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Complexity and Real Computation
TLDR
This chapter discusses decision problems and Complexity over a Ring and the Fundamental Theorem of Algebra: Complexity Aspects.Abstract:
1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem "P ? NP?".- 8 Newton's Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bezout's Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.read more
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Toward accurate polynomial evaluation in rounded arithmetic
TL;DR: The ultimate goal is to establish a decision procedure that, for any p and D, either exhibits an accurate algorithm or proves that none exists, and it is shown that some polynomials p are impossible to evaluate accurately.
Computations via experiments with kinematic systems 1
Edwin J. Beggs,John V. Tucker +1 more
TL;DR: It is argued that mechanics must be extended to include a formal theory for performing experiments, which includes the construction of systems, and conjecture that in such an extended mechanics the functions computed by experiments are precisely those computed by algorithms.
Posted Content
A Stochastic Complexity Perspective of Induction in Economics and Inference in Dynamics
TL;DR: A recasting of Jevon's sceptical vision of induction in the light of MDL; and a complexity interpretation of an undecidable question in dynamics are suggested.
Journal ArticleDOI
Natural computation measured as a reduction of complexity
Ruedi Stoop,Norbert Stoop +1 more
TL;DR: It is argued that the deeper nature of computation is to reduce the statistical obstruction against prediction, and an explicit measure of computation for general, artificial as well as natural, systems is derived.
Journal ArticleDOI
Computational Complexity of Quantum Satisfiability
TL;DR: The weak and strong satisfiability problem for quantum logic formulas is introduced, and both NP-complete in dimension two as well and the problem to be complete for the nondeterministic Blum-Shub-Smale model of real computation is shown.
References
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Book
Principles of Algebraic Geometry
Phillip Griffiths,Joe Harris +1 more
TL;DR: In this paper, a comprehensive, self-contained treatment of complex manifold theory is presented, focusing on results applicable to projective varieties, and including discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex.
Book
The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Proceedings ArticleDOI
The complexity of theorem-proving procedures
TL;DR: It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is a tautology.