On Oscillation-free ε-random Sequences

A Turing Machine (TM) is an abstract, synchronous, deterministic computer with a finite number of internal states. It operates on the set of infinite words, or tapes, over some finite alphabet, scanning exactly one symbol of the tape at a time. (Only a 2-symbol alphabet, consisting of “0” and “|”, will be considered here, and the scanned symbol of a tape will be distinguished by an underscore.) depending upon its internal state and the symbol under scan, it can perform one or more of the following operations: replace the scanned symbol with a new symbol, focus its attention on an adjacent square, and transfer control to a new state.

Two finite words u and v are called Abelian equivalent if each letter occurs equally many times in both u and v. The Abelian subshift \(\mathcal A_{\varvec{x}}\) of an infinite word \(\varvec{x}\) is the set of infinite words \(\varvec{y}\) such that, for each factor u of \(\varvec{y}\), there exists a factor v of \(\varvec{x}\) which is Abelian equivalent to u. The notion of Abelian subshift gives a characterization of Sturmian words: among binary uniformly recurrent words, Sturmian words are exactly those words for which \(\mathcal A_{\varvec{x}}\) equals the shift orbit closure \(\varOmega _{\varvec{x}}\). On the other hand, the Abelian subshift of the Thue-Morse word contains uncountably many minimal subshifts. In this paper we undertake a general study of Abelian subshifts. In particular, we characterize the Abelian subshifts of recurrent aperiodic balanced words and the Abelian subshifts of ternary words having factor complexity \(n+2\) for all \(n\ge 1\).

Complementation of finite automata on infinite words is not only a fundamental problem in automata theory, but also serves as a cornerstone for solving numerous decision problems in mathematical logic, model-checking, program analysis and verification. For Streett complementation, a significant gap exists between the current lower bound 2^{Omega(n*log(n*k))} and upper bound 2^{O(n*k*log(n*k))}, where n is the state size, k is the number of Streett pairs, and k can be as large as 2^{n}. Determining the complexity of Streett complementation has been an open question since the late 80's. In this paper we show a complementation construction with upper bound 2^{O(n*log(n)+n*k*log(k))} for k=O(n) and 2^{O(n^{2}*log(n))} for k=Omega(n), which matches well the lower bound obtained in the paper arXiv:1102.2963. We also obtain a tight upper bound 2^{O(n*log(n))} for parity complementation.

An {\omega}-language is a set of infinite words over a finite alphabet X. We consider the class of recursive {\omega}-languages, i.e. the class of {\omega}-languages accepted by Turing machines with a B\"uchi acceptance condition, which is also the class {\Sigma}11 of (effective) analytic subsets of X{\omega} for some finite alphabet X. We investigate here the notion of ambiguity for recursive {\omega}-languages with regard to acceptance by B\"uchi Turing machines. We first present in detail essentials on the literature on {\omega}-languages accepted by Turing Machines. Then we give a complete and broad view on the notion of ambiguity and unambiguity of B\"uchi Turing machines and of the {\omega}-languages they accept. To obtain our new results, we make use of results and methods of effective descriptive set theory.

Economically organized hierarchies in WordNet and the Oxford English Dictionary

Infinite Lyndon words

Repetition-free words

Deterministic Finite-Automata Abstractions of Time-Variant Sequential Behaviours

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsSolomon FefermanSOLOMON FEFERMAN is professor of Mathematics and Philosophy, and Suppes Family Professor in Humanities and Sciences at Stanford University. He is past President of the Association for Symbolic Logic and is a Fellow of the American Academy of Arts and Sciences. He received his Ph.D. at U.C. Berkeley in 1957 under the direction of Alfred Tarski. He is Editor-in-Chief of the Collected Works of Kurt Gödel, and the author of numerous articles on mathematical logic, the foundations of mathematics, and the history of modern logic.

A set X of partial words over a finite alphabet A is called unavoidable if every two-sided infinite word over A has a factor compatible with an element of X. Unlike the case of a set of words without holes, the problem of deciding whether or not a given finite set of n partial words over a k-letter alphabet is avoidable is NP-hard, even when we restrict to a set of partial words of uniform length. So classifying such sets, with parameters k and n, as avoidable or unavoidable becomes an interesting problem. In this paper, we work towards this classification problem by investigating the maximum number of holes we can fill in unavoidable sets of partial words of uniform length over an alphabet of any fixed size, while maintaining the unavoidability property.

This study addresses the relation between morphological productivity and semantic transparency. Using distributional semantics, we compare German word formation using particles with derivational word formation. We observed that derivational suffixes, but not particles, tend to make strong independent semantic contributions to their carrier words. In two-dimensional t-SNE maps, complex words show clustering by affix, but not by particle. Furthermore, the semantic vectors of suffixed words are predictable from their base words with higher accuracy than is possible for particle verbs. For particle verbs, but not affixed verbs, semantic similarity within the set of complex words correlated negatively with the number of types. Furthermore, only for particle verbs, a greater number of observed types predicted a reduced probability of observing unseen types. We propose that particle verbs primarily serve the onomasiological function of labeling, resulting in relatively idiosyncratic semantic vectors. By contrast, words sharing derivational affixes form distinct clusters in semantic space while maintaining strong and consistent semantic relations with their base words. This enables these words to serve not only as labels, but also allows them to be used with an anaphoric function in discourse.

This project aims to address pronunciation problems of English language learners. Current pronunciation software tools mainly address phonemic difficulties and give little or no analytic feedback. Achievement of near native fluency however involves the ability to reproduce English prosody in terms of pitch, intensity, and duration in addition to basic phonemic competence. This paper reports on the preliminary results a project which includes phonemic diagnosis but also takes the learner forward by analysing and giving feedback on prosody. The project is being carried out in an iterative and incremental software development approach with a focus on user experience design and evaluation, market research, and with a view to developing mobile apps. The first stage of evaluation initially focussed solely on prosody and investigated 10 subjects ( 5 intermediate to advanced, 5 beginners) , whose pronunciation of a set of predetermined words, phrases and sentences were bench-marked against a set of native speaker recordings in order to establish the following: • The extent to which the learner benefited from audio only or audio plus text as a prompt to pronouncing the words, in order to establish the feasibility of training ear and vocal apparatus without text-based prompts. • To what extent the learner's pronunciation was affected by their understanding of the words they were pronouncing in order to establish the most effective learning methodology when using de-contextualised speech segments. The first finding identified the use of audio as the prime focus but some learners expressed a preference for the written word. As a result, the option of accessing the text after the audio will be included in the design of the application. The second finding indicated that for all students it was beneficial for pronunciation to understand the speech segment. Additional findings (with the beginners group) indicate a requirement to devise an adaptive system for determining any individual phonemes that learners were struggling with in one or two syllable words before moving onto analysing prosody alongside phonemic diagnosis in more complex words and phrases. Other requirements identified were the need to provide contextualisation, e.g. through illustration or dictionary support, and to cater for different levels of students by creating sets of words appropriate to their lexical knowledge. The next stage of evaluation will involve a greater number of subjects – evaluating how to present the potentially complex feedback in such a way that learners both understand and engage and are motivated to improve. A set of simple visual symbols has been devised which represent pitch, intensity, duration, and give learners instant visual feedback on all three in a simple, clear manner that avoids information overload, such as can occur with existing methods of reproducing waveforms and spectrograms. Applications of this technology include mobile apps, e.g. for single words; call centre training, e.g. of customised scripts; and a children’s adventure game e.g. to engage children in mastering English pronunciation.

We investigate notions of randomness in the space \({\mathcal {C}}[2^{\mathbb {N}}]\) of nonempty closed subsets of {0,1}ℕ. A probability measure is given and a version of the Martin-Löf Test for randomness is defined. Π0 2 random closed sets exist but there are no random Π0 1 closed sets. It is shown that a random closed set is perfect, has measure 0, and has no computable elements. A closed subset of \(2^{{\mathbb N}}\) may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. This leads to some results on a Chaitin-style notion of randomness for closed sets.KeywordsComputabilityRandomnessΠ0 1 Classes

Weighted automata are nondeterministic automata with numerical weights on transitions. They can define quantitative languages L that assign to each word w a real number L(w). In the case of infinite words, the value of a run is naturally computed as the maximum, limsup, liminf, limit average, or discounted sum of the transition weights. We study expressiveness and closure questions about these quantitative languages. We first show that the set of words with value greater than a threshold can be non-omega-regular for deterministic limit-average and discounted-sum automata, while this set is always omega-regular when the threshold is isolated (i.e., some neighborhood around the threshold contains no word). In the latter case, we prove that the omega-regular language is robust against small perturbations of the transition weights. We next consider automata with transition weights 0 or 1 and show that they are as expressive as general weighted automata in the limit-average case, but not in the discounted-sum case. Third, for quantitative languages L1 and L2, we consider the operations max(L1, L2), min(L1, L2), and 1-L1, which generalize the boolean operations on languages, as well as the sum L1 + L2. We establish the closure properties of all classes of quantitative languages with respect to these four operations.

We consider the problem of designing DNA codes, namely sets of equi-length words over the alphabet [A, C, G, T] that satisfy certain combinatorial constraints. This problem is motivated by the task of reliably storing and retrieving information in synthetic DNA strands for use in DNA computing or as molecular bar codes in chemical libraries. The primary constraints that we consider, defined with respect to a parameter d, are as follows: for every pair of words w, x in a code, there are at least d mismatches between w and x if w not equal x and also between the reverse of w and the Watson-Crick complement of x. Extending classical results from coding theory, we present several upper and lower bounds on the maximum size of such DNA codes and give methods for constructing such codes. An additional constraint that is relevant to the design of DNA codes is that the free energies and enthalpies of the code words, and thus the melting temperatures, be similar. We describe dynamic programming algorithms that can (a) calculate the total number of words of length n whose free energy value, as approximated by a formula of Breslauer et al. (1986) falls in a given range, and (b) output a random such word. These algorithms are intended for use in heuristic algorithms for constructing DNA codes.