TY - JOUR

T1 - Coloring rules for finite trees, and probabilities of monadic second order sentences

AU - Woods, Alan

PY - 1997

Y1 - 1997

N2 - A system of coloring rules is a set of rules for coloring the vertices of any finite rooted tree, starting at its leaves, which has the property that the color assigned to each vertex depends only on how many of its immediate predecessors there are of each color. The asymptotic behavior of the fraction of n vertex tries with a given root color is investigated. As a primary application, if phi is any monadic second-order sentence, then the fractions mu(n)(phi), nu(n)(phi) of labeled and unlabeled rooted trees satisfying phi, converge to limiting probabilities. An extension of the idea shows that for a particular model of Boolean formulas in k variables, k fixed, the fraction of such formulas of size n which compute a given Boolean function approaches a positive limit as n --> infinity. (C) 1997 John Wiley & Sons, Inc.

AB - A system of coloring rules is a set of rules for coloring the vertices of any finite rooted tree, starting at its leaves, which has the property that the color assigned to each vertex depends only on how many of its immediate predecessors there are of each color. The asymptotic behavior of the fraction of n vertex tries with a given root color is investigated. As a primary application, if phi is any monadic second-order sentence, then the fractions mu(n)(phi), nu(n)(phi) of labeled and unlabeled rooted trees satisfying phi, converge to limiting probabilities. An extension of the idea shows that for a particular model of Boolean formulas in k variables, k fixed, the fraction of such formulas of size n which compute a given Boolean function approaches a positive limit as n --> infinity. (C) 1997 John Wiley & Sons, Inc.

U2 - 10.1002/(SICI)1098-2418(199707)10:4<453::AID-RSA3>3.0.CO;2-T

DO - 10.1002/(SICI)1098-2418(199707)10:4<453::AID-RSA3>3.0.CO;2-T

M3 - Article

VL - 10

SP - 453

EP - 485

JO - Random Structures & Algorithms

JF - Random Structures & Algorithms

ER -