PMC

A wavelet trie constructed for a tuple of bit vectors. Each node is labelled with a longest common prefix (LCP) α and an assignment vector β. During construction at a particular node, the LCP of the bit vectors is extracted and the next significant bit is used to assign the bit vector suffixes to that node’s children. A node becomes a leaf when all bit vectors assigned to it are equal. An example is given in bold. The sequence 0010101 results from the traversal along the dashed line from top to bottom. The index i being queried is updated by calling rank_0(·,i) (i) (traverse left) or rank_1(·,i) (i) (traverse right) on the β vectors

The conserved motifs of common bean NAC genes. The bit score indicates the information content for each position in the sequence

Most common conditions reported to be “quite a bit” limiting

Three-bit example. Consider again the theory described by the state space Ω of three bits, and all transformations on those bits. (a) All operations that change the third bit (of which id and not_C are labelled). (b) Equivalence classes built according to definition . These correspond to the view of an agent who can only distinguish the first two bits. The equivalence relation , which coarse-grains over the functions applied to the third bit, gives us the largest effective state space relative to which functions in are secret. (c) More coarse-grained equivalence classes [x]_A (vertical, yellow) and [x]_B (horizontal, blue), corresponding to an agent A who can only distinguish the first bit and an agent B who only sees the second bit, respectively. Operations in are still secret relative to these two agents. In addition, operations on the first bit are secret towards B and vice versa. These smaller effective state spaces correspond to equivalence relations on the effective state space (as in the nested agents of proposition ). The two-bit space is a common state space of A and B, including states that could be distinguished if the two agents could work together, with . (Online version in colour.)