In computer science, more precisely, in the theory of deterministic finite automata (DFA), a synchronizing word or reset sequence is a word in the input alphabet of the DFA that sends any state of the DFA to one and the same state. That is, if an ensemble of copies of the DFA are each started in different states, and all of the copies process the synchronizing word, they will all end up in the same state. Not every DFA has a synchronizing word; for instance, a DFA with two states, one for words of even length and one for words of odd length, can never be synchronized. == Existence == Given a DFA, the problem of determining if it has a synchronizing word can be solved in polynomial time using a theorem due to Ján Černý. A simple approach considers the power set of states of the DFA, and builds a directed graph where nodes belong to the power set, and a directed edge describes the action of the transition function. A path from the node of all states to a singleton state shows the existence of a synchronizing word. This algorithm is exponential in the number of states. A polynomial algorithm results however, due to a theorem of Černý that exploits the substructure of the problem, and shows that a synchronizing word exists if and only if every pair of states has a synchronizing word. == Length == The problem of estimating the length of synchronizing words has a long history and was posed independently by several authors, but it is commonly known as the Černý conjecture. In 1969, Ján Černý conjectured that (n − 1)2 is the upper bound for the length of the shortest synchronizing word for any n-state complete DFA (a DFA with complete state transition graph). If this is true, it would be tight: in his 1964 paper, Černý exhibited a class of automata (indexed by the number n of states) for which the shortest reset words have this length. The best upper bound known is 0.1654n3, far from the lower bound. For n-state DFAs over a k-letter input alphabet, an algorithm by David Eppstein finds a synchronizing word of length at most 11n3/48 + O(n2), and runs in time complexity O(n3+kn2). This algorithm does not always find the shortest possible synchronizing word for a given automaton; as Eppstein also shows, the problem of finding the shortest synchronizing word is NP-complete. However, for a special class of automata in which all state transitions preserve the cyclic order of the states, he describes a different algorithm with time O(kn2) that always finds the shortest synchronizing word, proves that these automata always have a synchronizing word of length at most (n − 1)2 (the bound given in Černý's conjecture), and exhibits examples of automata with this special form whose shortest synchronizing word has length exactly (n − 1)2. == Road coloring == The road coloring problem is the problem of labeling the edges of a regular directed graph with the symbols of a k-letter input alphabet (where k is the outdegree of each vertex) in order to form a synchronizable DFA. It was conjectured in 1970 by Benjamin Weiss and Roy Adler that any strongly connected and aperiodic regular digraph can be labeled in this way; their conjecture was proven in 2007 by Avraham Trahtman. == Related: transformation semigroups == A transformation semigroup is synchronizing if it contains an element of rank 1, that is, an element whose image is of cardinality 1. A DFA corresponds to a transformation semigroup with a distinguished generator set.
GazoPa
GazoPa was an image search engine that used features from an image to search for and identify similar images which closed in 2011. GazoPa began in TechCrunch50 in 2008 before launching into a state of open beta in 2009. GazoPa branched out and released a flower photo community site called "GazoPa Bloom" in 2010. This site was for exploring flower images and, if users need help identifying a flower, uploading images for other people try to identify them. Both sites closed to the public in 2011 when the company decided to focus on other areas of their business.
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IBM alignment models
The IBM alignment models are a sequence of increasingly complex models used in statistical machine translation to train a translation model and an alignment model, starting with lexical translation probabilities and moving to reordering and word duplication. They underpinned the majority of statistical machine translation systems for almost twenty years starting in the early 1990s, until neural machine translation began to dominate. These models offer principled probabilistic formulation and (mostly) tractable inference. The IBM alignment models were published in parts in 1988 and 1990, and the entire series is published in 1993. Every author of the 1993 paper subsequently went to the hedge fund Renaissance Technologies. The original work on statistical machine translation at IBM proposed five models, and a model 6 was proposed later. The sequence of the six models can be summarized as: Model 1: lexical translation Model 2: additional absolute alignment model Model 3: extra fertility model Model 4: added relative alignment model Model 5: fixed deficiency problem. Model 6: Model 4 combined with a HMM alignment model in a log linear way == Mathematical setup == The IBM alignment models translation as a conditional probability model. For each source-language ("foreign") sentence f {\displaystyle f} , we generate both a target-language ("English") sentence e {\displaystyle e} and an alignment a {\displaystyle a} . The problem then is to find a good statistical model for p ( e , a | f ) {\displaystyle p(e,a|f)} , the probability that we would generate English language sentence e {\displaystyle e} and an alignment a {\displaystyle a} given a foreign sentence f {\displaystyle f} . The meaning of an alignment grows increasingly complicated as the model version number grew. See Model 1 for the most simple and understandable version. == Model 1 == === Word alignment === Given any foreign-English sentence pair ( e , f ) {\displaystyle (e,f)} , an alignment for the sentence pair is a function of type { 1 , . , . . . , l e } → { 0 , 1 , . , . . . , l f } {\displaystyle \{1,.,...,l_{e}\}\to \{0,1,.,...,l_{f}\}} . That is, we assume that the English word at location i {\displaystyle i} is "explained" by the foreign word at location a ( i ) {\displaystyle a(i)} . For example, consider the following pair of sentences It will surely rain tomorrow -- 明日 は きっと 雨 だWe can align some English words to corresponding Japanese words, but not everyone:it -> ? will -> ? surely -> きっと rain -> 雨 tomorrow -> 明日This in general happens due to the different grammar and conventions of speech in different languages. English sentences require a subject, and when there is no subject available, it uses a dummy pronoun it. Japanese verbs do not have different forms for future and present tense, and the future tense is implied by the noun 明日 (tomorrow). Conversely, the topic-marker は and the grammar word だ (roughly "to be") do not correspond to any word in the English sentence. So, we can write the alignment as 1-> 0; 2 -> 0; 3 -> 3; 4 -> 4; 5 -> 1where 0 means that there is no corresponding alignment. Thus, we see that the alignment function is in general a function of type { 1 , . , . . . , l e } → { 0 , 1 , . , . . . , l f } {\displaystyle \{1,.,...,l_{e}\}\to \{0,1,.,...,l_{f}\}} . Future models will allow one English world to be aligned with multiple foreign words. === Statistical model === Given the above definition of alignment, we can define the statistical model used by Model 1: Start with a "dictionary". Its entries are of form t ( e i | f j ) {\displaystyle t(e_{i}|f_{j})} , which can be interpreted as saying "the foreign word f j {\displaystyle f_{j}} is translated to the English word e i {\displaystyle e_{i}} with probability t ( e i | f j ) {\displaystyle t(e_{i}|f_{j})} ". After being given a foreign sentence f {\displaystyle f} with length l f {\displaystyle l_{f}} , we first generate an English sentence length l e {\displaystyle l_{e}} uniformly in a range U n i f o r m [ 1 , 2 , . . . , N ] {\displaystyle Uniform[1,2,...,N]} . In particular, it does not depend on f {\displaystyle f} or l f {\displaystyle l_{f}} . Then, we generate an alignment uniformly in the set of all possible alignment functions { 1 , . , . . . , l e } → { 0 , 1 , . , . . . , l f } {\displaystyle \{1,.,...,l_{e}\}\to \{0,1,.,...,l_{f}\}} . Finally, for each English word e 1 , e 2 , . . . e l e {\displaystyle e_{1},e_{2},...e_{l_{e}}} , generate each one independently of every other English word. For the word e i {\displaystyle e_{i}} , generate it according to t ( e i | f a ( i ) ) {\displaystyle t(e_{i}|f_{a(i)})} . Together, we have the probability p ( e , a | f ) = 1 / N ( 1 + l f ) l e ∏ i = 1 l e t ( e i | f a ( i ) ) {\displaystyle p(e,a|f)={\frac {1/N}{(1+l_{f})^{l_{e}}}}\prod _{i=1}^{l_{e}}t(e_{i}|f_{a(i)})} IBM Model 1 uses very simplistic assumptions on the statistical model, in order to allow the following algorithm to have closed-form solution. === Learning from a corpus === If a dictionary is not provided at the start, but we have a corpus of English-foreign language pairs { ( e ( k ) , f ( k ) ) } k {\displaystyle \{(e^{(k)},f^{(k)})\}_{k}} (without alignment information), then the model can be cast into the following form: fixed parameters: the foreign sentences { f ( k ) } k {\displaystyle \{f^{(k)}\}_{k}} . learnable parameters: the entries of the dictionary t ( e i | f j ) {\displaystyle t(e_{i}|f_{j})} . observable variables: the English sentences { e ( k ) } k {\displaystyle \{e^{(k)}\}_{k}} . latent variables: the alignments { a ( k ) } k {\displaystyle \{a^{(k)}\}_{k}} In this form, this is exactly the kind of problem solved by expectation–maximization algorithm. Due to the simplistic assumptions, the algorithm has a closed-form, efficiently computable solution, which is the solution to the following equations: { max t ′ ∑ k ∑ i ∑ a ( k ) t ( a ( k ) | e ( k ) , f ( k ) ) ln t ( e i ( k ) | f a ( k ) ( i ) ( k ) ) ∑ x t ′ ( e x | f y ) = 1 ∀ y {\displaystyle {\begin{cases}\max _{t'}\sum _{k}\sum _{i}\sum _{a^{(k)}}t(a^{(k)}|e^{(k)},f^{(k)})\ln t(e_{i}^{(k)}|f_{a^{(k)}(i)}^{(k)})\\\sum _{x}t'(e_{x}|f_{y})=1\quad \forall y\end{cases}}} This can be solved by Lagrangian multipliers, then simplified. For a detailed derivation of the algorithm, see chapter 4 and. In short, the EM algorithm goes as follows:INPUT. a corpus of English-foreign sentence pairs { ( e ( k ) , f ( k ) ) } k {\displaystyle \{(e^{(k)},f^{(k)})\}_{k}} INITIALIZE. matrix of translations probabilities t ( e x | f y ) {\displaystyle t(e_{x}|f_{y})} .This could either be uniform or random. It is only required that every entry is positive, and for each y {\displaystyle y} , the probability sums to one: ∑ x t ( e x | f y ) = 1 {\displaystyle \sum _{x}t(e_{x}|f_{y})=1} . LOOP. until t ( e x | f y ) {\displaystyle t(e_{x}|f_{y})} converges: t ( e x | f y ) ← t ( e x | f y ) λ y ∑ k , i , j δ ( e x , e i ( k ) ) δ ( f y , f j ( k ) ) ∑ j ′ t ( e i ( k ) | f j ′ ( k ) ) {\displaystyle t(e_{x}|f_{y})\leftarrow {\frac {t(e_{x}|f_{y})}{\lambda _{y}}}\sum _{k,i,j}{\frac {\delta (e_{x},e_{i}^{(k)})\delta (f_{y},f_{j}^{(k)})}{\sum _{j'}t(e_{i}^{(k)}|f_{j'}^{(k)})}}} where each λ y {\displaystyle \lambda _{y}} is a normalization constant that makes sure each ∑ x t ( e x | f y ) = 1 {\displaystyle \sum _{x}t(e_{x}|f_{y})=1} .RETURN. t ( e x | f y ) {\displaystyle t(e_{x}|f_{y})} .In the above formula, δ {\displaystyle \delta } is the Dirac delta function -- it equals 1 if the two entries are equal, and 0 otherwise. The index notation is as follows: k {\displaystyle k} ranges over English-foreign sentence pairs in corpus; i {\displaystyle i} ranges over words in English sentences; j {\displaystyle j} ranges over words in foreign language sentences; x {\displaystyle x} ranges over the entire vocabulary of English words in the corpus; y {\displaystyle y} ranges over the entire vocabulary of foreign words in the corpus. === Limitations === There are several limitations to the IBM model 1. No fluency: Given any sentence pair ( e , f ) {\displaystyle (e,f)} , any permutation of the English sentence is equally likely: p ( e | f ) = p ( e ′ | f ) {\displaystyle p(e|f)=p(e'|f)} for any permutation of the English sentence e {\displaystyle e} into e ′ {\displaystyle e'} . No length preference: The probability of each length of translation is equal: ∑ e has length l p ( e | f ) = 1 N {\displaystyle \sum _{e{\text{ has length }}l}p(e|f)={\frac {1}{N}}} for any l ∈ { 1 , 2 , . . . , N } {\displaystyle l\in \{1,2,...,N\}} . Does not explicitly model fertility: some foreign words tend to produce a fixed number of English words. For example, for German-to-English translation, ja is usually omitted, and zum is usually translated to one of to the, for the, to a, for a. == Model 2 == Model 2 allows alignment to be conditional on sentence lengths. That is, we have a probability distribution p a ( j | i , l e , l f ) {\displaystyle
Web Dynpro
Web Dynpro (WD) is a web application technology developed by SAP SE that focuses on the development of server-side business applications. For modern releases (for instance as of NetWeaver 750, software layer SAP_UI) the user interface is rendered according to the HTML5 web standard. Since Netweaver 754 (software layer SAP_UI, ABAP Platform 1909) a touch enabled user interface is available. The newly released versions usually follow the SAP Fiori design principles. One of its main design features is that the user interface is defined in an entirely declarative manner. Web Dynpro applications can be developed using either the Java (Web Dynpro for Java, WDJ or WD4J) or ABAP (Web Dynpro ABAP, WDA or WD4A) development infrastructure. == Overview == The earliest version of Web Dynpro appeared in 2003 and was based on Java. This variant was released approximately 18 months before the ABAP variant. As of 2010, the Java variant of Web Dynpro was put into maintenance mode. WD follows a design architecture based on an interpretation of the MVC design pattern and uses a model driven development approach ("minimize coding, maximize design"). The Web Dynpro Framework is a server-side runtime environment into which many dedicated "hook methods" are available. The developer then places their own custom coding within these hook methods in order to implement the desired business functionality. These hook methods belong to one of the broad categories of either "life-cycle" and "round-trip"; that is, those methods that are concerned with the life-cycle of a software component (i.e. processing that takes place at start up and shut down etc.), and those methods that are concerned with processing the fixed sequence of events that take place during a client-initiated round trip to the server. Web Dynpro is aimed at the development of business applications that follow standardized UI principles, applications that connect to backend systems and which are scalable. Key Capabilities Declarative way of development: Web Dynpro offers a graphical and declarative means of UI development. UI controls, building blocks, views and windows are modeled, and the business logic can be coded separately. Separation of user interface and business logic: One advantage of Web Dynpro over SAP GUI is the separation between business logic and user interface, and the structured development process with less implementation effort. Support of stateful application: The state of the application is kept in the back-end. This leads to a reduced data transfer from ABAP server to browser and vice versa. Regarding Web Dynpro ABAP there is only one programming language (ABAP) and only one system necessary. Therefore, development can be easier and cost efficient.
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