  The degree of mixing is a fundamental property of a dynamical system. General
multi-dimensional shifts cannot be systematically determined. This work
introduces constructive and systematic methods for verifying the degree of
mixing, from topological mixing to strong specification (or strong
irreducibility) for two-dimensional shifts of finite type. First, transition
matrices on infinite strips of width $n$ are introduced for all $n\geq 2$. To
determine the primitivity of the transition matrices, connecting operators are
introduced to reduce the order of high-order transition matrices to yield
lower-order transition matrices. Two sufficient conditions for primitivity are
provided; they are invariant diagonal cycles and primitive commutative cycles
of connecting operators. After primitivity is established, the
corner-extendability and crisscross-extendability are used to demonstrate
topological mixing. In addition, the hole-filling condition yields the strong
specification. All mentioned conditions can be verified to apply in a finite
number of steps.
