The universal portfolio algorithm is a portfolio selection algorithm from the field of machine learning and information theory. The algorithm learns adaptively from historical data and maximizes log-optimal growth rate in the long run, per the Kelly criterion. It was introduced by the late Stanford University information theorist Thomas M. Cover. The algorithm rebalances the portfolio at the beginning of each trading period. At the beginning of the first trading period it starts with a naive diversification. In the following trading periods the portfolio composition depends on the historical total return of all possible constant-rebalanced portfolios. The universal portfolio algorithm is the predecessor of the various online portfolio selection methodologies.
Phrase structure grammar
The term phrase structure grammar was originally introduced by Noam Chomsky as the term for grammar studied previously by Emil Post and Axel Thue (Post canonical systems). Some authors, however, reserve the term for more restricted grammars in the Chomsky hierarchy: context-sensitive grammars or context-free grammars. In a broader sense, phrase structure grammars are also known as constituency grammars. The defining character of phrase structure grammars is thus their adherence to the constituency relation, as opposed to the dependency relation of dependency grammars. == History == In 1956, Chomsky wrote, "A phrase-structure grammar is defined by a finite vocabulary (alphabet) Vp, and a finite set Σ of initial strings in Vp, and a finite set F of rules of the form: X → Y, where X and Y are strings in Vp." == Constituency relation == In linguistics, phrase structure grammars are all those grammars that are based on the constituency relation, as opposed to the dependency relation associated with dependency grammars; hence, phrase structure grammars are also known as constituency grammars. Any of several related theories for the parsing of natural language qualify as constituency grammars, and most of them have been developed from Chomsky's work, including Government and binding theory Generalized phrase structure grammar Head-driven phrase structure grammar Lexical functional grammar The minimalist program Nanosyntax Further grammar frameworks and formalisms also qualify as constituency-based, although they may not think of themselves as having spawned from Chomsky's work, e.g. Arc pair grammar, and Categorial grammar.
NFA minimization
In automata theory (a branch of theoretical computer science), NFA minimization is the task of transforming a given nondeterministic finite automaton (NFA) into an equivalent NFA that has a minimum number of states, transitions, or both. While efficient algorithms exist for DFA minimization, NFA minimization is PSPACE-complete. No efficient (polynomial time) algorithms are known, and under the standard assumption that P ≠ PSPACE, none exist. The most efficient known algorithm is the Kameda–Weiner algorithm. == Non-uniqueness of minimal NFA == Unlike deterministic finite automata, minimal NFAs may not be unique. There may be multiple NFAs with the same number of states that accept the same regular language, but for which there is no equivalent NFA or DFA with fewer states.
Top 10 AI Voice Assistants Compared (2026)
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Finite-state transducer
A finite-state transducer (FST) is a finite-state machine with two memory tapes, following the terminology for Turing machines: an input tape and an output tape. This contrasts with an ordinary finite-state automaton, which has a single tape. An FST is a type of finite-state automaton (FSA) that maps between two sets of symbols. An FST is more general than an FSA. An FSA defines a formal language by defining a set of accepted strings, while an FST defines a relation between sets of strings. An FST will read a set of strings on the input tape and generate a set of relations on the output tape. An FST can be thought of as a translator or relater between strings in a set. In morphological parsing, an example would be inputting a string of letters into the FST, the FST would then output a string of morphemes. == Overview == An automaton can be said to recognize a string if we view the content of its tape as input. In other words, the automaton computes a function that maps strings into the set {0,1}. Alternatively, we can say that an automaton generates strings, which means viewing its tape as an output tape. On this view, the automaton generates a formal language, which is a set of strings. The two views of automata are equivalent: the function that the automaton computes is precisely the indicator function of the set of strings it generates. The class of languages generated by finite automata is known as the class of regular languages. The two tapes of a transducer are typically viewed as an input tape and an output tape. On this view, a transducer is said to transduce (i.e., translate) the contents of its input tape to its output tape, by accepting a string on its input tape and generating another string on its output tape. It may do so nondeterministically and it may produce more than one output for each input string. A transducer may also produce no output for a given input string, in which case it is said to reject the input. In general, a transducer computes a relation between two formal languages. Each string-to-string finite-state transducer relates the input alphabet Σ to the output alphabet Γ. Relations R on Σ×Γ that can be implemented as finite-state transducers are called rational relations. Rational relations that are partial functions, i.e. that relate every input string from Σ to at most one Γ, are called rational functions. Finite-state transducers are often used for phonological and morphological analysis in natural language processing research and applications. Pioneers in this field include Ronald Kaplan, Lauri Karttunen, Martin Kay and Kimmo Koskenniemi. A common way of using transducers is in a so-called "cascade", where transducers for various operations are combined into a single transducer by repeated application of the composition operator (defined below). == Formal construction == Formally, a finite transducer T is a 6-tuple (Q, Σ, Γ, I, F, δ) such that: Q is a finite set, the set of states; Σ is a finite set, called the input alphabet; Γ is a finite set, called the output alphabet; I is a subset of Q, the set of initial states; F is a subset of Q, the set of final states; and δ ⊆ Q × ( Σ ∪ { ϵ } ) × ( Γ ∪ { ϵ } ) × Q {\displaystyle \delta \subseteq Q\times (\Sigma \cup \{\epsilon \})\times (\Gamma \cup \{\epsilon \})\times Q} (where ε is the empty string) is the transition relation. We can view (Q, δ) as a labeled directed graph, known as the transition graph of T: the set of vertices is Q, and ( q , a , b , r ) ∈ δ {\displaystyle (q,a,b,r)\in \delta } means that there is a labeled edge going from vertex q to vertex r. We also say that a is the input label and b the output label of that edge. NOTE: This definition of finite transducer is also called letter transducer (Roche and Schabes 1997); alternative definitions are possible, but can all be converted into transducers following this one. Define the extended transition relation δ ∗ {\displaystyle \delta ^{}} as the smallest set such that: δ ⊆ δ ∗ {\displaystyle \delta \subseteq \delta ^{}} ; ( q , ϵ , ϵ , q ) ∈ δ ∗ {\displaystyle (q,\epsilon ,\epsilon ,q)\in \delta ^{}} for all q ∈ Q {\displaystyle q\in Q} ; and whenever ( q , x , y , r ) ∈ δ ∗ {\displaystyle (q,x,y,r)\in \delta ^{}} and ( r , a , b , s ) ∈ δ {\displaystyle (r,a,b,s)\in \delta } then ( q , x a , y b , s ) ∈ δ ∗ {\displaystyle (q,xa,yb,s)\in \delta ^{}} . The extended transition relation is essentially the reflexive transitive closure of the transition graph that has been augmented to take edge labels into account. The elements of δ ∗ {\displaystyle \delta ^{}} are known as paths. The edge labels of a path are obtained by concatenating the edge labels of its constituent transitions in order. The behavior of the transducer T is the rational relation [T] defined as follows: x [ T ] y {\displaystyle x[T]y} if and only if there exists i ∈ I {\displaystyle i\in I} and f ∈ F {\displaystyle f\in F} such that ( i , x , y , f ) ∈ δ ∗ {\displaystyle (i,x,y,f)\in \delta ^{}} . This is to say that T transduces a string x ∈ Σ ∗ {\displaystyle x\in \Sigma ^{}} into a string y ∈ Γ ∗ {\displaystyle y\in \Gamma ^{}} if there exists a path from an initial state to a final state whose input label is x and whose output label is y. === Weighted automata === Finite State Transducers can be weighted, where each transition is labelled with a weight in addition to the input and output labels. A Weighted Finite State Transducer (WFST) over a set K of weights can be defined similarly to an unweighted one as an 8-tuple T=(Q, Σ, Γ, I, F, E, λ, ρ), where: Q, Σ, Γ, I, F are defined as above; E ⊆ Q × ( Σ ∪ { ϵ } ) × ( Γ ∪ { ϵ } ) × Q × K {\displaystyle E\subseteq Q\times (\Sigma \cup \{\epsilon \})\times (\Gamma \cup \{\epsilon \})\times Q\times K} (where ε is the empty string) is the finite set of transitions; λ : I → K {\displaystyle \lambda :I\rightarrow K} maps initial states to weights; ρ : F → K {\displaystyle \rho :F\rightarrow K} maps final states to weights. In order to make certain operations on WFSTs well-defined, it is convenient to require the set of weights to form a semiring. Two typical semirings used in practice are the log semiring and tropical semiring: nondeterministic automata may be regarded as having weights in the Boolean semiring. Two weighted FST can be composed. == Operations on finite-state transducers == The following operations defined on finite automata also apply to finite transducers: Union. Given transducers T and S, there exists a transducer T ∪ S {\displaystyle T\cup S} such that x [ T ∪ S ] y {\displaystyle x[T\cup S]y} if and only if x [ T ] y {\displaystyle x[T]y} or x [ S ] y {\displaystyle x[S]y} . Concatenation. Given transducers T and S, there exists a transducer T ⋅ S {\displaystyle T\cdot S} such that x [ T ⋅ S ] y {\displaystyle x[T\cdot S]y} if and only if there exist x 1 , x 2 , y 1 , y 2 {\displaystyle x_{1},x_{2},y_{1},y_{2}} with x = x 1 x 2 , y = y 1 y 2 , x 1 [ T ] y 1 {\displaystyle x=x_{1}x_{2},y=y_{1}y_{2},x_{1}[T]y_{1}} and x 2 [ S ] y 2 . {\displaystyle x_{2}[S]y_{2}.} Kleene closure. Given a transducer T, there might exist a transducer T ∗ {\displaystyle T^{}} with the following properties: and x [ T ∗ ] y {\displaystyle x[T^{}]y} does not hold unless mandated by (k1) or (k2). Composition. Given a transducer T on alphabets Σ and Γ and a transducer S on alphabets Γ and Δ, there exists a transducer T ∘ S {\displaystyle T\circ S} on Σ and Δ such that x [ T ∘ S ] z {\displaystyle x[T\circ S]z} if and only if there exists a string y ∈ Γ ∗ {\displaystyle y\in \Gamma ^{}} such that x [ T ] y {\displaystyle x[T]y} and y [ S ] z {\displaystyle y[S]z} . This operation extends to the weighted case. This definition uses the same notation used in mathematics for relation composition. However, the conventional reading for relation composition is the other way around: given two relations T and S, ( x , z ) ∈ T ∘ S {\displaystyle (x,z)\in T\circ S} when there exist some y such that ( x , y ) ∈ S {\displaystyle (x,y)\in S} and ( y , z ) ∈ T . {\displaystyle (y,z)\in T.} Projection to an automaton. There are two projection functions: π 1 {\displaystyle \pi _{1}} preserves the input tape, and π 2 {\displaystyle \pi _{2}} preserves the output tape. The first projection, π 1 {\displaystyle \pi _{1}} is defined as follows: Given a transducer T, there exists a finite automaton π 1 T {\displaystyle \pi _{1}T} such that π 1 T {\displaystyle \pi _{1}T} accepts x if and only if there exists a string y for which x [ T ] y . {\displaystyle x[T]y.} :The second projection, π 2 {\displaystyle \pi _{2}} is defined similarly. Determinization. Given a transducer T, we want to build an equivalent transducer that has a unique initial state and such that no two transitions leaving any state share the same input label. The powerset construction can be extended to transducers, or even weighted transducers, but sometimes fails to halt; indeed, some non-deterministic transducers do not admit equivalent
Lexxe
Lexxe is an internet search engine that applies Natural Language Processing in its semantic search technology. Founded in 2005 by Dr. Hong Liang Qiao, Lexxe is based in Sydney, Australia. Today, Lexxe's key focus is on sentiment search with the launch of a news sentiment search site at News & Moods (www.newsandmoods.com). Lexxe has experienced several stages of change of focus in search technology: Lexxe launched its Alpha version in 2005, featuring Natural Language question answering (i.e. users could ask questions in English to the search engine apart from keyword searches — this feature has been suspended for redevelopment since 2010). It used only algorithms to extract answers from web pages, with no question-answer pair databases prepared in advance. In 2011, Lexxe launched a beta version with a new search technology called Semantic Key. Semantic Keys enable users to query with a conceptual keyword (or a keyword with a special meaning, hence the term Semantic Key) in order to find instances under the concept, e.g. price → $5.95 or €200, color → red, yellow, white. For example, “price: a pound of apples”, “color: ferrari”. With initial 500 Semantic Keys at the Beta launch, Lexxe became the first search engine in the world to offer this unique and useful search technology to the users. The cost of building Semantic Keys was too heavy though. In 2017, Lexxe launched News & Moods (www.newsandmoods.com), an open platform for news sentiment search, a first step towards sentiment search feature for the entire Internet search in Lexxe search engine. News & Moods also comes with smartphone apps in Android and iOS.
MedSLT
MedSLT is a medium-ranged open source spoken language translator developed by the University of Geneva. It is funded by the Swiss National Science Foundation. The system has been designed for the medical domain. It currently covers the doctor-patient diagnosis dialogues for the domains of headache, chest and abdominal pain in English, French, Japanese, Spanish, Catalan and Arabic. The vocabulary used ranges from 350 to 1000 words depending on the domain and language pair. == Motivation for creating MedSLT == With more than 6000 languages worldwide, language barriers become an increasing problem for healthcare. The lack of medical interpreters can lead to disastrous consequences. These range from prolonged hospital stays to wrong diagnosis and medication. A study found that only about half of the 23 million people with limited proficiency in English in the United States had been provided with a medical interpreter. Millions of refugees and immigrants worldwide face similar problems, although not always as severe. The gap between need and availability of language services might be closed with speech translation systems. == Challenges == The biggest challenge is and was to develop an ideal system, though it is not possible to do so at this moment. This system would fit the needs of doctors and the patients alike, and would provide accurate and flexible translation. A realisation of an ideal translation tool is impossible without the use of unrestricted language and a large vocabulary. Medical professionals demand high reliability from translation. This favours rule-based architectures over data-driven. The latter are more suitable for inexperienced users. Rule-based architectures achieve higher accuracy especially if used by experts. Though it is highly desirable to build a bidirectional system supporting a two-way dialogue, which concentrates on patient-centered communication, the patients will have difficult access to the system. Most patients have no experience with such systems. Less reliable results for translation from the patient-to-doctor direction are the outcome. To overcome this the system needs to provide either easy access or an integrated help tool to guide the users through the process. Although controlled rule-based systems achieve good results, they are brittle. To receive good translations the user needs to be familiar with the system and has to know what is covered by the grammar. Covering different sub-domains (headache, chest and abdominal pain) and language pairs presents additional problems. A shared structure and grammar for all subdomains and language pairs minimises development and maintenance costs. The integration of new doctor and patient languages is also a key challenge. Adding new languages should be quick and rather simple, because he system has to be used in many countries to cover multiple language pairs. Direct translation from source to target language proves to be rather difficult. Using interlingua for unidirectional translation instead of a bidirectional approach helps to simplify the translation process. On top of this, the system has to run on different platforms, because mobility is a key issue for many attending physicians. A portable version addresses these issues, but has to deal with the heavy load of the translation process. == The MedSLT system == The system's speech recognition is based on the Nuance 8.5 platform that supports grammar-based language models. All grammars used for recognition, analysis and generation are compiled from a small set of unification grammars. These core grammars are created by the open-source Regulus Grammar Compiler and are automatically specialised using corpus-driven methods. The specialisation considers both the task (recognition, analysis and generation) and the sub-domain (headache, chest and abdominal pain). The specialisation uses the explanation-based learning algorithm to create a treebank from the training corpus. These examples are divided into sets of subtrees by using domain- and grammar-specific rules (also known as "operationality criteria" in machine translation). The subtree rules are combined into a single rule, creating a specialised unification grammar. The grammar is compiled to an executable form, for analysis and generation by a parser or generator, and for recognition of a CFG grammar. A CFG grammar is required for the Nuance engine. Compilation by Nuance-specific criteria turns the grammar into speech recognition packages. The final step uses the training corpus again for statistical tuning of the language model. MedSLT translation processes are based on a rule-based interlingua. The interlingua is treated as an actual language (it is a very simple version of English) and is specified by a Regulus grammar. This grammar does not take account of complex surface syntax phenomena of real languages like movement or agreement. A set of rules is the base for translating the source language semantic representation to interlingua. Another set of rules covers the translation from interlingua to the target language. The semantic representations are converted to surface words using a target language grammar. Defining semantics for a specific domain enables the developers to specify interlingua with a small, tightly constraint semantic grammar. The translations based on interlingua match direct translations almost perfectly, because the development shifts to a decoupled monolingual architecture. A set of combined interlingua corpora, with one corpus per sub-domain, is the core of this architecture. All source language development corpora are translated to interlingua. These are sorted and grouped together with the corresponding source language examples. The interlingua forms are then translated into each target language, and the results are attached together. This organisation improves the translation process. There is no duplicated effort for multilingual regression testing, because each parsing and generation step is performed once. This allows more frequent testing. The representation language used for all forms is Almost Flat Functional semantics. AFF is derived from the Spoken Language Translator, the precursor of MEdSLT. SLT uses Quasi Logical Form, a logical based representation language. QLF is an expressive yet very complex language, causing high development and maintenance costs. A minimal solution was planned for the medical translator. Early versions of the system utilised a language using simple feature-value lists. These lists were supplemented with an optional level of nesting to represent subordinate clauses (i.e. embedded clauses). Determiners were not included, because they are hard to translate and it is difficult to reliably distinguish and recognise them. This way, translation rules became a lot simpler, because only a list of feature-value pairs had to be mapped to another list of pairs. The language turned out to be underconstrained. Adding natural sortal constraints to the grammar solved this problem, but also returned the language to a more expressive formalism. The newly created AFF combines elements of QLF and the feature-value list semantics. This version of flat semantics is enhanced with additional functional markings. This together with a relatively small vocabulary solved the ambiguity problem of the original flat representation language without creating overly complex rules. In addition, the syntactic structures are treated carefully by a compromise of linguistic and engineering traditions. The grammars are in fact retrieved from linguistically motivated resource, using corpus-based methods. They are driven by small sets of examples. This results in simpler and flatter domain-specific grammars. The semantics are less sophisticated and represent a minimal approach in the engineering tradition. Each lexical item contributes a set of feature-value pairs. This leads to simple-to-write translation rules. There are only lists of features-value pairs to map to other feature-value pairs. However, as a result the machine translation channel model becomes underspecified and is weakened, whereas the target language model is strengthened. An intelligent help module is integrated into the system to support users in utilising the full coverage of the grammars. This tool provides the user with examples as close as possible to the users original utterance. The output is based on a library. Each sub-domain and language pair has its own library. The contents are extracted from the combined interlingua corpora. The help module scans the corpus for the tagged source language form mapped with the corresponding target language form. Additionally a second statistical recogniser is used as backup. The results are used to select similar examples from the library. According to the generation preferences, one of the derived strings is picked and the target language string is realised as spoken language. Some statistical corpus based meth