1 Introduction to the Method.- 2 Importance of QE and CAD Algorithms.- 3 Alternative Approaches.- 4 Practical Issues.- Acknowledgments.- Quantifier Elimination by Cylindrical Algebraic Decomposition - Twenty Years of Progress.- 1 Introduction.- 2 Original Method.- 3 Adjacency and Clustering.- 4 Improved Projection.- 5 Partial CADs.- 6 Interactive Implementation.- 7 Solution Formula Construction.- 8 Equational Constraints.- 9 Subalgorithms.- 10 Future Improvements.- A Decision Method for Elementary Algebra and Geometry.- 1 Introduction.- 2 The System of Elementary Algebra.- 3 Decision Method for Elementary Algebra.- 4 Extensions to Related Systems.- 5 Notes.- 6 Supplementary Notes.- Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition.- 1 Introduction.- 2 Algebraic Foundations.- 3 The Main Algorithm.- 4 Algorithm Analysis.- 5 Observations.- Super-Exponential Complexity of Presburger Arithmetic.- 1 Introduction and Main Theorems.- 2 Algorithms.- 3 Method for Complexity Proofs.- 4 Proof of Theorem 3 (Real Addition).- 5 Proof of Theorem 4 (Lengths of Proofs for Real Addition).- 6 Proof of Theorems 1 and 2 (Presburger Arithmetic).- 7 Other Results.- Cylindrical Algebraic Decomposition I: The Basic Algorithm.- 1 Introduction.- 2 Definition of Cylindrical Algebraic Decomposition.- 3 The Cylindrical Algebraic Decomposition Algorithm: Projection Phase.- 4 The Cylindrical Algebraic Decomposition Algorithm: Base Phase.- 5 The Cylindrical Algebraic Decomposition Algorithm: Extension Phase.- 6 An Example.- Cylindrical Algebraic Decomposition II: An Adjacency Algorithm for the Plane.- 1 Introduction.- 2 Adjacencies in Proper Cylindrical Algebraic Decompositions.- 3 Determination of Section-Section Adjacencies.- 4 Construction of Proper Cylindrical Algebraic Decompositions.- 5 An Example.- An Improvement of the Projection Operator in Cylindrical Algebraic Decomposition.- 1 Introduction.- 2 Idea.- 3 Analysis.- 4 Empirical Results.- Partial Cylindrical Algebraic Decomposition for Quantifier Elimination.- 1 Introduction.- 2 Main Idea.- 3 Partial CAD Construction Algorithm.- 4 Strategy for Cell Choice.- 5 Illustration..- 6 Empirical Results.- 7 Conclusion.- Simple Solution Formula Construction in Cylindrical Algebraic Decomposition Based Quantifier Elimination.- 1 Introduction.- 2 Problem Statement.- 3 (Complex) Solution Formula Construction.- 4 Simplification of Solution Formulas.- 5 Experiments.- Recent Progress on the Complexity of the Decision Problem for the Reals.- 1 Some Terminology.- 2 Some Complexity Highlights.- 3 Discussion of Ideas Behind the Algorithms.- An Improved Projection Operation for Cylindrical Algebraic Decomposition.- 1 Introduction..- 2 Background Material..- 3 Statements of Theorems about Improved Projection Map.- 4 Proof of Theorem 3 (and Lemmas).- 5 Proof of Theorem 4 (and Lemmas).- 6 CAD Construction Using Improved Projection.- 7 Examples.- 8 Appendix.- Algorithms for Polynomial Real Root Isolation.- 1 Introduction.- 2 Preliminary Mathematics.- 3 Algorithms.- 4 Computing Time Analysis.- 5 Empirical Computing Times.- Sturm- Twenty Years of Progress.- 1 Introduction.- 2 Original Method.- 3 Adjacency and Clustering.- 4 Improved Projection.- 5 Partial CADs.- 6 Interactive Implementation.- 7 Solution Formula Construction.- 8 Equational Constraints.- 9 Subalgorithms.- 10 Future Improvements.- A Decision Method for Elementary Algebra and Geometry.- 1 Introduction.- 2 The System of Elementary Algebra.- 3 Decision Method for Elementary Algebra.- 4 Extensions to Related Systems.- 5 Notes.- 6 Supplementary Notes.- Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition.- 1 Introduction.- 2 Algebraic Foundations.- 3 The Main Algorithm.- 4 Algorithm Analysis.- 5 Observations.- Super-Exponential Complexity of Presburger Arithmetic.- 1 Introduction and Main Theorems.- 2 Algorithms.- 3 Method for Complexity Proofs.- 4 Proof of Theorem 3 (Real Addition).- 5 Proof of Theorem 4 (Lengths of Proofs for Real Addition).- 6 Proof of Theorems 1 and 2 (Presburger Arithmetic).- 7 Other Results.- Cylindrical Algebraic Decomposition I: The Basic Algorithm.- 1 Introduction.- 2 Definition of Cylindrical Algebraic Decomposition.- 3 The Cylindrical Algebraic Decomposition Algorithm: Projection Phase.- 4 The Cylindrical Algebraic Decomposition Algorithm: Base Phase.- 5 The Cylindrical Algebraic Decomposition Algorithm: Extension Phase.- 6 An Example.- Cylindrical Algebraic Decomposition II: An Adjacency Algorithm for the Plane.- 1 Introduction.- 2 Adjacencies in Proper Cylindrical Algebraic Decompositions.- 3 Determination of Section-Section Adjacencies.- 4 Construction of Proper Cylindrical Algebraic Decompositions.- 5 An Example.- An Improvement of the Projection Operator in Cylindrical Algebraic Decomposition.- 1 Introduction.- 2 Idea.- 3 Analysis.- 4 Empirical Results.- Partial Cylindrical Algebraic Decomposition for Quantifier Elimination.- 1 Introduction.- 2 Main Idea.- 3 Partial CAD Construction Algorithm.- 4 Strategy for Cell Choice.- 5 Illustration..- 6 Empirical Results.- 7 Conclusion.- Simple Solution Formula Construction in Cylindrical Algebraic Decomposition Based Quantifier Elimination.- 1 Introduction.- 2 Problem Statement.- 3 (Complex) Solution Formula Construction.- 4 Simplification of Solution Formulas.- 5 Experiments.- Recent Progress on the Complexity of the Decision Problem for the Reals.- 1 Some Terminology.- 2 Some Complexity Highlights.- 3 Discussion of Ideas Behind the Algorithms.- An Improved Projection Operation for Cylindrical Algebraic Decomposition.- 1 Introduction..- 2 Background Material..- 3 Statements of Theorems about Improved Projection Map.- 4 Proof of Theorem 3 (and Lemmas).- 5 Proof of Theorem 4 (and Lemmas).- 6 CAD Construction Using Improved Projection.- 7 Examples.- 8 Appendix.- Algorithms for Polynomial Real Root Isolation.- 1 Introduction.- 2 Preliminary Mathematics.- 3 Algorithms.- 4 Computing Time Analysis.- 5 Empirical Computing Times.- Sturm-Habicht Sequences, Determinants and Real Roots of Univariate Polynomials.- 1 Introduction.- 2 Algebraic Properties of Sturm-Habicht Sequences.- 3 Sturm-Habicht Sequences and Real Roots of Polynomial.- 4 Sturm-Habicht Sequences and Hankel Forms.- 5 Applications and Examples.- Characterizations of the Macaulay Matrix and Their Algorithmic Impact.- 1 Introduction.- 2 Notation.- 3 Definitions of the Macaulay Matrix.- 4 Extraneous Factor and First Properties of the Macaulay Determinant.- 5 Characterization of the Macaulay Matrix.- 6 Characterization of the Macaulay Matrix, if It Is Used to Calculate the u-Resultant.- 7 Two Sorts of Homogenization.- 8 Characterization of the Matrix of the Extraneous Factor.- 9 Conclusion.- Computation of Variant Resultants.- 1 Introduction.- 2 Problem Statement.- 3 Review of Determinant Based Method.- 4 Quotient Based Method.- 5 Modular Methods..- 6 Theoretical Computing Time Analysis.- 7 Experiments.- A New Algorithm to Find a Point in Every Cell Defined by a Family of Polynomials.- 1 Introduction.- 2 Proof of the Theorem.- Local Theories and Cylindrical Decomposition.- 1 Introduction.- 2 Infinitesimal Sectors at the Origin.- 3 Neighborhoods of Infinity.- 4 Exponential Polynomials in Two Variables.- A Combinatorial Algorithm Solving Some Quantifier Elimination Problems.- 1 Introduction.- 2 Sturm-Habicht Sequence.- 3 The Algorithms.- 4 Conclusions.- A New Approach to Quantifier Elimination for Real Algebra.- 1 Introduction.- 2 The Quantifier Elimination Problem for the Elementary Theory of the Reals.- 3 Counting Real Zeros Using Quadratic Forms.- 4 Comprehensive Grobner Bases.- 5 Steps of the Quantifier Elimination Method.- 6 Examples.- References.

Quantifier elimination and cylindrical algebraic decomposition

The model-checking problem for a logic L on a class C of structures asks whether a given L-sentence holds in a given structure in C. In this paper, we give super-exponential lower bounds for fixed-parameter tractable model-checking problems for first-order and monadic second-order logic. We show that unless PTIME=NP, the model-checking problem for monadic second-order logic on finite words is not solvable in time f(k)/spl middot/p(n), for any elementary function f and any polynomial p. Here k denotes the size of the input sentence and n the size of the input word. We prove the same result for first-order logic under a stronger complexity theoretic assumption from parameterized complexity theory. Furthermore, we prove that the model-checking problems for first-order logic on structures of degree 2 and of bounded degree d/spl ges/3 are not solvable in time 2(2/sup o(k)/)/spl middot/p(n) (for degree 2) and 2(2/sup 2o(k)/)/spl middot/p(n) (for degree d), for any polynomial p, again under an assumption from parameterized complexity theory. We match these lower bounds by corresponding upper bounds.

/pdf/the-complexity-of-first-order-and-monadic-second-order-logic-1hzvlyhaae.pdf

The complexity of first-order and monadic second-order logic revisited

https://www.cs.ox.ac.uk/files/295/dtlf.pdf

Domain theory in logical form

The mathematical framework of Stone duality is used to synthesise a number of hitherto separate developments in theoretical computer science. Domain theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. The theory of concurrency and systems behavior developed by Milner, Hennesy et al. based on operational semantics. Logics of programs

Two of the most influential works on C-algebras from the mid-seventies ‐ Brown, Douglas and Fillmore’s [6] and Elliott’s [21] ‐ both contain uniqueness and existence results in the now standard sense which we shall outline below These papers served as keystones for two separate theories ‐ KK-theory and the classification program ‐ which for many years parted ways with only moderate interaction But with this common origin in mind, it is not surprising that recent years have seen a fruitful interaction which has been one of the main engines behind rapid progress in the classification program In the present paper we take this interaction even further Combining a concept of quasidiagonality for representations and a new characterization of equivalence in the Cuntz picture of KK-theory, we achieve a general uniqueness result And from the quasidiagonality notion we derive a corresponding general existence result by comparing well-known realizations of the KK-groups These results are then employed to obtain new classification results for certain classes of quasidiagonal C-algebras introduced by H Lin An important novel feature of these classes is that they are defined by a certain local approximation property, rather than by an inductive limit construction We emphasize that our fundamental uniqueness result does not depend on the universal coefficient theorem (UCT), nor on nuclearity of the C-algebras involved On the other hand, when we refine the uniqueness result for use in classification, we need to gradually add such conditions on the C-algebras It is to be expected that requirements of this type are necessary, but we have found that holding them back as long as possible leads to more conceptual proofs while clarifying the role of the UCT and nuclearity in classification Further, since uniqueness results answer fundamental questions about KK- and K-theory, they are of interest in themselves We have pursued this theme in [14], and defer to this paper the proof of the new characterization of KK-theory which lies behind our uniqueness results Our existence and uniqueness results are in the spirit of the classical Ext-theory from [6] The main complication overcome in the paper is to control the stabilization which is necessary when one works with finite C-algebras In the infinite case, where programs of this type have already been successfully carried out, stabilization is unnecessary Yet, our methods are sufficiently versatile to allow us to reprove, from a handful of basic results, the classification of purely infinite nuclear C-algebras of Kirchberg and Phillips

/pdf/on-the-classification-of-nuclear-c-algebras-im3od29rdk.pdf

On the classification of nuclear C*-algebras

Let f be a unitary polynomial over F. Previously, the concept of a separant of a polynomial f was defined for the case where f has no multiple roots. The notion of a separant turned out to be very useful for generalizations of Hensel’s lemma. We propose a generalization of this concept to the case where a polynomial may have multiple roots. This allows us to extend Hensel’s lemma to this case as well.

Separant of an Arbitrary Polynomial

The d-Rank of an α-Space Does Not Exceed 1

EACH FAMILY OF SUBSETS OF PRAELEMENTS GENERATES AN ADMISSIBLE SET* Yu. L. Ershov UDC 510.5 The aim of the present article is to establish the result formulated in the title~ It corroborates a conjecture put forward by the author in [I]. We assume an acquaintance with this article which contains the precise definitions and motivation of the problems. Set-theoretic techniques which can also be used to prove [i, Theorem i] more simply lie at the basis of the proof. Let A0 and A~ be two admissible sets. We will say that AI is an Z-extension of Ao (A0 ~AI) if AI is an end extension of Ao (A0 ~dAl)(see [2]) and for each Z-formula (x 0 ..... x n) and arbitrary a0 .... , annA0 it follows from A!~(a0, ..., a~)that A0~

Each family of subsets of praelements generates an admissible set

We are concerned with a class of valued fields, called stable. We propound an extension of a notion in the monograph by S. Bosch, U. Guntzer, and R. Remmert (Non-Archimedean Analysis. A Systematic Approach to Rigid Analytic Geometry, Springer, Berlin (1984)), namely, that of a (ultrametric) norm on groups, rings, algebras, and vector spaces, to the case where the value of the norm is taken from an arbitrary (not necessarily Archimedean) linearly ordered Abelian group (using — as in the general theory of valuations — the version of a logarithmic norm). Our main result extends Proposition 6 in the cited monograph to the general case, thereby making it possible to use the technique of Cartesian spaces to deliver further results on stable valued fields.

Stable valued fields

http://iopscience.iop.org/article/10.1070/RM2013v068n04ABEH004853/pdf

Vladimir Petrovich Shunkov (obituary)

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Yu. L. Ershov

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