On the model in general..

[rihardsk]

To my understanding of it, the new model simply "shifts" (parts of) the complexity of boolean computation to the complexity of the associated arithmetic of p-adic numbers. It seems to me that all the presented examples just reuse the idea of the first example - computing bitwise AND. To briefly outline: In the classical query model, one has to read all the input bits in the worst case to determine the outcome of AND. In the suggested new model, one simply leaves the work to the associated arithmetic which "sums" all the input bits in one step, and then the decision is simple. If this is the correct understanding of the (undefined) new concept, then what is the real benefit of doing this? In my eyes, this new concept would be useful if it brought new complexity results for a "fixed-complexity" arithmetic (which would prevent shifting of the problem complexity to this arithmetic). But, maybe,
it is just my wrong understanding of the proposed model, and with proper definitions the benefits would become more clear.
The authors should definitely keep this in mind in their next submission