Formal languages over GF(2) Bakinova, Ekaterina; Basharin, Artem; Batmanov, Igor ...
Information and computation,
02/2022, Volume:
283
Journal Article
Peer reviewed
Variants of the union and concatenation operations on formal languages are investigated, in which Boolean logic in the definitions (that is, conjunction and disjunction) is replaced with the ...operations in the two-element field GF(2) (conjunction and exclusive OR). Union is thus replaced with symmetric difference, whereas concatenation gives rise to a new GF(2)-concatenation operation, which is notable for being invertible. All operations preserve regularity, and for a pair of languages recognized by an m-state and an n-state DFA, their GF(2)-concatenation is recognized by a DFA with m⋅2n states, and this number of states is in the worst case necessary. Similarly, the state complexity of GF(2)-inverse is 2n+1. Next, a new class of formal grammars based on GF(2)-operations is defined, and it is shown to have the same computational complexity as ordinary grammars with union and concatenation: in particular, simple parsing in time O(n3), fast parsing in the time of matrix multiplication, and parsing in NC2.
We develop a new method HTBB for the multidimensional black-box approximation and gradient-free optimization, which is based on the low-rank hierarchical Tucker decomposition with the use of the ...MaxVol indices selection procedure. Numerical experiments for 14 complex model problems demonstrate the robustness of the proposed method for dimensions up to 1000, while it shows significantly more accurate results than classical gradient-free optimization methods, as well as approximation and optimization methods based on the popular tensor train decomposition, which represents a simpler case of a tensor network.
