This paper presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u(x)) in the inhomogenenous quasi-linear parabolic equation u(t)(x, t) = (k(u(x))u(x)(x, t))(x) + F(u) with the Dirichlet boundary conditions u(0, t)=psi(0),u(1, t)=psi(1) and source function F(u). The main purpose of this paper is to investigate the distinguishability of the input-output mappings Phi[.] : K -> C-1[0, T], psi[.]: K -> C-1[0,T] via sernigroup theory. Copyright (C) 2009 John Wiley & Sons, Ltd.