761465
New results on quantifier elimination over real closed fields and applications to constraint databases
In this paper, 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 [Basu et al. 1996; Renegar 1992; Heintz et al. 1990], the combinatorial part of the complexity (the part depending on the number of polynomials in the input) of this new algorithm is independent of the number of free variables. Moreover, under the assumption that each polynomial in the input depends only on a constant number of the free variables, the algebraic part of the complexity(the part depending on the degrees of the input polynomials) can also be made independent of the number of free variables. This new feature of our algorithm allow 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.
Using tthe uniform quantifier elimination algorithm, we give an algorithm for solving a problem arising in the field of constraint databases with real polynomial constraints. We give an algorithm for converting a query with natural domain semantics to an equivalent one with active domain semantics. A nonconstructive version of this result was proved in Benedikt et al. [1998]. Very recently, a constructive proof was also given independently in Benedikt and Libkin [1997]. However, complexity issues were not considered and no algorithm with a 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. This leads to simpler and constructive proofs for these inexpressibility results than the ones known before.
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.
8143891
Different bounds on the different Betti numbers of semi-algebraic sets
A classic result in real algebraic geometry due to Oleinik-Petrovsky, Thom and Milnor, bounds the {\em 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.
In this paper we prove separate bounds on the different Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic hypersurfaces. These are the first results in this direction.
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. >
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.
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.
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.
37245904
Model checking the Java meta-locking algorithm
We apply the XMC model checker to the Java meta-locking algorithm, a highly optimized technique for ensuring mutually exclusive access by threads to object monitor queues. Our abstract specification of the meta-locking algorithm is fully parameterized, both on M, the number of threads, and N, the number of objects. Using XMC, we show that for a variety of values of M and N, the algorithm indeed provides mutual exclusion and freedom from lockout.
2629561
Compositional Analysis for Verification of Parameterized Systems
Many safety-critical systems that have been considered by the verification community are parameterized by the number of concurrent components in the system, and hence describe an infinite family of systems. Traditional model checking techniques can only be used to verify specific instances of this family. In this paper, we present a technique based on compositional model checking and program analysis for automatic verification of infinite families of systems. The technique views a parameterized system as an expression in a process algebra (CCS) and interprets this expression over a domain of formulas (modal mu-calculus), considering a process as a property transformer. The transformers are constructed using partial model checking techniques. At its core, our technique solves the verification problem by finding the limit of a chain of formulas. We present a widening operation to find such a limit for properties expressible in a subset of modal mu-calculus. We describe the verification of a number of parameterized systems using our technique to demonstrate its utility.
17912290
An environment for enabling interactive grids
Traditional use of grid computing allows a user to submit batch jobs in a grid environment. We believe, next generation grids will extend the application domain to include interactive graphical sessions. We term such grids interactive grids. In this paper, we describe some of the challenges involved in building interactive grids. These include fine grain access control, QoS guarantees, and dynamic account management. In order to architect interactive grids, we propose and describe I-GENV, an environment for enabling interactive grids. I-GENV consists of GISH-'Grid Interactive Shell', Controlled Desktop, SAC-'Session Admission Control' module, GMMA-'Grid Monitoring and Management Agents', System Policies, and Dynamic Account Manager. We also present our testbed implementation of I-GENV using and extending Globus Toolkit 2.0 for the Grid middleware infrastructure, and VNC as the remote display technology.
8152994
Localizing Program Errors for Cimple Debugging
We present automated techniques for the explanation of counter-examples, where a counter-example should be understood as a sequence of program statements. Our approach is based on variable dependency analysis and is applicable to programs written in Cimple, an expressive subset of the C programming language. Central to our approach is the derivation of a focus-statement sequence (FSS) from a given counter-example: a subsequence of the counter-example containing only those program statements that directly or indirectly affect the variable valuation leading to the program error behind the counter-example. We develop a ranking procedure for FSSs where FSSs of higher rank are conceptually easier to understand and correct than those of lower rank. We also analyze constraints over uninitialized variables in order to localize program errors to specific program segments; this often allows the user to subsequently take appropriate debugging measures. We have implemented our techniques in the FocusCheck model checker, which efficiently checks for assertion violations in Cimple programs on a per-procedure basis. The practical utility of our approach is illustrated by its successful application to a fast, linear-time median identification algorithm commonly used in statistical analysis and in the Resolution Advisory module of the Traffic Collision Avoidance System.