Results 1  10
of
18
On the convergence of Newton’s method for monotone systems of polynomial equations
 In Proceedings of STOC
, 2007
"... kiefersn, luttenml, esparza o ..."
Newtonian Program Analysis
, 2010
"... This article presents a novel generic technique for solving dataflow equations in interprocedural dataflow analysis. The technique is obtained by generalizing Newton’s method for computing a zero of a differentiable function to ωcontinuous semirings. Complete semilattices, the common program analy ..."
Abstract

Cited by 15 (5 self)
 Add to MetaCart
This article presents a novel generic technique for solving dataflow equations in interprocedural dataflow analysis. The technique is obtained by generalizing Newton’s method for computing a zero of a differentiable function to ωcontinuous semirings. Complete semilattices, the common program analysis framework, are a special class of ωcontinuous semirings. We show that our generalized method always converges to the solution, and requires at most as many iterations as current methods based on Kleene’s fixedpoint theorem. We also show that, contrary to Kleene’s method, Newton’s method always terminates for arbitrary idempotent and commutative semirings. More precisely, in the latter setting the number of iterations required to solve a system of n equations is at most n.
Bounded Underapproximations
"... We show a new and constructive proof of the following languagetheoretic result: for every contextfree language L, there is a bounded contextfree language L ′ ⊆ L which has the same Parikh (commutative) image as L. Bounded languages, introduced by Ginsburg and Spanier, are subsets of regular lang ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
(Show Context)
We show a new and constructive proof of the following languagetheoretic result: for every contextfree language L, there is a bounded contextfree language L ′ ⊆ L which has the same Parikh (commutative) image as L. Bounded languages, introduced by Ginsburg and Spanier, are subsets of regular languages of the form w ∗ 1w ∗ 2 · · · w ∗ m for some w1,..., wm ∈ Σ ∗. In particular bounded contextfree languages have nice structural and decidability properties. Our proof proceeds in two parts. First, we give a new construction that shows that each context free language L has a subset LN that has the same Parikh image as L and that can be represented as a sequence of substitutions on a linear language. Second, we inductively construct a Parikhequivalent bounded contextfree subset of LN. We show two applications of this result in model checking: to underapproximate the reachable state space of multithreaded procedural programs and to underapproximate the reachable state space of recursive counter programs. The bounded language constructed above provides a decidable underapproximation for the original
Newton’s method for ωcontinuous semirings
, 2008
"... Fixed point equations X = f(X) over ωcontinuous semirings are a natural mathematical foundation of interprocedural program analysis. Generic algorithms for solving these equations are based on Kleene’s theorem, which states that the sequence 0, f(0), f(f(0)),... converges to the least fixed point. ..."
Abstract

Cited by 12 (6 self)
 Add to MetaCart
Fixed point equations X = f(X) over ωcontinuous semirings are a natural mathematical foundation of interprocedural program analysis. Generic algorithms for solving these equations are based on Kleene’s theorem, which states that the sequence 0, f(0), f(f(0)),... converges to the least fixed point. However, this approach is often inefficient. We report on recent work in which we extend Newton’s method, the wellknown technique from numerical mathematics, to arbitrary ωcontinuous semirings, and analyze its convergence speed in the real semiring.
Analyzing Asynchronous Programs with Preemption
"... ABSTRACT. Multiset pushdown systems have been introduced by Sen and Viswanathan as an adequate model for asynchronous programs where some procedure calls can be stored as tasks to be processed later. The model is a pushdown system supplied with a multiset of pending tasks. Tasks may be added to the ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
ABSTRACT. Multiset pushdown systems have been introduced by Sen and Viswanathan as an adequate model for asynchronous programs where some procedure calls can be stored as tasks to be processed later. The model is a pushdown system supplied with a multiset of pending tasks. Tasks may be added to the multiset at each transition, whereas a task is taken from the multiset only when the stack is empty. In this paper, we consider an extension of these models where tasks may be of different priority level, and can be preempted at any point of their execution by tasks of higher priority. We investigate the control point reachability problem for these models. Our main result is that this problem is decidable by reduction to the reachability problem for a decidable class of Petri nets with inhibitor arcs. We also identify two subclasses of these models for which the control point reachability problem is reducible respectively to the reachability problem and to the coverability problem for Petri nets (without inhibitor arcs). 1
Solving FixedPoint Equations by Derivation Tree Analysis ⋆
"... Abstract. Systems of equations over ωcontinuous semirings can be mapped to contextfree grammars in a natural way. We show how an analysis of the derivation trees of the grammar yields new algorithms for approximating and even computing exactly the least solution of the system. 1 ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Abstract. Systems of equations over ωcontinuous semirings can be mapped to contextfree grammars in a natural way. We show how an analysis of the derivation trees of the grammar yields new algorithms for approximating and even computing exactly the least solution of the system. 1
Convergence of Newton’s Method over Commutative Semirings ⋆
"... Abstract. We give a lower bound on the speed at which Newton’s method (as defined in [5, 6]) converges over arbitrary ωcontinuous commutative semirings. From this result, we deduce that Newton’s method converges within a finite number of iterations over any semiring which is “collapsed at some k ∈ ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We give a lower bound on the speed at which Newton’s method (as defined in [5, 6]) converges over arbitrary ωcontinuous commutative semirings. From this result, we deduce that Newton’s method converges within a finite number of iterations over any semiring which is “collapsed at some k ∈ N ” (i.e. k = k + 1 holds) in the sense of [1]. We apply these results to (1) obtain a generalization of Parikh’s theorem, (2) to compute the provenance of Datalog queries, and (3) to analyze weighted pushdown systems. We further show how to compute Newton’s method over any ωcontinuous semiring. 1
Putting Newton into Practice: A Solver for Polynomial Equations over Semirings ⋆
"... Abstract. We present the first implementation of Newton’s method for solving systems of equations over ωcontinuous semirings (based on [5,11]). For instance, such equation systems arise naturally in the analysis of interprocedural programs or the provenance computation for Datalog. Our implementati ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We present the first implementation of Newton’s method for solving systems of equations over ωcontinuous semirings (based on [5,11]). For instance, such equation systems arise naturally in the analysis of interprocedural programs or the provenance computation for Datalog. Our implementation provides an attractive alternative for computing their exact least solution in some cases where the ascending chain condition is not met and hence, standard fixedpoint iteration needs to be combined with some overapproximation (e.g., widening techniques) to terminate. We present a generic C++ library along with the main algorithms and analyze their complexity. Furthermore, we describe our implementation of the counting semiring based on semilinear sets. Finally, we discuss motivating examples as well as performance benchmarks. 1
Interprocedural Dataflow Analysis over Weight Domains with Infinite Descending Chains
 in "Proceedings of the 12th International Conference on Foundations of Software Science and Computational Structures
"... Abstract. We study generalized fixedpoint equations over idempotent semirings and provide an efficient algorithm for the detection whether a sequence of Kleene’s iterations stabilizes after a finite number of steps. Previously known approaches considered only bounded semirings where there are no in ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We study generalized fixedpoint equations over idempotent semirings and provide an efficient algorithm for the detection whether a sequence of Kleene’s iterations stabilizes after a finite number of steps. Previously known approaches considered only bounded semirings where there are no infinite descending chains. The main novelty of our work is that we deal with semirings without the boundedness restriction. Our study is motivated by several applications from interprocedural dataflow analysis. We demonstrate how the reachability problem for weighted pushdown automata can be reduced to solving equations in the framework mentioned above and we describe a few applications to demonstrate its usability. 1
Derivation Tree Analysis for Accelerated FixedPoint Computation
"... Abstract. We show that for several classes of idempotent semirings the least fixedpoint of a polynomial system of equations X = f(X) is equal to the least fixedpoint of a linear system obtained by “linearizing ” the polynomials of f in a certain way. Our proofs rely on derivation tree analysis, a ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We show that for several classes of idempotent semirings the least fixedpoint of a polynomial system of equations X = f(X) is equal to the least fixedpoint of a linear system obtained by “linearizing ” the polynomials of f in a certain way. Our proofs rely on derivation tree analysis, a proof principle that combines methods from algebra, calculus, and formal language theory, and was first used in [5] to show that Newton’s method over commutative and idempotent semirings converges in a linear number of steps. Our results lead to efficient generic algorithms for computing the least fixedpoint. We use these algorithms to derive several consequences, including an O(N 3) algorithm for computing the throughput of a contextfree grammar (obtained by speeding up the O(N 4) algorithm of [2]), and a generalization of Courcelle’s result stating that the downwardclosed image of a contextfree language is regular [3]. 1