Open AccessBook
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
Citations
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Journal ArticleDOI
The Critic as Artist: Oscar Wilde’s Prolegomena to Shape Grammars
TL;DR: Shape grammars include Wilde's aesthetic (critical) method as discussed by the authors, which can calculate with shapes as in themselves they really are not, and makes this possible with schemas and rules that are "superb in [their] changes and contradictions".
Journal ArticleDOI
Grammatical compression: compressed equivalence and other problems
TL;DR: This work proposes some methods for reducing Compressed Word Problem from an algebra to another and considers inclusion problems for context-free languages, proving how in some cases efficient algorithms for these problems benefit from the ability to work with compressed data.
Book ChapterDOI
Embodying Computation at Higher Types
TL;DR: Since Turing’s early speculations, progress with the building of intelligent machines has been slow, with obstacles tending to be dealt with in ad hoc rather than theoretical ways.
Book ChapterDOI
Query Languages for Constraint Databases: First-Order Logic, Fixed-Points, and Convex Hulls
TL;DR: This paper defines various extensions of first-order logic on linear as well as polynomial constraint databases and shows this logic, FO(conv), to be closed and to have PTIME data-complexity and proves the equivalence between this language and the multiplication part of PFOL.
Proceedings ArticleDOI
Effective Subdivision Algorithm for Isolating Zeros of Real Systems of Equations, with Complexity Analysis
TL;DR: In this paper, a subdivision-based algorithm for isolating the simple zeros of a function within a box B_0 subseteq B_1 subset of a BigFloat number is presented.
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.