this question is inspired by a particular somewhat natural physical system specifically constructed to mimic another complex highly-studied physical computing system. (some may astutely guess at least half of this analogy correctly; the full details are involved but more details will be given if there is significant interest.)consider a computing system of c-bits or circle-bits and a single gate that operates/combines them. a circle-bit is a single polar coordinate on the unit circle, eg denoted by a number in the range 0-360 (degrees).the sole gate operator $z=mid(x,y)$ takes two c-bits $x,y$ as input and computes as output $z$ the c-bit that is the midpoint of the arc between them.now consider a c-bit circuit composed of inputs, mid gates, and a single output. the question is related to whether the c-bit circuit can convert binary functions with a binary-to-cbit adapter/converter on the binary input wires, and a cbit-to-binary adapter/converter on a single final c-bit output gate. call this overall construction a c-bit sandwich circuit.the binary-to-cbit adapter/converter is just a mapping of 0,1 bits (false,true) onto 0,180 respectively.a single output function (ie cbit-to-binary adapter/converter) $test_{a,b}(x)$ on a c-bit $x$ is given where $a,b$ are c-bit constants and it returns true iff $a \leq x \leq b$.is the c-bit sandwich circuit functionally complete for boolean functions, or not? ie can it compute all boolean functions?if not, what is a characterization of the functions it can compute?looking for any refs to the nearest problem considered in published literature elsewhere.(there are natural generalizations of this problem & may ask about those depending on response.)note: know there is ambiguity on the c-bit gate where the c-bits are 180 apart. the computing scheme solution may choose a consistent rule in that case. 
computing with gates on polar coordinates, functionally complete wrt boolean functions?
reference request;boolean functions;arithmetic circuits;natural computing