The following is a comparison of cloud-computing software and providers. == IaaS (Infrastructure as a service) == === Providers === ==== General ==== == SaaS (Software as a Service) == === General === === Supported hosts === === Supported guests === == PaaS (Platform as a service) == === Providers === === Providers on IaaS === PaaS providers which can run on IaaS providers ("itself" means the provider is both PaaS and IaaS):
Molecular graphics
Molecular graphics is the discipline and philosophy of studying molecules and their properties through graphical representation. IUPAC limits the definition to representations on a "graphical display device". Ever since Dalton's atoms and Kekulé's benzene, there has been a rich history of hand-drawn atoms and molecules, and these representations have had an important influence on modern molecular graphics. Colour molecular graphics are often used on chemistry journal covers artistically. == History == Prior to the use of computer graphics in representing molecular structure, Robert Corey and Linus Pauling developed a system for representing atoms or groups of atoms from hard wood on a scale of 1 inch = 1 angstrom connected by a clamping device to maintain the molecular configuration. These early models also established the CPK coloring scheme that is still used today to differentiate the different types of atoms in molecular models (e.g. carbon = black, oxygen = red, nitrogen = blue, etc). This early model was improved upon in 1966 by W.L. Koltun and are now known as Corey-Pauling-Koltun (CPK) models. The earliest efforts to produce models of molecular structure was done by Project MAC using wire-frame models displayed on a cathode ray tube in the mid 1960s. In 1965, Carroll Johnson distributed the Oak Ridge thermal ellipsoid plot (ORTEP) that visualized molecules as a ball-and-stick model with lines representing the bonds between atoms and ellipsoids to represent the probability of thermal motion. Thermal ellipsoid plots quickly became the de facto standard used in the display of X-ray crystallography data, and are still in wide use today. The first practical use of molecular graphics was a simple display of the protein myoglobin using a wireframe representation in 1966 by Cyrus Levinthal and Robert Langridge working at Project MAC. Among the milestones in high-performance molecular graphics was the work of Nelson Max in "realistic" rendering of macromolecules using reflecting spheres. Initially much of the technology concentrated on high-performance 3D graphics. During the 1970s, methods for displaying 3D graphics using cathode ray tubes were developed using continuous tone computer graphics in combination with electro-optic shutter viewing devices. The first devices used an active shutter 3D system, generating different perspective views for the left and right channel to provide the illusion of three-dimensional viewing. Stereoscopic viewing glasses were designed using lead lanthanum zirconate titanate (PLZT) ceramics as electronically controlled shutter elements. Active 3D glasses require batteries and work in concert with the display to actively change the presentation by the lenses to the wearer's eyes. Many modern 3D glasses use a passive, polarized 3D system that enables the wearer to visualize 3D effects based on their own perception. Passive 3D glasses are more common today since they are less expensive. The requirements of macromolecular crystallography also drove molecular graphics because the traditional techniques of physical model-building could not scale. The first two protein structures solved by molecular graphics without the aid of the Richards' Box were built with Stan Swanson's program FIT on the Vector General graphics display in the laboratory of Edgar Meyer at Texas A&M University: First Marge Legg in Al Cotton's lab at A&M solved a second, higher-resolution structure of staph. nuclease (1975) and then Jim Hogle solved the structure of monoclinic lysozyme in 1976. A full year passed before other graphics systems were used to replace the Richards' Box for modelling into density in 3-D. Alwyn Jones' FRODO program (and later "O") were developed to overlay the molecular electron density determined from X-ray crystallography and the hypothetical molecular structure. === Timeline === == Types == === Ball-and-stick models === In the ball-and-stick model, atoms are drawn as small sphered connected by rods representing the chemical bonds between them. === Space-filling models === In the space-filling model, atoms are drawn as solid spheres to suggest the space they occupy, in proportion to their van der Waals radii. Atoms that share a bond overlap with each other. === Surfaces === In some models, the surface of the molecule is approximated and shaded to represent a physical property of the molecule, such as electronic charge density. === Ribbon diagrams === Ribbon diagrams are schematic representations of protein structure and are one of the most common methods of protein depiction used today. The ribbon shows the overall path and organization of the protein backbone in 3D, and serves as a visual framework on which to hang details of the full atomic structure, such as the balls for the oxygen atoms bound to the active site of myoglobin in the adjacent image. Ribbon diagrams are generated by interpolating a smooth curve through the polypeptide backbone. α-helices are shown as coiled ribbons or thick tubes, β-strands as arrows, and non-repetitive coils or loops as lines or thin tubes. The direction of the polypeptide chain is shown locally by the arrows, and may be indicated overall by a colour ramp along the length of the ribbon.
Markov partition
A Markov partition in mathematics is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be applied to the study of hyperbolic dynamics. By using a Markov partition, the system can be made to resemble a discrete-time Markov process, with the long-term dynamical characteristics of the system represented as a Markov shift. The appellation 'Markov' is appropriate because the resulting dynamics of the system obeys the Markov property. The Markov partition thus allows standard techniques from symbolic dynamics to be applied, including the computation of expectation values, correlations, topological entropy, topological zeta functions, Fredholm determinants and the like. == Motivation == Let ( M , φ ) {\displaystyle (M,\varphi )} be a discrete dynamical system. A basic method of studying its dynamics is to find a symbolic representation: a faithful encoding of the points of M {\displaystyle M} by sequences of symbols such that the map φ {\displaystyle \varphi } becomes the shift map. Suppose that M {\displaystyle M} has been divided into a number of pieces E 1 , E 2 , … , E r {\displaystyle E_{1},E_{2},\ldots ,E_{r}} which are thought to be as small and localized, with virtually no overlaps. The behavior of a point x {\displaystyle x} under the iterates of φ {\displaystyle \varphi } can be tracked by recording, for each n {\displaystyle n} , the part E i {\displaystyle E_{i}} which contains φ n ( x ) {\displaystyle \varphi ^{n}(x)} . This results in an infinite sequence on the alphabet { 1 , 2 , … , r } {\displaystyle \{1,2,\ldots ,r\}} which encodes the point. In general, this encoding may be imprecise (the same sequence may represent many different points) and the set of sequences which arise in this way may be difficult to describe. Under certain conditions, which are made explicit in the rigorous definition of a Markov partition, the assignment of the sequence to a point of M {\displaystyle M} becomes an almost one-to-one map whose image is a symbolic dynamical system of a special kind called a shift of finite type. In this case, the symbolic representation is a powerful tool for investigating the properties of the dynamical system ( M , φ ) {\displaystyle (M,\varphi )} . == Formal definition == A Markov partition is a finite cover of the invariant set of the manifold by a set of curvilinear rectangles { E 1 , E 2 , … , E r } {\displaystyle \{E_{1},E_{2},\ldots ,E_{r}\}} such that For any pair of points x , y ∈ E i {\displaystyle x,y\in E_{i}} , that W s ( x ) ∩ W u ( y ) ∈ E i {\displaystyle W_{s}(x)\cap W_{u}(y)\in E_{i}} Int E i ∩ Int E j = ∅ {\displaystyle \operatorname {Int} E_{i}\cap \operatorname {Int} E_{j}=\emptyset } for i ≠ j {\displaystyle i\neq j} If x ∈ Int E i {\displaystyle x\in \operatorname {Int} E_{i}} and φ ( x ) ∈ Int E j {\displaystyle \varphi (x)\in \operatorname {Int} E_{j}} , then φ [ W u ( x ) ∩ E i ] ⊃ W u ( φ x ) ∩ E j {\displaystyle \varphi \left[W_{u}(x)\cap E_{i}\right]\supset W_{u}(\varphi x)\cap E_{j}} φ [ W s ( x ) ∩ E i ] ⊂ W s ( φ x ) ∩ E j {\displaystyle \varphi \left[W_{s}(x)\cap E_{i}\right]\subset W_{s}(\varphi x)\cap E_{j}} Here, W u ( x ) {\displaystyle W_{u}(x)} and W s ( x ) {\displaystyle W_{s}(x)} are the unstable and stable manifolds of x, respectively, and Int E i {\displaystyle \operatorname {Int} E_{i}} simply denotes the interior of E i {\displaystyle E_{i}} . These last two conditions can be understood as a statement of the Markov property for the symbolic dynamics; that is, the movement of a trajectory from one open cover to the next is determined only by the most recent cover, and not the history of the system. It is this property of the covering that merits the 'Markov' appellation. The resulting dynamics is that of a Markov shift; that this is indeed the case is due to theorems by Yakov Sinai (1968) and Rufus Bowen (1975), thus putting symbolic dynamics on a firm footing. Variants of the definition are found, corresponding to conditions on the geometry of the pieces E i {\displaystyle E_{i}} . == Examples == Markov partitions have been constructed in several situations. Anosov diffeomorphisms of the torus. Dynamical billiards, in which case the covering is countable. Markov partitions make homoclinic and heteroclinic orbits particularly easy to describe. The system ( [ 0 , 1 ) , x ↦ 2 x m o d 1 ) {\displaystyle ([0,1),x\mapsto 2x\ mod\ 1)} has the Markov partition E 0 = ( 0 , 1 / 2 ) , E 1 = ( 1 / 2 , 1 ) {\displaystyle E_{0}=(0,1/2),E_{1}=(1/2,1)} , and in this case the symbolic representation of a real number in [ 0 , 1 ) {\displaystyle [0,1)} is its binary expansion. For example: x ∈ E 0 , T x ∈ E 1 , T 2 x ∈ E 1 , T 3 x ∈ E 1 , T 4 x ∈ E 0 ⇒ x = ( 0.01110... ) 2 {\displaystyle x\in E_{0},Tx\in E_{1},T^{2}x\in E_{1},T^{3}x\in E_{1},T^{4}x\in E_{0}\Rightarrow x=(0.01110...)_{2}} . The assignment of points of [ 0 , 1 ) {\displaystyle [0,1)} to their sequences in the Markov partition is well defined except on the dyadic rationals - morally speaking, this is because ( 0.01111 … ) 2 = ( 0.10000 … ) 2 {\displaystyle (0.01111\dots )_{2}=(0.10000\dots )_{2}} , in the same way as 1 = 0.999 … {\displaystyle 1=0.999\dots } in decimal expansions.
Salvatore J. Stolfo
Salvatore J. Stolfo is an academic and professor of computer science at Columbia University, specializing in computer security. == Early life == Born in Brooklyn, New York, Stolfo received a Bachelor of Science degree in Computer Science and Mathematics from Brooklyn College in 1974. He received his Ph.D. from NYU Courant Institute in 1979 and has been on the faculty of Columbia ever since, where he's taught courses in Artificial Intelligence, Intrusion and Anomaly Detection Systems, Introduction to Programming, Fundamental Algorithms, Data Structures, and Knowledge-Based Expert Systems. == Academic research == While at Columbia, Stolfo has received close to $50M in funding for research that has broadly focused on Security, Intrusion Detection, Anomaly Detection, Machine Learning and includes early work in parallel computing and artificial intelligence. He has published or co-authored over 250 papers and has over 46,000 citations with an H-index of 102. In 1996 he proposed a project with DARPA that applies machine learning to behavioral patterns to detect fraud or intrusion in networks. DADO, developed by in part by Stolfo, introduced the parallel computing primitive: “Broadcast, Resolve, Report”, a hardwire implemented mechanism that today is called MapReduce. Among his earliest work, Stolfo along with colleague Greg Vesonder of Bell Labs, developed a large-scale expert data analysis system, called ACE (Automated Cable Expertise) for the nation's phone system. AT&T Bell Labs distributed ACE to a number of telephone wire centers to improve the management and scheduling of repairs in the local loop. Stolfo coined the term FOG computing (not to be confused with fog computing) where technology is used “to launch disinformation attacks against malicious insiders, preventing them from distinguishing the real sensitive customer data from fake worthless data.” In 2005 Stolfo received funding from the Army Research Office to conduct a workshop to bring together a group of researchers to help identify a research program to focus on insider threats. He was elevated to IEEE Fellow in 2018 "for his contributions to machine learning based cybersecurity." He was elected as an ACM Fellow in 2019 "for contributions to machine-learning-based cybersecurity and parallel hardware for database inference systems". == Career == Founded in 2011, Red Balloon Security (or RBS) is a cyber security company founded by Dr Sal Stolfo and Dr Ang Cui. A spinout from the IDS lab, RBS developed a symbiote technology called FRAK as a host defense for embedded systems under the sponsorship of DARPA's Cyber Fast Track program. Created based on their IDS lab research for the DARPA Active Authentication and the Anomaly Detection at Multiple Scales program, Dr Sal Stolfo and Dr. Angelos Keromytis founded Allure Security Technologies. Using active behavioral authentication and decoy technology Stolfo pioneered and patented in 1996. Founded in 2009, Allure Security Technology was created based on work done under DARPA sponsorship in Columbia's IDS lab based on DARPA prompts to research how to detect hackers once they are inside an organization's perimeter and how to continuously authenticate a user without a password. Stolfo's company Electronic Digital Documents produced a “DataBlade” technology, which Informix marketed during their strategy of acquisition and development in the mid 80's. Stolfo's patented merge/purge technology called EDD DataCleanser DataBlade was licensed by Informix. Since its acquisition by IBM in 2005, IBM Informix is one of the world's most widely used database servers, with users ranging from the world's largest corporations to startups. System Detection was one of the companies founded by Prof. Stolfo to commercialize the Anomaly Detection technology developed in the IDS lab. The company ultimately reorganized and was rebranded as Trusted Computer Solutions. That company was recently acquired by Raytheon. Recently a jury awarded Columbia University $185 million for patent infringement for one of Prof. Stolfo's inventions, the Application Communities technology. https://news.columbia.edu/news/columbia-university-awarded-185-million-patent-infringement-nortonlifelock-inc. The final order from the judge applied nearly treble damages: https://www.reuters.com/legal/litigation/gen-digital-owes-columbia-481-mln-us-patent-fight-judge-says-2023-10-02/
General Internet Corpus of Russian
General Internet Corpus of Russian (GICR) is a corpus of Russian internet texts that has been accessible on request through an online query interface since 2013. The corpus includes rich text materials from the blogosphere, social networks, major news sources and literary magazines. == Goals of the project == The project has the status of an educational and scientific one, and many tasks of computational linguistics are solved by independent researchers and research groups with the materials obtained by GICR. While other corpus projects of Russian are focused on fiction and edited texts, General Internet Corpus provides linguists timely opportunity to learn the language as it is, with all the slang and regional peculiarities. Corpus gives the opportunity to carry out research in Linguistic research of a wide range: dialectological research, study of word distribution, study of the language of the social networks, study of the influence of gender, age and other factors on the language, frequency of words, fixed expressions and different constructions, stylistic features of texts of different segments of the Internet, etc. Social media analysis Corpus-based machine learning for evaluating automatic tagging At various times, student papers and independent researches were carried out on the project material by students, graduates and employees of MSU, MIPT, Russian State Humanitarian University, Novosibirsk State University, Higher School of Economics, Russian Academy of Sciences, SFU, CSU, SGMP, IAAS of MSU. Scientific project leaders: Belikov V. - RSUH, Moscow, Russia Selegey V. - RSUH, ABBYY, Moscow, Russia Sharoff S. - RSUH, Moscow, Russia; University of Leeds, UK The organizations involved in support of GICR: Russian State University of Humanities ABBYY Company Moscow Institute of Physics and Technology Skolkovo Institute of Science and Technology == Size and content of the corpus == Corpus size for the summer 2016 is 19.8 billion tokens, of which 49% are from VKontakte, 40% are from LiveJournal, another 4% - from Mail.ru Blogs and News, and 2% - from Russian Magazine Hall. The sources collected in news segment are: RIA Novosti, Regnum, Lenta.ru, Rosbalt. Texts are provided with metamarkup (by date of creation of the text, sex, place and year of birth of the author, Internet genre, etc.); all texts are provided with automatic morphological tagging and lemmatization. Most of the texts collected are of 2013–2014 years of creation, although in some segments, such as in Russian Magazine Hall, there are some texts collected since 1994. GICR is one of the few mega-corpora projects nowadays, which means its available size is reaching several billion of words. == Access == Currently the interface of GICR is in beta stage, so access to the search in the corpora is provided and is free, but is available for researchers on request.
IBM ALP
IBM Assembly Language Processor (ALP) is an assembler written by IBM for 32-bit OS/2 Warp (OS/2 3.0), which was released in 1994. ALP accepts source programs compatible with Microsoft Macro Assembler (MASM) version 5.1, which was originally used to build many of the device drivers included with OS/2. For OS/2 versions 3 and 4, ALP was distributed, along with other tools and documentation, as part of the Device Driver Kit (DDK). The DDK was withdrawn in 2004 as part of IBM's discontinuance of OS/2.
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