halting Problem which is essentially the question of debugging a computer program: Will a given program eventu ally halt? Computational complexity today addresses the quantitative aspects of the solutions obtained: Is the problem to be solved tractable? But how does one measure the intractability of computation? Several ideas were proposed: A. Cobham [Cob65] raised the question of what is the right model in order to measure a "computation step" , M. Rabin [Rab60] proposed the introduction of axioms that a complexity measure should satisfy, and C. Shannon [Sha49] suggested the boolean circuit that computes a boolean function. However, an important question remains: What is the nature of computa tion? In 1957, John von Neumann [vN58] wrote in his notes for the Silliman Lectures concerning the nature of computation and the human brain that . . . logics and statistics should be primarily, although not exclusively, viewed as the basic tools of 'information theory'. Also, that body of experience which has grown up around the planning, evaluating, and coding of complicated logical and mathematical automata will be the focus of much of this information theory. The most typical, but not the only, such automata are, of course, the large electronic computing machines. |

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### Contents

I | xiii |

III | xiv |

IV | 5 |

V | 6 |

VI | 9 |

VII | 10 |

VIII | 15 |

IX | 19 |

LXXVII | 233 |

LXXVIII | 234 |

LXXIX | 236 |

LXXX | 237 |

LXXXI | 239 |

LXXXII | 240 |

LXXXIII | 241 |

LXXXIV | 245 |

X | 22 |

XI | 28 |

XII | 29 |

XIII | 30 |

XIV | 32 |

XV | 33 |

XVI | 37 |

XVII | 43 |

XVIII | 44 |

XIX | 46 |

XX | 51 |

XXI | 52 |

XXIII | 53 |

XXIV | 54 |

XXV | 59 |

XXVII | 61 |

XXVIII | 63 |

XXIX | 66 |

XXX | 75 |

XXXI | 76 |

XXXII | 88 |

XXXIII | 93 |

XXXIV | 94 |

XXXV | 97 |

XXXVI | 100 |

XXXVII | 105 |

XXXVIII | 108 |

XXXIX | 122 |

XLI | 127 |

XLII | 130 |

XLIV | 133 |

XLV | 135 |

XLVI | 138 |

XLVII | 139 |

XLVIII | 141 |

XLIX | 143 |

L | 144 |

LI | 146 |

LII | 148 |

LIII | 153 |

LIV | 154 |

LV | 160 |

LVI | 162 |

LVII | 166 |

LVIII | 170 |

LIX | 172 |

LX | 176 |

LXI | 182 |

LXII | 183 |

LXIII | 192 |

LXV | 205 |

LXVI | 207 |

LXVII | 210 |

LXVIII | 211 |

LXIX | 212 |

LXX | 215 |

LXXI | 216 |

LXXII | 222 |

LXXIII | 224 |

LXXIV | 227 |

LXXV | 228 |

LXXVI | 230 |

LXXXV | 247 |

LXXXVI | 253 |

LXXXVII | 255 |

LXXXVIII | 257 |

LXXXIX | 265 |

XC | 266 |

XCI | 269 |

XCII | 277 |

XCIII | 283 |

XCIV | 289 |

XCV | 294 |

XCVI | 298 |

XCVII | 304 |

XCVIII | 306 |

XCIX | 314 |

C | 322 |

CI | 324 |

CII | 330 |

CIII | 335 |

CIV | 341 |

CV | 343 |

CVI | 346 |

CVII | 351 |

CVIII | 353 |

CIX | 357 |

CX | 364 |

CXI | 368 |

CXII | 370 |

CXIII | 391 |

CXIV | 396 |

CXV | 401 |

CXVI | 403 |

CXVII | 404 |

CXVIII | 411 |

CXIX | 413 |

CXX | 422 |

CXXI | 425 |

CXXII | 431 |

CXXIII | 432 |

CXXIV | 435 |

CXXV | 436 |

CXXVI | 448 |

CXXVII | 456 |

CXXVIII | 463 |

CXXIX | 468 |

CXXX | 476 |

CXXXI | 485 |

CXXXII | 486 |

CXXXIII | 487 |

CXXXIV | 495 |

CXXXVI | 500 |

CXXXVII | 509 |

CXXXVIII | 525 |

CXXXIX | 552 |

CXL | 553 |

CXLI | 554 |

CXLII | 556 |

CXLIII | 562 |

CXLIV | 563 |

CXLV | 567 |

589 | |

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### Common terms and phrases

algorithm ALOGTIME assume binary boolean circuit boolean formula boolean function bounded recursion clauses CNF formula completes the proof conjunctive normal form constant depth cut-free cut-free proof decision tree Definition denote derivation disjunction edge elements encoding equation Exercise exists f is defined fan-in finite follows Frege proof Frege systems function f gates given graph hence hypercube i-th inequality input integer labeled length Let f linear literals monotonic real circuits node Nullstellensatz obtained OCRAM oracle oracle Turing machine orbits output permutation group pigeonhole principle polynomial-size primitive recursive functions processors Proof of Claim proof of Theorem proof system propositional formula prove recursion on notation representation resolution refutation resp restriction result satisfies sequence sequent calculus st-connectivity subformula subset Suppose Switching Lemma tape term threshold tree-like truth assignment Turing machine undirected graph unsatisfiable upper bound vertex vertices

### Popular passages

Page v - Neumann sought a theory of the organization of automata which would be based on "that body of experience which has grown up around the planning, evaluating, and coding of complicated logical and mathematical automata" *) and which would have applications in the design and programming of digital computers. He outlined the general nature of this proposed automata theory : its materials, some of its problems, what it would be like, and the form of its mathematics. He began a comparative study of artificial...

Page 572 - MT Chao and J. Franco, Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k satisfiable problem, Inf.

### References to this book

Complexity Theory: Exploring the Limits of Efficient Algorithms Ingo Wegener No preview available - 2005 |