Complexity and Real Computation

Abstract

Library of Congress Cataloging-in-Publication Data Complexity and real computation / Lenore Blum ... [et al.].p. cm. IncIudes bibliographical references and index.

The Physical Church–Turing Thesis: Modest or Bold?

“…Gödel's remark: Tarski has stressed in his lecture (and I think justly) the great importance of the concept of general recursiveness (or Turing's computability). It seems to me that this importance is largely due to the fact that with this concept one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion, i.e., one not depending on the formalism chosen (Gödel 1946, 84, emphasis added Blum et al 1998) is (C) Any system of equations describing a physical system gives rise to computable solutions (cf. Earman 1986, Pour-El 1999 (Pitowsky 1990).…”

On the Ultimate Complexity of Factorials

“…Firstly, we compute a number S = 2 e 1 × 3 e 2 · · · × p es s , where p s is the maximal prime less than or equal to L p [1] and for every 1 ≤ i ≤ s, p e i i is the least p i -power greater than…”

“…There are at least Now we proceed to the third step. We randomly choose w = L p [1] curves E 1 , · · · , E w from D. We call the step successful if for any prime 7B ≤ q ≤ p, q does not divide the discriminant of at least one of the curves in {E 1 , · · · , E w } and the reduction of this curve at q has a L p [1]-smooth order over F q . In the other words, in graph…”

“…mod m is to find a short straight-line program for n!. This problem relates to the algebraic model of computation [2,1,3]. In the algebraic model, it makes sense to ask whether the factorial problem has a polynomial time algorithm, because in this context, to estimate the time complexity, we only count the number of ring operations used to compute n!…”

On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract)

“…A related (and in a sense more powerful) problem is the PosSLP problem: given a division-free straight-line program, or equivalently, an arithmetic circuit with operations +, −, * and inputs 0 and 1, and a designated output gate, determine whether the integer N that is the output of the circuit is positive. As shown in [1], the class P PosSLP , i.e., decision problems that can be solved in polynomial time using an oracle for PosSLP, is equal to the Boolean part (restriction to inputs over {0, 1}) of decision problems over the reals that can be solved in polynomial time in the Blum-Shub-Smale model of real computation [4] using algebraic numbers as constants. This is a powerful model, which is equivalent to the unit cost algebraic RAM model (operations on arbitrary numbers take unit time); in particular the SQRT-SUM problem can be decided in polynomial time on this model [40].…”