Probablistic assembly
A probablistic assembly (as opposed to a deterministic assembly) is a system which can potentially build multiple shapes, but will build one of a desired class of shapes with high probabliity. To study this model we can think of the assembly process as a markov chain where each producible supertile is a state and transitions occur with non-zero probability from supertile A to each B <m>\in</m> A <m>\to_{T}</m>. For each B <m>\in</m> A <m>\to_{T}</m> let <m>t_{B}</m> denote the tile added to A to get B. The transition probability from A to B is defined to be
The probability that a tile system T terminally assembles a supertile A is thus defined to be the probability that the Markov chain ends in state A. Further, the probability that a system terminally assembles a shape <m>\Upsilon</m> is the probability the chain ends in a supertile state of shape <m>\Upsilon</m>.
