Randomized generation of error control codes with automata and transducers

Abstract

We introduce the concept of an f -maximal error-detecting block code, for some parameter f in (0,1), in order to formalize the situation where a block code is close to maximal with respect to being error-detecting. Our motivation for this is that it is computationally hard to decide whether an error-detecting block code is maximal. We present an output-polynomial time randomized algorithm that takes as input two positive integers N, and a specification of the errors permitted in some application, and generates an error-detecting, or error-correcting, block code of length that is 99%-maximal, or contains N words with a high likelihood. We model error specifications as (nondeterministic) transducers, which allow one to represent any rational combination of substitution and synchronization errors. We also present some elements of our implementation of various error-detecting properties and their associated methods. Then, we show several tests of the implemented randomized algorithm on various error specifications. A methodological contribution is the presentation of how various desirable error combinations can be expressed formally and processed algorithmically.Mathematics Subject Classification. 68Q45, 68W20, 94A40, 94B25.

Approximate NFA Universality and Related Problems Motivated by Information Theory

“…s , where ℓ is the word length of b}: Deciding whether a given block NFA of some word length ℓ accepts all words of length ℓ is a coNP-complete problem, [14].…”

“…The following definition is inspired from the "approximate" algorithmic solution of [14] for the task of generating an error-detecting code of N codewords, for given N , if possible, or an error-detecting code of less than N codewords which is "close to" maximal. -A(x, ε) works within polynomial time w.r.t.…”

“…Our motivation for defining the concept of approximate universality comes from the problem of generating codes (whether variable length codes, or fixed length error control codes) that are maximal, where on the one hand the question of deciding maximality is hard, but on the other hand it is acceptable to generate codes that are maximal within a tolerance ε, [6,14]. For infinite languages, we define approximate universality relative to some probability distribution on the set of words.…”

Applications of Transducers in Independent Languages, Word Distances, Codes

“…Some results from [9,20,26] are the following. (ii) given input-preserving transducer t and NFA a, return as above, either (None, None), or a pair (u, v) depending on whether (1) is satisfied.…”

“…2. The decision version of the maximality question can be coNP-hard, [26], PSPACE-hard or undecidable, [20]. Specifically, if the given transducer t is input-preserving, or input-altering, or θ-input-altering 3 , then deciding whether L(a) is maximal satisfying, respectively, (1) or (2) or (3), is PSPACEhard.…”