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State-space explosion is the rapid growth of a model's possible global configurations as its components are combined. If $n$ independent components each have two local states, their Cartesian product has $2^n$ global states. Thirty such components already permit $1073741824$ combinations, before accounting for additional data values.

Model checking must establish a property over relevant reachable behavior, so checking one representative execution is insufficient. The reachable subset can be much smaller than the full product, and symbolic representations can compactly describe many configurations at once. However, compactness depends on structure; it is not a universal removal of exponential difficulty. The practical aim is to exploit shared structure without omitting behavior needed for the claimed formal verification.^[ref]

Symbolic model checking verifies a Transition system by representing sets of states and transitions as formulas or decision structures rather than enumerating each state separately. Encode a current state by bits $x$ and its successor by bits $x^{\prime }$. A formula $T(x,x^{\prime })$ describes the transition relation. If $R(x)$ denotes currently reachable states, the next reachable set isIterating from the initial-state formula $I$ reaches a fixed point in a finite system. Intersecting that set with a bad-state formula checks a safety property.

Binary decision diagrams can make these operations compact when the represented Boolean structure is favorable. A system with $n$ Boolean state components has up to $2^n$ states, so sharing structure can be decisive. The representation can nevertheless become exponentially large, and variable ordering strongly affects a decision diagram's size. Symbolic representation mitigates state-space explosion; it does not abolish it.