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On Fixed Point Equations over Commutative Semirings [chapter]

Javier Esparza, Stefan Kiefer, Michael Luttenberger
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.  ... 
doi:10.1007/978-3-540-70918-3_26 dblp:conf/stacs/EsparzaKL07 fatcat:6q7gte455zarvfyrzjkf76fo6i

A Fully Equational Proof of Parikh's Theorem

Luca Aceto, Zoltán Ésik, Anna Ingólfsdóttir
2002 RAIRO - Theoretical Informatics and Applications  
We show that the validity of Parikh's theorem for contextfree languages depends only 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 ⊕ .  ... 
doi:10.1051/ita:2002007 fatcat:5lnzsfced5c33k5r6fqtsuco3q

A Fully Equational Proof of Parikh's Theorem

Luca Aceto, Zoltán Ésik, Anna Ingólfsdóttir
2001 BRICS Report Series  
We show that the validity of Parikh's theorem for context-free languages depends only 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 ⊕ .  ... 
doi:10.7146/brics.v8i28.21688 fatcat:hzh2q7dgznhqxmqjmqjitnimfe

Multi-linear iterative K-Sigma-semialgebras [article]

Zoltan Esik
2011 arXiv   pre-print
When each finite system of guarded polynomial 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.  ... 
arXiv:1102.3106v2 fatcat:pkdsdhgz6rbzvh3r77far6wcci

Putting Newton into Practice: A Solver for Polynomial Equations over Semirings [chapter]

Maximilian Schlund, Michał Terepeta, Michael Luttenberger
2013 Lecture Notes in Computer Science  
We present the first implementation of Newton's method for solving systems of 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.  ... 
doi:10.1007/978-3-642-45221-5_48 fatcat:w7drmoqflfdejakz2ni64se2ci

Multi-Linear Iterative K-Σ-Semialgebras

Z. Ésik
2011 Electronical Notes in Theoretical Computer Science  
When each finite system of guarded polynomial 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.  ... 
doi:10.1016/j.entcs.2011.09.020 fatcat:x4l4eirubne4tpklllhwkwtuvu

On ∗–λ-semirings

Feng Feng, Young Bae Jun, Xian Zhong Zhao
2007 Information Sciences  
A * -k-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.  ... 
doi:10.1016/j.ins.2007.05.019 fatcat:nkelquwrprbrhptqz6knyvwlvm

Regular Expressions [chapter]

Mark Pilgrim
2009 Dive Into Python 3  
We apply these results to provenance problems: Semi-naive 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  ...  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.  ... 
doi:10.1007/978-1-4302-2416-7_5 fatcat:547ubf4etnesblpwdvrbv2iluu

Newtonian program analysis

Javier Esparza, Stefan Kiefer, Michael Luttenberger
2010 Journal of the ACM  
As a consequence, we also replace greatest 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]  ... 
doi:10.1145/1857914.1857917 fatcat:dl4e6x6qbbg7pp27pqfvb4poka

Solving Fixed-Point Equations by Derivation Tree Analysis [chapter]

Javier Esparza, Michael Luttenberger
2011 Lecture Notes in Computer Science  
Systems of 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.  ... 
doi:10.1007/978-3-642-22944-2_2 fatcat:zgedfzbkzjfu3m5o6etpys7ai4

Free inductive K-semialgebras [article]

Zoltan Esik, Werner Kuich
2011 arXiv   pre-print
We consider rational power series 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.  ... 
arXiv:1009.4820v2 fatcat:hjj3f3vukbhbzhgkj2d4hnhpte

*-μ-semirings and *-λ-semirings

Feng Feng, Xianzhong Zhao, Young Bae Jun
2005 Theoretical Computer Science  
This gives a positive answer to 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  ... 
doi:10.1016/j.tcs.2005.08.010 fatcat:ehwou3fhbvfehdrho6g6ijs3jy

Universal numerical algorithms and their software implementation [article]

Grigori Litvinov, Elena Maslova
2001 arXiv   pre-print
Particular emphasis is placed 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.  ... 
arXiv:math/0102114v1 fatcat:gtb537zj5rfrfavrbipfkj2pgi

Axiomatizing rational power series over natural numbers

S.L. Bloom, Z. Ésik
2009 Information and Computation  
adding a 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.  ... 
doi:10.1016/j.ic.2009.02.003 fatcat:tvqp6ttherh4zep2fotkpsuzou

Continuous Semiring-Semimodule Pairs and Mixed Algebraic Systems

Zoltán Ésik, Werner Kuich
2017 Acta Cybernetica  
⟫, and a function from When S is also an ω-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.  ... 
doi:10.14232/actacyb.23.1.2017.5 fatcat:fxhjg7u7g5adrfvcy4jozooxpa
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