Integrated Software Systems Corporation (ISSCO), doing business as ISSCO Graphics, was an American software developer and publisher based in San Diego, California, and active from 1970 to 1986. They were best known for their enterprise graphics software packages, including Tellagraf, CueChart and Disspla. == History == ISSCO Graphics had considered acquiring Breakthrough Software, whose software focus involved PC DOS, as a means of getting into the PC arena, but backed off when Computer Associates made an offer to acquire ISSCO. By early 1987 it was reported that "Issco users breathe sigh of relief" that all was well. The ISSCO User's Group was founded in 1976. ISSCO, which was founded in 1970 by Peter Preuss, was acquired by Computer Associates in 1986. == Notable products == === Tellagraf === ISSCO's Tellagraf is an early software package designed to allow end-users to "turn out full color, professional quality charts" with initial results displayed on a screen, modified as needed, and then "a final 'hard-copy' can be made .. or made into 35mm color transparencies for projection onto a screen." Users of Tellagraf often had access to CueChart and Disspla software. Often computer sites having one had all three. Terminals with varying degrees of graphics, such as the DEC's VT100 and Tektronix's Tektronix 4xxx family of text and graphics terminals. were supported, and the software ran on popular computing platforms. Four years are important to Tellagraf's early history: 1978: ease of use 1980: graphic-artist quality 1982: introduction of CueChart, and recognition by IEEE. 1983: "quality graphics enters the mainstream of data processing with ..." Tellegraf was eventually acquired by Computer Associates and renamed CA-Tellegraf. SAS users found it helpful. Universities, research institutes and financial services firms were among early users. === Disspla === Disspla is a package of data plotting subroutines that can be used from high level languages. It was also acquired by Computer Associates. === Tellaplan === In 1983 ISSCO introduced Tellaplan, "a project planning, report and schedule charting system for Tell-A- Graf users in IBM MVS or CMS or Digital Equipment Corp. VAX computers" atop which they built "two visual project management software packages" three years later.
Protocol engineering
Protocol engineering is the application of systematic methods to the development of communication protocols. It uses many of the principles of software engineering, but it is specific to the development of distributed systems. == History == When the first experimental and commercial computer networks were developed in the 1970s, the concept of protocols was not yet well developed. These were the first distributed systems. In the context of the newly adopted layered protocol architecture (see OSI model), the definition of the protocol of a specific layer should be such that any entity implementing that specification in one computer would be compatible with any other computer containing an entity implementing the same specification, and their interactions should be such that the desired communication service would be obtained. On the other hand, the protocol specification should be abstract enough to allow different choices for the implementation on different computers. It was recognized that a precise specification of the expected service provided by the given layer was important. It is important for the verification of the protocol, which should demonstrate that the communication service is provided if both protocol entities implement the protocol specification correctly. This principle was later followed during the standardization of the OSI protocol stack, in particular for the transport layer. It was also recognized that some kind of formalized protocol specification would be useful for the verification of the protocol and for developing implementations, as well as test cases for checking the conformance of an implementation against the specification. While initially mainly finite-state machine were used as (simplified) models of a protocol entity, in the 1980s three formal specification languages were standardized, two by ISO and one by ITU. The latter, called SDL, was later used in industry and has been merged with UML state machines. == Principles == The following are the most important principles for the development of protocols: Layered architecture: A protocol layer at the level n consists of two (or more) entities that have a service interface through which the service of the layer is provided to the users of the protocol, and which uses the service provided by a local entity of level (n-1). The service specification of a layer describes, in an abstract and global view, the behavior of the layer as visible at the service interfaces of the layer. The protocol specification defines the requirements that should be satisfied by each entity implementation. Protocol verification consists of showing that two (or more) entities satisfying the protocol specification will provide at their service interfaces the specified service of that layer. The (verified) protocol specification is used mainly for the following two activities: The development of an entity implementation. Note that the abstract properties of the service interface are defined by the service specification (and also used by the protocol specification), but the detailed nature of the interface can be chosen during the implementation process, separately for each entity. Test suite development for conformance testing. Protocol conformance testing checks that a given entity implementation conforms to the protocol specification. The conformance test cases are developed based on the protocol specification and are applicable to all entity implementations. Therefore standard conformance test suites have been developed for certain protocol standards. == Methods and tools == Tools for the activities of protocol verification, entity implementation and test suite development can be developed when the protocol specification is written in a formalized language which can be understood by the tool. As mentioned, formal specification languages have been proposed for protocol specification, and the first methods and tools where based on finite-state machine models. Reachability analysis was proposed to understand all possible behaviors of a distributed system, which is essential for protocol verification. This was later complemented with model checking. However, finite-state descriptions are not powerful enough to describe constraints between message parameters and the local variables in the entities. Such constraints can be described by the standardized formal specification languages mentioned above, for which powerful tools have been developed. It is in the field of protocol engineering that model-based development was used very early. These methods and tools have later been used for software engineering as well as hardware design, especially for distributed and real-time systems. On the other hand, many methods and tools developed in the more general context of software engineering can also be used of the development of protocols, for instance model checking for protocol verification, and agile methods for entity implementations. == Constructive methods for protocol design == Most protocols are designed by human intuition and discussions during the standardization process. However, some methods have been proposed for using constructive methods possibly supported by tools to automatically derive protocols that satisfy certain properties. The following are a few examples: Semi-automatic protocol synthesis: The user defines all message sending actions of the entities, and the tool derives all necessary reception actions (even if several messages are in transit). Synchronizing protocol: The state transitions of one protocol entity are given by the user, and the method derives the behavior of the other entity such that it remains in states that correspond to the former entity. Protocol derived from service specification: The service specification is given by the user and the method derives a suitable protocol for all entities. Protocol for control applications: The specification of one entity (called the plant - which must be controlled) is given, and the method derives a specification of the other entity such that certain fail states of the plant are never reached and certain given properties of the plant's service interactions are satisfied. This is a case of supervisory control. == Books == Ming T. Liu, Protocol Engineering, Advances in Computers, Volume 29, 1989, Pages 79–195. G.J. Holzmann, Design and Validation of Computer Protocols, Prentice Hall, 1991. H. König, Protocol Engineering, Springer, 2012. M. Popovic, Communication Protocol Engineering, CRC Press, 2nd Ed. 2018. P. Venkataram, S.S. Manvi, B.S. Babu, Communication Protocol Engineering, 2014.
Sepp Hochreiter
Josef "Sepp" Hochreiter (born 14 February 1967) is a German computer scientist. Since 2018 he has led the Institute for Machine Learning at the Johannes Kepler University of Linz after having led the Institute of Bioinformatics from 2006 to 2018. In 2017 he became the head of the Linz Institute of Technology (LIT) AI Lab. Hochreiter is also a founding director of the Institute of Advanced Research in Artificial Intelligence (IARAI). Previously, he was at Technische Universität Berlin, at University of Colorado Boulder, and at the Technical University of Munich. He is a chair of the Critical Assessment of Massive Data Analysis (CAMDA) conference. Hochreiter has made contributions in the fields of machine learning, deep learning and bioinformatics, most notably the development of the long short-term memory (LSTM) neural network architecture, but also in meta-learning, reinforcement learning and biclustering with application to bioinformatics data. == Scientific career == === Long short-term memory (LSTM) === Hochreiter developed the long short-term memory (LSTM) neural network architecture in his diploma thesis in 1991 leading to the main publication in 1997. LSTM overcomes the problem of numerical instability in training recurrent neural networks (RNNs) that prevents them from learning from long sequences (vanishing or exploding gradient). In 2007, Hochreiter and others successfully applied LSTM with an optimized architecture to very fast protein homology detection without requiring a sequence alignment. LSTM networks have also been used in Google Voice for transcription and search, and in the Google Allo chat app for generating response suggestion with low latency. === Other machine learning contributions === Beyond LSTM, Hochreiter has developed "Flat Minimum Search" to increase the generalization of neural networks and introduced rectified factor networks (RFNs) for sparse coding which have been applied in bioinformatics and genetics. Hochreiter introduced modern Hopfield networks with continuous states and applied them to the task of immune repertoire classification. Hochreiter worked with Jürgen Schmidhuber in the field of reinforcement learning on actor-critic systems that learn by "backpropagation through a model". Hochreiter has been involved in the development of factor analysis methods with application to bioinformatics, including FABIA for biclustering, HapFABIA for detecting short segments of identity by descent and FARMS for preprocessing and summarizing high-density oligonucleotide DNA microarrays to analyze RNA gene expression. In 2006, Hochreiter and others proposed an extension of the support vector machine (SVM), the "Potential Support Vector Machine" (PSVM), which can be applied to non-square kernel matrices and can be used with kernels that are not positive definite. Hochreiter and his collaborators have applied PSVM to feature selection, including gene selection for microarray data. == Awards == Hochreiter was awarded the IEEE CIS Neural Networks Pioneer Prize in 2021 for his work on LSTM.
Separating words problem
In theoretical computer science, the separating words problem is the problem of finding the smallest deterministic finite automaton that behaves differently on two given strings, meaning that it accepts one of the two strings and rejects the other string. It is an open problem how large such an automaton must be, in the worst case, as a function of the length of the input strings. == Example == The two strings 0010 and 1000 may be distinguished from each other by a three-state automaton in which the transitions from the start state go to two different states, both of which are terminal in the sense that subsequent transitions from these two states always return to the same state. The state of this automaton records the first symbol of the input string. If one of the two terminal states is accepting and the other is rejecting, then the automaton will accept only one of the strings 0010 and 1000. However, these two strings cannot be distinguished by any automaton with fewer than three states. == Simplifying assumptions == For proving bounds on this problem, it may be assumed without loss of generality that the inputs are strings over a two-letter alphabet. For, if two strings over a larger alphabet differ then there exists a string homomorphism that maps them to binary strings of the same length that also differ. Any automaton that distinguishes the binary strings can be translated into an automaton that distinguishes the original strings, without any increase in the number of states. It may also be assumed that the two strings have equal length. For strings of unequal length, there always exists a prime number p whose value is logarithmic in the smaller of the two input lengths, such that the two lengths are different modulo p. An automaton that counts the length of its input modulo p can be used to distinguish the two strings from each other in this case. Therefore, strings of unequal lengths can always be distinguished from each other by automata with few states. == History and bounds == The problem of bounding the size of an automaton that distinguishes two given strings was first formulated by Goralčík & Koubek (1986), who showed that the automaton size is always sublinear. Later, Robson (1989) proved the upper bound O(n2/5(log n)3/5) on the automaton size that may be required. This was improved by Chase (2020) to O(n1/3(log n)7). There exist pairs of inputs that are both binary strings of length n for which any automaton that distinguishes the inputs must have size Ω(log n). Closing the gap between this lower bound and Chase's upper bound remains an open problem. Jeffrey Shallit has offered a prize of 100 British pounds for any improvement to Robson's upper bound. == Special cases == Several special cases of the separating words problem are known to be solvable using few states: If two binary words have differing numbers of zeros or ones, then they can be distinguished from each other by counting their Hamming weights modulo a prime of logarithmic size, using a logarithmic number of states. More generally, if a pattern of length k appears a different number of times in the two words, they can be distinguished from each other using O(k log n) states. If two binary words differ from each other within their first or last k positions, they can be distinguished from each other using k + O(1) states. This implies that almost all pairs of binary words can be distinguished from each other with a logarithmic number of states, because only a polynomially small fraction of pairs have no difference in their initial O(log n) positions. If two binary words have Hamming distance d, then there exists a prime p with p = O(d log n) and a position i at which the two strings differ, such that i is not equal modulo p to the position of any other difference. By computing the parity of the input symbols at positions congruent to i modulo p, it is possible to distinguish the words using an automaton with O(d log n) states.
Nabil Ali
Nabil Ali Mohammed Abd AL Azeez (Arabic:نبيل علي) (3 January 1938 – 27 January 2016) was an Egyptian scientist, writer, and intellectual who worked in the field of natural language processing and computational linguistics. Ali is considered a pioneer of Arabic language computing, making significant innovations in early computational linguistics. == Education and career == Ali earned a bachelor's degree in Aeronautical Engineering in 1960, and a master's degree in 1967. In 1971, he earned a PhD in Aeronautics. From 1961 to 1972 Ali worked as an engineering officer in the Egyptian Air Force, specializing in maintenance and training. In 1972, he shifted focus to computing, and from 1972 to 1977 he worked as a computer manager at Egyptair. While in this position, Ali introduced the first automated reservation system for airlines in the Arab world. He later held various computing positions in Egypt, Kuwait, Europe, Canada and the US. Ali started working for Sakhr Software, an Arabic language technology company, in 1983. From 1985 to 1999, he was vice president of Sakhr's council for Research and Development. As a director of the Multilingual Advanced Systems Foundation and project manager at the Egyptian National Company for Scientific and Technical Information, Ali did extensive research on information culture and artificial intelligence relating to the Arabic language. Over the course of his career, Ali developed more than 20 educational programs relating to computational linguistics. He developed the first Arabic lexical database and the first knowledge base for Arabic poetry, as well as many other pieces of Arabic language software. == Awards == 1994: General Book Authority Award for Best Book (in the field of future studies). 2003: General Book Authority Award for Best Culture Book (in the field of "Challenges of the Information Age"). 2007: General Book Authority "Innovation in Information Technology" Award. 2012: King Faisal International Award, with Professor Ali Helmy Mousa, in the field of computer processing of the Arabic Language. == Works == Arabic Language and Computer (Research study), Dar Localization, 1988. Al Arab and the Information Age, Knowledge World Series No. 184, April 1994. Arab Culture and the Information Age: A Vision for the Future of Arab Culture Discourse, World of Knowledge Series, No. 265 January 2001. The Digital Gap: an Arab Vision for a Knowledge Society (in partnership with Dr. Nadia Hegazy), World of Knowledge Series, No. 318 August 2005. The Arab Mind and the Knowledge Society: Manifestations of the Crisis and Suggestions for Solutions, Part 1, The World of Knowledge Series, No. 369, November 2009. The Arab Mind and the Knowledge Society: Manifestations of the Crisis and Suggestions for Solutions, Part 2, The World of Knowledge Series, No. 370, December 2009. == Tribute == On 3 January 2020, Google Doodle celebrated Nabil Ali Mohamed's 82nd Birthday.
Highway network
In machine learning, the Highway Network was the first working very deep feedforward neural network with hundreds of layers, much deeper than previous neural networks. It uses skip connections modulated by learned gating mechanisms to regulate information flow, inspired by long short-term memory (LSTM) recurrent neural networks. The advantage of the Highway Network over other deep learning architectures is its ability to overcome or partially prevent the vanishing gradient problem, thus improving its optimization. Gating mechanisms are used to facilitate information flow across the many layers ("information highways"). Highway Networks have found use in text sequence labeling and speech recognition tasks. In 2014, the state of the art was training deep neural networks with 20 to 30 layers. Stacking too many layers led to a steep reduction in training accuracy, known as the "degradation" problem. In 2015, two techniques were developed to train such networks: the Highway Network (published in May), and the residual neural network, or ResNet (December). ResNet behaves like an open-gated Highway Net. == Model == The model has two gates in addition to the H ( W H , x ) {\displaystyle H(W_{H},x)} gate: the transform gate T ( W T , x ) {\displaystyle T(W_{T},x)} and the carry gate C ( W C , x ) {\displaystyle C(W_{C},x)} . The latter two gates are non-linear transfer functions (specifically sigmoid by convention). The function H {\displaystyle H} can be any desired transfer function. The carry gate is defined as: C ( W C , x ) = 1 − T ( W T , x ) {\displaystyle C(W_{C},x)=1-T(W_{T},x)} while the transform gate is just a gate with a sigmoid transfer function. == Structure == The structure of a hidden layer in the Highway Network follows the equation: y = H ( x , W H ) ⋅ T ( x , W T ) + x ⋅ C ( x , W C ) = H ( x , W H ) ⋅ T ( x , W T ) + x ⋅ ( 1 − T ( x , W T ) ) {\displaystyle {\begin{aligned}y=H(x,W_{H})\cdot T(x,W_{T})+x\cdot C(x,W_{C})\\=H(x,W_{H})\cdot T(x,W_{T})+x\cdot (1-T(x,W_{T}))\end{aligned}}} == Related work == Sepp Hochreiter analyzed the vanishing gradient problem in 1991 and attributed to it the reason why deep learning did not work well. To overcome this problem, Long Short-Term Memory (LSTM) recurrent neural networks have residual connections with a weight of 1.0 in every LSTM cell (called the constant error carrousel) to compute y t + 1 = F ( x t ) + x t {\textstyle y_{t+1}=F(x_{t})+x_{t}} . During backpropagation through time, this becomes the residual formula y = F ( x ) + x {\textstyle y=F(x)+x} for feedforward neural networks. This enables training very deep recurrent neural networks with a very long time span t. A later LSTM version published in 2000 modulates the identity LSTM connections by so-called "forget gates" such that their weights are not fixed to 1.0 but can be learned. In experiments, the forget gates were initialized with positive bias weights, thus being opened, addressing the vanishing gradient problem. As long as the forget gates of the 2000 LSTM are open, it behaves like the 1997 LSTM. The Highway Network of May 2015 applies these principles to feedforward neural networks. It was reported to be "the first very deep feedforward network with hundreds of layers". It is like a 2000 LSTM with forget gates unfolded in time, while the later Residual Nets have no equivalent of forget gates and are like the unfolded original 1997 LSTM. If the skip connections in Highway Networks are "without gates," or if their gates are kept open (activation 1.0), they become Residual Networks. The residual connection is a special case of the "short-cut connection" or "skip connection" by Rosenblatt (1961) and Lang & Witbrock (1988) which has the form x ↦ F ( x ) + A x {\displaystyle x\mapsto F(x)+Ax} . Here the randomly initialized weight matrix A does not have to be the identity mapping. Every residual connection is a skip connection, but almost all skip connections are not residual connections. The original Highway Network paper not only introduced the basic principle for very deep feedforward networks, but also included experimental results with 20, 50, and 100 layers networks, and mentioned ongoing experiments with up to 900 layers. Networks with 50 or 100 layers had lower training error than their plain network counterparts, but no lower training error than their 20 layers counterpart (on the MNIST dataset, Figure 1 in ). No improvement on test accuracy was reported with networks deeper than 19 layers (on the CIFAR-10 dataset; Table 1 in ). The ResNet paper, however, provided strong experimental evidence of the benefits of going deeper than 20 layers. It argued that the identity mapping without modulation is crucial and mentioned that modulation in the skip connection can still lead to vanishing signals in forward and backward propagation (Section 3 in ). This is also why the forget gates of the 2000 LSTM were initially opened through positive bias weights: as long as the gates are open, it behaves like the 1997 LSTM. Similarly, a Highway Net whose gates are opened through strongly positive bias weights behaves like a ResNet. The skip connections used in modern neural networks (e.g., Transformers) are dominantly identity mappings.
Noisy channel model
The noisy channel model is a framework used in spell checkers, question answering, speech recognition, and machine translation. In this model, the goal is to find the intended word given a word where the letters have been scrambled in some manner. == In spell-checking == See Chapter B of. Given an alphabet Σ {\displaystyle \Sigma } , let Σ ∗ {\displaystyle \Sigma ^{}} be the set of all finite strings over Σ {\displaystyle \Sigma } . Let the dictionary D {\displaystyle D} of valid words be some subset of Σ ∗ {\displaystyle \Sigma ^{}} , i.e., D ⊆ Σ ∗ {\displaystyle D\subseteq \Sigma ^{}} . The noisy channel is the matrix Γ w s = Pr ( s | w ) {\displaystyle \Gamma _{ws}=\Pr(s|w)} , where w ∈ D {\displaystyle w\in D} is the intended word and s ∈ Σ ∗ {\displaystyle s\in \Sigma ^{}} is the scrambled word that was actually received. The goal of the noisy channel model is to find the intended word given the scrambled word that was received. The decision function σ : Σ ∗ → D {\displaystyle \sigma :\Sigma ^{}\to D} is a function that, given a scrambled word, returns the intended word. Methods of constructing a decision function include the maximum likelihood rule, the maximum a posteriori rule, and the minimum distance rule. In some cases, it may be better to accept the scrambled word as the intended word rather than attempt to find an intended word in the dictionary. For example, the word schönfinkeling may not be in the dictionary, but might in fact be the intended word. === Example === Consider the English alphabet Σ = { a , b , c , . . . , y , z , A , B , . . . , Z , . . . } {\displaystyle \Sigma =\{a,b,c,...,y,z,A,B,...,Z,...\}} . Some subset D ⊆ Σ ∗ {\displaystyle D\subseteq \Sigma ^{}} makes up the dictionary of valid English words. There are several mistakes that may occur while typing, including: Missing letters, e.g., leter instead of letter Accidental letter additions, e.g., misstake instead of mistake Swapping letters, e.g., recieved instead of received Replacing letters, e.g., fimite instead of finite To construct the noisy channel matrix Γ {\displaystyle \Gamma } , we must consider the probability of each mistake, given the intended word ( Pr ( s | w ) {\displaystyle \Pr(s|w)} for all w ∈ D {\displaystyle w\in D} and s ∈ Σ ∗ {\displaystyle s\in \Sigma ^{}} ). These probabilities may be gathered, for example, by considering the Damerau–Levenshtein distance between s {\displaystyle s} and w {\displaystyle w} or by comparing the draft of an essay with one that has been manually edited for spelling. == In machine translation == One naturally wonders if the problem of translation could conceivably be treated as a problem in cryptography. When I look at an article in Russian, I say: 'This is really written in English, but it has been coded in some strange symbols. I will now proceed to decode. See chapter 1, and chapter 25 of. Suppose we want to translate a foreign language to English, we could model P ( E | F ) {\displaystyle P(E|F)} directly: the probability that we have English sentence E given foreign sentence F, then we pick the most likely one E ^ = arg max E P ( E | F ) {\displaystyle {\hat {E}}=\arg \max _{E}P(E|F)} . However, by Bayes law, we have the equivalent equation: E ^ = argmax E ∈ English P ( F ∣ E ) ⏞ translation model P ( E ) ⏞ language model {\displaystyle {\hat {E}}={\underset {E\in {\text{ English }}}{\operatorname {argmax} }}\overbrace {P(F\mid E)} ^{\text{translation model }}\overbrace {P(E)} ^{\text{language model}}} The benefit of the noisy-channel model is in terms of data: If collecting a parallel corpus is costly, then we would have only a small parallel corpus, so we can only train a moderately good English-to-foreign translation model, and a moderately good foreign-to-English translation model. However, we can collect a large corpus in the foreign language only, and a large corpus in the English language only, to train two good language models. Combining these four models, we immediately get a good English-to-foreign translator and a good foreign-to-English translator. The cost of noisy-channel model is that using Bayesian inference is more costly than using a translation model directly. Instead of reading out the most likely translation by arg max E P ( E | F ) {\displaystyle \arg \max _{E}P(E|F)} , it would have to read out predictions by both the translation model and the language model, multiply them, and search for the highest number. == In speech recognition == Speech recognition can be thought of as translating from a sound-language to a text-language. Consequently, we have T ^ = argmax T ∈ Text P ( S ∣ T ) ⏞ speech model P ( T ) ⏞ language model {\displaystyle {\hat {T}}={\underset {T\in {\text{ Text }}}{\operatorname {argmax} }}\overbrace {P(S\mid T)} ^{\text{speech model }}\overbrace {P(T)} ^{\text{language model}}} where P ( S | T ) {\displaystyle P(S|T)} is the probability that a speech sound S is produced if the speaker is intending to say text T. Intuitively, this equation states that the most likely text is a text that's both a likely text in the language, and produces the speech sound with high probability. The utility of the noisy-channel model is not in capacity. Theoretically, any noisy-channel model can be replicated by a direct P ( T | S ) {\displaystyle P(T|S)} model. However, the noisy-channel model factors the model into two parts which are appropriate for the situation, and consequently it is generally more well-behaved. When a human speaks, it does not produce the sound directly, but first produces the text it wants to speak in the language centers of the brain, then the text is translated into sound by the motor cortex, vocal cords, and other parts of the body. The noisy-channel model matches this model of the human, and so it is appropriate. This is justified in the practical success of noisy-channel model in speech recognition. === Example === Consider the sound-language sentence (written in IPA for English) S = aɪ wʊd laɪk wʌn tuː. There are three possible texts T 1 , T 2 , T 3 {\displaystyle T_{1},T_{2},T_{3}} : T 1 = {\displaystyle T_{1}=} I would like one to. T 2 = {\displaystyle T_{2}=} I would like one too. T 3 = {\displaystyle T_{3}=} I would like one two. that are equally likely, in the sense that P ( S | T 1 ) = P ( S | T 2 ) = P ( S | T 3 ) {\displaystyle P(S|T_{1})=P(S|T_{2})=P(S|T_{3})} . With a good English language model, we would have P ( T 2 ) > P ( T 1 ) > P ( T 3 ) {\displaystyle P(T_{2})>P(T_{1})>P(T_{3})} , since the second sentence is grammatical, the first is not quite, but close to a grammatical one (such as "I would like one to [go]."), while the third one is far from grammatical. Consequently, the noisy-channel model would output T 2 {\displaystyle T_{2}} as the best transcription.