Audio mining

Audio mining is a technique by which the content of an audio signal can be automatically analyzed and searched. It is most commonly used in the field of automatic speech recognition, where the analysis tries to identify any speech within the audio. The term audio mining is sometimes used interchangeably with audio indexing, phonetic searching, phonetic indexing, speech indexing, audio analytics, speech analytics, word spotting, and information retrieval. Audio indexing, however, is mostly used to describe the pre-process of audio mining, in which the audio file is broken down into a searchable index of words. == History == Academic research on audio mining began in the late 1970s in schools like Carnegie Mellon University, Columbia University, the Georgia Institute of Technology, and the University of Texas. Audio data indexing and retrieval began to receive attention and demand in the early 1990s, when multimedia content started to develop and the volume of audio content significantly increased. Before audio mining became the mainstream method, written transcripts of audio content were created and manually analyzed. == Process == Audio mining is typically split into four components: audio indexing, speech processing and recognition systems, feature extraction and audio classification. The audio will typically be processed by a speech recognition system in order to identify word or phoneme units that are likely to occur in the spoken content. This information may either be used immediately in pre-defined searches for keywords or phrases (a real-time "word spotting" system), or the output of the speech recognizer may be stored in an index file. One or more audio mining index files can then be loaded at a later date in order to run searches for keywords or phrases. The results of a search will normally be in terms of hits, which are regions within files that are good matches for the chosen keywords. The user may then be able to listen to the audio corresponding to these hits in order to verify if a correct match was found. === Audio Indexing === In audio, there is the main problem of information retrieval - there is a need to locate the text documents that contain the search key. Unlike humans, a computer is not able to distinguish between the different types of audios such as speed, mood, noise, music or human speech - an effective searching method is needed. Hence, audio indexing allows efficient search for information by analyzing an entire file using speech recognition. An index of content is then produced, bearing words and their locations done through content-based audio retrieval, focusing on extracted audio features. It is done through mainly two methods: Large Vocabulary Continuous Speech Recognition (LVCSR) and Phonetic-based Indexing. ==== Large Vocabulary Continuous Speech Recognizers (LVCSR) ==== In text-based indexing or large vocabulary continuous speech recognition (LVCSR), the audio file is first broken down into recognizable phonemes. It is then run through a dictionary that can contain several hundred thousand entries and matched with words and phrases to produce a full text transcript. A user can then simply search a desired word term and the relevant portion of the audio content will be returned. If the text or word could not be found in the dictionary, the system will choose the next most similar entry it can find. The system uses a language understanding model to create a confidence level for its matches. If the confidence level be below 100 percent, the system will provide options of all the found matches. ===== Advantages and disadvantages ===== The main draw of LVCSR is its high accuracy and high searching speed. In LVCSR, statistical methods are used to predict the likelihood of different word sequences, hence the accuracy is much higher than the single word lookup of a phonetic search. If the word can be found, the probability of the word spoken is very high. Meanwhile, while initial processing of audio takes a fair bit of time, searching is quick as just a simple test to text matching is needed. On the other hand, LVCSR is susceptible to common issues of speech recognition. The inherent random nature of audio and problems of external noise all affect the accuracies of text-based indexing. Another problem with LVCSR is its over reliance on its dictionary database. LVCSR only recognizes words that are found in their dictionary databases, and these dictionaries and databases are unable to keep up with the constant evolving of new terminology, names and words. Should the dictionary not contain a word, there is no way for the system to identify or predict it. This reduces the accuracy and reliability of the system. This is named the Out-of-vocabulary (OOV) problem. Audio mining systems try to cope with OOV by continuously updating the dictionary and language model used, but the problem still remains significant and has probed a search for alternatives. Additionally, due to the need to constantly update and maintain task-based knowledge and large training databases to cope with the OOV problem, high computational costs are incurred. This makes LVCSR an expensive approach to audio mining. ==== Phonetic-based Indexing ==== Phonetic-based indexing also breaks the audio file into recognizable phonemes, but instead of converting them to a text index, they are kept as they are and analyzed to create a phonetic-based index. The process of phonetic-based indexing can be split into two phases. The first phase is indexing. It begins by converting the input media into a standard audio representation format (PCM). Then, an acoustic model is applied to the speech. This acoustic model represents characteristics of both an acoustic channel (an environment in which the speech was uttered and a transducer through which it was recorded) and a natural language (in which human beings expressed the input speech). This produces a corresponding phonetic search track, or phonetic audio track (PAT), a highly compressed representation of the phonetic content of the input media. The second phase is searching. The user's search query term is parsed into a possible phoneme string using a phonetic dictionary. Then, multiple PAT files can be scanned at high speed during a single search for likely phonetic sequences that closely match corresponding strings of phonemes in the query term. ===== Advantages and disadvantages ===== Phonetic indexing is most attractive as it is largely unaffected by linguistic issues such as unrecognized words and spelling errors. Phonetic preprocessing maintains an open vocabulary that does not require updating. That makes it particularly useful for searching specialized terminology or words in foreign languages that do not commonly appear in dictionaries. It is also more effective for searching audio files with disruptive background noise and/or unclear utterances as it can compile results based on the sounds it can discern, and should the user wish to, they can search through the options until they find the desired item. Furthermore, in contrast to LVCSR, it can process audio files very quickly as there are very few unique phonemes between languages. However, phonemes cannot be effectively indexed like an entire word, thus searching on a phonetic-based system is slow. An issue with phonetic indexing is its low accuracy. Phoneme-based searches result in more false matches than text-based indexing. This is especially prevalent for short search terms, which have a stronger likelihood of sounding similar to other words or being part of bigger words. It could also return irrelevant results from other languages. Unless the system recognizes exactly the entire word, or understands phonetic sequences of languages, it is difficult for phonetic-based indexing to return accurate findings. === Speech processing and recognition system === Deemed as the most critical and complex component of audio mining, speech recognition requires the knowledge of human speech production system and its modeling. To correspond the Human speech production system, the electrical speech production system is developed to consist of: Speech generation Speech perception Voiced & unvoiced speech Model of human speech The electrical speech production system converts acoustic signal into corresponding representation of the spoken through the acoustic models in their software where all phonemes are represented. A statistical language model aids in the process by identifying how likely words are to follow each other in certain languages. Put together with a complex probability analysis, the speech recognition system is capable of taking an unknown speech signal and transcribing it into words based on the program's dictionary. ASR (automatic speech recognition) system includes: Acoustic analysis: input sound waveform is transformed into a feature Acoustic model: establishes relationship between speech signal and phonemes, pronunciation model and lang

Data item

A data item describes an atomic state of a particular object concerning a specific property at a certain time point. A collection of data items for the same object at the same time forms an object instance (or table row). Any type of complex information can be broken down to elementary data items (atomic state). Data items are identified by object (o), property (p) and time (t), while the value (v) is a function of o, p and t: v = F(o,p,t). Values typically are represented by symbols like numbers, texts, images, sounds or videos. Values are not necessarily atomic. A value's complexity depends on the complexity of the property and time component. When looking at databases or XML files, the object is usually identified by an object name or other type of object identifier, which is part of the "data". Properties are defined as columns (table row), properties (object instance) or tags (XML). Often, time is not explicitly expressed and is an attribute applying to the complete data set. Other data collections provide time on the instance level (time series), column level, or even attribute/property level.

General Regionally Annotated Corpus of Ukrainian

General Regionally Annotated Corpus of the Ukrainian Language (GRAC, Ukrainian: Генеральний регіонально анотований корпус української мови, romanized: Heneralnyi rehionalno anotovanyi korpus ukrainskoi movy, ГРАК, Ukrainian грак for rook) is a text corpus of the Ukrainian language comprising more than 2 billion tokens, intended for linguistic research in grammar, vocabulary, and the history of the Ukrainian literary language, as well as for use in compiling dictionaries and grammars. The corpus can be used for language study and also for preparing teaching materials, textbooks, learner’s dictionaries, and exercises using examples from real texts, taking into account frequency and collocational patterns, and so on. The corpus is not a model of standard Ukrainian: it may contain words and combinations that do not match current norms of the literary language. The corpus covers the period from 1816 to 2025, and as of 29 November 2025 it contains more than 812,000 texts by about 35,000 authors. == Composition of the corpus == In the 10th version of the corpus, available for searching from 20 October 2020, 35% consists of fiction. Some fiction genres are highlighted separately: children’s literature, folklore, dramatic works, and scripts. Among non-fiction texts: journalistic writing, including newspaper collections from 1888–1893, 1905, 1913–1918, 1919–1943, modern newspapers from different regions, and texts from online news/information sites; memoirs, letters, and diaries, including a sizeable corpus of Facebook texts representing blogs by people from all regions of Ukraine and the diaspora; scholarly and educational texts: monographs, dissertations, academic articles, textbooks; large subcorpora of academic literature in history, ethnography, philosophy, and law are singled out separately; religious texts, including two Ukrainian translations of the Bible; speeches and interviews. Some dictionaries that include phrasal examples and phraseology have also been incorporated, including the Ukrainian dictionary by Borys Hrinchenko and the Russian-Ukrainian idiomatic dictionary by I. Vyrhan and M. Pylynska. Using the corpus tools, these dictionaries can be searched not only for words, but also for lexico-grammatical patterns within examples and phraseological expressions. About 20% of the texts in the corpus are translations. The corpus includes translations from more than 80 languages, most of all from English and Russian. == Dating == Texts in the corpus are dated by the year of writing, or by the latest year in which a work could have been written; translated texts are dated by the year the translation was produced. A publication year may also be indicated, corresponding to the edition from which the text is taken. == Regional annotation == The corpus’s regional annotation is based on the modern administrative division of Ukraine. The corpus includes texts from all oblasts of Ukraine and from Crimea. A single text may belong to several regional subcorpora (if the author or translator was born, studied, or lived for a long time in different regions). In addition to regional subcorpora, there are subcorpora of works by authors of the Ukrainian diaspora (USA, Canada, Poland, Germany, the United Kingdom, France, etc.). These are mostly texts by emigrants of the 1940s, and to a lesser extent of 1917–1920s. == Morphological annotation == GRAC is based on the morphological analysis system nlp_uk, developed by specialists from the r2u group. The program analyzes the text and, for each word form, determines the lemma (lexeme) and tags (grammatical features). == Research based on the corpus == Research on the Ukrainian language has been carried out using the corpus, including studies of the historical dynamics of language norms, and letter and letter-combination frequencies for font development.

Hanna Hajishirzi

Hannaneh Hajishirzi is an Iranian-American computer scientist specializing in natural language processing. She is Torode Family Professor in Computer Science & Engineering in the Paul G. Allen School of Computer Science and Engineering at the University of Washington, head of the H2Lab in the Allen School, and a senior director of natural language processing in the Allen Institute for AI. == Education and career == After a bachelor's degree from the Sharif University of Technology, Hajishirzi completed her Ph.D. in computer science in 2011, at the University of Illinois Urbana-Champaign. Her dissertation, Action-Centered Reasoning for Probabilistic Dynamic Systems, was supervised by Eyal Amir. After postdoctoral research at Disney Research in Pittsburgh, Hajishirzi joined the University of Washington in 2012, as a research scientist in electrical engineering. In 2015 she became a research assistant professor in electrical engineering. She obtained a regular-rank assistant professorship in 2018, at the same time becoming an AI Fellow in the Allen Institute for AI, where she became a senior director of research in 2021. She was promoted to associate professor in 2022 and to full professor in 2025. == Recognition == Hajishirzi was named as a Fellow of the Association for Computational Linguistics in 2025, "for significant contributions to question answering, scientific applications, multimodal artificial intelligence, and fully open language models". == Personal life == Hajishirzi is married to Ali Farhadi, the CEO of the Allen Institute for AI.

Top 10 AI Virtual Assistants Compared (2026)

Looking for the best AI virtual assistant? An AI virtual assistant is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI virtual assistant slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.

Semantic analytics

Semantic analytics, also termed semantic relatedness, is the use of ontologies to analyze content in web resources. This field of research combines text analytics and Semantic Web technologies like RDF. Semantic analytics measures the relatedness of different ontological concepts. Some academic research groups that have active project in this area include Kno.e.sis Center at Wright State University among others. == History == An important milestone in the beginning of semantic analytics occurred in 1996, although the historical progression of these algorithms is largely subjective. In his seminal study publication, Philip Resnik established that computers have the capacity to emulate human judgement. Spanning the publications of multiple journals, improvements to the accuracy of general semantic analytic computations all claimed to revolutionize the field. However, the lack of a standard terminology throughout the late 1990s was the cause of much miscommunication. This prompted Budanitsky & Hirst to standardize the subject in 2006 with a summary that also set a framework for modern spelling and grammar analysis. In the early days of semantic analytics, obtaining a large enough reliable knowledge bases was difficult. In 2006, Strube & Ponzetto demonstrated that Wikipedia could be used in semantic analytic calculations. The usage of a large knowledge base like Wikipedia allows for an increase in both the accuracy and applicability of semantic analytics. == Methods == Given the subjective nature of the field, different methods used in semantic analytics depend on the domain of application. No singular methods is considered correct, however one of the most generally effective and applicable method is explicit semantic analysis (ESA). ESA was developed by Evgeniy Gabrilovich and Shaul Markovitch in the late 2000s. It uses machine learning techniques to create a semantic interpreter, which extracts text fragments from articles into a sorted list. The fragments are sorted by how related they are to the surrounding text. Latent semantic analysis (LSA) is another common method that does not use ontologies, only considering the text in the input space. == Applications == Entity linking Ontology building / knowledge base population Search and query tasks Natural language processing Spoken dialog systems (e.g., Amazon Alexa, Google Assistant, Microsoft's Cortana) Artificial intelligence Knowledge management The application of semantic analysis methods generally streamlines organizational processes of any knowledge management system. Academic libraries often use a domain-specific application to create a more efficient organizational system. By classifying scientific publications using semantics and Wikipedia, researchers are helping people find resources faster. Search engines like Semantic Scholar provide organized access to millions of articles.

Büchi automaton

In computer science and automata theory, a deterministic Büchi automaton is a theoretical machine which either accepts or rejects infinite inputs. Such a machine has a set of states and a transition function, which determines which state the machine should move to from its current state when it reads the next input character. Some states are accepting states and one state is the start state. The machine accepts an input if and only if it will pass through an accepting state infinitely many times as it reads the input. A non-deterministic Büchi automaton, later referred to just as a Büchi automaton, has a transition function which may have multiple outputs, leading to many possible paths for the same input; it accepts an infinite input if and only if some possible path is accepting. Deterministic and non-deterministic Büchi automata generalize deterministic finite automata and nondeterministic finite automata to infinite inputs. Each are types of ω-automata. Büchi automata recognize the ω-regular languages, the infinite word version of regular languages. They are named after the Swiss mathematician Julius Richard Büchi, who invented them in 1962. Büchi automata are often used in model checking as an automata-theoretic version of a formula in linear temporal logic. == Formal definition == Formally, a deterministic Büchi automaton is a tuple A = ( Q , Σ , δ , q 0 , F ) {\textstyle A=(Q,\Sigma ,\delta ,q_{0},\mathbf {F} )} that consists of the following components: Q {\textstyle Q} is a finite set. The elements of Q {\textstyle Q} are called the states of A {\textstyle A} . Σ {\textstyle \Sigma } is a finite set called the alphabet of A {\textstyle A} . δ : Q × Σ → Q {\textstyle \delta \colon Q\times \Sigma \to Q} is a function, called the transition function of A {\textstyle A} . q 0 {\textstyle q_{0}} is an element of Q {\textstyle Q} , called the initial state of A {\textstyle A} . F ⊆ Q {\textstyle \mathbf {F} \subseteq Q} is the acceptance condition. A run i _ = i 0 i 1 i 2 ⋯ ∈ Σ ω {\displaystyle {\underline {i}}=i_{0}i_{1}i_{2}\cdots \in \Sigma ^{\omega }} is an infinite string of inputs of A {\displaystyle A} . By calling δ {\displaystyle \delta } recursively, we can extend it to a function δ ω : Σ ω → Q ω {\displaystyle \delta ^{\omega }:\Sigma ^{\omega }\to Q^{\omega }} . A state q ∈ Q {\displaystyle q\in Q} is said to occur infinitely often for a run i _ {\displaystyle {\underline {i}}} when the set { n ∈ N ∣ δ ω ( i _ ) n = q } {\displaystyle \{n\in \mathbb {N} \mid \delta ^{\omega }({\underline {i}})_{n}=q\}} is infinite. Let I n f ( i _ ) {\displaystyle \mathrm {Inf} ({\underline {i}})} be the set of states occurring infinitely often for i _ {\displaystyle {\underline {i}}} . The language of A {\displaystyle A} is then the set of runs of A {\displaystyle A} in which at least one of the infinitely-often occurring states is in F {\textstyle \mathbf {F} } ; in symbols: L ( A ) = { i _ ∈ Σ ω ∣ I n f ( i _ ) ∩ F ≠ ∅ } . {\displaystyle L(A)=\{{\underline {i}}\in \Sigma ^{\omega }\mid \mathrm {Inf} ({\underline {i}})\cap \mathbf {F} \neq \varnothing \}.} In a (non-deterministic) Büchi automaton, the transition function δ {\textstyle \delta } is replaced with a transition relation Δ {\textstyle \Delta } that returns a set of states, and the single initial state q 0 {\textstyle q_{0}} is replaced by a set I {\textstyle I} of initial states. Generally, the term Büchi automaton without qualifier refers to non-deterministic Büchi automata. For more comprehensive formalism see also ω-automaton. == Closure properties == The set of Büchi automata is closed under the following operations. Let A = ( Q A , Σ , Δ A , I A , F A ) {\displaystyle A=(Q_{A},\Sigma ,\Delta _{A},I_{A},{F}_{A})} and B = ( Q B , Σ , Δ B , I B , F B ) {\displaystyle B=(Q_{B},\Sigma ,\Delta _{B},I_{B},{F}_{B})} be Büchi automata and C = ( Q C , Σ , Δ C , I C , F C ) {\displaystyle C=(Q_{C},\Sigma ,\Delta _{C},I_{C},{F}_{C})} be a finite automaton. Union: There is a Büchi automaton that recognizes the language L ( A ) ∪ L ( B ) . {\displaystyle L(A)\cup L(B).} Proof: If we assume, w.l.o.g., Q A ∩ Q B {\displaystyle Q_{A}\cap Q_{B}} is empty then L ( A ) ∪ L ( B ) {\displaystyle L(A)\cup L(B)} is recognized by the Büchi automaton ( Q A ∪ Q B , Σ ∪ Σ , Δ A ∪ Δ B , I A ∪ I B , F A ∪ F B ) . {\displaystyle (Q_{A}\cup Q_{B},\Sigma \cup \Sigma ,\Delta _{A}\cup \Delta _{B},I_{A}\cup I_{B},{F}_{A}\cup {F}_{B}).} Intersection: There is a Büchi automaton that recognizes the language L ( A ) ∩ L ( B ) . {\displaystyle L(A)\cap L(B).} Proof: The Büchi automaton A ′ = ( Q ′ , Σ , Δ ′ , I ′ , F ′ ) {\displaystyle A'=(Q',\Sigma ,\Delta ',I',F')} recognizes L ( A ) ∩ L ( B ) , {\displaystyle L(A)\cap L(B),} where Q ′ = Q A × Q B × { 1 , 2 } {\displaystyle Q'=Q_{A}\times Q_{B}\times \{1,2\}} Δ ′ = Δ 1 ∪ Δ 2 {\displaystyle \Delta '=\Delta _{1}\cup \Delta _{2}} Δ 1 = { ( ( q A , q B , 1 ) , a , ( q A ′ , q B ′ , i ) ) | ( q A , a , q A ′ ) ∈ Δ A and ( q B , a , q B ′ ) ∈ Δ B and if q A ∈ F A then i = 2 else i = 1 } {\displaystyle \Delta _{1}=\{((q_{A},q_{B},1),a,(q'_{A},q'_{B},i))|(q_{A},a,q'_{A})\in \Delta _{A}{\text{ and }}(q_{B},a,q'_{B})\in \Delta _{B}{\text{ and if }}q_{A}\in F_{A}{\text{ then }}i=2{\text{ else }}i=1\}} Δ 2 = { ( ( q A , q B , 2 ) , a , ( q A ′ , q B ′ , i ) ) | ( q A , a , q A ′ ) ∈ Δ A and ( q B , a , q B ′ ) ∈ Δ B and if q B ∈ F B then i = 1 else i = 2 } {\displaystyle \Delta _{2}=\{((q_{A},q_{B},2),a,(q'_{A},q'_{B},i))|(q_{A},a,q'_{A})\in \Delta _{A}{\text{ and }}(q_{B},a,q'_{B})\in \Delta _{B}{\text{ and if }}q_{B}\in F_{B}{\text{ then }}i=1{\text{ else }}i=2\}} I ′ = I A × I B × { 1 } {\displaystyle I'=I_{A}\times I_{B}\times \{1\}} F ′ = { ( q A , q B , 2 ) | q B ∈ F B } {\displaystyle F'=\{(q_{A},q_{B},2)|q_{B}\in F_{B}\}} By construction, r ′ = ( q A 0 , q B 0 , i 0 ) , ( q A 1 , q B 1 , i 1 ) , … {\displaystyle r'=(q_{A}^{0},q_{B}^{0},i^{0}),(q_{A}^{1},q_{B}^{1},i^{1}),\dots } is a run of automaton A' on input word w {\textstyle w} if r A = q A 0 , q A 1 , … {\displaystyle r_{A}=q_{A}^{0},q_{A}^{1},\dots } is run of A {\textstyle A} on w {\textstyle w} and r B = q B 0 , q B 1 , … {\displaystyle r_{B}=q_{B}^{0},q_{B}^{1},\dots } is run of B {\textstyle B} on w {\textstyle w} . r A {\textstyle r_{A}} is accepting and r B {\textstyle r_{B}} is accepting if r ′ {\textstyle r'} is concatenation of an infinite series of finite segments of 1-states (states with third component 1) and 2-states (states with third component 2) alternatively. There is such a series of segments of r ′ {\textstyle r'} if r ′ {\textstyle r'} is accepted by A ′ {\textstyle A'} . Concatenation: There is a Büchi automaton that recognizes the language L ( C ) ⋅ L ( A ) . {\displaystyle L(C)\cdot L(A).} Proof: If we assume, w.l.o.g., Q C ∩ Q A {\displaystyle Q_{C}\cap Q_{A}} is empty then the Büchi automaton A ′ = ( Q C ∪ Q A , Σ , Δ ′ , I ′ , F A ) {\displaystyle A'=(Q_{C}\cup Q_{A},\Sigma ,\Delta ',I',F_{A})} recognizes L ( C ) ⋅ L ( A ) {\displaystyle L(C)\cdot L(A)} , where Δ ′ = Δ A ∪ Δ C ∪ { ( q , a , q ′ ) | q ′ ∈ I A and ∃ f ∈ F C . ( q , a , f ) ∈ Δ C } {\displaystyle \Delta '=\Delta _{A}\cup \Delta _{C}\cup \{(q,a,q')|q'\in I_{A}{\text{ and }}\exists f\in F_{C}.(q,a,f)\in \Delta _{C}\}} if I C ∩ F C is empty then I ′ = I C otherwise I ′ = I C ∪ I A {\displaystyle {\text{ if }}I_{C}\cap F_{C}{\text{ is empty then }}I'=I_{C}{\text{ otherwise }}I'=I_{C}\cup I_{A}} ω-closure: If L ( C ) {\displaystyle L(C)} does not contain the empty word then there is a Büchi automaton that recognizes the language L ( C ) ω . {\displaystyle L(C)^{\omega }.} Proof: The Büchi automaton that recognizes L ( C ) ω {\displaystyle L(C)^{\omega }} is constructed in two stages. First, we construct a finite automaton A ′ {\textstyle A'} such that A ′ {\textstyle A'} also recognizes L ( C ) {\displaystyle L(C)} but there are no incoming transitions to initial states of A ′ {\textstyle A'} . So, A ′ = ( Q C ∪ { q new } , Σ , Δ ′ , { q new } , F C ) , {\displaystyle A'=(Q_{C}\cup \{q_{\text{new}}\},\Sigma ,\Delta ',\{q_{\text{new}}\},F_{C}),} where Δ ′ = Δ C ∪ { ( q new , a , q ′ ) | ∃ q ∈ I C . ( q , a , q ′ ) ∈ Δ C } . {\displaystyle \Delta '=\Delta _{C}\cup \{(q_{\text{new}},a,q')|\exists q\in I_{C}.(q,a,q')\in \Delta _{C}\}.} Note that L ( C ) = L ( A ′ ) {\displaystyle L(C)=L(A')} because L ( C ) {\displaystyle L(C)} does not contain the empty string. Second, we will construct the Büchi automaton A ″ {\textstyle A''} that recognize L ( C ) ω {\displaystyle L(C)^{\omega }} by adding a loop back to the initial state of A ′ {\textstyle A'} . So, A ″ = ( Q C ∪ { q new } , Σ , Δ ″ , { q new } , { q new } ) {\displaystyle A''=(Q_{C}\cup \{q_{\text{new}}\},\Sigma ,\Delta '',\{q_{\text{new}}\},\{q_{\text{new}}\})} , where Δ ″ = Δ ′ ∪ { ( q , a , q new ) | ∃ q ′ ∈ F C . ( q , a , q ′ ) ∈ Δ ′ } . {\displaystyle \Delta ''=\Delta '\cup \{(q,a,q_{\text{new}})|\exists q'\in F_{C}.(q,a,q')\in \Delta '\}.} Complementation: