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14
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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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
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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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 ..."
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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
FPsolve: A Generic Solver for Fixpoint Equations over Semirings
 In Proceedings of CIAA 2014
, 2014
"... Abstract. We introduce FPsolve, an implementation of generic algorithms for solving fixpoint equations over semirings. We first illustrate the interest of generic solvers by means of a scenario. We then succinctly describe some of the algorithms implemented in the tool, and provide some implementat ..."
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Abstract. We introduce FPsolve, an implementation of generic algorithms for solving fixpoint equations over semirings. We first illustrate the interest of generic solvers by means of a scenario. We then succinctly describe some of the algorithms implemented in the tool, and provide some implementation details. 1
Regular Expressions for Provenance
"... As noted by Green et al. several provenance analyses can be considered a special case of the general problem of computing formal polynomials resp. powerseries as solutions of an algebraic system. Specific provenance is then obtained by means of evaluating the formal polynomial under a suitable hom ..."
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As noted by Green et al. several provenance analyses can be considered a special case of the general problem of computing formal polynomials resp. powerseries as solutions of an algebraic system. Specific provenance is then obtained by means of evaluating the formal polynomial under a suitable homomorphism. Recently, we presented the idea of approximating the least solution of such algebraic systems by means of unfolding the system into a sequence of simpler algebraic systems. Similar ideas are at the heart of the seminaive evaluation algorithm for datalog. We apply these results to provenance problems: Seminaive evaluation can be seen as a particular implementation of fixed point iteration which can only be used to compute (finite) provenance polynomials. Other unfolding schemes, e.g. based on Newton’s method, allow us to compute a regular expression which yields a finite representation of (possibly infinite) provenance power series in the case of commutative and idempotent semirings. For specific semirings (e.g. Why(X)) we can then, in a second step, transform these regular expressions resp. power series into polynomials which capture the provenance. Using techniques like subterm sharing both the regular expressions and the polynomials can be succinctly represented.
c © World Scientific Publishing Company FPsolve: A Generic Solver for Fixpoint Equations over Semirings∗
"... Communicated by (xxxxxxxxxx) We introduce FPsolve, an implementation of generic algorithms for solving fixpoint equations over semirings. We first illustrate the interest of generic solvers by means of a scenario. We then succinctly describe some of the algorithms implemented in the tool, and provid ..."
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Communicated by (xxxxxxxxxx) We introduce FPsolve, an implementation of generic algorithms for solving fixpoint equations over semirings. We first illustrate the interest of generic solvers by means of a scenario. We then succinctly describe some of the algorithms implemented in the tool, and provide some implementation details.
Pointsto Analysis as a System of Linear Equations
"... Abstract. We propose a novel formulation of the pointsto analysis as a system of linear equations. With this, the efficiency of the pointsto analysis can be significantly improved by leveraging the advances in solution procedures for solving the systems of linear equations. However, such a formula ..."
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Abstract. We propose a novel formulation of the pointsto analysis as a system of linear equations. With this, the efficiency of the pointsto analysis can be significantly improved by leveraging the advances in solution procedures for solving the systems of linear equations. However, such a formulation is nontrivial and becomes challenging due to various facts, namely, multiple pointer indirections, addressof operators and multiple assignments to the same variable. Further, the problem is exacerbated by the need to keep the transformed equations linear. Despite this, we successfully model all the pointer operations. We propose a novel inclusionbased contextsensitive pointsto analysis algorithm based on prime factorization, which can model all the pointer operations. Experimental evaluation on SPEC 2000 benchmarks and two large open source programs reveals that our approach is competitive to the stateoftheart algorithms. With an average memory requirement of mere 21MB, our contextsensitive pointsto analysis algorithm analyzes each benchmark in 55 seconds on an average. 1