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11
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 ..."
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Cited by 15 (5 self)
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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 ..."
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Cited by 13 (2 self)
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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
Approximating Petri net reachability along contextfree traces
 In FSTTCS, volume 13 of LIPIcs
, 2011
"... ABSTRACT. We investigate the problem asking whether the intersection of a contextfree language (CFL) and a Petri net language (PNL) is empty. Our contribution to solve this longstanding problem which relates, for instance, to the reachability analysis of recursive programs over unbounded data doma ..."
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Cited by 6 (2 self)
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ABSTRACT. We investigate the problem asking whether the intersection of a contextfree language (CFL) and a Petri net language (PNL) is empty. Our contribution to solve this longstanding problem which relates, for instance, to the reachability analysis of recursive programs over unbounded data domain, is to identify a class of CFLs called the finiteindex CFLs for which the problem is decidable. The kindex approximation of a CFL can be obtained by discarding all the words that cannot be derived within a budget k on the number of occurrences of nonterminals. A finiteindex CFL is thus a CFL which coincides with its kindex approximation for some k. We decide whether the intersection of a finiteindex CFL and a PNL is empty by reducing it to the reachability problem of Petri nets with weak inhibitor arcs, a class of systems with infinitely many states for which reachability is known to be decidable. Conversely, we show that the reachability problem for a Petri net with weak inhibitor arcs reduces to the emptiness problem of a finiteindex CFL intersected with a PNL. 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 ..."
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Cited by 4 (2 self)
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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 ∈ ..."
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Cited by 4 (3 self)
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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
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 ..."
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Cited by 3 (1 self)
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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 ..."
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Cited by 3 (3 self)
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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
Spaceefficient scheduling of stochastically generated tasks
"... We study the problem of scheduling tasks for execution by a processor when the tasks can stochastically generate new tasks. Tasks can be of different types, and each type has a fixed, known probability of generating other tasks. We present results on the random variable S σ modeling the maximal sp ..."
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Cited by 1 (1 self)
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We study the problem of scheduling tasks for execution by a processor when the tasks can stochastically generate new tasks. Tasks can be of different types, and each type has a fixed, known probability of generating other tasks. We present results on the random variable S σ modeling the maximal space needed by the processor to store the currently active tasks when acting under the scheduler σ. We obtain tail bounds for the distribution of S σ for both offline and online schedulers, and investigate the expected value E[S σ].
Weighted Dynamic Pushdown Networks
, 2010
"... We develop a generic framework for the analysis of programs with recursive procedures and dynamic process creation. To this end we combine the approach of weighted pushdown systems (WPDS) with the model of dynamic pushdown networks (DPN). The resulting model, weighted dynamic pushdown networks (WDPN ..."
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Cited by 1 (0 self)
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We develop a generic framework for the analysis of programs with recursive procedures and dynamic process creation. To this end we combine the approach of weighted pushdown systems (WPDS) with the model of dynamic pushdown networks (DPN). The resulting model, weighted dynamic pushdown networks (WDPN), describes processes running in parallel, each of them being able to perform pushdown actions, that may spawn new processes as a side effect. As with WPDS, transitions are labelled by weights to carry additional information. Starting from techniques for WPDS and DPN, we derive a method to determine meetoverallpaths values for the paths between regular sets of configurations of a WDPN. Using this method we are able to solve basic dataflow analysis problems in a parallel context.
Fast and Accurate Unlexicalized Parsing via Structural Annotations
"... We suggest a new annotation scheme for unlexicalized PCFGs that is inspired by formal language theory and only depends on the structure of the parse trees. We evaluate this scheme on the TüBaD/Z treebank w.r.t. several metrics and show that it improves both parsing accuracy and parsing speed consi ..."
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We suggest a new annotation scheme for unlexicalized PCFGs that is inspired by formal language theory and only depends on the structure of the parse trees. We evaluate this scheme on the TüBaD/Z treebank w.r.t. several metrics and show that it improves both parsing accuracy and parsing speed considerably. We also show that our strategy can be fruitfully combined with known ones like parent annotation to achieve accuracies of over 90 % labeled F1 and leafancestor score. Despite increasing the size of the grammar, our annotation allows for parsing more than twice as fast as the PCFG baseline. 1