3,793 research outputs found
Quotient Complexities of Atoms in Regular Ideal Languages
A (left) quotient of a language by a word is the language
. The quotient complexity of a regular language
is the number of quotients of ; it is equal to the state complexity of ,
which is the number of states in a minimal deterministic finite automaton
accepting . An atom of is an equivalence class of the relation in which
two words are equivalent if for each quotient, they either are both in the
quotient or both not in it; hence it is a non-empty intersection of
complemented and uncomplemented quotients of . A right (respectively, left
and two-sided) ideal is a language over an alphabet that satisfies
(respectively, and ). We
compute the maximal number of atoms and the maximal quotient complexities of
atoms of right, left and two-sided regular ideals.Comment: 17 pages, 4 figures, two table
Families of locally separated Hamilton paths
We improve by an exponential factor the lower bound of K¨orner and Muzi for the cardinality of the largest family of Hamilton paths in a complete graph of n vertices in which the union of any two paths has maximum degree 4. The improvement is through an explicit construction while the previous bound was obtained by a greedy algorithm. We solve a similar problem for permutations up to an exponential factor
Families of locally separated Hamilton paths
We improve by an exponential factor the lower bound of K¨orner and Muzi for the cardinality of the largest family of Hamilton paths in a complete graph of n vertices in which the union of any two paths has maximum degree 4. The improvement is through an explicit construction while the previous bound was obtained by a greedy algorithm. We solve a similar problem for permutations up to an exponential factor
Signal Propagation, with Application to a Lower Bound on the Depth of Noisy Formulas
We study the decay of an information signal propagating through a series of noisy channels. We obtain exact bounds on such decay, and as a result provide a new lower bound on the depth of formulas with noisy components. This improves upon previous work of N. Pippenger and significantly decreases the gap between his lower bound and the classical upper bound of von Neumann. We also discuss connections between our work and the study of mixing rates of Markov chains
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