Zig-zag tile assembly system

Zig-zag systems are those that grow one horizontal row at a time, alternating left-to-right and right-to-left growth, always adding new rows to the north and never growing south. Zig-zag systems are capable of simulating universal Turing machines, and thus universal computation.

Compact zig-zag systems are zig-zag system that only extend the width of each row by one over the length of the previous row, and only grow upward by one tile before continuing the horizontal growth. Compact zig-zag systems are capable of simulating universal Turing machines.

formal definition

A tile assembly system <m>\tau= (T,\sigma,2)</m> is called a zig-zag tile assembly system if (1) <m>\tau</m> is directed, (2) there is a single assembly sequence <m>\alpha</m> in <m>\tau</m>, with <m>A[\tau] = \{\alpha\}</m>, and (3) for every <m>x \in</m> dom <m>\alpha</m>, (0,1) <m>\notin</m> IN<m>^{\alpha}(x)</m>.

Additionally, we say that <m>\tau</m> is compact if <m>\tau</m> is a zig-zag system with <m>A[\tau] = \{\alpha\}</m> and for every <m>x \in</m> dom <m>\alpha</m> and ever <m>u \in U_{2},str_{\alpha(x)}(u)+str_{\alpha(x)}(-u)<4</m>.