Fourteenth Annual IEEE Symposium on

Logic in Computer Science (LICS 1999)

Paper: Parikh's Theorem in Commutative Kleene Algebra (at LICS 1999)

Authors: Mark W. Hopkins Dexter C. Kozen


Parikh's Theorem says that the commutative image of every context free language is the commutative image of some regular set. Pilling has shown that this theorem is essentially a statement about least solutions of polynomial inequalities. We prove the following general theorem of commutative Kleene algebra, of which Parikh's theorem is a special case: Every system of polynomial inequalities fi(x1,...,xn) \mathxi, 1 \mathi \math, over a commutative Kleene algebra K has a unique least solution in Kn; moreover, the components of the solution are given by polynomials in the coefficients of the fi. We also give a closed-form solution in terms of the Jacobian of the system.


    author = 	 {Mark W. Hopkins and Dexter C. Kozen},
    title = 	 {Parikh's Theorem in Commutative Kleene Algebra},
    booktitle =  {Proceedings of the Fourteenth Annual IEEE Symp. on Logic in Computer Science, {LICS} 1999},
    year =	 1999,
    editor =	 {Giuseppe Longo},
    month =	 {July}, 
    pages =      {394--401},
    location =   {Trento, Italy}, 
    publisher =	 {IEEE Computer Society Press}