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On Fixed Point Equations over Commutative Semirings
[chapter]

*
STACS 2007
*

*Fixed*

*point*

*equations*x = f (x)

*over*ω-continuous

*semirings*can be seen as the mathematical foundation of interprocedural program analysis. ... The sequence 0, f (0), f 2 (0), . . . converges to the least

*fixed*

*point*µf . The convergence can be accelerated if the underlying

*semiring*is

*commutative*. ... A program can be mapped (in a syntax-driven way) to a system of

*fixed*

*point*

*equations*

*over*an abstract

*semiring*containing

*one*

*equation*for each program

*point*. ...

##
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A Fully Equational Proof of Parikh's Theorem

2002
*
RAIRO - Theoretical Informatics and Applications
*

We show that the validity of Parikh's theorem for contextfree languages depends only

doi:10.1051/ita:2002007
fatcat:5lnzsfced5c33k5r6fqtsuco3q
*on*a few*equational*properties of least pre-*fixed**points*. ... Moreover, we exhibit an infinite basis of µ-term*equations*of continuous*commutative*idempotent*semirings*. Mathematics Subject Classification. 03C05, 16Y60, 68Q70. ... of a system of "polynomial*fixed**point**equations*"*over*L Σ ⊕ . ...##
###
A Fully Equational Proof of Parikh's Theorem

2001
*
BRICS Report Series
*

We show that the validity of Parikh's theorem for context-free languages depends only

doi:10.7146/brics.v8i28.21688
fatcat:hzh2q7dgznhqxmqjmqjitnimfe
*on*a few*equational*properties of least pre-*fixed**points*. ... Moreover, we exhibit an infinite basis of mu-term*equations*of continuous*commutative*idempotent*semirings*. ... of a system of "polynomial*fixed**point**equations*"*over*L Σ ⊕ . ...##
###
Multi-linear iterative K-Sigma-semialgebras
[article]

2011
*
arXiv
*
pre-print

When each finite system of guarded polynomial

arXiv:1102.3106v2
fatcat:pkdsdhgz6rbzvh3r77far6wcci
*fixed**point**equations*has a unique solution*over*such an algebra, then we call it an iterative multi-linear K-Σ-semialgebra. ... We show that for many*commutative**semirings*K, the rational Σ-tree series*over*A with coefficients in K form the free multi-linear iterative K-Σ-semialgebra*on*A. ... Proper*commutative**semirings*include N, all locally finite*commutative**semirings*and all*commutative*rings. ...##
###
Putting Newton into Practice: A Solver for Polynomial Equations over Semirings
[chapter]

2013
*
Lecture Notes in Computer Science
*

We present the first implementation of Newton's method for solving systems of

doi:10.1007/978-3-642-45221-5_48
fatcat:w7drmoqflfdejakz2ni64se2ci
*equations**over*ω-continuous*semirings*(based*on*[5, 11] ). ... 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*fixed*-*point*iteration needs ... Thus, Newton's method allows to compute precise solutions of*equation*systems*over*many domains where the standard*fixed*-*point*iteration does not terminate. ...##
###
Multi-Linear Iterative K-Σ-Semialgebras

2011
*
Electronical Notes in Theoretical Computer Science
*

When each finite system of guarded polynomial

doi:10.1016/j.entcs.2011.09.020
fatcat:x4l4eirubne4tpklllhwkwtuvu
*fixed**point**equations*has a unique solution*over*such an algebra, then we call it an iterative multi-linear K-Σ-semialgebra. ... We show that for many*commutative**semirings*K, the rational Σ-tree series*over*A with coefficients in K form the free multi-linear iterative K-Σ-semialgebra*on*A. ... However, it is not iterative, since*fixed**point**equations*may have many solutions. Some constructions Let K denote a*commutative**semiring*. ...##
###
On ∗–λ-semirings

2007
*
Information Sciences
*

A * -k-

doi:10.1016/j.ins.2007.05.019
fatcat:nkelquwrprbrhptqz6knyvwlvm
*semiring*is an ordered*semiring*S equipped with a star operation such that for any a, b 2 S, a * b is the least*fixed**point*of the linear mapping x # ax + b*over*S. ... Some of these results can be seen as generalizations of relevant results*on*inductive * -*semirings*. ... Moreover, context free grammars are indeed systems of*fixed**point**equations**over*the*semiring*of languages. ...##
###
Regular Expressions
[chapter]

2009
*
Dive Into Python 3
*

We apply these results to provenance problems: Semi-naive evaluation can be seen as a particular implementation of

doi:10.1007/978-1-4302-2416-7_5
fatcat:547ubf4etnesblpwdvrbv2iluu
*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*... Complexity For algebraic systems with n variables*over*a*commutative*idempotent*semiring*we have shown that the n+1-st unfolding X <n+1 is already equal to the least*fixed*-*point*. ...##
###
Newtonian program analysis

2010
*
Journal of the ACM
*

As a consequence, we also replace greatest

doi:10.1145/1857914.1857917
fatcat:dl4e6x6qbbg7pp27pqfvb4poka
*fixed**points*by least*fixed**points*, meet-*over*-all-paths by join-*over*-all-paths, etc. ... 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. ... Using Kleene's*fixed*-*point*theorem it is easy to show that systems of*equations**over*ω-continuous*semirings*still have a least*fixed**point*with respect to the partial order ⊑ (see for instance [Kui97] ...##
###
Solving Fixed-Point Equations by Derivation Tree Analysis
[chapter]

2011
*
Lecture Notes in Computer Science
*

Systems of

doi:10.1007/978-3-642-22944-2_2
fatcat:zgedfzbkzjfu3m5o6etpys7ai4
*equations**over*ω-continuous*semirings*can be mapped to context-free grammars in a natural way. ... Acknowledgments Many thanks to Volker Diekert for his help with Theorem 4, to Rupak Majumdar for*pointing*us to applications of*semirings*to the provenance problem in databases [GKT07] , and to Pierre ... In computer science, a prominent example of*fixed*-*point**equations*are dataflow*equations*. ...##
###
Free inductive K-semialgebras
[article]

2011
*
arXiv
*
pre-print

We consider rational power series

arXiv:1009.4820v2
fatcat:hjj3f3vukbhbzhgkj2d4hnhpte
*over*an alphabet Σ with coefficients in a ordered*commutative**semiring*K and characterize them as the free ordered K-semialgebras in various classes of ordered K-semialgebras ... equipped with a star operation satisfying the least pre-*fixed**point*rule and/or its dual. ...*point*identity and*one*or both forms of the least pre-*fixed**point*rule. ...##
###
*-μ-semirings and *-λ-semirings

2005
*
Theoretical Computer Science
*

This gives a positive answer to

doi:10.1016/j.tcs.2005.08.010
fatcat:ehwou3fhbvfehdrho6g6ijs3jy
*one*of Ésik and Kuich's open problems. ... , -*semiring*, * --*semiring*] is a weak inductive * -semirng [ -*semiring*, -*semiring*, * --*semiring*, respectively]. ... An ordered * -*semiring*is called an weak inductive * -*semiring*[4] if it satisfies the*fixed**point**equation*, the sum-star*equation*and the weak*fixed**point*induction rule ax + b = x ⇒ a * b x. (8) In ...##
###
Universal numerical algorithms and their software implementation
[article]

2001
*
arXiv
*
pre-print

Particular emphasis is placed

arXiv:math/0102114v1
fatcat:gtb537zj5rfrfavrbipfkj2pgi
*on*universal algorithms of linear algebra*over**semirings*. ... UNIVERSAL ALGORITHMS AND ACCURACY OF COMPUTATIONS Calculations*on*computers usually are based*on*a floating-*point*arithmetic with a mantissa of a*fixed*length; i.e., computations are performed with*fixed*... R min ; various modifications of the Bellman*equation*, i.e., the basic*equation*of optimization theory, are also linear*over*appropriate idempotent*semirings*. ...##
###
Axiomatizing rational power series over natural numbers

2009
*
Information and Computation
*

adding a

doi:10.1016/j.ic.2009.02.003
fatcat:tvqp6ttherh4zep2fotkpsuzou
*point*at infinity. ... Iteration*semirings*are Conway*semirings*satisfying Conway's group identities. ... The unique*fixed**point*rule is a sentence in the first-order language of *-*semirings*. Because of the extra condition*on*a, the unique*fixed**point*rule is not a quasi-identity. ...##
###
Continuous Semiring-Semimodule Pairs and Mixed Algebraic Systems

2017
*
Acta Cybernetica
*

⟫, and a function from When S is also an ω-

doi:10.14232/actacyb.23.1.2017.5
fatcat:fxhjg7u7g5adrfvcy4jozooxpa
*semiring*(equipped with an infinite product operation), then we define a*fixed**point*operation*over*our category and show that it satisfies all identities of ... We then use this*fixed**point*operation to give semantics to recursion schemes defining series of finite and infinite words. ... It is known that s * is the least solution of the*fixed**point**equation*x = sx + 1 (and also of x = xs + 1)*over*S. ...
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