A Fully Equational Proof of Parikh's Theorem

Aceto, Luca; Ésik, Zoltán; Ingólfsdóttir, Anna

doi:10.7146/brics.v8i28.21688

Cited by 7 publications

(8 citation statements)

References 12 publications

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“…Moreover, we have proved that the algorithm requires at most n iterations for a system of n equations, a tight bound that improves on the O(3 n ) bound presented in Hopkins and Kozen [1999]. From a theoretical point of view, giving a purely algebraic proof of this fact along the lines of Hopkins and Kozen [1999] and Aceto et al [2001] is an interesting challenge.…”

Section: Newtonian Program Analysis 33:39mentioning

confidence: 99%

Newtonian program analysis

Esparza

Kiefer

Luttenberger

2010

J. ACM

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Abstract. 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 ω-continuous semirings. Complete semilattices, the common program analysis framework, are a special class of ω-continuous 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 fixed-point 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.

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Section: Newtonian Program Analysis 33:39mentioning

confidence: 99%

Newtonian program analysis

Esparza

Kiefer

Luttenberger

2010

J. ACM

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“…In a previous work [9], we proposed to extend that investigation by considering the classical notion of Parikh equivalence [10], which has been extensively studied in the literature (e.g., [1,6]) even for the connections with semilinear sets [7] and with other fields such as Presburger Arithmetics [5], Petri Nets [3], logical formulas [13], and formal verification [12]. We remind the reader that two words over a same alphabet Σ are Parikh equivalent if and only if they are equal up to a permutation of their symbols or, equivalently, for each letter a ∈ Σ, the number of occurrences of a in the two words is the same (the vector ψ(w) consisting of these numbers is also called Parikh image of a word w ∈ Σ * ).…”

Section: Introductionmentioning

confidence: 99%

Operational State Complexity under Parikh Equivalence

Lavado

Pighizzini

Seki

2014

Lecture Notes in Computer Science

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Abstract. We investigate, under Parikh equivalence, the state complexity of some language operations which preserve regularity. For union, concatenation, Kleene star, complement, intersection, shuffle, and reversal, we obtain a polynomial state complexity over any fixed alphabet, in contrast to the intrinsic exponential state complexity of some of these operations in the classical version. For projection we prove a superpolynomial state complexity, which is lower than the exponential one of the corresponding classical operation. We also prove that for each two deterministic automata A and B it is possible to obtain a deterministic automaton with a polynomial number of states whose accepted language has as Parikh image the intersection of the Parikh images of the languages accepted by A and B.

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“…A prominent example of the top-down approach is the equational axioms of Piling [7], which can be used to convert any regular expression t to an equivalent regular expression t , modulo commutativity, such that t is of star-height at most one. See [1] for an excellent overview. These equational axioms however do not immediately lend themselves to a terminating algorithm for computing semi-linear Parikh images.…”

Section: Contributionsmentioning

confidence: 99%

“…the width of produced Parikh images, the equational axioms of [7,1] cannot be directly compared to the inductive method and the reduction system. This is because, in [7,1], depending on the axioms which are chosen for simplification, and their order, the resulting images may have different widths.…”

Section: Contributionsmentioning

confidence: 99%