The Kruskal count (also known as Kruskal's principle, Dynkin–Kruskal count, Dynkin's counting trick, Dynkin's card trick, coupling card trick or shift coupling) is a probabilistic concept originally demonstrated by the Russian mathematician Evgenii Borisovich Dynkin in the 1950s or 1960s discussing coupling effects and rediscovered as a card trick by the American mathematician Martin David Kruskal in the early 1970s as a side-product while working on another problem. It was published by Kruskal's friend Martin Gardner and magician Karl Fulves in 1975. This is related to a similar trick published by magician Alexander F. Kraus in 1957 as Sum total and later called Kraus principle. Besides uses as a card trick, the underlying phenomenon has applications in cryptography, code breaking, software tamper protection, code self-synchronization, control-flow resynchronization, design of variable-length codes and variable-length instruction sets, web navigation, object alignment, and others. == Card trick == The trick is performed with cards, but is more a magical-looking effect than a conventional magic trick. The magician has no access to the cards, which are manipulated by members of the audience. Thus sleight of hand is not possible. Rather the effect is based on the mathematical fact that the output of a Markov chain, under certain conditions, is typically independent of the input. A simplified version using the hands of a clock performed by David Copperfield is as follows. A volunteer picks a number from one to twelve and does not reveal it to the magician. The volunteer is instructed to start from 12 on the clock and move clockwise by a number of spaces equal to the number of letters that the chosen number has when spelled out. This is then repeated, moving by the number of letters in the new number. The output after three or more moves does not depend on the initially chosen number and therefore the magician can predict it.
Local-first software
Local-first software is a software engineering approach in which an application stores its data primarily on the user's own device rather than on remote servers. Users can read and write data without an Internet connection, and changes are synchronized across devices in the background when connectivity is available. The approach differs from conventional cloud-based applications, where the server holds the authoritative copy of user data and the client acts as a thin client. The term was coined in a 2019 paper published by researchers at Ink & Switch, an independent research lab, and presented at the Onward! conference at ACM SIGPLAN. The paper, sometimes referred to as a manifesto, was authored by Martin Kleppmann, Adam Wiggins, Peter van Hardenberg, and Mark McGranaghan. == Background == Before the widespread adoption of Internet-connected software in the 2000s, most desktop applications stored data as files on the user's local disk. Users had direct access to their files and could copy, back up, or delete them at will. The rise of software as a service (SaaS) and cloud-based applications like Google Docs shifted data storage to centralized servers. While cloud applications made real-time collaboration across devices straightforward, they introduced a dependency on the service provider: if the provider discontinued the service or experienced an outage, users could lose access to their data. A related concept, "offline-first," emerged in the early 2010s and focused on making web applications resilient to network interruptions. The local-first approach built on these earlier efforts while placing greater emphasis on long-term data ownership and end-to-end encryption. == Origins == === Ink & Switch manifesto === Ink & Switch is an industrial research lab co-founded by Adam Wiggins, who had earlier co-founded Heroku. Martin Kleppmann, an associate professor in the Department of Computer Science and Technology at the University of Cambridge, was a co-author of the 2019 paper. The manifesto proposed seven "ideals" for local-first software: Fast — Operations respond without network round-trips. Multi-device — Data synchronizes across a user's devices. Offline — Users can read and write data without a network connection. Collaboration — Multiple users can work on the same data concurrently. Longevity — Data remains accessible even if the software vendor ceases operation. Privacy — End-to-end encryption protects user data. User control — The vendor cannot restrict how users access or use their data. The paper surveyed existing approaches to data storage and collaboration — ranging from email attachments and Dropbox-style file synchronization to web applications and mobile backends — and argued that none of them satisfied all seven ideals simultaneously. === Role of CRDTs === The manifesto identified conflict-free replicated data types (CRDTs) as a promising technical foundation for local-first applications. CRDTs are data structures that allow multiple replicas to be edited independently and then merged without conflicts, a property first formalized in research by Marc Shapiro and colleagues around 2011. Kleppmann and collaborators at Ink & Switch developed Automerge, an open-source CRDT library for JSON documents, to make these algorithms available to application developers. == Adoption and community == Developer interest in the local-first approach grew after the 2019 paper spread on Hacker News and at developer conferences In August 2023, Wired published a feature article on the movement, describing it as an effort to reduce reliance on large cloud providers. The first Local-First Conf took place on 30 May 2024 in Berlin, with talks by Kleppmann and developers from companies including Linear and Anytype. The community has continued to expand, with regular "LoFi" meetups, a podcast (localfirst.fm), and a third edition of the conference planned for Berlin in July 2026. == Criticisms and limitations == Developers and commentators have pointed out practical difficulties with the local-first approach. Synchronizing data between multiple devices that may be offline for extended periods introduces complexity that cloud-based architectures avoid. Conflict resolution, even with CRDTs, can produce results that are technically consistent but semantically unexpected to users. Schema migrations across thousands of client devices running different application versions pose another difficulty that does not arise with server-side databases. Web browsers impose storage limits and may evict locally stored data. Safari, for instance, has been reported to clear IndexedDB data after seven days of inactivity on a given site, which undermines the assumption that local data is persistent. There is also disagreement within the local-first community about whether a fully decentralized architecture is required. The original manifesto described decentralization as the "logical end goal," but a number of products that identify as local-first still depend on centralized servers for authentication, backup, or synchronization. In a talk at Local-First Conf 2024, Kleppmann said the seven ideals are better understood as a "gradient" rather than a strict checklist.
Sammon mapping
Sammon mapping or Sammon projection is an algorithm that maps a high-dimensional space to a space of lower dimensionality (see multidimensional scaling) by trying to preserve the structure of inter-point distances in high-dimensional space in the lower-dimension projection. It is particularly suited for use in exploratory data analysis. The method was proposed by John W. Sammon in 1969. It is considered a non-linear approach as the mapping cannot be represented as a linear combination of the original variables as possible in techniques such as principal component analysis, which also makes it more difficult to use for classification applications. Denote the distance between ith and jth objects in the original space by d i j ∗ {\displaystyle \scriptstyle d_{ij}^{}} , and the distance between their projections by d i j {\displaystyle \scriptstyle d_{ij}^{}} . Sammon's mapping aims to minimize the following error function, which is often referred to as Sammon's stress or Sammon's error: E = 1 ∑ i < j d i j ∗ ∑ i < j ( d i j ∗ − d i j ) 2 d i j ∗ . {\displaystyle E={\frac {1}{\sum \limits _{i In decision tree learning, ID3 (Iterative Dichotomiser 3) is a greedy algorithm invented by Ross Quinlan used to generate a decision tree from a dataset. ID3 is the precursor to the C4.5 algorithm. The 3 in the name is meant to signify that this was Quinlan's third attempt at a model based on entropy-based splitting, and the term dichotimser is a misnomer as it implies a binary split, but the ID3 algorithm can split on multi-valued attributes. == Algorithm == The ID3 algorithm begins with the original set S {\displaystyle S} as the root node. On each iteration of the algorithm, it iterates through every unused attribute of the set S {\displaystyle S} and calculates the entropy H ( S ) {\displaystyle \mathrm {H} {(S)}} or the information gain I G ( S ) {\displaystyle IG(S)} of that attribute. It then selects the attribute which has the smallest entropy (or largest information gain) value. The set S {\displaystyle S} is then split or partitioned by the selected attribute to produce subsets of the data. (For example, a node can be split into child nodes based upon the subsets of the population whose ages are less than 50, between 50 and 100, and greater than 100.) The algorithm continues to recurse on each subset, considering only attributes never selected before. Recursion on a subset may stop in one of these cases: every element in the subset belongs to the same class; in which case the node is turned into a leaf node and labelled with the class of the examples. there are no more attributes to be selected, but the examples still do not belong to the same class. In this case, the node is made a leaf node and labelled with the most common class of the examples in the subset. there are no examples in the subset, which happens when no example in the parent set was found to match a specific value of the selected attribute. An example could be the absence of a person among the population with age over 100 years. Then a leaf node is created and labelled with the most common class of the examples in the parent node's set. Throughout the algorithm, the decision tree is constructed with each non-terminal node (internal node) representing the selected attribute on which the data was split, and terminal nodes (leaf nodes) representing the class label of the final subset of this branch. === Summary === Calculate the entropy of every attribute a {\displaystyle a} of the data set S {\displaystyle S} . Partition ("split") the set S {\displaystyle S} into subsets using the attribute for which the resulting entropy after splitting is minimized; or, equivalently, information gain is maximum. Make a decision tree node containing that attribute. Recurse on subsets using the remaining attributes. === Properties === ID3 does not guarantee an optimal solution. It can converge upon local optima. It uses a greedy strategy by selecting the locally best attribute to split the dataset on each iteration. The algorithm's optimality can be improved by using backtracking during the search for the optimal decision tree at the cost of possibly taking longer. ID3 can overfit the training data. To avoid overfitting, smaller decision trees should be preferred over larger ones. This algorithm usually produces small trees, but it does not always produce the smallest possible decision tree. ID3 is harder to use on continuous data than on factored data (factored data has a discrete number of possible values, thus reducing the possible branch points). If the values of any given attribute are continuous, then there are many more places to split the data on this attribute, and searching for the best value to split by can be time-consuming. === Usage === The ID3 algorithm is used by training on a data set S {\displaystyle S} to produce a decision tree which is stored in memory. At runtime, this decision tree is used to classify new test cases (feature vectors) by traversing the decision tree using the features of the datum to arrive at a leaf node. == The ID3 metrics == === Entropy === Entropy H ( S ) {\displaystyle \mathrm {H} {(S)}} is a measure of the amount of uncertainty in the (data) set S {\displaystyle S} (i.e. entropy characterizes the (data) set S {\displaystyle S} ). H ( S ) = ∑ x ∈ X − p ( x ) log 2 p ( x ) {\displaystyle \mathrm {H} {(S)}=\sum _{x\in X}{-p(x)\log _{2}p(x)}} Where, S {\displaystyle S} – The current dataset for which entropy is being calculated This changes at each step of the ID3 algorithm, either to a subset of the previous set in the case of splitting on an attribute or to a "sibling" partition of the parent in case the recursion terminated previously. X {\displaystyle X} – The set of classes in S {\displaystyle S} p ( x ) {\displaystyle p(x)} – The proportion of the number of elements in class x {\displaystyle x} to the number of elements in set S {\displaystyle S} When H ( S ) = 0 {\displaystyle \mathrm {H} {(S)}=0} , the set S {\displaystyle S} is perfectly classified (i.e. all elements in S {\displaystyle S} are of the same class). In ID3, entropy is calculated for each remaining attribute. The attribute with the smallest entropy is used to split the set S {\displaystyle S} on this iteration. Entropy in information theory measures how much information is expected to be gained upon measuring a random variable; as such, it can also be used to quantify the amount to which the distribution of the quantity's values is unknown. A constant quantity has zero entropy, as its distribution is perfectly known. In contrast, a uniformly distributed random variable (discretely or continuously uniform) maximizes entropy. Therefore, the greater the entropy at a node, the less information is known about the classification of data at this stage of the tree; and therefore, the greater the potential to improve the classification here. As such, ID3 is a greedy heuristic performing a best-first search for locally optimal entropy values. Its accuracy can be improved by preprocessing the data. === Information gain === Information gain I G ( A ) {\displaystyle IG(A)} is the measure of the difference in entropy from before to after the set S {\displaystyle S} is split on an attribute A {\displaystyle A} . In other words, how much uncertainty in S {\displaystyle S} was reduced after splitting set S {\displaystyle S} on attribute A {\displaystyle A} . I G ( S , A ) = H ( S ) − ∑ t ∈ T p ( t ) H ( t ) = H ( S ) − H ( S | A ) . {\displaystyle IG(S,A)=\mathrm {H} {(S)}-\sum _{t\in T}p(t)\mathrm {H} {(t)}=\mathrm {H} {(S)}-\mathrm {H} {(S|A)}.} Where, H ( S ) {\displaystyle \mathrm {H} (S)} – Entropy of set S {\displaystyle S} T {\displaystyle T} – The subsets created from splitting set S {\displaystyle S} by attribute A {\displaystyle A} such that S = ⋃ t ∈ T t {\displaystyle S=\bigcup _{t\in T}t} p ( t ) {\displaystyle p(t)} – The proportion of the number of elements in t {\displaystyle t} to the number of elements in set S {\displaystyle S} H ( t ) {\displaystyle \mathrm {H} (t)} – Entropy of subset t {\displaystyle t} In ID3, information gain can be calculated (instead of entropy) for each remaining attribute. The attribute with the largest information gain is used to split the set S {\displaystyle S} on this iteration. Generalized blockmodeling of binary networks (also relational blockmodeling) is an approach of generalized blockmodeling, analysing the binary network(s). As most network analyses deal with binary networks, this approach is also considered as the fundamental approach of blockmodeling. This is especially noted, as the set of ideal blocks, when used for interpretation of blockmodels, have binary link patterns, which precludes them to be compared with valued empirical blocks. When analysing the binary networks, the criterion function is measuring block inconsistencies, while also reporting the possible errors. The ideal block in binary blockmodeling has only three types of conditions: "a certain cell must be (at least) 1, a certain cell must be 0 and the f {\displaystyle f} over each row (or column) must be at least 1". It is also used as a basis for developing the generalized blockmodeling of valued networks. The Journal of Machine Learning Research is a peer-reviewed open access scientific journal covering machine learning. It was established in 2000 and the first editor-in-chief was Leslie Kaelbling. The current editors-in-chief are Francis Bach (Inria) and David Blei (Columbia University). == History == The journal was established as an open-access alternative to the journal Machine Learning. In 2001, forty editorial board members of Machine Learning resigned, saying that in the era of the Internet, it was detrimental for researchers to continue publishing their papers in expensive journals with pay-access archives. The open access model employed by the Journal of Machine Learning Research allows authors to publish articles for free and retain copyright, while archives are freely available online. Print editions of the journal were published by MIT Press until 2004 and by Microtome Publishing thereafter. From its inception, the journal received no revenue from the print edition and paid no subvention to MIT Press or Microtome Publishing. In response to the prohibitive costs of arranging workshop and conference proceedings publication with traditional academic publishing companies, the journal launched a proceedings publication arm in 2007 and now publishes proceedings for several leading machine learning conferences, including the International Conference on Machine Learning, COLT, AISTATS, and workshops held at the Conference on Neural Information Processing Systems. Language identification in the limit is a formal model for inductive inference of formal languages, mainly by computers (see machine learning and induction of regular languages). It was introduced by E. Mark Gold in a technical report and a journal article with the same title. In this model, a teacher provides to a learner some presentation (i.e. a sequence of strings) of some formal language. The learning is seen as an infinite process. Each time the learner reads an element of the presentation, it should provide a representation (e.g. a formal grammar) for the language. Gold defines that a learner can identify in the limit a class of languages if, given any presentation of any language in the class, the learner will produce only a finite number of wrong representations, and then stick with the correct representation. However, the learner need not be able to announce its correctness; and the teacher might present a counterexample to any representation arbitrarily long after. Gold defined two types of presentations: Text (positive information): an enumeration of all strings the language consists of. Complete presentation (positive and negative information): an enumeration of all possible strings, each with a label indicating if the string belongs to the language or not. == Learnability == This model is an early attempt to formally capture the notion of learnability. Gold's journal article introduces for contrast the stronger models Finite identification (where the learner has to announce correctness after a finite number of steps), and Fixed-time identification (where correctness has to be reached after an apriori-specified number of steps). A weaker formal model of learnability is the Probably approximately correct learning (PAC) model, introduced by Leslie Valiant in 1984. == Examples == It is instructive to look at concrete examples (in the tables) of learning sessions the definition of identification in the limit speaks about. A fictitious session to learn a regular language L over the alphabet {a,b} from text presentation:In each step, the teacher gives a string belonging to L, and the learner answers a guess for L, encoded as a regular expression. In step 3, the learner's guess is not consistent with the strings seen so far; in step 4, the teacher gives a string repeatedly. After step 6, the learner sticks to the regular expression (ab+ba). If this happens to be a description of the language L the teacher has in mind, it is said that the learner has learned that language.If a computer program for the learner's role would exist that was able to successfully learn each regular language, that class of languages would be identifiable in the limit. Gold has shown that this is not the case. A particular learning algorithm always guessing L to be just the union of all strings seen so far:If L is a finite language, the learner will eventually guess it correctly, however, without being able to tell when. Although the guess didn't change during step 3 to 6, the learner couldn't be sure to be correct.Gold has shown that the class of finite languages is identifiable in the limit, however, this class is neither finitely nor fixed-time identifiable. Learning from complete presentation by telling:In each step, the teacher gives a string and tells whether it belongs to L (green) or not (red, struck-out). Each possible string is eventually classified in this way by the teacher. Learning from complete presentation by request:The learner gives a query string, the teacher tells whether it belongs to L (yes) or not (no); the learner then gives a guess for L, followed by the next query string. In this example, the learner happens to query in each step just the same string as given by the teacher in example 3.In general, Gold has shown that each language class identifiable in the request-presentation setting is also identifiable in the telling-presentation setting, since the learner, instead of querying a string, just needs to wait until it is eventually given by the teacher. == Gold's theorem == More formally, a language L {\displaystyle L} is a nonempty set, and its elements are called sentences. a language family is a set of languages. a language-learning environment E {\displaystyle E} for a language L {\displaystyle L} is a stream of sentences from L {\displaystyle L} , such that each sentence in L {\displaystyle L} appears at least once. a language learner is a function f {\displaystyle f} that sends a list of sentences to a language. This is interpreted as saying that, after seeing sentences a 1 , a 2 . . . , a n {\displaystyle a_{1},a_{2}...,a_{n}} in that order, the language learner guesses that the language that produces the sentences should be f ( a 1 , . . . , a n ) {\displaystyle f(a_{1},...,a_{n})} . Note that the learner is not obliged to be correct — it could very well guess a language that does not even contain a 1 , . . . , a n {\displaystyle a_{1},...,a_{n}} . a language learner f {\displaystyle f} learns a language L {\displaystyle L} in environment E = ( a 1 , a 2 , . . . ) {\displaystyle E=(a_{1},a_{2},...)} if the learner always guesses L {\displaystyle L} after seeing enough examples from the environment. a language learner f {\displaystyle f} learns a language L {\displaystyle L} if it learns L {\displaystyle L} in any environment E {\displaystyle E} for L {\displaystyle L} . a language family is learnable if there exists a language learner that can learn all languages in the family. Notes: In the context of Gold's theorem, sentences need only be distinguishable. They need not be anything in particular, such as finite strings (as usual in formal linguistics). Learnability is not a concept for individual languages. Any individual language L {\displaystyle L} could be learned by a trivial learner that always guesses L {\displaystyle L} . Learnability is not a concept for individual learners. A language family is learnable if, and only if, there exists some learner that can learn the family. It does not matter how well the learner performs for learning languages outside the family. Gold's theorem is easily bypassed if negative examples are allowed. In particular, the language family { L 1 , L 2 , . . . , L ∞ } {\displaystyle \{L_{1},L_{2},...,L_{\infty }\}} can be learned by a learner that always guesses L ∞ {\displaystyle L_{\infty }} until it receives the first negative example ¬ a n {\displaystyle \neg a_{n}} , where a n ∈ L n + 1 ∖ L n {\displaystyle a_{n}\in L_{n+1}\setminus L_{n}} , at which point it always guesses L n {\displaystyle L_{n}} . == Learnability characterization == Dana Angluin gave the characterizations of learnability from text (positive information) in a 1980 paper. If a learner is required to be effective, then an indexed class of recursive languages is learnable in the limit if there is an effective procedure that uniformly enumerates tell-tales for each language in the class (Condition 1). It is not hard to see that if an ideal learner (i.e., an arbitrary function) is allowed, then an indexed class of languages is learnable in the limit if each language in the class has a tell-tale (Condition 2). == Language classes learnable in the limit == The table shows which language classes are identifiable in the limit in which learning model. On the right-hand side, each language class is a superclass of all lower classes. Each learning model (i.e. type of presentation) can identify in the limit all classes below it. In particular, the class of finite languages is identifiable in the limit by text presentation (cf. Example 2 above), while the class of regular languages is not. Pattern Languages, introduced by Dana Angluin in another 1980 paper, are also identifiable by normal text presentation; they are omitted in the table, since they are above the singleton and below the primitive recursive language class, but incomparable to the classes in between. == Sufficient conditions for learnability == Condition 1 in Angluin's paper is not always easy to verify. Therefore, people come up with various sufficient conditions for the learnability of a language class. See also Induction of regular languages for learnable subclasses of regular languages. === Finite thickness === A class of languages has finite thickness if every non-empty set of strings is contained in at most finitely many languages of the class. This is exactly Condition 3 in Angluin's paper. Angluin showed that if a class of recursive languages has finite thickness, then it is learnable in the limit. A class with finite thickness certainly satisfies MEF-condition and MFF-condition; in other words, finite thickness implies M-finite thickness. === Finite elasticity === A class of languages is said to have finite elasticity if for every infinite sequence of strings s 0 , s 1 , . . . {\displaystyle s_{0},s_{1},...} and every infinite sequence of languages in the class L 1 , L 2 , . . . {\displaystyle L_{1},L_{2},...} , there exists a finite number n such ID3 algorithm
Generalized blockmodeling of binary networks
Journal of Machine Learning Research
Language identification in the limit