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Solvable vertex models in statistical mechanics give rise to soliton cellular automata at q=0 in a ferromagnetic regime. By means of the crystal base theory we study a class of such automata associated with non-exceptional quantum affine algebras U'_q(\hat{\geh}_n). Let B_l be the crystal of the U'_q(\hat{\geh}_n)-module corresponding to the l-fold symmetric fusion of the vector representation. For any crystal of the form B = B_{l_1} \otimes ... \otimes B_{l_N}, we prove that the combinatorial R matrix B_M \otimes B \xrightarrow{\sim} B \otimes B_M is factorized into a product of Weyl group operators in a certain domain if M is sufficiently large. It implies the factorization of certain transfer matrix at q=0, hence the time evolution in the associated cellular automata. The result generalizes the ball-moving algorithm in the box-ball systems., Comment: 19pages, LaTeX2e, no figure. For proceedings of The Baxter revolution in mathematical physics. (revised version)

We consider a category of finite crystals of a quantum affine algebra whose objects are not necessarily perfect, and set of paths, semi-infinite tensor product of an object of this category with a certain boundary condition. It is shown that the set of paths is isomorphic to a direct sum of infinitely many, in general, crystals of integrable highest weight modules. We present examples from C_n^{(1)} and A_{n-1}^{(1)}, in which the direct sum becomes a tensor product as suggested from the Bethe Ansatz., Comment: 15 pages, LaTeX2e, submitted to the proceedings of the RIMS98 program "Combinatorial Methods in Representation Theory"