42203358
Different Bounds on the Different Betti Numbers of Semi-Algebraic Sets 


Abstract. A classic result in real algebraic geometry due to Oleinik—Petrovskii, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers. We prove better bounds on the individual Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic hypersurfaces. As a corollary we obtain a polynomial bound on the highest Betti numbers of basic semi-algebraic sets defined by quadratic inequalities.
8618090
Computing the first Betti number and the connected components of semi-algebraic sets
In this paper we describe the first singly exponential algorithm for computing the first Betti number of a given semi-algebraic set. We also describe algorithms for obtaining semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti number, and the Euler-Poincaré characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti numbers other than the zero-th one.
18808745
An improved algorithm for quantifier elimination over real closed fields
We give a new algorithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms the combinatorial part of the complexity of this new algorithm is independent of the number of free variables. Moreover, under the assumption that each polynomial in the input depend only on a constant number of the free variables, the algebraic part of the complexity can also be made independent of the number of free variables. This new feature of our algorithm allows us to obtain a new algorithm for a variant of the quantifier elimination problem. We give an almost optimal algorithm for this new problem, which we call the uniform quantifier elimination problem and apply it to solve a problem arising in the field of constraint databases. No algorithm with reasonable complexity bound was known for this latter problem till now. We also point out interesting logical consequences of this algorithmic result, concerning the expressive power of a constraint query language over the reals. Moreover, our improved algorithm for performing quantifier elimination immediately leads to improved algorithms for several problems for which quantifier elimination is a basic step, for example, the problem of computing the closure of a given semi-algebraic set.
24764914
Computing the Betti numbers of arrangements via spectral sequences
In this paper, we consider the problem of computing the Betti numbers of an arrangement of n compact semi-algebraic sets, S1,...,Sn ⊂ Rk, where each Si is described using a constant number of polynomials with degrees bounded by a constant. Such arrangements are ubiquitous in computational geometry. We give an algorithm for computing lth Betti number, β(∪i=1n Si), 0 ≤ l ≤ k - 1 using O(nl+2) algebraic operations. Additionally, one has to perform linear algebra on integer matrices of size bounded by O(nl+2). All previous algorithms for computing the Betti numbers of arrangements, triangulated the whole arrangement giving rise to a complex of size O(n2k) in the worst case. Thus, the complexity of computing the Betti numbers (other than the zeroth one) for these algorithms was O(n2k). To our knowledge this is the first algorithm for computing βl(∪i=1n Si) that does not rely on such a global triangulation, and has a graded complexity which depends on l.
33624885
On the combinatorial and topological complexity of a single cell
The problem of bounding the combinatorial complexity of a single connected component (a single cell) of the complement of a set of a geometric objects in R/sup k/, each object of constant description complexity, is an important problem in computational geometry which has attracted much attention over the past decade. It has been conjectured that the combinatorial complexity of a single cell is bounded by a function much closer to O(n/sup k-1/) rather than O(n/sup k/) which is the bound for the combinatorial complexity of the whole arrangement. Till now, this was known to be rule only for k/spl les/3 and only for some special cases in higher dimensions. A classic result in real algebraic geometry due to Oleinik-Petrovsky, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. However, till now no better bounds were known if we restricted attention to a single connected component of a basic semi-algebraic set. In this paper, we show how these two problems are related. We prove a new bound on the sum of the Betti numbers of one connected component of a basic semi-algebraic set which is an improvement over the Oleinik-Petrovsky-Thom-Milnor bound. This also implies that the topological complexity of a single cell, measured by the sum of the Betti numbers, is bounded by O(n/sup k-1/).
32812813
Design of CAECC-Cellular Automata Based Error Correcting Code
A new scheme for designing error detecting and error correcting codes around cellular automata (CA) is reported. A simple and efficient scheme for generating SEC-DED codes is presented which can also be extended for generating codes with higher distances. A CA-based hardware scheme for very fast decoding (and correcting) of the codewords is also reported. >
15323130
MoSCoE: A Framework for Modeling Web Service Composition and Execution
Development of sound approaches and software tools for specification, assembly, and deployment of composite Web services from independently developed components promises to enhance collaborative software design and reuse. In this context, the proposed research introduces a new incremental approach to service composition, MoSCoE (Modeling Web Service Composition and Execution), based on the three steps of abstraction, composition and refinement. Abstraction refers to the high-level description of the service desired (goal) by the user, which drives the identification of an appropriate composition strategy. In the event that such a composition is not realizable, MoSCoE guides the user through successive refinements of the specification towards a realizable goal service that meets the user requirements.
12323199
On Context-Specific Substitutability of Web Services
Web service substitution refers to the problem of identifying a service that can replace another service in the context of a composition with a specified functionality. Existing solutions to this problem rely on detecting the functional and behavioral equivalence of a particular service to be replaced and candidate services that could replace it. We introduce the notion of context-specific substitutability, where context refers to the overall functionality of the composition that is required to be maintained after replacement of its constituents. Using the context information, we investigate two variants of the substitution problem, namely environment-independent and environment- dependent, where environment refers to the constituents of a composition and show how the substitutability criteria can be relaxed within this model. We provide a logical formulation of the resulting criteria based on model checking techniques as well as prove the soundness and completeness of the proposed approach.
8225775
A global lower bound on parameter estimation error with periodic distortion functions
We present a global Ziv-Zakai (1969) type lower bound on the mean square error for estimation of signal parameter vectors, where some components of the distortion function may be periodic. Periodic distortion functions arise naturally in the context of direction of arrival or phase estimation problems. The bound is applied to an image registration problem, and compared to the performance of the maximum-likelihood estimator (MLE).
121785
Generation of All Counter-Examples for Push-Down Systems
We present a new, on-the-fly algorithm that given a push-down model representing a sequential program with (recursive) procedure calls and an extended finite-state automaton representing (the negation of) a safety property, produces a succinct, symbolic representation of all counter-examples; i.e., traces of system behaviors that violate the property. The class of what we call minimum-recursion loop-free counter-examples can then be generated from this representation on an as-needed basis and presented to the user. Our algorithm is also applicable, without modification, to finite-state system models. Simultaneous consideration of multiple counter-examples can minimize the number of model-checking runs needed to recognize common root causes of property violations. We illustrate the use of our techniques via application to a Java-Tar utility and an FTP-server program, and discuss a prototype tool implementation which offers several abstraction techniques for easy-viewing of generated counter-examples.
9852130
Parallel Web Service Composition in MoSCoE: A Choreography-Based Approach
We present a goal-driven approach to model a choreographer for realizing composite Web services. In this framework, the users start with an abstract, and possibly incomplete functional specification of a desired goal service. This specification is used to compose a choreographer that allows communication between the client and the set of available component services, and is functionally equivalent to the goal service. However, if such a composition cannot be realized, the proposed approach identifies the cause(s) for the failure of composition. This information can be used by the user to minimally reformulate the goal to reduce the `gap' between the desired functionality. The process can be iterated until a feasible composition is realized or the user decides to abort. The approach ensures that (i) a choreographer, if one is produced by our composition algorithm, in fact realizes the user-specified goal functionality; and (ii) the algorithm is guaranteed to find a composition that meets the user needs as captured in the goal specifications (whenever such a composition exists)