Applications of Transducers in Independent Languages, Word Distances, Codes

Abstract

A (nondeterministic) transducer t is an operator mapping an input word to a set of possible output words. A few types of transducers are important in this work: input-altering, input-preserving, and inputdecreasing. Two words are t-dependent, if one is the output of t when the other one is used as input. A t-independent language is one containing no two t-dependent words. Examples of independent languages are found in noiseless coding theory, noisy coding theory and DNA computing. We discuss how the above transducer types can provide elegant solutions to some cases of the following broad problems: (i) computing two minimum distance witness words of a given regular language; (ii) computing witness words for the non-satisfaction, or non-maximality, of a given regular language with respect to the independence specified by a given transducer t; (iii) computing, for any given t and language L, a maximal t-independent language containing L; (iv) computing, for any given positive integer n and transducer t, a t-independent language of n words. The descriptional complexity cost of converting between transducer types is discussed, when this conversion is possible. We also explore methods of defining more independences in a way that some of the above problems can still be computed.

Partitioning a Symmetric Rational Relation into Two Asymmetric Rational Relations

“…If R is irreflexive and realized by some transducer t then, [5], the above condition is equivalent to t(L)∩L = ∅. In any case, we have that L is R-independent, if and only if it is (R∪R −1 )-independent; and of course the relation (R ∪R −1 ) is always symmetric.…”

Regular Expressions and Transducers over Alphabet-Invariant and User-Defined Labels

“…If the transducer t is input-altering then, [15], the above condition is equivalent to t(L) ∩ L = ∅.…”

“…The witness version of the property satisfaction question for inputaltering transducers ŝ (see Eq. ( 29)) can be answered in time O(|ŝ| • |â| 2 ), where â is the given ε-NFA accepting L (see [15]). This can be done using the function call nonEmptyW(ŝ ↓ â ↑ â).…”

“…First, L(B ∩ F ) ≥ 3. Then R = R(A/G) and, by(15),Q ⊆ R. Now let (a, g) ∈ Rand pick any b ∈ L(B ∩ F ) \ {a, g}. Then, (a, g) = (a, b) • (b, g) ∈ R(A/B =) • R(F/G =).…”