ID - 127346 TI - Algebraic complexity theory AU - Bürgisser, Peter AU - Clausen, Michael AU - Lickteig, Thomas AU - Shokrollahi, Mohammad Amin PY - 1997 VL - 315 SN - 00727830 SN - 3540605827 3642082289 3662033380 9783540605829 PB - Berlin: Springer, DB - UniCat KW - Numerical solutions of algebraic equations KW - Computer science KW - Computational Complexity KW - Computational complexity KW - Computational complexity. KW - Complexité de calcul (Informatique) KW - Combinatorics. KW - Computers. KW - Mathematical logic. KW - Algorithms. KW - Algebraic geometry. KW - Theory of Computation. KW - Mathematical Logic and Foundations. KW - Algorithm Analysis and Problem Complexity. KW - Algebraic Geometry. KW - Algebraic geometry KW - Geometry KW - Algorism KW - Algebra KW - Arithmetic KW - Algebra of logic KW - Logic, Universal KW - Mathematical logic KW - Symbolic and mathematical logic KW - Symbolic logic KW - Mathematics KW - Algebra, Abstract KW - Metamathematics KW - Set theory KW - Syllogism KW - Automatic computers KW - Automatic data processors KW - Computer hardware KW - Computing machines (Computers) KW - Electronic brains KW - Electronic calculating-machines KW - Electronic computers KW - Hardware, Computer KW - Computer systems KW - Cybernetics KW - Machine theory KW - Calculators KW - Cyberspace KW - Combinatorics KW - Mathematical analysis KW - Foundations UR - https://www.unicat.be/uniCat?func=search&query=sysid:127346 AB - The algorithmic solution of problems has always been one of the major concerns of mathematics. For a long time such solutions were based on an intuitive notion of algorithm. It is only in this century that metamathematical problems have led to the intensive search for a precise and sufficiently general formalization of the notions of computability and algorithm. In the 1930s, a number of quite different concepts for this purpose were pro­ posed, such as Turing machines, WHILE-programs, recursive functions, Markov algorithms, and Thue systems. All these concepts turned out to be equivalent, a fact summarized in Church's thesis, which says that the resulting definitions form an adequate formalization of the intuitive notion of computability. This had and continues to have an enormous effect. First of all, with these notions it has been possible to prove that various problems are algorithmically unsolvable. Among of group these undecidable problems are the halting problem, the word problem theory, the Post correspondence problem, and Hilbert's tenth problem. Secondly, concepts like Turing machines and WHILE-programs had a strong influence on the development of the first computers and programming languages. In the era of digital computers, the question of finding efficient solutions to algorithmically solvable problems has become increasingly important. In addition, the fact that some problems can be solved very efficiently, while others seem to defy all attempts to find an efficient solution, has called for a deeper under­ standing of the intrinsic computational difficulty of problems. ER -