640 research outputs found
An introspective algorithm for the integer determinant
We present an algorithm computing the determinant of an integer matrix A. The
algorithm is introspective in the sense that it uses several distinct
algorithms that run in a concurrent manner. During the course of the algorithm
partial results coming from distinct methods can be combined. Then, depending
on the current running time of each method, the algorithm can emphasize a
particular variant. With the use of very fast modular routines for linear
algebra, our implementation is an order of magnitude faster than other existing
implementations. Moreover, we prove that the expected complexity of our
algorithm is only O(n^3 log^{2.5}(n ||A||)) bit operations in the dense case
and O(Omega n^{1.5} log^2(n ||A||) + n^{2.5}log^3(n||A||)) in the sparse case,
where ||A|| is the largest entry in absolute value of the matrix and Omega is
the cost of matrix-vector multiplication in the case of a sparse matrix.Comment: Published in Transgressive Computing 2006, Grenade : Espagne (2006
Resolution over Linear Equations and Multilinear Proofs
We develop and study the complexity of propositional proof systems of varying
strength extending resolution by allowing it to operate with disjunctions of
linear equations instead of clauses. We demonstrate polynomial-size refutations
for hard tautologies like the pigeonhole principle, Tseitin graph tautologies
and the clique-coloring tautologies in these proof systems. Using the
(monotone) interpolation by a communication game technique we establish an
exponential-size lower bound on refutations in a certain, considerably strong,
fragment of resolution over linear equations, as well as a general polynomial
upper bound on (non-monotone) interpolants in this fragment.
We then apply these results to extend and improve previous results on
multilinear proofs (over fields of characteristic 0), as studied in
[RazTzameret06]. Specifically, we show the following:
1. Proofs operating with depth-3 multilinear formulas polynomially simulate a
certain, considerably strong, fragment of resolution over linear equations.
2. Proofs operating with depth-3 multilinear formulas admit polynomial-size
refutations of the pigeonhole principle and Tseitin graph tautologies. The
former improve over a previous result that established small multilinear proofs
only for the \emph{functional} pigeonhole principle. The latter are different
than previous proofs, and apply to multilinear proofs of Tseitin mod p graph
tautologies over any field of characteristic 0.
We conclude by connecting resolution over linear equations with extensions of
the cutting planes proof system.Comment: 44 page
On computing Belyi maps
We survey methods to compute three-point branched covers of the projective
line, also known as Belyi maps. These methods include a direct approach,
involving the solution of a system of polynomial equations, as well as complex
analytic methods, modular forms methods, and p-adic methods. Along the way, we
pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French
abstract; revised according to referee's suggestion
Hyperdescent and \'etale K-theory
We study the \'etale sheafification of algebraic K-theory, called \'etale
K-theory. Our main results show that \'etale K-theory is very close to a
noncommutative invariant called Selmer K-theory, which is defined at the level
of categories. Consequently, we show that \'etale K-theory has surprisingly
well-behaved properties, integrally and without finiteness assumptions. A key
theoretical ingredient is the distinction, which we investigate in detail,
between sheaves and hypersheaves of spectra on \'etale sites.Comment: 89 pages, v3: various corrections and edit
Naming the largest number: Exploring the boundary between mathematics and the philosophy of mathematics
What is the largest number accessible to the human imagination? The question
is neither entirely mathematical nor entirely philosophical. Mathematical
formulations of the problem fall into two classes: those that fail to fully
capture the spirit of the problem, and those that turn it back into a
philosophical problem
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
A digital computer is generally believed to be an efficient universal
computing device; that is, it is believed able to simulate any physical
computing device with an increase in computation time of at most a polynomial
factor. This may not be true when quantum mechanics is taken into
consideration. This paper considers factoring integers and finding discrete
logarithms, two problems which are generally thought to be hard on a classical
computer and have been used as the basis of several proposed cryptosystems.
Efficient randomized algorithms are given for these two problems on a
hypothetical quantum computer. These algorithms take a number of steps
polynomial in the input size, e.g., the number of digits of the integer to be
factored.Comment: 28 pages, LaTeX. This is an expanded version of a paper that appeared
in the Proceedings of the 35th Annual Symposium on Foundations of Computer
Science, Santa Fe, NM, Nov. 20--22, 1994. Minor revisions made January, 199
Programmability of Chemical Reaction Networks
Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a well-stirred solution according to standard chemical kinetics equations. SCRNs have been widely used for describing naturally occurring (bio)chemical systems, and with the advent of synthetic biology they become a promising language for the design of artificial biochemical circuits. Our interest here is the computational power of SCRNs and how they relate to more conventional models of computation. We survey known connections and give new connections between SCRNs and Boolean Logic Circuits, Vector Addition Systems, Petri Nets, Gate Implementability, Primitive Recursive Functions, Register Machines, Fractran, and Turing Machines. A theme to these investigations is the thin line between decidable and undecidable questions about SCRN behavior
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