  This article deals with the approximate synchronization of two coupled
multi-valued logical networks. According to the initial state set from which
both systems start, two kinds of approximate synchronization problem, local
approximate synchronization and global approximate synchronization, are
proposed for the first time. Three new notions: approximate synchronization
state set (ASSS), the maximum approximate synchronization basin (MASB) and the
shortest approximate synchronization time (SAST) are introduced and analyzed.
Based on ASSS, several necessary and sufficient conditions are obtained for
approximate synchronization. MASB, the set of all possible initial states, from
which the systems are approximately synchronous, is investigated combining with
the maximum invariant subset. And the calculation method of the SAST,
associated with transient period, is presented. By virtue of MASB, pinning
control scheme is investigated to make two coupled systems achieve global
approximate synchronization. Furthermore, the related theories are also applied
to the complete synchronization problem of $k$-valued ($k\geq2$) logical
networks. Finally, four examples are given to illustrate the obtained results.
