Figure 2 - uploaded by Stavros Konstantinidis
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The σ-automaton corresponding to the automaton in Fig. 1. It accepts all edit strings that transform a word of (10 + 010) * 0 into a different word of (10 + 010) * 0 using only substitution errors.
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A (combinatorial) channel is a set of pairs of words describing all the possible input- output channel situations. We introduce the concept "maximal error-detecting capa- bility" of a given language, with respect to a certain class of channels, which is simply a maximal channel for which the given language is error-detecting. The new concept is int...
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... states of A. As the detailed view of these transitions is not needed here, we simplify the notation by using expressions of the form (¯ p, (x/y), ¯ q) for the transitions of A σ . We also note that A σ consists of at most 2s 2 A states, where s A is the number of states in A, and contains only states that are reachable from the start state -see Fig. 2. The next statement refers specifically to thin languages, as they are special when it comes to error-detection for channels of the form σ(m, l). A language K is called thin if any two different words of K have different lengths [8]. Obviously a thin language is error-detecting for σ(m, l), for every possible values of the parameters ...
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Citations
... From a logical point of view (see Lemma 3 below) errordetection subsumes the concept of error-correction. This connection is stated already in [6] but without making use of it there. Here we add the fact that maximal error-detection subsumes maximal error-correction. ...
... Note: The operation '•' between two transducers (channels) t and s is called composition and returns a new transducer s • t such that z ∈ (s • t)(x) if and only if y ∈ t(x) and z ∈ t(y), for some y. Proof: The first statement is already in [6]. For the second statement, first assume that C is maximal er-correcting and consider any word w ∈ A \ C. If C ∪ {w} were (er • er −1 )detecting then C∪{w} would also be er-correcting and, hence, C would be non-maximal; a contradiction. ...
We present a randomized algorithm that takes as input two positive integers $N,\ell$ and a channel (=specification of the errors permitted), and computes an error-detecting, or -correcting, block code having up to $N$ codewords of length $\ell$. The channel could allow any rational combination of substitution and synchronization errors. Moreover, if the algorithm finds less than $N$ codewords then those codewords constitute a code that, with high probability, is close to maximal (in a certain precise sense defined here). We also present some components of an open source Python package in which several code related concepts have been implemented. A methodological contribution is the presentation of how various error combinations can be expressed formally and processed algorithmically.
... The error-detection case follows from the previous theorem when we observe that the "error-detection for R(t)" property is the input-preserving transducer property P pr t , i.e., the property obtained using exactlyt as the defining transducer. The error-correction case follows from the fact that a language is error-correcting for a channel γ iff it is errordetecting for γ −1 • γ, [27], and the fact that transducers are effectively closed under inverse and composition [3,42]. ...
The branch of coding theory that is based on formal languages has produced several methods for defining code properties, including word relations, dependence systems, implicational conditions, trajectories, and language inequations. Of those, the latter three can be viewed as formal methods in the sense that a certain formal expression can be used to denote a code property. Here we present a formal method which is based on transducers. Each transducer of a certain type defines/describes a desired code property. The method provides simple and uniform decision procedures for the basic questions of property satisfaction and maximality for regular languages. Our work includes statements about the hardness of deciding some of the problems involved. It turns out that maximality can be hard to decide even for "classical" code properties of finite languages. We also present an initial implementation of a LAnguage SERver capable of deciding the satisfaction problem for a given transducer code property and regular language.
... Intuitively, C contains the possible channels that appear to model the error situations arising in some application where information needs to be transmitted or stored. In [8] we introduced the concept of maximal error-detecting capability of a language L, with respect to a certain error model C. This is a channel γ in C such that L is error-detecting for γ and L is not error-detecting for γ ′ , for any channel γ ′ in C that properly contains γ. ...
... This is a channel γ in C such that L is error-detecting for γ and L is not error-detecting for γ ′ , for any channel γ ′ in C that properly contains γ. In [8] we posed the question of computing the maximal error-detecting capabilities of a given regular language for various error models, including the rational channels as well as various models of SID channels. Rational channels are exactly those realized by finite transducers. ...
... The symbol ⊙ is used simply as a connective for the simpler types. In [8] we showed how to compute all maximal error-detecting capabilities of a given regular language with respect to the error-model C τ = {τ (m, l) : for any m and l with 0 ≤ m < l} when τ = σ, and we left open the problem when τ = σ. In this paper we settle this problem. ...
A (combinatorial) channel γ consists of pairs of words representing all possible input-output situations of the channel. In an earlier paper, [5], we formalized the intuitive concept of "largest set of errors" detectable by a given language L by defining the maximal error-detecting capabilities of L with respect to a given class of channels, and we showed how to compute all maximal error-detecting capabilities of a given regular language with respect to the class of rational channels and a class of channels involving only the substitution-error type. In this paper we resolve the problem for channels involving errors of any combination of the basic types substitution, insertion, deletion. We also consider the problem of finding the inverses of these channels, in view of the fact that L is error-detecting for γ if and only if it is error-detecting for the inverse of γ.
The neighbourhood of a language L consists of all strings that are within a given distance from a string of L. For example, additive distances or the prefix-distance are regularity preserving in the sense that the neighbourhood of a regular language is always regular. For error detection and error correction applications an important question is to determine the size of the minimal deterministic finite automaton (DFA) needed to recognize the neighbourhood of a language recognized by an n state DFA. This paper surveys recent work on the state complexity of neighbourhoods of regularity preserving distances.
The neighbourhood of a language L with respect to an additive distance consists of all strings that have distance at most the given radius from some string of L. We show that the worst case (deterministic) state complexity of a radius r neighbourhood of a language recognized by an n state nondeterministic finite automaton A is \((r+2)^n\). The lower bound construction uses an alphabet of size linear in n. We show that the worst case state complexity of the set of strings that contain a substring within distance r from a string recognized by A is \((r+2)^{n-2} + 1\).
We survey the actual and potential rôles of automata in the modelling of information transmission systems and, in particular, in the encoder, channel and decoder components of such systems. Our focus is on applications of codes in such systems and on the relevance of automaton theoretic methods to these applications. We discuss, for example, the issues of error-detection, fault-tolerance and error-correction for variable-length codes. Beyond reviewing known work in a possibly new setting, we also present some recent results on fault-tolerant decoders for systems in which synchronization errors are likely. We conclude with a kind of research programme, a list of rather general open problems requiring solutions.
A (combinatorial) channel consists of pairs of words representing all possible input-output channel situations. In a past paper, we formalized the intuitive concept of “largest amount of errors” detectable by a given language L, by defining the maximal error-detecting capabilities of L with respect to a given class of channels, and we showed how to compute all maximal error-detecting capabilities (channels) of a given regular language with respect to the class of rational channels and a class of channels involving only the substitution-error type. In this paper we resolve the problem for channels involving any combination of the basic error types: substitution, insertion, deletion. Moreover, we consider the problem of finding the inverses of these channels, in view of the fact that L is error-detecting for γ if and only if it is error-detecting for the inverse of γ. We also discuss a natural method of reducing the problem of computing (inner) distances of a given regular language L to the problem of computing maximal error-detecting capabilities of L.
