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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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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.
Semilinear Parikh Images of Regular Expressions via Reduction
"... Abstract. A reduction system for regular expressions is presented. For a regular expression t, the reduction system is proved to terminate in a state where the mostreduced expression readily yields a semilinear representation for the Parikh image of the language of t. 1 ..."
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Abstract. A reduction system for regular expressions is presented. For a regular expression t, the reduction system is proved to terminate in a state where the mostreduced expression readily yields a semilinear representation for the Parikh image of the language of t. 1
KarpMiller trees for a branching extension of VASS. Research Report LSV043
 Available at http://www.lsv.enscachan.fr/Publis/RAPPORTS LSV/ PS/rrlsv20043.rr.ps. Kumar Neeraj Verma and Jean GoubaultLarrecq
, 2004
"... transitions that merge two configurations. Runs in BVASS are treelike structures instead of linear ones as for VASS. We show that the construction of KarpMiller trees for VASS can be extended to BVASS. This entails that the coverability set for BVASS is computable. This allows us to obtain decidab ..."
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transitions that merge two configurations. Runs in BVASS are treelike structures instead of linear ones as for VASS. We show that the construction of KarpMiller trees for VASS can be extended to BVASS. This entails that the coverability set for BVASS is computable. This allows us to obtain decidability results for certain classes of equational tree automata with an associativecommutative symbol. Recent independent work by de Groote et al. implies that decidability of reachability in BVASS is equivalent to decidability of provability in MELL (multiplicative exponential linear logic), which is still an open problem. Hence our results are also a step towards answering this question in the affirmative. Keywords: branching vector addition systems, KarpMiller trees, coverability, multiplicative exponential linear logic, equational tree automata. 1
Parikh’s Theorem: A simple and direct automaton construction
"... Parikh’s theorem states that the Parikh image of a contextfree language is semilinear or, equivalently, that every contextfree language has the same Parikh image as some regular language. We present a very simple construction that, given a contextfree grammar, produces a finite automaton recogniz ..."
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Parikh’s theorem states that the Parikh image of a contextfree language is semilinear or, equivalently, that every contextfree language has the same Parikh image as some regular language. We present a very simple construction that, given a contextfree grammar, produces a finite automaton recognizing such a regular language. The Parikh image of a word w over an alphabet {a1,..., an} is the vector (v1,..., vn) ∈ Nn such that vi is the number of occurrences of ai in w. For example, the Parikh image of a1a1a2a2 over the alphabet {a1, a2, a3} is (2, 2, 0). The Parikh image of a language is the set of Parikh images of its words. Parikh images are named after Rohit Parikh, who in 1966 proved a classical theorem of formal language theory which also carries his name. Parikh’s theorem [1] states that the Parikh image of any contextfree language is semilinear. Since semilinear sets coincide with the Parikh images of regular languages, the theorem is equivalent to the statement that every contextfree language has the same Parikh image as some regular language. For instance, the language {anbn  n ≥ 0} has the same Parikh image as (ab)∗. This statement is also often referred to as Parikh’s theorem, see e.g. [10], and in fact it has been considered a more natural formulation [14]. Parikh’s proof of the theorem, as many other subsequent proofs [8, 14, 13, 9, 10, 2], is constructive: given a contextfree grammar G, the proof produces (at least implicitly) an automaton or regular expression whose language has the same Parikh image as L(G). However, these constructions are relatively complicated, not given explicitly, or yield crude upper bounds: automata of size O(nn) for grammars in Chomsky normal form with n variables (see Section 4 for a detailed discussion). In this note we present an explicit and very simple construction yielding an automaton with O(4n) states, for a lower bound of 2n.