FPSOLVE: A Generic Solver for Fixpoint Equations Over Semirings

Esparza, Javier; Luttenberger, Michael; Schlund, Maximilian

doi:10.1142/s0129054115400018

Cited by 3 publications

(2 citation statements)

References 14 publications

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“…Newton-Raphson iteration has been extended to Kleene algebras or more generally power series over semi-rings [25,26,37,41] where systems of equations in this setting can represent contextfree grammars, data-flow equations, authorization problems, datalog queries etc. Newton iteration always converges for power series over -continuous semi-rings and if we restrict to idempotent semi-rings, then it was shown to converge after a finite number of steps [25].…”

Section: Newton-raphson Iteration Schemementioning

confidence: 99%

Combining fixpoint and differentiation theory

Galal,

Pacaud Lemay

2024

Proceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science

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Interactions between derivatives and fixpoints have many important applications in both computer science and mathematics. In this paper, we provide a categorical framework to combine fixpoints with derivatives by studying Cartesian differential categories with a fixpoint operator. We introduce an additional axiom relating the derivative of a fixpoint with the fixpoint of the derivative. We show how the standard examples of Cartesian differential categories where we can compute fixpoints provide canonical models of this notion. We also consider when the fixpoint operator is a Conway operator, or when the underlying category is closed. As an application, we show how this framework is a suitable setting to formalize the Newton-Raphson optimization for fast approximation of fixpoints and extend it to higher order languages. CCS CONCEPTS• Theory of computation → Categorical semantics; • Mathematics of computing → Differential calculus.

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Section: Newton-raphson Iteration Schemementioning

confidence: 99%

Combining fixpoint and differentiation theory

Galal,

Pacaud Lemay

2024

Proceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science

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“…We implemented the inequivalence check in an extension 1 of the FPsolve tool [6]. The additional code comprises roughly 1800 lines of C++ and uses libfa 2 to handle finite automata.…”

Section: Implementation and Experimentsmentioning

confidence: 99%

Finite Automata for the Sub- and Superword Closure of CFLs: Descriptional and Computational Complexity

Bachmeier

Luttenberger

Schlund

2015

Lecture Notes in Computer Science

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We answer two open questions by (Gruber, Holzer, Kutrib, 2009) on the state-complexity of representing sub-or superword closures of context-free grammars (CFGs): (1) We prove a (tight) upper bound of 2 O(n) on the size of nondeterministic finite automata (NFAs) representing the subword closure of a CFG of size n. (2) We present a family of CFGs for which the minimal deterministic finite automata representing their subword closure matches the upper-bound of 2 2 O(n) following from (1). Furthermore, we prove that the inequivalence problem for NFAs representing sub-or superword-closed languages is only NP-complete as opposed to PSPACE-complete for general NFAs. Finally, we extend our results into an approximation method to attack inequivalence problems for CFGs.This work was partially funded by the DFG project "Polynomial Systems on Semirings: Foundations, Algorithms, Applications".

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