20 research outputs found
On Czerwinski's " relative to a -complete oracle"
In this paper, we take a closer look at Czerwinski's "
relative to a -complete oracle" [Cze23]. There are (uncountably)
infinitely-many relativized worlds where and differ, and
it is well-known that for any -complete problem , . The paper defines two sets and and builds the purported proof of their main theorem
on the claim that an oracle Turing machine with as its
oracle and that accepts must make queries to
the oracle. We invalidate the latter by proving that there is an oracle Turing
machine with as its oracle that accepts
and yet only makes one query to the oracle. We thus conclude that Czerwinski's
paper [Cze23] fails to establish that
A Note on Universal Measures for Weak Implicit Computational Complexity
Abstract. This note is a case study for finding universal measures for weak implicit computational complexity. We will instantiate âuniver-sal measures â by âdynamic ordinalsâ, and âweak implicit computational complexity â by âbounded arithmeticâ. Concretely, we will describe the connection between dynamic ordinals and witness oracle Turing ma-chines for bounded arithmetic theories
Complexity of certificates, heuristics, and counting types , with applications to cryptography and circuit theory
In dieser Habilitationsschrift werden Struktur und Eigenschaften von KomplexitÀtsklassen wie P und NP untersucht, vor allem im Hinblick auf: ZertifikatkomplexitÀt, Einwegfunktionen, Heuristiken gegen NP-VollstÀndigkeit und ZÀhlkomplexitÀt. Zum letzten Punkt werden speziell untersucht: (a) die KomplexitÀt von ZÀhleigenschaften von Schaltkreisen, (b) Separationen von ZÀhlklassen mit ImmunitÀt und (c) die KomplexitÀt des ZÀhlens der Lösungen von ,,tally`` NP-Problemen
Why Philosophers Should Care About Computational Complexity
One might think that, once we know something is computable, how efficiently
it can be computed is a practical question with little further philosophical
importance. In this essay, I offer a detailed case that one would be wrong. In
particular, I argue that computational complexity theory---the field that
studies the resources (such as time, space, and randomness) needed to solve
computational problems---leads to new perspectives on the nature of
mathematical knowledge, the strong AI debate, computationalism, the problem of
logical omniscience, Hume's problem of induction, Goodman's grue riddle, the
foundations of quantum mechanics, economic rationality, closed timelike curves,
and several other topics of philosophical interest. I end by discussing aspects
of complexity theory itself that could benefit from philosophical analysis.Comment: 58 pages, to appear in "Computability: G\"odel, Turing, Church, and
beyond," MIT Press, 2012. Some minor clarifications and corrections; new
references adde
On lower bounds for circuit complexity and algorithms for satisfiability
This work is devoted to explore the novel method of proving circuit lower bounds for the class NEXP by Ryan Williams. Williams is able to show two circuit lower bounds: A conditional lower bound which says that NEXP does not have polynomial size circuits if there exists better-than-trivial algorithms for CIRCUIT SAT and an inconditional lower bound which says that NEXP does not have polynomial size circuits of the class ACC^0. We put special emphasis on the first result by exposing, in as much as of a self-contained manner as possible, all the results from complexity theory that Williams use in his proof. In particular, the focus is put in an efficient reduction from non-deterministic computations to satisfiability of Boolean formulas. The second result is also studied, although not as thoroughly, and some pointers with regards to the relationship of Williams' method and the known complexity theory barriers are given
Scalable Task Schedulers for Many-Core Architectures
This thesis develops schedulers for many-cores with different optimization objectives. The proposed schedulers are designed to be scale up as the number of cores in many-cores increase while continuing to provide guarantees on the quality of the schedule