AI App Gemini

AI App Gemini — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

• 

  Zhura

  Zhura ( ZUR-ə) is a free, web-based screenwriting software application for writing and formatting screenplays to the film industry standard, as well as other formats. Zhura allows users to collaborate on scripts in public or private groups and uses Creative Commons Licensing for all work in the public workspace. On March 29, 2010, Zhura announced its merger with Scripped. Scripped's CEO, Sunil Rajaraman, remains the company's Chief Executive Officer (CEO) as of 2022. The Zhura CEO was Eric MacDonald, a former Cascade Communications engineer. Scripped later closed on April 1, 2015 after a catastrophic, irrecoverable data loss. == Script editor == Screenplay Template – The script editor provides a built-in screenplay template which formats the document to a standard for scripts as recommended by the AMPAS. The screenplay document is composed of seven elements: scene, action, character, dialogue, parenthetical, transition, and shot (see image). Each element has a specific style to which the script editor conforms as you type.Script Formats – Other major script formats for stage play, sitcom, audio drama and comic book are also supported as well as the ability to switch between them.Auto-Complete – Characters, scene headings and custom transitions are “remembered” as they are written and “recalled” with tab-completion when a writer starts a new character, scene heading or transition, respectively.Multiple Editors – With a collaborative editing model comparable to Google Docs, two or more users can edit the same script simultaneously, regardless of having a different operating system or web browser. Import/Export – A screenplay written in another program can be imported into the script editor and automatically conformed to the screenplay template. The closer the original script has adhered to the standard format, the better it will appear when imported. Supported import/export formats include Text (.txt) Word (.doc) Rich Text (.rtf) and OpenDocument (.odt). Scripts can also be exported as a PDF file with additional options.Tracking Changes – Similar to the “tracking” feature in Microsoft Word, a user can review all changes made to a script in the revision history as well as highlight the contributions of each writer. Offline Mode – The Google Gears-based offline functionality is in the process of being updated and is not available for new subscribers, according to the company founders. == Community == Scripped supports typical social networking features such as discussion boards, comments, user profiles, public and private writing groups, internal web mail and instant messaging within the script editor. There is also the option to share scripts with others outside of Scripped by making scripts externally viewable. Scripped is made up entirely of user-generated scripts that other users can share, critique and edit, offering creative support to a community of writers. == Licensing of user-created work == There are three types of work-spaces on Scripped (personal, group and public) with unique copyright and licensing management for the work created in each area. Any work a user originates may be moved from the personal area to a public or group area at any time. Once another user edits a script, however, it cannot be moved into the originator’s personal area. Personal Workspace – Any script created or video uploaded in the user’s personal workspace remains copyrighted to that user. Until the user moves that script or video from their personal area into a group or public area, no other user shares a copyright or license to that work. Private Group Workspace – The copyright to any script created or video uploaded in a private group workspace is allocated by the individual members of the group, however they see fit. Public Workspace – Any script created or video uploaded in the public workspace is assigned a Creative Commons license by the originator of that work. The originator of a script may select one of four Creative Commons licenses before introducing that script to the public. The selection of the license is determined by what the author wants to allow others to do with the work. Below is a list of Creative Commons licenses available for all scripts and videos in the public workspace. Share Alike (BY-SA) This license lets others remix, tweak, and build upon your work even for commercial reasons, as long as they credit the original user and license their new creations under the identical terms. This license is often compared to open source software licenses. All new works based on the original user's will carry the same license, so any derivatives will also allow commercial use. No Derivatives (BY-ND) This license allows for redistribution, commercial and non-commercial, as long as it is passed along unchanged and in whole, with credit to the original user. Non-Commercial, No Derivatives (BY-NC-ND) This license is the most restrictive of the four licenses, allowing redistribution. This license is often called the "free advertising" license because it allows others to download the original user work and share them with others as long as they mention the original user and link back to them, but they can't change them in any way or use them commercially. Non-Commercial, Share Alike (BY-NC-SA) This license lets others remix, tweak, and build upon the original user's work non-commercially, as long as they credit the original user and license their new creations under the identical terms. Others can download and redistribute the original user's work just like the BY-NC-ND license, but they can also translate, make remixes, and produce new stories based on the original user's work. All new work based on the original user's work will carry the same license, so any derivatives will also be non-commercial in nature. == Events == In April 2008, Zhura partnered with Improv Asylum, a comedy troupe in Boston, Massachusetts to produce a live sketch comedy show called "You Wrote It, Live" entirely written by the public on Zhura. Another show was produced in June.

  Read more →
• 

  UIMA

  UIMA ( yoo-EE-mə), short for Unstructured Information Management Architecture, is an OASIS standard for content analytics, originally developed at IBM. It provides a component software architecture for the development, discovery, composition, and deployment of multi-modal analytics for the analysis of unstructured information and integration with search technologies. == Structure == The UIMA architecture can be thought of in four dimensions: It specifies component interfaces in an analytics pipeline. It describes a set of design patterns. It suggests two data representations: an in-memory representation of annotations for high-performance analytics and an XML representation of annotations for integration with remote web services. It suggests development roles allowing tools to be used by users with diverse skills. == Implementations and uses == Apache UIMA, a reference implementation of UIMA, is maintained by the Apache Software Foundation. UIMA is used in a number of software projects: IBM Research's Watson uses UIMA for analyzing unstructured data. The Clinical Text Analysis and Knowledge Extraction System (Apache cTAKES) is a UIMA-based system for information extraction from medical records. DKPro Core is a collection of reusable UIMA components for general-purpose natural language processing.

  Read more →
• 

  Policy gradient method

  Policy gradient methods are a class of reinforcement learning algorithms and a sub-class of policy optimization methods. Unlike value-based methods which learn a value function to derive a policy, policy optimization methods directly learn a policy function π {\displaystyle \pi } that selects actions without consulting a value function. For policy gradient to apply, the policy function π θ {\displaystyle \pi _{\theta }} is parameterized by a differentiable parameter θ {\displaystyle \theta } . == Overview == In policy-based RL, the actor is a parameterized policy function π θ {\displaystyle \pi _{\theta }} , where θ {\displaystyle \theta } are the parameters of the actor. The actor takes as argument the state of the environment s {\displaystyle s} and produces a probability distribution π θ ( ⋅ ∣ s ) {\displaystyle \pi _{\theta }(\cdot \mid s)} . If the action space is discrete, then ∑ a π θ ( a ∣ s ) = 1 {\displaystyle \sum _{a}\pi _{\theta }(a\mid s)=1} . If the action space is continuous, then ∫ a π θ ( a ∣ s ) d a = 1 {\displaystyle \int _{a}\pi _{\theta }(a\mid s)\mathrm {d} a=1} . The goal of policy optimization is to find some θ {\displaystyle \theta } that maximizes the expected episodic reward J ( θ ) {\displaystyle J(\theta )} : J ( θ ) = E π θ [ ∑ t = 0 T γ t R t | S 0 = s 0 ] {\displaystyle J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\gamma ^{t}R_{t}{\Big |}S_{0}=s_{0}\right]} where γ {\displaystyle \gamma } is the discount factor, R t {\displaystyle R_{t}} is the reward at step t {\displaystyle t} , s 0 {\displaystyle s_{0}} is the starting state, and T {\displaystyle T} is the time-horizon (which can be infinite). The policy gradient is defined as ∇ θ J ( θ ) {\displaystyle \nabla _{\theta }J(\theta )} . Different policy gradient methods stochastically estimate the policy gradient in different ways. The goal of any policy gradient method is to iteratively maximize J ( θ ) {\displaystyle J(\theta )} by gradient ascent. Since the key part of any policy gradient method is the stochastic estimation of the policy gradient, they are also studied under the title of "Monte Carlo gradient estimation". == REINFORCE == === Policy gradient === The REINFORCE algorithm, introduced by Ronald J. Williams in 1992, was the first policy gradient method. It is based on the identity for the policy gradient ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t ∣ S t ) ∑ t = 0 T ( γ t R t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})\;\sum _{t=0}^{T}(\gamma ^{t}R_{t}){\Big |}S_{0}=s_{0}\right]} which can be improved via the "causality trick" ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t ∣ S t ) ∑ τ = t T ( γ τ R τ ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau }){\Big |}S_{0}=s_{0}\right]} Thus, we have an unbiased estimator of the policy gradient: ∇ θ J ( θ ) ≈ 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t , n ∣ S t , n ) ∑ τ = t T ( γ τ − t R τ , n ) ] {\displaystyle \nabla _{\theta }J(\theta )\approx {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t,n}\mid S_{t,n})\sum _{\tau =t}^{T}(\gamma ^{\tau -t}R_{\tau ,n})\right]} where the index n {\displaystyle n} ranges over N {\displaystyle N} rollout trajectories using the policy π θ {\displaystyle \pi _{\theta }} . The score function ∇ θ ln ⁡ π θ ( A t ∣ S t ) {\displaystyle \nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})} can be interpreted as the direction in the parameter space that increases the probability of taking action A t {\displaystyle A_{t}} in state S t {\displaystyle S_{t}} . The policy gradient, then, is a weighted average of all possible directions to increase the probability of taking any action in any state, but weighted by reward signals, so that if taking a certain action in a certain state is associated with high reward, then that direction would be highly reinforced, and vice versa. === Algorithm === The REINFORCE algorithm is a loop: Rollout N {\displaystyle N} trajectories in the environment, using π θ t {\displaystyle \pi _{\theta _{t}}} as the policy function. Compute the policy gradient estimation: g i ← 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ t ln ⁡ π θ ( A t , n ∣ S t , n ) ∑ τ = t T ( γ τ R τ , n ) ] {\displaystyle g_{i}\leftarrow {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta _{t}}\ln \pi _{\theta }(A_{t,n}\mid S_{t,n})\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau ,n})\right]} Update the policy by gradient ascent: θ i + 1 ← θ i + α i g i {\displaystyle \theta _{i+1}\leftarrow \theta _{i}+\alpha _{i}g_{i}} Here, α i {\displaystyle \alpha _{i}} is the learning rate at update step i {\displaystyle i} . == Variance reduction == REINFORCE is an on-policy algorithm, meaning that the trajectories used for the update must be sampled from the current policy π θ {\displaystyle \pi _{\theta }} . This can lead to high variance in the updates, as the returns R ( τ ) {\displaystyle R(\tau )} can vary significantly between trajectories. Many variants of REINFORCE have been introduced, under the title of variance reduction. === REINFORCE with baseline === A common way for reducing variance is the REINFORCE with baseline algorithm, based on the following identity: ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t | S t ) ( ∑ τ = t T ( γ τ R τ ) − b ( S t ) ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })-b(S_{t})\right){\Big |}S_{0}=s_{0}\right]} for any function b : States → R {\displaystyle b:{\text{States}}\to \mathbb {R} } . This can be proven by applying the previous lemma. The algorithm uses the modified gradient estimator g i ← 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ t ln ⁡ π θ ( A t , n | S t , n ) ( ∑ τ = t T ( γ τ R τ , n ) − b i ( S t , n ) ) ] {\displaystyle g_{i}\leftarrow {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta _{t}}\ln \pi _{\theta }(A_{t,n}|S_{t,n})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau ,n})-b_{i}(S_{t,n})\right)\right]} and the original REINFORCE algorithm is the special case where b i ≡ 0 {\displaystyle b_{i}\equiv 0} . === Actor-critic methods === If b i {\textstyle b_{i}} is chosen well, such that b i ( S t ) ≈ ∑ τ = t T ( γ τ R τ ) = γ t V π θ i ( S t ) {\textstyle b_{i}(S_{t})\approx \sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })=\gamma ^{t}V^{\pi _{\theta _{i}}}(S_{t})} , this could significantly decrease variance in the gradient estimation. That is, the baseline should be as close to the value function V π θ i ( S t ) {\displaystyle V^{\pi _{\theta _{i}}}(S_{t})} as possible, approaching the ideal of: ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t | S t ) ( ∑ τ = t T ( γ τ R τ ) − γ t V π θ ( S t ) ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })-\gamma ^{t}V^{\pi _{\theta }}(S_{t})\right){\Big |}S_{0}=s_{0}\right]} Note that, as the policy π θ t {\displaystyle \pi _{\theta _{t}}} updates, the value function V π θ i ( S t ) {\displaystyle V^{\pi _{\theta _{i}}}(S_{t})} updates as well, so the baseline should also be updated. One common approach is to train a separate function that estimates the value function, and use that as the baseline. This is one of the actor-critic methods, where the policy function is the actor and the value function is the critic. The Q-function Q π {\displaystyle Q^{\pi }} can also be used as the critic, since ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T γ t ∇ θ ln ⁡ π θ ( A t | S t ) ⋅ Q π θ ( S t , A t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\sum _{0\leq t\leq T}\gamma ^{t}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\cdot Q^{\pi _{\theta }}(S_{t},A_{t}){\Big |}S_{0}=s_{0}\right]} by a similar argument using the tower law. Subtracting the value function as a baseline, we find that the advantage function A π ( S , A ) = Q π ( S , A ) − V π ( S ) {\displaystyle A^{\pi }(S,A)=Q^{\pi }(S,A)-V^{\pi }(S)} can be used as the critic as well: ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T γ t ∇ θ ln ⁡ π θ ( A t | S t ) ⋅ A π θ ( S t , A t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\sum _{0\leq t\leq T}\gamma ^{t}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\cdot A^{\pi _{\theta }}(S_{t},A_{t}){\Big |}S_{0}=s_{0}\right]} In summary, there are many unbiased estimators for ∇ θ J θ {\textstyle \nabla _{\theta }J_{\theta }} , all in the form of: ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T ∇ θ ln ⁡ π θ ( A t | S t ) ⋅ Ψ t | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\su

  Read more →
• 

  Absorbing Markov chain

  In the mathematical theory of probability, an absorbing Markov chain is a Markov chain in which every state can reach an absorbing state. An absorbing state is a state that, once entered, cannot be left. Like general Markov chains, there can be continuous-time absorbing Markov chains with an infinite state space. However, this article concentrates on the discrete-time discrete-state-space case. == Formal definition == A Markov chain is an absorbing chain if there is at least one absorbing state and it is possible to go from any state to at least one absorbing state in a finite number of steps. In an absorbing Markov chain, a state that is not absorbing is called transient. === Canonical form === Let an absorbing Markov chain with transition matrix P have t transient states and r absorbing states. The rows of P represent sources, while columns represent destinations. By ordering the transient states before the absorbing states, it can be assumed that P has the form P = [ Q R 0 I r ] , {\displaystyle P={\begin{bmatrix}Q&R\\\mathbf {0} &I_{r}\end{bmatrix}},} where Q is a t-by-t matrix, R is a nonzero t-by-r matrix, 0 is an r-by-t zero matrix, and Ir is the r-by-r identity matrix. Thus, Q describes the probability of transitioning from some transient state to another while R describes the probability of transitioning from some transient state to some absorbing state. The probability of transitioning from i to j in exactly k steps is the (i,j)-entry of Pk, further computed below. When considering only transient states, the probability is found in the upper left of Pk, the (i,j)-entry of Qk. == Fundamental matrix == === Expected number of visits to a transient state === A basic property about an absorbing Markov chain is the expected number of visits to a transient state j starting from a transient state i (before being absorbed). This can be established to be given by the (i, j) entry of so-called fundamental matrix N, obtained by summing Qk for all k (from 0 to ∞). It can be proven that N := ∑ k = 0 ∞ Q k = ( I t − Q ) − 1 , {\displaystyle N:=\sum _{k=0}^{\infty }Q^{k}=(I_{t}-Q)^{-1},} where It is the t-by-t identity matrix. The computation of this formula is the matrix equivalent of the geometric series of scalars, ∑ k = 0 ∞ q k = 1 1 − q {\displaystyle {\textstyle \sum }_{k=0}^{\infty }q^{k}={\tfrac {1}{1-q}}} . With the matrix N in hand, also other properties of the Markov chain are easy to obtain. === Expected number of steps before being absorbed === The expected number of steps before being absorbed in any absorbing state, when starting in transient state i can be computed via a sum over transient states. The value is given by the ith entry of the vector t := N 1 , {\displaystyle \mathbf {t} :=N\mathbf {1} ,} where 1 is a length-t column vector whose entries are all 1. === Absorbing probabilities === By induction, P k = [ Q k ( I t − Q k ) N R 0 I r ] . {\displaystyle P^{k}={\begin{bmatrix}Q^{k}&(I_{t}-Q^{k})NR\\\mathbf {0} &I_{r}\end{bmatrix}}.} The probability of eventually being absorbed in the absorbing state j when starting from transient state i is given by the (i,j)-entry of the matrix B := N R {\displaystyle B:=NR} . The number of columns of this matrix equals the number of absorbing states r. An approximation of those probabilities can also be obtained directly from the (i,j)-entry of P k {\displaystyle P^{k}} for a large enough value of k, when i is the index of a transient, and j the index of an absorbing state. This is because ( lim k → ∞ P k ) i , t + j = B i , j {\displaystyle \left(\lim _{k\to \infty }P^{k}\right)_{i,t+j}=B_{i,j}} . === Transient visiting probabilities === The probability of visiting transient state j when starting at a transient state i is the (i,j)-entry of the matrix H := ( N − I t ) ( N dg ) − 1 , {\displaystyle H:=(N-I_{t})(N_{\operatorname {dg} })^{-1},} where Ndg is the diagonal matrix with the same diagonal as N. === Variance on number of transient visits === The variance on the number of visits to a transient state j with starting at a transient state i (before being absorbed) is the (i,j)-entry of the matrix N 2 := N ( 2 N dg − I t ) − N sq , {\displaystyle N_{2}:=N(2N_{\operatorname {dg} }-I_{t})-N_{\operatorname {sq} },} where Nsq is the Hadamard product of N with itself (i.e. each entry of N is squared). === Variance on number of steps === The variance on the number of steps before being absorbed when starting in transient state i is the ith entry of the vector ( 2 N − I t ) t − t sq , {\displaystyle (2N-I_{t})\mathbf {t} -\mathbf {t} _{\operatorname {sq} },} where tsq is the Hadamard product of t with itself (i.e., as with Nsq, each entry of t is squared). == Examples == === String generation === Consider the process of repeatedly flipping a fair coin until the sequence (heads, tails, heads) appears. This process is modeled by an absorbing Markov chain with transition matrix P = [ 1 / 2 1 / 2 0 0 0 1 / 2 1 / 2 0 1 / 2 0 0 1 / 2 0 0 0 1 ] . {\displaystyle P={\begin{bmatrix}1/2&1/2&0&0\\0&1/2&1/2&0\\1/2&0&0&1/2\\0&0&0&1\end{bmatrix}}.} The first state represents the empty string, the second state the string "H", the third state the string "HT", and the fourth state the string "HTH". Although in reality, the coin flips cease after the string "HTH" is generated, the perspective of the absorbing Markov chain is that the process has transitioned into the absorbing state representing the string "HTH" and, therefore, cannot leave. For this absorbing Markov chain, the fundamental matrix is N = ( I − Q ) − 1 = ( [ 1 0 0 0 1 0 0 0 1 ] − [ 1 / 2 1 / 2 0 0 1 / 2 1 / 2 1 / 2 0 0 ] ) − 1 = [ 1 / 2 − 1 / 2 0 0 1 / 2 − 1 / 2 − 1 / 2 0 1 ] − 1 = [ 4 4 2 2 4 2 2 2 2 ] . {\displaystyle {\begin{aligned}N&=(I-Q)^{-1}=\left({\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}-{\begin{bmatrix}1/2&1/2&0\\0&1/2&1/2\\1/2&0&0\end{bmatrix}}\right)^{-1}\\[4pt]&={\begin{bmatrix}1/2&-1/2&0\\0&1/2&-1/2\\-1/2&0&1\end{bmatrix}}^{-1}={\begin{bmatrix}4&4&2\\2&4&2\\2&2&2\end{bmatrix}}.\end{aligned}}} The expected number of steps starting from each of the transient states is t = N 1 = [ 4 4 2 2 4 2 2 2 2 ] [ 1 1 1 ] = [ 10 8 6 ] . {\displaystyle \mathbf {t} =N\mathbf {1} ={\begin{bmatrix}4&4&2\\2&4&2\\2&2&2\end{bmatrix}}{\begin{bmatrix}1\\1\\1\end{bmatrix}}={\begin{bmatrix}10\\8\\6\end{bmatrix}}.} Therefore, the expected number of coin flips before observing the sequence (heads, tails, heads) is 10, the entry for the state representing the empty string. === Games of chance === Games based entirely on chance can be modeled by an absorbing Markov chain. A classic example of this is the ancient Indian board game Snakes and Ladders. The graph on the left plots the probability mass in the lone absorbing state that represents the final square as the transition matrix is raised to larger and larger powers. To determine the expected number of turns to complete the game, compute the vector t as described above and examine tstart, which is approximately 39.2. === Infectious disease testing === Infectious disease testing, either of blood products or in medical clinics, is often taught as an example of an absorbing Markov chain. The public U.S. Centers for Disease Control and Prevention (CDC) model for HIV and for hepatitis B, for example, illustrates the property that absorbing Markov chains can lead to the detection of disease, versus the loss of detection through other means. In the standard CDC model, the Markov chain has five states, a state in which the individual is uninfected, then a state with infected but undetectable virus, a state with detectable virus, and absorbing states of having quit/been lost from the clinic, or of having been detected (the goal). The typical rates of transition between the Markov states are the probability p per unit time of being infected with the virus, w for the rate of window period removal (time until virus is detectable), q for quit/loss rate from the system, and d for detection, assuming a typical rate λ {\displaystyle \lambda } at which the health system administers tests of the blood product or patients in question. It follows that we can "walk along" the Markov model to identify the overall probability of detection for a person starting as undetected, by multiplying the probabilities of transition to each next state of the model as: p ( p + q ) w ( w + q ) d ( d + q ) {\displaystyle {\frac {p}{(p+q)}}{\frac {w}{(w+q)}}{\frac {d}{(d+q)}}} . The subsequent total absolute number of false negative tests—the primary CDC concern—would then be the rate of tests, multiplied by the probability of reaching the infected but undetectable state, times the duration of staying in the infected undetectable state: p ( p + q ) 1 ( w + q ) λ {\displaystyle {\frac {p}{(p+q)}}{\frac {1}{(w+q)}}\lambda } .

  Read more →
• 

  Baby Bundle (app)

  Baby Bundle is a parenting mobile app for iPhone and iPad. It was designed to help new parents through pregnancy and the first two years of parenthood. Developed in collaboration with medical experts, it helps track and record the child's development and growth, offers parental advice, manages vaccinations and health check-ups, stores photos and provides baby monitoring services. == History == Baby Bundle was founded in the United Kingdom by brothers, Nick and Anthony von Christierson. Each worked in investment banking prior to developing Baby Bundle, Nick at Greenhill & Co., and Anthony at Goldman Sachs. The idea for the app came when a friend's wife voiced her frustration over having multiple parenting apps on her smartphone. Nick and Anthony left their jobs to create a single app that would include all those features. They conducted market research by interviewing more than 500 parents in the UK and US. It took them a year to build the app, which was named by their mother. Looking for endorsement, they first went to the US in 2013 and partnered with parenting expert and pediatrician Dr. Jennifer Trachtenberg. Baby Bundle was launched in the US and Canadian App Stores in April 2014. In the same month, it became the #1 parenting app in iTunes and was featured by Apple as the #1 Editor's pick across all categories. Mashable called it one of the "Top 5 Can’t Miss Apps." Baby Bundle raised $1.8m seed round in March 2015 to fund development. The money came from a range of angel investors from across the US, UK and Asia. The von Christierson brothers have signed a deal to co-brand the app in the Middle East and expect to launch in Europe and Africa. == Features == Baby Bundle is an app for both the iPhone or iPad and provides smart monitoring tools and trackers for pregnancy and child development. It acts as a growth and daily activity tracker and offers parental advice, manages vaccinations and health check-ups. It has a parenting guide with tips and advice on what to expect when the baby arrives. An interactive forum also lets parents ask questions from others in the community. The app is free and also include paid premium features like the ability to turn two iPhones running into a baby monitor, a cloud service to share the child's data with a spouse and the ability to store data on more than one baby.

  Read more →
• 

  NSynth

  NSynth (a portmanteau of "Neural Synthesis") is a WaveNet-based autoencoder for synthesizing audio, outlined in a paper in April 2017. == Overview == The model generates sounds through a neural network based synthesis, employing a WaveNet-style autoencoder to learn its own temporal embeddings from four different sounds. Google then released an open source hardware interface for the algorithm called NSynth Super, used by notable musicians such as Grimes and YACHT to generate experimental music using artificial intelligence. The research and development of the algorithm was part of a collaboration between Google Brain, Magenta and DeepMind. == Technology == === Dataset === The NSynth dataset is composed of 305,979 one-shot instrumental notes featuring a unique pitch, timbre, and envelope, sampled from 1,006 instruments from commercial sample libraries. For each instrument the dataset contains four-second 16 kHz audio snippets by ranging over every pitch of a standard MIDI piano, as well as five different velocities. The dataset is made available under a Creative Commons Attribution 4.0 International (CC BY 4.0) license. === Machine learning model === A spectral autoencoder model and a WaveNet autoencoder model are publicly available on GitHub. The baseline model uses a spectrogram with fft_size 1024 and hop_size 256, MSE loss on the magnitudes, and the Griffin-Lim algorithm for reconstruction. The WaveNet model trains on mu-law encoded waveform chunks of size 6144. It learns embeddings with 16 dimensions that are downsampled by 512 in time. == NSynth Super == In 2018 Google released a hardware interface for the NSynth algorithm, called NSynth Super, designed to provide an accessible physical interface to the algorithm for musicians to use in their artistic production. Design files, source code and internal components are released under an open source Apache License 2.0, enabling hobbyists and musicians to freely build and use the instrument. At the core of the NSynth Super there is a Raspberry Pi, extended with a custom printed circuit board to accommodate the interface elements. == Influence == Despite not being publicly available as a commercial product, NSynth Super has been used by notable artists, including Grimes and YACHT. Grimes reported using the instrument in her 2020 studio album Miss Anthropocene. YACHT announced an extensive use of NSynth Super in their album Chain Tripping. Claire L. Evans compared the potential influence of the instrument to the Roland TR-808. The NSynth Super design was honored with a D&AD Yellow Pencil award in 2018.

  Read more →
• 

  Independent component analysis

  In signal processing, independent component analysis (ICA) is a computational method for separating a multivariate signal into additive subcomponents. This is done by assuming that at most one subcomponent is Gaussian and that the subcomponents are statistically independent from each other. ICA was invented by Jeanny Hérault and Christian Jutten in 1985. ICA is a special case of blind source separation. A common example application of ICA is the "cocktail party problem" of listening in on one person's speech in a noisy room. == Introduction == Independent component analysis attempts to decompose a multivariate signal into independent non-Gaussian signals. As an example, sound is usually a signal that is composed of the numerical addition, at each time t, of signals from several sources. The question then is whether it is possible to separate these contributing sources from the observed total signal. When the statistical independence assumption is correct, blind ICA separation of a mixed signal gives very good results. It is also used for signals that are not supposed to be generated by mixing for analysis purposes. A simple application of ICA is the "cocktail party problem", where the underlying speech signals are separated from a sample data consisting of people talking simultaneously in a room. Usually the problem is simplified by assuming no time delays or echoes. Note that a filtered and delayed signal is a copy of a dependent component, and thus the statistical independence assumption is not violated. Mixing weights for constructing the M {\textstyle M} observed signals from the N {\textstyle N} components can be placed in an M × N {\textstyle M\times N} matrix. An important thing to consider is that if N {\textstyle N} sources are present, at least N {\textstyle N} observations (e.g. microphones if the observed signal is audio) are needed to recover the original signals. When there are an equal number of observations and source signals, the mixing matrix is square ( M = N {\textstyle M=N} ). Other cases of underdetermined ( M < N {\textstyle M N {\textstyle M>N} ) have been investigated. The success of ICA separation of mixed signals relies on two assumptions and three effects of mixing source signals. Two assumptions: The source signals are independent of each other. The values in each source signal have non-Gaussian distributions. Three effects of mixing source signals: Independence: As per assumption 1, the source signals are independent; however, their signal mixtures are not. This is because the signal mixtures share the same source signals. Normality: According to the Central Limit Theorem, the distribution of a sum of independent random variables with finite variance tends towards a Gaussian distribution.Loosely speaking, a sum of two independent random variables usually has a distribution that is closer to Gaussian than any of the two original variables. Here we consider the value of each signal as the random variable. Complexity: The temporal complexity of any signal mixture is greater than that of its simplest constituent source signal. Those principles contribute to the basic establishment of ICA. If the signals extracted from a set of mixtures are independent and have non-Gaussian distributions or have low complexity, then they must be source signals. Another common example is image steganography, where ICA is used to embed one image within another. For instance, two grayscale images can be linearly combined to create mixed images in which the hidden content is visually imperceptible. ICA can then be used to recover the original source images from the mixtures. This technique underlies digital watermarking, which allows the embedding of ownership information into images, as well as more covert applications such as undetected information transmission. The method has even been linked to real-world cyberespionage cases. In such applications, ICA serves to unmix the data based on statistical independence, making it possible to extract hidden components that are not apparent in the observed data. Steganographic techniques, including those potentially involving ICA-based analysis, have been used in real-world cyberespionage cases. In 2010, the FBI uncovered a Russian spy network known as the "Illegals Program" (Operation Ghost Stories), where agents used custom-built steganography tools to conceal encrypted text messages within image files shared online. In another case, a former General Electric engineer, Xiaoqing Zheng, was convicted in 2022 for economic espionage. Zheng used steganography to exfiltrate sensitive turbine technology by embedding proprietary data within image files for transfer to entities in China. == Defining component independence == ICA finds the independent components (also called factors, latent variables or sources) by maximizing the statistical independence of the estimated components. We may choose one of many ways to define a proxy for independence, and this choice governs the form of the ICA algorithm. The two broadest definitions of independence for ICA are Minimization of mutual information Maximization of non-Gaussianity The Minimization-of-Mutual information (MMI) family of ICA algorithms uses measures like Kullback-Leibler Divergence and maximum entropy. The non-Gaussianity family of ICA algorithms, motivated by the central limit theorem, uses kurtosis and negentropy. Typical algorithms for ICA use centering (subtract the mean to create a zero mean signal), whitening (usually with the eigenvalue decomposition), and dimensionality reduction as preprocessing steps in order to simplify and reduce the complexity of the problem for the actual iterative algorithm. == Mathematical definitions == Linear independent component analysis can be divided into noiseless and noisy cases, where noiseless ICA is a special case of noisy ICA. Nonlinear ICA should be considered as a separate case. === General Derivation === In the classical ICA model, it is assumed that the observed data x i ∈ R m {\displaystyle \mathbf {x} _{i}\in \mathbb {R} ^{m}} at time t i {\displaystyle t_{i}} is generated from source signals s i ∈ R m {\displaystyle \mathbf {s} _{i}\in \mathbb {R} ^{m}} via a linear transformation x i = A s i {\displaystyle \mathbf {x} _{i}=A\mathbf {s} _{i}} , where A {\displaystyle A} is an unknown, invertible mixing matrix. To recover the source signals, the data is first centered (zero mean), and then whitened so that the transformed data has unit covariance. This whitening reduces the problem from estimating a general matrix A {\displaystyle A} to estimating an orthogonal matrix V {\displaystyle V} , significantly simplifying the search for independent components. If the covariance matrix of the centered data is Σ x = A A ⊤ {\displaystyle \Sigma _{x}=AA^{\top }} , then using the eigen-decomposition Σ x = Q D Q ⊤ {\displaystyle \Sigma _{x}=QDQ^{\top }} , the whitening transformation can be taken as D − 1 / 2 Q ⊤ {\displaystyle D^{-1/2}Q^{\top }} . This step ensures that the recovered sources are uncorrelated and of unit variance, leaving only the task of rotating the whitened data to maximize statistical independence. This general derivation underlies many ICA algorithms and is foundational in understanding the ICA model. ==== Reduced Mixing Problem ==== Independent component analysis (ICA) addresses the problem of recovering a set of unobserved source signals s i = ( s i 1 , s i 2 , … , s i m ) T {\displaystyle s_{i}=(s_{i1},s_{i2},\dots ,s_{im})^{T}} from observed mixed signals x i = ( x i 1 , x i 2 , … , x i m ) T {\displaystyle x_{i}=(x_{i1},x_{i2},\dots ,x_{im})^{T}} , based on the linear mixing model: x i = A s i , {\displaystyle x_{i}=A\,s_{i},} where the A {\displaystyle A} is an m × m {\displaystyle m\times m} invertible matrix called the mixing matrix, s i {\displaystyle s_{i}} represents the m‑dimensional vector containing the values of the sources at time t i {\displaystyle t_{i}} , and x i {\displaystyle x_{i}} is the corresponding vector of observed values at time t i {\displaystyle t_{i}} . The goal is to estimate both A {\displaystyle A} and the source signals { s i } {\displaystyle \{s_{i}\}} solely from the observed data { x i } {\displaystyle \{x_{i}\}} . After centering, the Gram matrix is computed as: ( X ∗ ) T X ∗ = Q D Q T , {\displaystyle (X^{})^{T}X^{}=Q\,D\,Q^{T},} where D is a diagonal matrix with positive entries (assuming X ∗ {\displaystyle X^{}} has maximum rank), and Q is an orthogonal matrix. Writing the SVD of the mixing matrix A = U Σ V T {\displaystyle A=U\Sigma V^{T}} and comparing with A A T = U Σ 2 U T {\displaystyle AA^{T}=U\Sigma ^{2}U^{T}} the mixing A has the form A = Q D 1 / 2 V T . {\displaystyle A=Q\,D^{1/2}\,V^{T}.} So, the normalized source values satisfy s i ∗ = V y i ∗ {\displaystyle s_{i}^{}=V\,y_{i}^{}} , where y i ∗ = D − 1 2 Q T x i ∗ . {\displaystyle y_{i}^{}=D^{-{\tfrac {1}{2}}}Q^{T}x_{i}^{}.} Thus, ICA reduces

  Read more →
• 

  Multi-label classification

  In machine learning, multi-label classification or multi-output classification is a variant of the classification problem where multiple nonexclusive labels may be assigned to each instance. Multi-label classification is a generalization of multiclass classification, which is the single-label problem of categorizing instances into precisely one of several (greater than or equal to two) classes. In the multi-label problem the labels are nonexclusive and there is no constraint on how many of the classes the instance can be assigned to. The formulation of multi-label learning was first introduced by Shen et al. in the context of Semantic Scene Classification, and later gained popularity across various areas of machine learning. Formally, multi-label classification is the problem of finding a model that maps inputs x to binary vectors y; that is, it assigns a value of 0 or 1 for each element (label) in y. == Problem transformation methods == Several problem transformation methods exist for multi-label classification, and can be roughly broken down into: === Transformation into binary classification problems === The baseline approach, called the binary relevance method, amounts to independently training one binary classifier for each label. Given an unseen sample, the combined model then predicts all labels for this sample for which the respective classifiers predict a positive result. Although this method of dividing the task into multiple binary tasks may resemble superficially the one-vs.-all (OvA) and one-vs.-rest (OvR) methods for multiclass classification, it is essentially different from both, because a single classifier under binary relevance deals with a single label, without any regard to other labels whatsoever. A classifier chain is an alternative method for transforming a multi-label classification problem into several binary classification problems. It differs from binary relevance in that labels are predicted sequentially, and the output of all previous classifiers (i.e. positive or negative for a particular label) are input as features to subsequent classifiers. Classifier chains have been applied, for instance, in HIV drug resistance prediction. Bayesian network has also been applied to optimally order classifiers in Classifier chains. In case of transforming the problem to multiple binary classifications, the likelihood function reads L = ∏ i = 1 n ( ∏ k ( ∏ j k ( p k , j k ( x i ) δ y i , k , j k ) ) ) {\displaystyle L=\prod _{i=1}^{n}(\prod _{k}(\prod _{j_{k}}(p_{k,j_{k}}(x_{i})^{\delta _{y_{i,k},j_{k}}})))} where index i {\displaystyle i} runs over the samples, index k {\displaystyle k} runs over the labels, j k {\displaystyle j_{k}} indicates the binary outcomes 0 or 1, δ a , b {\displaystyle \delta _{a,b}} indicates the Kronecker delta, y i , k ∈ 0 , 1 {\displaystyle y_{i,k}\in {0,1}} indicates the multiple hot encoded labels of sample i {\displaystyle i} . === Transformation into multi-class classification problem === The label powerset (LP) transformation creates one binary classifier for every label combination present in the training set. For example, if possible labels for an example were A, B, and C, the label powerset representation of this problem is a multi-class classification problem with the classes [0 0 0], [1 0 0], [0 1 0], [0 0 1], [1 1 0], [1 0 1], [0 1 1], and [1 1 1] where for example [1 0 1] denotes an example where labels A and C are present and label B is absent. === Ensemble methods === A set of multi-class classifiers can be used to create a multi-label ensemble classifier. For a given example, each classifier outputs a single class (corresponding to a single label in the multi-label problem). These predictions are then combined by an ensemble method, usually a voting scheme where every class that receives a requisite percentage of votes from individual classifiers (often referred to as the discrimination threshold) is predicted as a present label in the multi-label output. However, more complex ensemble methods exist, such as committee machines. Another variation is the random k-labelsets (RAKEL) algorithm, which uses multiple LP classifiers, each trained on a random subset of the actual labels; label prediction is then carried out by a voting scheme. A set of multi-label classifiers can be used in a similar way to create a multi-label ensemble classifier. In this case, each classifier votes once for each label it predicts rather than for a single label. == Adapted algorithms == Some classification algorithms/models have been adapted to the multi-label task, without requiring problem transformations. Examples of these including for multi-label data are k-nearest neighbors: the ML-kNN algorithm extends the k-NN classifier to multi-label data. decision trees: "Clare" is an adapted C4.5 algorithm for multi-label classification; the modification involves the entropy calculations. MMC, MMDT, and SSC refined MMDT, can classify multi-labeled data based on multi-valued attributes without transforming the attributes into single-values. They are also named multi-valued and multi-labeled decision tree classification methods. kernel methods for vector output neural networks: BP-MLL is an adaptation of the popular back-propagation algorithm for multi-label learning. == Learning paradigms == Based on learning paradigms, the existing multi-label classification techniques can be classified into batch learning and online machine learning. Batch learning algorithms require all the data samples to be available beforehand. It trains the model using the entire training data and then predicts the test sample using the found relationship. The online learning algorithms, on the other hand, incrementally build their models in sequential iterations. In iteration t, an online algorithm receives a sample, xt and predicts its label(s) ŷt using the current model; the algorithm then receives yt, the true label(s) of xt and updates its model based on the sample-label pair: (xt, yt). == Multi-label stream classification == Data streams are possibly infinite sequences of data that continuously and rapidly grow over time. Multi-label stream classification (MLSC) is the version of multi-label classification task that takes place in data streams. It is sometimes also called online multi-label classification. The difficulties of multi-label classification (exponential number of possible label sets, capturing dependencies between labels) are combined with difficulties of data streams (time and memory constraints, addressing infinite stream with finite means, concept drifts). Many MLSC methods resort to ensemble methods in order to increase their predictive performance and deal with concept drifts. Below are the most widely used ensemble methods in the literature: Online Bagging (OzaBagging)-based methods: Observing the probability of having K many of a certain data point in a bootstrap sample is approximately Poisson(1) for big datasets, each incoming data instance in a data stream can be weighted proportional to Poisson(1) distribution to mimic bootstrapping in an online setting. This is called Online Bagging (OzaBagging). Many multi-label methods that use Online Bagging are proposed in the literature, each of which utilizes different problem transformation methods. EBR, ECC, EPS, EBRT, EBMT, ML-Random Rules are examples of such methods. ADWIN Bagging-based methods: Online Bagging methods for MLSC are sometimes combined with explicit concept drift detection mechanisms such as ADWIN (Adaptive Window). ADWIN keeps a variable-sized window to detect changes in the distribution of the data, and improves the ensemble by resetting the components that perform poorly when there is a drift in the incoming data. Generally, the letter 'a' is used as a subscript in the name of such ensembles to indicate the usage of ADWIN change detector. EaBR, EaCC, EaHTPS are examples of such multi-label ensembles. GOOWE-ML-based methods: Interpreting the relevance scores of each component of the ensemble as vectors in the label space and solving a least squares problem at the end of each batch, Geometrically-Optimum Online-Weighted Ensemble for Multi-label Classification (GOOWE-ML) is proposed. The ensemble tries to minimize the distance between the weighted prediction of its components and the ground truth vector for each instance over a batch. Unlike Online Bagging and ADWIN Bagging, GOOWE-ML utilizes a weighted voting scheme where better performing components of the ensemble are given more weight. The GOOWE-ML ensemble grows over time, and the lowest weight component is replaced by a new component when it is full at the end of a batch. GOBR, GOCC, GOPS, GORT are the proposed GOOWE-ML-based multi-label ensembles. Multiple Windows : Here, BR models that use a sliding window are replaced with two windows for each label, one for relevant and one for non-relevant examples. Instances are oversampled or undersampled according to a load factor that is kept

  Read more →
• 

  Corel VideoStudio

  Corel VideoStudio (formerly Ulead VideoStudio) is a video editing software package for Microsoft Windows. == Features == === Basic editing === The software allows storyboard and timeline-oriented editing. Various formats are supported for source clips, and the resulting video can be exported to a video file. DVD and AVCHD DVD authoring capabilities are included, and Blu-ray authoring is available via a plug-in. VideoStudio supports direct DV and HDV capture and burning. === Overlay === Users can overlay videos, images, and text. Using the overlay track, up to 50 clips can be displayed simultaneously. It can handle videos in MOV and AVI formats, including alpha channel, and images in PSP, PSD, PNG, and GIF formats. Clips that do not contain an alpha channel can have specific colours removed from the overlay video so that the required background or image is displayed in the foreground. === Proxy video files === VideoStudio supports high-definition video. Proxy files are smaller versions of the video source that stand in for the full-resolution source during editing to improve performance. === Plug-ins/bundles === VideoStudio supports VFX-type plug-ins from providers, including NewBlue and proDAD. proDAD plug-ins Roto-Pen, Script, Vitascene, and Mercalli-Stabilizer are bundled with X4 and later Ultimate Editions. == Version history == Ulead VideoStudio 4 (1999) Ulead VideoStudio 5 (2001) Ulead VideoStudio 6 (2002) Ulead VideoStudio 7 (2003) Ulead VideoStudio 8 (2004) Ulead VideoStudio 9 (2005) Ulead VideoStudio 10 plus. (2006) Corel Ulead VideoStudio 11 plus. (2007) Corel VideoStudio Pro X2 (v12, 2008) Corel VideoStudio Pro X3 (v13, 2010) 2011: Corel VideoStudio Pro X4 (v14, 2011) Adds support for stop motion animation, time-lapse mode photography, 3D movies, and 2nd generation Intel Core. Corel VideoStudio Pro X5 (v15, March 9, 2012): Adds HTML5 export (Comparison of HTML5 and Flash). Corel VideoStudio Pro X6 (v16, April 25, 2013): Windows 8 compatible. Adds UHD 4K support. Corel VideoStudio Pro X7 (v17, March 5, 2014): Software becomes 64-bit. Corel VideoStudio Pro X8 (v18, May 8, 2015): Several improvements. Corel VideoStudio Pro X9 (v19, February 16, 2016): Windows 10 compatible. Adds H.265 support, Multi-Camera Editor, and Match moving. Corel VideoStudio Pro X10 (v20, February 15, 2017): Adds Mask Creator, Track Transparency, and 360-degree video support. Corel VideoStudio Pro 2018 (v21, February 13, 2018): Adds split screen Video, Lens Correction, and 3D Title Editor. Corel VideoStudio Pro 2019 (v22, February 12, 2019): Adds Color Grading, Morph Transitions, and MultiCam Capture Lite. Corel VideoStudio Pro 2020 (v23, February 25, 2020). Corel VideoStudio Pro 2021 (v24, March 26, 2021): Adds Instant Project Templates, AR Stickers, and performance improvements (particularly regarding hardware acceleration). Corel VideoStudio Pro 2022 (v25, March 6, 2022): Adds face effects, GIF Creator, transitions for Camera Movements, a speech to text converter, and ProRes Smart Proxy.

  Read more →
• 

  Chromosome (evolutionary algorithm)

  A chromosome or genotype in evolutionary algorithms (EA) is a set of parameters which define a proposed solution of the problem that the evolutionary algorithm is trying to solve. The set of all solutions, also called individuals according to the biological model, is known as the population. The genome of an individual consists of one, more rarely of several, chromosomes and corresponds to the genetic representation of the task to be solved. A chromosome is composed of a set of genes, where a gene consists of one or more semantically connected parameters, which are often also called decision variables. They determine one or more phenotypic characteristics of the individual or at least have an influence on them. In the basic form of genetic algorithms, the chromosome is represented as a binary string, while in later variants and in EAs in general, a wide variety of other data structures are used. == Chromosome design == When creating the genetic representation of a task, it is determined which decision variables and other degrees of freedom of the task should be improved by the EA and possible additional heuristics and how the genotype-phenotype mapping should look like. The design of a chromosome translates these considerations into concrete data structures for which an EA then has to be selected, configured, extended, or, in the worst case, created. Finding a suitable representation of the problem domain for a chromosome is an important consideration, as a good representation will make the search easier by limiting the search space; similarly, a poorer representation will allow a larger search space. In this context, suitable mutation and crossover operators must also be found or newly defined to fit the chosen chromosome design. An important requirement for these operators is that they not only allow all points in the search space to be reached in principle, but also make this as easy as possible. The following requirements must be met by a well-suited chromosome: It must allow the accessibility of all admissible points in the search space. Design of the chromosome in such a way that it covers only the search space and no additional areas. so that there is no redundancy or only as little redundancy as possible. Observance of strong causality: small changes in the chromosome should only lead to small changes in the phenotype. This is also called locality of the relationship between search and problem space. Designing the chromosome in such a way that it excludes prohibited regions in the search space completely or as much as possible. While the first requirement is indispensable, depending on the application and the EA used, one usually only has to be satisfied with fulfilling the remaining requirements as far as possible. The evolutionary search is supported and possibly considerably accelerated by a fulfillment as complete as possible. == Examples of chromosomes == === Chromosomes for binary codings === In their classical form, GAs use bit strings and map the decision variables to be optimized onto them. An example for one Boolean and three integer decision variables with the value ranges 0 ≤ D 1 ≤ 60 {\displaystyle 0\leq D_{1}\leq 60} , 28 ≤ D 2 ≤ 30 {\displaystyle 28\leq D_{2}\leq 30} and − 12 ≤ D 3 ≤ 14 {\displaystyle -12\leq D_{3}\leq 14} may illustrate this: Note that the negative number here is given in two's complement. This straight forward representation uses five bits to represent the three values of D 2 {\displaystyle D_{2}} , although two bits would suffice. This is a significant redundancy. An improved alternative, where 28 is to be added for the genotype-phenotype mapping, could look like this: with D 2 = 28 + D 2 ′ = 29 {\displaystyle D_{2}=28+D'_{2}=29} . === Chromosomes with real-valued or integer genes === For the processing of tasks with real-valued or mixed-integer decision variables, EAs such as the evolution strategy or the real-coded GAs are suited. In the case of mixed-integer values, rounding is often used, but this represents some violation of the redundancy requirement. If the necessary precisions of the real values can be reasonably narrowed down, this violation can be remedied by using integer-coded GAs. For this purpose, the valid digits of real values are mapped to integers by multiplication with a suitable factor. For example, 12.380 becomes the integer 12380 by multiplying by 1000. This must of course be taken into account in genotype-phenotype mapping for evaluation and result presentation. A common form is a chromosome consisting of a list or an array of integer or real values. === Chromosomes for permutations === Combinatorial problems are mainly concerned with finding an optimal sequence of a set of elementary items. As an example, consider the problem of the traveling salesman who wants to visit a given number of cities exactly once on the shortest possible tour. The simplest and most obvious mapping onto a chromosome is to number the cities consecutively, to interpret a resulting sequence as permutation and to store it directly in a chromosome, where one gene corresponds to the ordinal number of a city. Then, however, the variation operators may only change the gene order and not remove or duplicate any genes. The chromosome thus contains the path of a possible tour to the cities. As an example the sequence 3 , 5 , 7 , 1 , 4 , 2 , 9 , 6 , 8 {\displaystyle 3,5,7,1,4,2,9,6,8} of nine cities may serve, to which the following chromosome corresponds: In addition to this encoding frequently called path representation, there are several other ways of representing a permutation, for example the ordinal representation or the matrix representation. === Chromosomes for co-evolution === When a genetic representation contains, in addition to the decision variables, additional information that influences evolution and/or the mapping of the genotype to the phenotype and is itself subject to evolution, this is referred to as co-evolution. A typical example is the evolution strategy (ES), which includes one or more mutation step sizes as strategy parameters in each chromosome. Another example is an additional gene to control a selection heuristic for resource allocation in a scheduling tasks. This approach is based on the assumption that good solutions are based on an appropriate selection of strategy parameters or on control gene(s) that influences genotype-phenotype mapping. The success of the ES gives evidence to this assumption. === Chromosomes for complex representations === The chromosomes presented above are well suited for processing tasks of continuous, mixed-integer, pure-integer or combinatorial optimization. For a combination of these optimization areas, on the other hand, it becomes increasingly difficult to map them to simple strings of values, depending on the task. The following extension of the gene concept is proposed by the EA GLEAM (General Learning Evolutionary Algorithm and Method) for this purpose: A gene is considered to be the description of an element or elementary trait of the phenotype, which may have multiple parameters. For this purpose, gene types are defined that contain as many parameters of the appropriate data type as are required to describe the particular element of the phenotype. A chromosome now consists of genes as data objects of the gene types, whereby, depending on the application, each gene type occurs exactly once as a gene or can be contained in the chromosome any number of times. The latter leads to chromosomes of dynamic length, as they are required for some problems. The gene type definitions also contain information on the permissible value ranges of the gene parameters, which are observed during chromosome generation and by corresponding mutations, so they cannot lead to lethal mutations. For tasks with a combinatorial part, there are suitable genetic operators that can move or reposition genes as a whole, i.e. with their parameters. A scheduling task is used as an illustration, in which workflows are to be scheduled that require different numbers of heterogeneous resources. A workflow specifies which work steps can be processed in parallel and which have to be executed one after the other. In this context, heterogeneous resources mean different processing times at different costs in addition to different processing capabilities. Each scheduling operation therefore requires one or more parameters that determine the resource selection, where the value ranges of the parameters depend on the number of alternative resources available for each work step. A suitable chromosome provides one gene type per work step and in this case one corresponding gene, which has one parameter for each required resource. The order of genes determines the order of scheduling operations and, therefore, the precedence in case of allocation conflicts. The exemplary gene type definition of work step 15 with two resources, for which there are four and seven alternatives respectively

  Read more →
• 

  Receptron

  The receptron (short for "reservoir perceptron") is a neuromorphic data processing model — specifically neuromorphic computing — that generalizes the traditional perceptron, by incorporating non-linear interactions between inputs. Unlike classical perceptron, which rely on linearly independent weights, the receptron leverages complexity in physical substrates, such as the electric conduction properties of nanostructured materials or optical speckle fields, to perform classification tasks. The receptron bridges unconventional computing and neural network principles, enabling solutions that do not require the training approaches typical of artificial neural networks based on the perceptron model. == Algorithm == The receptron is an algorithm for supervised learning of binary classifiers, so a classification algorithm that makes its predictions based on a predictor function, combining a set of weights with the feature vector. The mathematical model is based on the sum of inputs with non-linear interactions: S = ∑ k = 1 n x j w ~ j ( x → ) | S ∈ R {\displaystyle S=\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})|S\in R} (1) where j ∈ [ 1 , n ] {\displaystyle j\in [1,n]} and w ~ j {\displaystyle {\widetilde {w}}_{j}} are non-linear weight functions depending on the inputs, x → {\displaystyle {\vec {x}}} . Nonlinearity will typically make the system extremely complex, and allowing for the solution of problems not solvable through the simpler rules of a linear system, such as the perceptron or McCulloch Pitts neurons, which is based on the sum of linearly independent weights: S = ∑ k = 1 n x j w j p {\displaystyle S=\sum _{k=1}^{n}x_{j}w_{j}^{p}} (2) where w j {\displaystyle w_{j}} are constant real values. A consequence of this simplicity is the limitation to linearly separable functions, which necessitates multi-layer architectures and training algorithms like backpropagation As in the perceptron case, the summation in Eq. 1 origins the activation of the receptron output through the thresholding process, Y ( x 1 , . . . , x n ) = { 1 if S > th 0 if S ≤ th {\displaystyle Y(x_{1},...,x_{n})={\begin{cases}1&{\text{if }}S>{\text{th}}\\0&{\text{if }}S\leq {\text{th}}\end{cases}}} (3) where th is a constant threshold parameter. Equation 3 can be written by using the Heaviside step function. The weight functions w ~ ( x → ) {\displaystyle {\widetilde {w}}({\vec {x}})} can be written with a finite number of parameters w j 1 . . . j n {\displaystyle w_{j_{1}...j_{n}}} , simplifying the model representation. One can Taylor-expand w ~ ( x → ) {\displaystyle {\widetilde {w}}({\vec {x}})} and use the idempotency of Boolean variables ( x j ) q = x j ∀ q ≥ 1 {\displaystyle (x_{j})^{q}=x_{j}\forall q\geq 1} such that S ′ = b + ∑ k = 1 n x j w ~ j ( x → ) {\displaystyle S'=b+\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})} can be written as S ′ ( x → ) = b + ∑ j w j x j + ∑ j < k w j k x j x k + ∑ j < k < l w j k l x j x k x l + . . . {\displaystyle S'({\vec {x}})=b+\sum _{j}w_{j}x_{j}+\sum _{j Read more →

• 

  Defining length

  In the field of genetic algorithms, a schema (plural: schemata) is a template that represents a subset of potential solutions. These templates use fixed symbols (e.g., `0` or `1`) for specific positions and a wildcard or "don't care" symbol (often `#` or ``) for others. The defining length of a schema, denoted as L(H), measures the distance between the outermost fixed positions in the template. According to the Schema theorem, a schema with a shorter defining length is less likely to be disrupted by the genetic operator of crossover. As a result, short schemata are considered more robust and are more likely to be propagated to the next generation. In genetic programming, where solutions are often represented as trees, the defining length is the number of links in the minimum tree fragment that includes all the non-wildcard symbols within a schema H. == Example == The defining length is calculated by subtracting the position of the first fixed symbol from the position of the last one. Using 1-based indexing for a string of length 5: The schema `1##0#` has its first fixed symbol (`1`) at position 1 and its last fixed symbol (`0`) at position 4. Its defining length is 4 − 1 = 3. The schema `00##0` has its first fixed symbol at position 1 and its last at position 5. Its defining length is 5 − 1 = 4. The schema `##0##` has only one fixed symbol at position 3. The first and last fixed positions are the same, so its defining length is 3 − 3 = 0.

  Read more →
• 

  Turret lathe

  A turret lathe is a form of metalworking lathe that is used for repetitive production of duplicate parts, which by the nature of their cutting process are usually interchangeable. It evolved from earlier lathes with the addition of the turret, which is an indexable toolholder that allows multiple cutting operations to be performed, each with a different cutting tool, in easy, rapid succession, with no need for the operator to perform set-up tasks in between (such as installing or uninstalling tools) or to control the toolpath. The latter is due to the toolpath's being controlled by the machine, either in jig-like fashion, via the mechanical limits placed on it by the turret's slide and stops, or via digitally-directed servomechanisms for computer numerical control lathes. The name derives from the way early turrets took the general form of a flattened cylindrical block mounted to the lathe's cross-slide, capable of rotating about the vertical axis and with toolholders projecting out to all sides, and thus vaguely resembled a swiveling gun turret. Capstan lathe is the usual name in the UK and Commonwealth, though the two terms are also used in contrast: see below, Capstan versus turret. == History == Turret lathes became indispensable to the production of interchangeable parts and for mass production. The first turret lathe was built by Stephen Fitch in 1845 to manufacture screws for pistol percussion parts. In the mid-nineteenth century, the need for interchangeable parts for Colt revolvers enhanced the role of turret lathes in achieving this goal as part of the "American system" of manufacturing arms. Clock-making and bicycle manufacturing had similar requirements. Christopher Spencer invented the first fully automated turret lathe in 1873, which led to designs using cam action or hydraulic mechanisms. From the late-19th through mid-20th centuries, turret lathes, both manual and automatic (i.e., screw machines and chuckers), were one of the most important classes of machine tools for mass production. They were used extensively in the mass production for the war effort in World War II. The U.S. company Warner & Swasey was one of the premier brands in heavy turret lathes between the 1910s and 1960s; it became the world's largest manufacturer of such lathes by 1928. During World War II, it employed 7,000 people and produced half of the turret lathes manufactured in the United States. == Types == There are many variants of the turret lathe. They can be most generally classified by size (small, medium, or large); method of control (manual, automated mechanically, or automated via computer (numerical control (NC) or computer numerical control (CNC)); and bed orientation (horizontal or vertical). === Archetypical: horizontal, manual === In the late 1830s a "capstan lathe" with a turret was patented in Britain. The first American turret lathe was invented by Stephen Fitch in 1845. The archetypical turret lathe, and the first in order of historical appearance, is the horizontal-bed, manual turret lathe. The term "turret lathe" without further qualification is still understood to refer to this type. The formative decades for this class of machine were the 1840s through 1860s, when the basic idea of mounting an indexable turret on a bench lathe or engine lathe was born, developed, and disseminated from the originating shops to many other factories. Some important tool-builders in this development were Stephen Fitch; Gay, Silver & Co.; Elisha K. Root of Colt; J.D. Alvord of the Sharps Armory; Frederick W. Howe, Richard S. Lawrence, and Henry D. Stone of Robbins & Lawrence; J.R. Brown of Brown & Sharpe; and Francis A. Pratt of Pratt & Whitney. Various designers at these and other firms later made further refinements. === Semi-automatic === Sometimes machines similar to those above, but with power feeds and automatic turret-indexing at the end of the return stroke, are called "semi-automatic turret lathes". This nomenclature distinction is blurry and not consistently observed. The term "turret lathe" encompasses them all. During the 1860s, when semi-automatic turret lathes were developed, they were sometimes called "automatic". What we today would call "automatics", that is, fully automatic machines, had not been developed yet. During that era both manual and semi-automatic turret lathes were sometimes called "screw machines", although we today reserve that term for fully automatic machines. === Automatic === During the 1870s through 1890s, the mechanically automated "automatic" turret lathe was developed and disseminated. These machines can execute many part-cutting cycles without human intervention. Thus the duties of the operator, which were already greatly reduced by the manual turret lathe, were even further reduced, and productivity increased. These machines use cams to automate the sliding and indexing of the turret and the opening and closing of the chuck. Thus, they execute the part-cutting cycle somewhat analogously to the way in which an elaborate cuckoo clock performs an automated theater show. Small- to medium-sized automatic turret lathes are usually called "screw machines" or "automatic screw machines", while larger ones are usually called "automatic chucking lathes", "automatic chuckers", or "chuckers". Such machine tools of the "automatic" variety, which in the pre-computer era meant mechanically automated, had already reached a highly advanced state by World War I. === Computer numerical control === When World War II ended, the digital computer was poised to develop from a colossal laboratory curiosity into a practical technology that could begin to disseminate into business and industry. The advent of computer-based automation in machine tools via numerical control (NC) and then computer numerical control (CNC) displaced to a large extent, but not at all completely, the previously existing manual and mechanically automated machines. Numerically controlled turrets allow automated selection of tools on a turret. CNC lathes may be horizontal or vertical in orientation and mount six separate tools on one or more turrets. Such machine tools can work in two axes per turret, with up to six axes being feasible for complex work. === Vertical === Vertical turret lathes have the workpiece held vertically, which allows the headstock to sit on the floor and the faceplate to become a horizontal rotating table, analogous to a huge potter's wheel. This is useful for the handling of very large, heavy, short workpieces. Vertical lathes in general are also called "vertical boring mills" or often simply "boring mills"; therefore a vertical turret lathe is a vertical boring mill equipped with a turret. == Other variations == === Capstan versus turret === The term "capstan lathe" overlaps in sense with the term "turret lathe" to a large extent. In many times and places, it has been understood to be synonymous with "turret lathe". In other times and places it has been held in technical contradistinction to "turret lathe", with the difference being in whether the turret's slide is fixed to the bed (ram-type turret) or slides on the bed's ways (saddle-type turret). The difference in terminology is mostly a matter of United Kingdom and Commonwealth usage versus United States usage. === Flat === A subtype of horizontal turret lathe is the flat-turret lathe. Its turret is flat (and analogous to a rotary table), allowing the turret to pass beneath the part. Patented by James Hartness of Jones & Lamson, and first disseminated in the 1890s, it was developed to provide more rigidity via requiring less overhang in the tool setup, especially when the part is relatively long. === Hollow-hexagon === Hollow-hexagon turret lathes competed with flat-turret lathes by taking the conventional hexagon turret and making it hollow, allowing the part to pass into it during the cut, analogously to how the part would pass over the flat turret. In both cases, the main idea is to increase rigidity by allowing a relatively long part to be turned without the tool overhang that would be needed with a conventional turret, which is not flat or hollow. === Monitor lathe === The term "monitor lathe" formerly (1860s–1940s) referred to the class of small- to medium-sized manual turret lathes used on relatively small work. The name was inspired by the monitor-class warships, which the monitor lathe's turret resembled. Today, lathes of such appearance, such as the Hardinge DSM-59 and its many clones, are still common, but the name "monitor lathe" is no longer current in the industry. === Toolpost turrets and tailstock turrets === Turrets can be added to non-turret lathes (bench lathes, engine lathes, toolroom lathes, etc.) by mounting them on the toolpost, tailstock, or both. Often these turrets are not as large as a turret lathe's, and they usually do not offer the sliding and stopping that a turret lathe's turret does; but they do offer the ability to index through successive tool

  Read more →
• 

  Neocognitron

  The neocognitron is a hierarchical, multilayered artificial neural network proposed by Kunihiko Fukushima in 1979. It has been used for Japanese handwritten character recognition and other pattern recognition tasks, and served as the inspiration for convolutional neural networks. Previously in 1969, he published a similar architecture, but with hand-designed kernels inspired by convolutions in mammalian vision. In 1975 he improved it to the Cognitron, and in 1979 he improved it to the neocognitron, which learns all convolutional kernels by unsupervised learning (in his terminology, "self-organized by 'learning without a teacher'"). The neocognitron was inspired by the model proposed by Hubel & Wiesel in 1959. They found two types of cells in the visual primary cortex called simple cell and complex cell, and also proposed a cascading model of these two types of cells for use in pattern recognition tasks. The neocognitron is a natural extension of these cascading models. The neocognitron consists of multiple types of cells, the most important of which are called S-cells and C-cells. The local features are extracted by S-cells, and these features' deformation, such as local shifts, are tolerated by C-cells. Local features in the input are integrated gradually and classified in the higher layers. The idea of local feature integration is found in several other models, such as the Convolutional Neural Network model, the SIFT method, and the HoG method. There are various kinds of neocognitron. For example, some types of neocognitron can detect multiple patterns in the same input by using backward signals to achieve selective attention.

  Read more →
• 

  Induction of regular languages

  In computational learning theory, induction of regular languages refers to the task of learning a formal description (e.g. grammar) of a regular language from a given set of example strings. Although E. Mark Gold has shown that not every regular language can be learned this way (see language identification in the limit), approaches have been investigated for a variety of subclasses. They are sketched in this article. For learning of more general grammars, see Grammar induction. == Definitions == A regular language is defined as a (finite or infinite) set of strings that can be described by one of the mathematical formalisms called "finite automaton", "regular grammar", or "regular expression", all of which have the same expressive power. Since the latter formalism leads to shortest notations, it shall be introduced and used here. Given a set Σ of symbols (a.k.a. alphabet), a regular expression can be any of ∅ (denoting the empty set of strings), ε (denoting the singleton set containing just the empty string), a (where a is any character in Σ; denoting the singleton set just containing the single-character string a), r + s (where r and s are, in turn, simpler regular expressions; denoting their set's union) r ⋅ s (denoting the set of all possible concatenations of strings from r's and s's set), r + (denoting the set of n-fold repetitions of strings from r's set, for any n ≥ 1), or r (similarly denoting the set of n-fold repetitions, but also including the empty string, seen as 0-fold repetition). For example, using Σ = {0,1}, the regular expression (0+1+ε)⋅(0+1) denotes the set of all binary numbers with one or two digits (leading zero allowed), while 1⋅(0+1)⋅0 denotes the (infinite) set of all even binary numbers (no leading zeroes). Given a set of strings (also called "positive examples"), the task of regular language induction is to come up with a regular expression that denotes a set containing all of them. As an example, given {1, 10, 100}, a "natural" description could be the regular expression 1⋅0, corresponding to the informal characterization "a 1 followed by arbitrarily many (maybe even none) 0's". However, (0+1) and 1+(1⋅0)+(1⋅0⋅0) is another regular expression, denoting the largest (assuming Σ = {0,1}) and the smallest set containing the given strings, and called the trivial overgeneralization and undergeneralization, respectively. Some approaches work in an extended setting where also a set of "negative example" strings is given; then, a regular expression is to be found that generates all of the positive, but none of the negative examples. == Lattice of automata == Dupont et al. have shown that the set of all structurally complete finite automata generating a given input set of example strings forms a lattice, with the trivial undergeneralized and the trivial overgeneralized automaton as bottom and top element, respectively. Each member of this lattice can be obtained by factoring the undergeneralized automaton by an appropriate equivalence relation. For the above example string set {1, 10, 100}, the picture shows at its bottom the undergeneralized automaton Aa,b,c,d in grey, consisting of states a, b, c, and d. On the state set {a,b,c,d}, a total of 15 equivalence relations exist, forming a lattice. Mapping each equivalence E to the corresponding quotient automaton language L(Aa,b,c,d / E) obtains the partially ordered set shown in the picture. Each node's language is denoted by a regular expression. The language may be recognized by quotient automata w.r.t. different equivalence relations, all of which are shown below the node. An arrow between two nodes indicates that the lower node's language is a proper subset of the higher node's. If both positive and negative example strings are given, Dupont et al. build the lattice from the positive examples, and then investigate the separation border between automata that generate some negative example and such that do not. Most interesting are those automata immediately below the border. In the picture, separation borders are shown for the negative example strings 11 (green), 1001 (blue), 101 (cyan), and 0 (red). Coste and Nicolas present an own search method within the lattice, which they relate to Mitchell's version space paradigm. To find the separation border, they use a graph coloring algorithm on the state inequality relation induced by the negative examples. Later, they investigate several ordering relations on the set of all possible state fusions. Kudo and Shimbo use the representation by automaton factorizations to give a unique framework for the following approaches (sketched below): k-reversible languages and the "tail clustering" follow-up approach, Successor automata and the predecessor-successor method, and pumping-based approaches (framework-integration challenged by Luzeaux, however). Each of these approaches is shown to correspond to a particular kind of equivalence relations used for factorization. == Approaches == === k-reversible languages === Angluin considers so-called "k-reversible" regular automata, that is, deterministic automata in which each state can be reached from at most one state by following a transition chain of length k. Formally, if Σ, Q, and δ denote the input alphabet, the state set, and the transition function of an automaton A, respectively, then A is called k-reversible if: ∀a0, ..., ak ∈ Σ ∀s1, s2 ∈ Q: δ(s1, a0...ak) = δ(s2, a0...ak) ⇒ s1 = s2, where δ means the homomorphic extension of δ to arbitrary words. Angluin gives a cubic algorithm for learning of the smallest k-reversible language from a given set of input words; for k = 0, the algorithm has even almost linear complexity. The required state uniqueness after k + 1 given symbols forces unifying automaton states, thus leading to a proper generalization different from the trivial undergeneralized automaton. This algorithm has been used to learn simple parts of English syntax; later, an incremental version has been provided. Another approach based on k-reversible automata is the tail clustering method. === Successor automata === From a given set of input strings, Vernadat and Richetin build a so-called successor automaton, consisting of one state for each distinct character and a transition between each two adjacent characters' states. For example, the singleton input set {aabbaabb} leads to an automaton corresponding to the regular expression (a+⋅b+). An extension of this approach is the predecessor-successor method which generalizes each character repetition immediately to a Kleene + and then includes for each character the set of its possible predecessors in its state. Successor automata can learn exactly the class of local languages. Since each regular language is the homomorphic image of a local language, grammars from the former class can be learned by lifting, if an appropriate (depending on the intended application) homomorphism is provided. In particular, there is such a homomorphism for the class of languages learnable by the predecessor-successor method. The learnability of local languages can be reduced to that of k-reversible languages. === Early approaches === Chomsky and Miller (1957) used the pumping lemma: they guess a part v of an input string uvw and try to build a corresponding cycle into the automaton to be learned; using membership queries they ask, for appropriate k, which of the strings uw, uvvw, uvvvw, ..., uvkw also belongs to the language to be learned, thereby refining the structure of their automaton. In 1959, Solomonoff generalized this approach to context-free languages, which also obey a pumping lemma. === Cover automata === Câmpeanu et al. learn a finite automaton as a compact representation of a large finite language. Given such a language F, they search a so-called cover automaton A such that its language L(A) covers F in the following sense: L(A) ∩ Σ≤ l = F, where l is the length of the longest string in F, and Σ≤ l denotes the set of all strings not longer than l. If such a cover automaton exists, F is uniquely determined by A and l. For example, F = {ad, read, reread } has l = 6 and a cover automaton corresponding to the regular expression (r⋅e)⋅a⋅d. For two strings x and y, Câmpeanu et al. define x ~ y if xz ∈ F ⇔ yz ∈ F for all strings z of a length such that both xz and yz are not longer than l. Based on this relation, whose lack of transitivity causes considerable technical problems, they give an O(n4) algorithm to construct from F a cover automaton A of minimal state count. Moreover, for union, intersection, and difference of two finite languages they provide corresponding operations on their cover automata. Păun et al. improve the time complexity to O(n2). === Residual automata === For a set S of strings and a string u, the Brzozowski derivative u−1S is defined as the set of all rest-strings obtainable from a string in S by cutting off its prefix u (if possible), formally: u−1S = {v ∈ Σ: uv ∈ S}, cf. picture. Denis et al. define a

  Read more →