Topologies for Error-Detecting Variable-Length Codes

Given a finite alphabet $A$, a quasi-metric $d$ over $A^*$, and a non-negative integer $k$, we introduce the relation $\tau_{d,k}\subseteq A^*\times A^*$ such that $(x,y)\in\tau_{d,k}$ holds whenever $d(x,y)\le k$. The error detection capability of variable-length codes is expressed in term of conditions over $\tau_{d,k}$. With respect to the prefix metric, the factor one, and any quasi-metric associated with some free monoid (anti-)automorphism, we prove that one can decide whether a given regular variable-length code satisfies any of those error detection constraints.Comment: arXiv admin note: text overlap with arXiv:2208.1468

A system for describing and deciding properties of regular languages using input altering transducers

ii, 94 leaves : ill. ; 29 cm.Includes abstract.Includes bibliographical references (leaves 92-94).We present a formal method for describing and deciding code related properties of regular languages using input altering transducers. We also provide an implementation of that method in the form of a web application. We introduce the concept of an input altering transducer. We show how to use such transducers to describe properties of languages and present examples of transducers describing some well known properties (like suffix codes, prefix codes, infix codes, solid codes, and others). We discuss some limitations of our method. In particular, all properties that can be described using input altering transducers are 3-independence properties. We also give an example of a 3-independence property that cannot be represented using a transducer. We explain how our method is a specialisation of a more general method based on language in-equations. We also discuss the relation between our method and a method that uses sets of trajectories to describe properties. In particular, we show how, for any given set of trajectories describing some property, to build an input altering transducer describing the same property. We introduce the concept of counterexample, which is a pair of words that, if a given language does not belong to a given property, illustrate that fact. We show how we can incorporate extracting such counterexample into our method. Finally, we provide some details on the implementation and usage of the web application that was built as a part of this research

Application and implementation of transducer tools in answering certain questions about regular languages

121 leaves : ill. ; 29 cm.Includes abstract.Includes bibliographical references (leaves 117-121).In this research, we investigate, refine, and implement algorithmic tools that allow us to answer decision questions about regular languages. We provide a thorough presentation of existing algorithmic tools to answer the satisfaction questions of whether a given language satisfies a given property described by an input-preserving transducer, which is equivalent to the question of whether a given language is error-detecting for the channel realized by the same input-preserving transducer; whether a given language is error-correcting for the channel realized by an input-preserving transducer; whether a given regular language satisfies the code property. In the process, we give a thorough presentation of an existing algorithm to decide whether a transducer is functional and an algorithm about how to translate a normal form transducer into a real-time transducer. We also introduce our method to provide counterexamples in cases where the answers to the satisfaction questions are negative. In addition, we discuss our new method to estimate the edit distance of a regular language by the error-correction property, which is much faster than the existing method of computing the edit distance via error-detection. Finally, we deliver an open implementation of these algorithms and methods via a web interface – I-LaSer, and add the implementation of transducer classes into our copy of the FAdo libraries

Finite maximal solid codes

AbstractSolid codes, a special class of bifix codes, were introduced recently in the connection with formal languages. However, they have a much earlier history and more important motivation in information transmission dating back to the 1960s. In this paper, they are studied as an independent subject in the theory of variable-length codes. It is shown that every finite solid code is contained in a finite maximal one; based on further analysis of the structure of finite maximal solid codes, an algorithm is proposed to construct all of them starting from the most simple and evident ones