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  MultiValue database

  A MultiValue database is a type of NoSQL and multidimensional database. It is typically considered synonymous with PICK, a database originally developed as the Pick operating system. MultiValue databases include commercial products from Rocket Software, Revelation, InterSystems, Northgate Information Solutions, ONgroup, and other companies. These databases differ from a relational database in that they have features that support and encourage the use of attributes which can take a list of values, rather than all attributes being single-valued. They are often categorized with MUMPS within the category of post-relational databases, although the data model actually pre-dates the relational model. Unlike SQL-DBMS tools, most MultiValue databases can be accessed both with or without SQL. == History == Don Nelson designed the MultiValue data model in the early to mid-1960s. Dick Pick, a developer at TRW, worked on the first implementation of this model for the US Army in 1965. Pick considered the software to be in the public domain because it was written for the military, this was but the first dispute regarding MultiValue databases that was addressed by the courts. Ken Simms wrote DataBASIC, sometimes known as S-BASIC, in the mid-1970s. It was based on Dartmouth BASIC, but had enhanced features for data management. Simms played a lot of Star Trek (a text-based early computer game originally written in Dartmouth BASIC) while developing the language, to ensure that DataBASIC functioned to his satisfaction. Three of the implementations of MultiValue - PICK version R77, Microdata Reality 3.x, and Prime Information 1.0 - were very similar. In spite of attempts to standardize, particularly by International Spectrum and the Spectrum Manufacturers Association, who designed a logo for all to use, there are no standards across MultiValue implementations. Subsequently, these flavors diverged, although with some cross-over. These streams of MultiValue database development could be classified as one stemming from PICK R83, one from Microdata Reality, and one from Prime Information. Because of the differences, some implementations have provisions for supporting several flavors of the languages. An attempt to document the similarities and differences can be found at the Post-Relational Database Reference (PRDB). One reasonable hypothesis for this data model lasting 50 years, with new database implementations of the model even in the 21st century is that it provides inexpensive database solutions. == Data model example == In a MultiValue database system: a database or schema is called an "account" a table or collection is called a "file" a column or field is called a field or an "attribute", which is composed of "multi-value attributes" and "sub-value attributes" to store multiple values in the same attribute. a row or document is called a "record" or "item" Data is stored using two separate files: a "file" to store raw data and a "dictionary" to store the format for displaying the raw data. For example, assume there's a file (table) called "PERSON". In this file, there is an attribute called "eMailAddress". The eMailAddress field can store a variable number of email address values in a single record. The list [[email protected], [email protected], [email protected]] can be stored and accessed via a single query when accessing the associated record. Achieving the same (one-to-many) relationship within a traditional relational database system would include creating an additional table to store the variable number of email addresses associated with a single "PERSON" record. However, modern relational database systems support this multi-value data model too. For example, in PostgreSQL, a column can be an array of any base type. == MultiValue Basic Language == Multivalue Basic (now commonly styled as mvBasic) is a family of programming languages more or less common (and portable) to all the multivalue databases derived from the original Pick Operating System. The variations between implementations are known as flavours. The language originates from Dartmouth Basic and the earliest implementation of PickBASIC (now D3 FlashBasic). Over time various customisations and extensions have been added to take advantage of capabilities added to the different flavours while staying mainly in sync. mvBasic statements and functions are designed to access and take advantage of the multivalue database model and providing the usual capabilities of most modern languages. For example, cryptography and communications. mvBasic is typeless and lends itself to structured programming techniques. Example code is available but limited. Whilst there are commercial applications and tools available, the multivalue database community has not embraced the open source library/package model to the degree seen with other languages. The typical mvBasic compiler compiles program source to a P-code executable object and runs in an interpreter, with D3 FlashBasic and jBASE being notable exceptions. == MultiValue Query Language == Known as ENGLISH, ACCESS, AQL, UniQuery, Retrieve, CMQL, and by many other names over the years, corresponding to the different MultiValue implementations, the MultiValue query language differs from SQL in several respects. Each query is issued against a single dictionary within the schema, which could be understood as a virtual file or a portal to the database through which to view the data. LIST PEOPLE LAST_NAME FIRST_NAME EMAIL_ADDRESSES WITH LAST_NAME LIKE "Van..." The above statement would list all e-mail addresses for each person whose last name starts with "Van". A single entry would be output for each person, with multiple lines showing the multiple e-mail addresses (without repeating other data about the person).

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  Baum–Welch algorithm

  In electrical engineering, statistical computing and bioinformatics, the Baum–Welch algorithm is a special case of the expectation–maximization algorithm used to find the unknown parameters of a hidden Markov model (HMM). It makes use of the forward-backward algorithm to compute the statistics for the expectation step. The Baum–Welch algorithm, the primary method for inference in hidden Markov models, is numerically unstable due to its recursive calculation of joint probabilities. As the number of variables grows, these joint probabilities become increasingly small, leading to the forward recursions rapidly approaching values below machine precision. == History == The Baum–Welch algorithm was named after its inventors Leonard E. Baum and Lloyd R. Welch. The algorithm and the Hidden Markov models were first described in a series of articles by Baum and his peers at the IDA Center for Communications Research, Princeton in the late 1960s and early 1970s. One of the first major applications of HMMs was to the field of speech processing. In the 1980s, HMMs were emerging as a useful tool in the analysis of biological systems and information, and in particular genetic information. They have since become an important tool in the probabilistic modeling of genomic sequences. == Description == A hidden Markov model describes the joint probability of a collection of "hidden" and observed discrete random variables. It relies on the assumption that the i-th hidden variable given the (i − 1)-th hidden variable is independent of previous hidden variables, and the current observation variables depend only on the current hidden state. The Baum–Welch algorithm uses the well known EM algorithm to find the maximum likelihood estimate of the parameters of a hidden Markov model given a set of observed feature vectors. Let X t {\displaystyle X_{t}} be a discrete hidden random variable with N {\displaystyle N} possible values (i.e. We assume there are N {\displaystyle N} states in total). We assume the P ( X t ∣ X t − 1 ) {\displaystyle P(X_{t}\mid X_{t-1})} is independent of time t {\displaystyle t} , which leads to the definition of the time-independent stochastic transition matrix A = { a i j } = P ( X t = j ∣ X t − 1 = i ) . {\displaystyle A=\{a_{ij}\}=P(X_{t}=j\mid X_{t-1}=i).} The initial state distribution (i.e. when t = 1 {\displaystyle t=1} ) is given by π i = P ( X 1 = i ) . {\displaystyle \pi _{i}=P(X_{1}=i).} The observation variables Y t {\displaystyle Y_{t}} can take one of K {\displaystyle K} possible values. We also assume the observation given the "hidden" state is time independent. The probability of a certain observation y i {\displaystyle y_{i}} at time t {\displaystyle t} for state X t = j {\displaystyle X_{t}=j} is given by b j ( y i ) = P ( Y t = y i ∣ X t = j ) . {\displaystyle b_{j}(y_{i})=P(Y_{t}=y_{i}\mid X_{t}=j).} Taking into account all the possible values of Y t {\displaystyle Y_{t}} and X t {\displaystyle X_{t}} , we obtain the N × K {\displaystyle N\times K} matrix B = { b j ( y i ) } {\displaystyle B=\{b_{j}(y_{i})\}} where b j {\displaystyle b_{j}} belongs to all the possible states and y i {\displaystyle y_{i}} belongs to all the observations. An observation sequence is given by Y = ( Y 1 = y 1 , Y 2 = y 2 , … , Y T = y T ) {\displaystyle Y=(Y_{1}=y_{1},Y_{2}=y_{2},\ldots ,Y_{T}=y_{T})} . Thus we can describe a hidden Markov chain by θ = ( A , B , π ) {\displaystyle \theta =(A,B,\pi )} . The Baum–Welch algorithm finds a local maximum for θ ∗ = a r g m a x θ ⁡ P ( Y ∣ θ ) {\displaystyle \theta ^{}=\operatorname {arg\,max} _{\theta }P(Y\mid \theta )} (i.e. the HMM parameters θ {\displaystyle \theta } that maximize the probability of the observation). === Algorithm === Set θ = ( A , B , π ) {\displaystyle \theta =(A,B,\pi )} with random initial conditions. They can also be set using prior information about the parameters if it is available; this can speed up the algorithm and also steer it toward the desired local maximum. ==== Forward procedure ==== Let α i ( t ) = P ( Y 1 = y 1 , … , Y t = y t , X t = i ∣ θ ) {\displaystyle \alpha _{i}(t)=P(Y_{1}=y_{1},\ldots ,Y_{t}=y_{t},X_{t}=i\mid \theta )} , the probability of seeing the observations y 1 , y 2 , … , y t {\displaystyle y_{1},y_{2},\ldots ,y_{t}} and being in state i {\displaystyle i} at time t {\displaystyle t} . This is found recursively: α i ( 1 ) = π i b i ( y 1 ) , {\displaystyle \alpha _{i}(1)=\pi _{i}b_{i}(y_{1}),} α i ( t + 1 ) = b i ( y t + 1 ) ∑ j = 1 N α j ( t ) a j i . {\displaystyle \alpha _{i}(t+1)=b_{i}(y_{t+1})\sum _{j=1}^{N}\alpha _{j}(t)a_{ji}.} Since this series converges exponentially to zero, the algorithm will numerically underflow for longer sequences. However, this can be avoided in a slightly modified algorithm by scaling α {\displaystyle \alpha } in the forward and β {\displaystyle \beta } in the backward procedure below. ==== Backward procedure ==== Let β i ( t ) = P ( Y t + 1 = y t + 1 , … , Y T = y T ∣ X t = i , θ ) {\displaystyle \beta _{i}(t)=P(Y_{t+1}=y_{t+1},\ldots ,Y_{T}=y_{T}\mid X_{t}=i,\theta )} that is the probability of the ending partial sequence y t + 1 , … , y T {\displaystyle y_{t+1},\ldots ,y_{T}} given starting state i {\displaystyle i} at time t {\displaystyle t} . We calculate β i ( t ) {\displaystyle \beta _{i}(t)} as, β i ( T ) = 1 , {\displaystyle \beta _{i}(T)=1,} β i ( t ) = ∑ j = 1 N β j ( t + 1 ) a i j b j ( y t + 1 ) . {\displaystyle \beta _{i}(t)=\sum _{j=1}^{N}\beta _{j}(t+1)a_{ij}b_{j}(y_{t+1}).} ==== Update ==== We can now calculate the temporary variables, according to Bayes' theorem: γ i ( t ) = P ( X t = i ∣ Y , θ ) = P ( X t = i , Y ∣ θ ) P ( Y ∣ θ ) = α i ( t ) β i ( t ) ∑ j = 1 N α j ( t ) β j ( t ) , {\displaystyle \gamma _{i}(t)=P(X_{t}=i\mid Y,\theta )={\frac {P(X_{t}=i,Y\mid \theta )}{P(Y\mid \theta )}}={\frac {\alpha _{i}(t)\beta _{i}(t)}{\sum _{j=1}^{N}\alpha _{j}(t)\beta _{j}(t)}},} which is the probability of being in state i {\displaystyle i} at time t {\displaystyle t} given the observed sequence Y {\displaystyle Y} and the parameters θ {\displaystyle \theta } ξ i j ( t ) = P ( X t = i , X t + 1 = j ∣ Y , θ ) = P ( X t = i , X t + 1 = j , Y ∣ θ ) P ( Y ∣ θ ) = α i ( t ) a i j β j ( t + 1 ) b j ( y t + 1 ) ∑ k = 1 N ∑ w = 1 N α k ( t ) a k w β w ( t + 1 ) b w ( y t + 1 ) , {\displaystyle \xi _{ij}(t)=P(X_{t}=i,X_{t+1}=j\mid Y,\theta )={\frac {P(X_{t}=i,X_{t+1}=j,Y\mid \theta )}{P(Y\mid \theta )}}={\frac {\alpha _{i}(t)a_{ij}\beta _{j}(t+1)b_{j}(y_{t+1})}{\sum _{k=1}^{N}\sum _{w=1}^{N}\alpha _{k}(t)a_{kw}\beta _{w}(t+1)b_{w}(y_{t+1})}},} which is the probability of being in state i {\displaystyle i} and j {\displaystyle j} at times t {\displaystyle t} and t + 1 {\displaystyle t+1} respectively given the observed sequence Y {\displaystyle Y} and parameters θ {\displaystyle \theta } . The denominators of γ i ( t ) {\displaystyle \gamma _{i}(t)} and ξ i j ( t ) {\displaystyle \xi _{ij}(t)} are the same ; they represent the probability of making the observation Y {\displaystyle Y} given the parameters θ {\displaystyle \theta } . The parameters of the hidden Markov model θ {\displaystyle \theta } can now be updated: π i ∗ = γ i ( 1 ) , {\displaystyle \pi _{i}^{}=\gamma _{i}(1),} which is the expected frequency spent in state i {\displaystyle i} at time 1 {\displaystyle 1} . a i j ∗ = ∑ t = 1 T − 1 ξ i j ( t ) ∑ t = 1 T − 1 γ i ( t ) , {\displaystyle a_{ij}^{}={\frac {\sum _{t=1}^{T-1}\xi _{ij}(t)}{\sum _{t=1}^{T-1}\gamma _{i}(t)}},} which is the expected number of transitions from state i to state j compared to the expected total number of transitions starting in state i, including from state i to itself. The number of transitions starting in state i is equivalent to the number of times state i is observed in the sequence from t = 1 to t = T − 1. b i ∗ ( v k ) = ∑ t = 1 T 1 y t = v k γ i ( t ) ∑ t = 1 T γ i ( t ) , {\displaystyle b_{i}^{}(v_{k})={\frac {\sum _{t=1}^{T}1_{y_{t}=v_{k}}\gamma _{i}(t)}{\sum _{t=1}^{T}\gamma _{i}(t)}},} where 1 y t = v k = { 1 if y t = v k , 0 otherwise {\displaystyle 1_{y_{t}=v_{k}}={\begin{cases}1&{\text{if }}y_{t}=v_{k},\\0&{\text{otherwise}}\end{cases}}} is an indicator function, and b i ∗ ( v k ) {\displaystyle b_{i}^{}(v_{k})} is the expected number of times the output observations have been equal to v k {\displaystyle v_{k}} while in state i {\displaystyle i} over the expected total number of times in state i {\displaystyle i} . These steps are now repeated iteratively until a desired level of convergence. Note: It is possible to over-fit a particular data set. That is, P ( Y ∣ θ final ) > P ( Y ∣ θ true ) {\displaystyle P(Y\mid \theta _{\text{final}})>P(Y\mid \theta _{\text{true}})} . The algorithm also does not guarantee a global maximum. ==== Multiple sequences ==== The algorithm described thus far assumes a single observed sequence Y = y 1 , … , y T {\displaystyle Y=y_{1},\ldots ,y_{T}} . However, in many situations, there are several sequences observed: Y 1 ,

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  Extended affix grammar

  In computer science, extended affix grammars (EAGs) are a formal grammar formalism for describing the context free and context sensitive syntax of language, both natural language and programming languages. EAGs are a member of the family of two-level grammars; more specifically, a restriction of Van Wijngaarden grammars with the specific purpose of making parsing feasible. Like Van Wijngaarden grammars, EAGs have hyperrules that form a context-free grammar except in that their nonterminals may have arguments, known as affixes, the possible values of which are supplied by another context-free grammar, the metarules. EAGs were introduced and studied by D.A. Watt in 1974; recognizers were developed at the University of Nijmegen between 1985 and 1995. The EAG compiler developed there will generate either a recogniser, a transducer, a translator, or a syntax directed editor for a language described in the EAG formalism. The formalism is quite similar to Prolog, to the extent that it borrowed its cut operator. EAGs have been used to write grammars of natural languages such as English, Spanish, and Hungarian. The aim was to verify the grammars by making them parse corpora of text (corpus linguistics); hence, parsing had to be sufficiently practical. However, the parse tree explosion problem that ambiguities in natural language tend to produce in this type of approach is worsened for EAGs because each choice of affix value may produce a separate parse, even when several different values are equivalent. The remedy proposed was to switch to the much simpler Affix Grammar over a Finite Lattice (AGFL) instead, in which metagrammars can only produce simple finite languages.

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  AI Virtual Assistants: Free vs Paid (2026)

  Trying to pick the best AI virtual assistant? An AI virtual assistant is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI virtual assistant slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.

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  A Logical Calculus of the Ideas Immanent in Nervous Activity

  "A Logical Calculus of the Ideas Immanent in Nervous Activity" is a 1943 paper written by Warren Sturgis McCulloch and Walter Pitts, published in the journal The Bulletin of Mathematical Biophysics. The paper proposed a mathematical model of the nervous system as a network of simple logical elements, later known as artificial neurons, or McCulloch–Pitts neurons. These neurons receive inputs, perform a weighted sum, and fire an output signal based on a threshold function. By connecting these units in various configurations, McCulloch and Pitts demonstrated that their model could perform all logical functions. It is a seminal work in cognitive science, computational neuroscience, computer science, and artificial intelligence. It was a foundational result in automata theory. John von Neumann cited it as a significant result. == Mathematics == The artificial neuron used in the original paper is slightly different from the modern version. They considered neural networks that operate in discrete steps of time t = 0 , 1 , … {\displaystyle t=0,1,\dots } . The neural network contains a number of neurons. Let the state of a neuron i {\displaystyle i} at time t {\displaystyle t} be N i ( t ) {\displaystyle N_{i}(t)} . The state of a neuron can either be 0 or 1, standing for "not firing" and "firing". Each neuron also has a firing threshold θ {\displaystyle \theta } , such that it fires if the total input exceeds the threshold. Each neuron can connect to any other neuron (including itself) with positive synapses (excitatory) or negative synapses (inhibitory). That is, each neuron can connect to another neuron with a weight w {\displaystyle w} taking an integer value. A peripheral afferent is a neuron with no incoming synapses. We can regard each neural network as a directed graph, with the nodes being the neurons, and the directed edges being the synapses. A neural network has a circle or a circuit if there exists a directed circle in the graph. Let w i j ( t ) {\displaystyle w_{ij}(t)} be the connection weight from neuron j {\displaystyle j} to neuron i {\displaystyle i} at time t {\displaystyle t} , then its next state is N i ( t + 1 ) = H ( ∑ j = 1 n w i j ( t ) N j ( t ) − θ i ( t ) ) , {\displaystyle N_{i}(t+1)=H\left(\sum _{j=1}^{n}w_{ij}(t)N_{j}(t)-\theta _{i}(t)\right),} where H {\displaystyle H} is the Heaviside step function (outputting 1 if the input is greater than or equal to 0, and 0 otherwise). === Symbolic logic === The paper used, as a logical language for describing neural networks, "Language II" from The Logical Syntax of Language by Rudolf Carnap with some notations taken from Principia Mathematica by Alfred North Whitehead and Bertrand Russell. Language II covers substantial parts of classical mathematics, including real analysis and portions of set theory. To describe a neural network with peripheral afferents N 1 , N 2 , … , N p {\displaystyle N_{1},N_{2},\dots ,N_{p}} and non-peripheral afferents N p + 1 , N p + 2 , … , N n {\displaystyle N_{p+1},N_{p+2},\dots ,N_{n}} they considered logical predicate of form P r ( N 1 , N 2 , … , N p , t ) {\displaystyle Pr(N_{1},N_{2},\dots ,N_{p},t)} where P r {\displaystyle Pr} is a first-order logic predicate function (a function that outputs a boolean), N 1 , … , N p {\displaystyle N_{1},\dots ,N_{p}} are predicates that take t {\displaystyle t} as an argument, and t {\displaystyle t} is the only free variable in the predicate. Intuitively speaking, N 1 , … , N p {\displaystyle N_{1},\dots ,N_{p}} specifies the binary input patterns going into the neural network over all time, and P r ( N 1 , N 2 , … , N n , t ) {\displaystyle Pr(N_{1},N_{2},\dots ,N_{n},t)} is a function that takes some binary input patterns, and constructs an output binary pattern P r ( N 1 , N 2 , … , N n , 0 ) , P r ( N 1 , N 2 , … , N n , 1 ) , … {\displaystyle Pr(N_{1},N_{2},\dots ,N_{n},0),Pr(N_{1},N_{2},\dots ,N_{n},1),\dots } . A logical sentence P r ( N 1 , N 2 , … , N n , t ) {\displaystyle Pr(N_{1},N_{2},\dots ,N_{n},t)} is realized by a neural network iff there exists a time-delay T ≥ 0 {\displaystyle T\geq 0} , a neuron i {\displaystyle i} in the network, and an initial state for the non-peripheral neurons N p + 1 ( 0 ) , … , N n ( 0 ) {\displaystyle N_{p+1}(0),\dots ,N_{n}(0)} , such that for any time t {\displaystyle t} , the truth-value of the logical sentence is equal to the state of the neuron i {\displaystyle i} at time t + T {\displaystyle t+T} . That is, ∀ t = 0 , 1 , 2 , … , P r ( N 1 , N 2 , … , N p , t ) = N i ( t + T ) {\displaystyle \forall t=0,1,2,\dots ,\quad Pr(N_{1},N_{2},\dots ,N_{p},t)=N_{i}(t+T)} === Equivalence === In the paper, they considered some alternative definitions of artificial neural networks, and have shown them to be equivalent, that is, neural networks under one definition realizes precisely the same logical sentences as neural networks under another definition. They considered three forms of inhibition: relative inhibition, absolute inhibition, and extinction. The definition above is relative inhibition. By "absolute inhibition" they meant that if any negative synapse fires, then the neuron will not fire. By "extinction" they meant that if at time t {\displaystyle t} , any inhibitory synapse fires on a neuron i {\displaystyle i} , then θ i ( t + j ) = θ i ( 0 ) + b j {\displaystyle \theta _{i}(t+j)=\theta _{i}(0)+b_{j}} for j = 1 , 2 , 3 , … {\displaystyle j=1,2,3,\dots } , until the next time an inhibitory synapse fires on i {\displaystyle i} . It is required that b j = 0 {\displaystyle b_{j}=0} for all large j {\displaystyle j} . Theorem 4 and 5 state that these are equivalent. They considered three forms of excitation: spatial summation, temporal summation, and facilitation. The definition above is spatial summation (which they pictured as having multiple synapses placed close together, so that the effect of their firing sums up). By "temporal summation" they meant that the total incoming signal is ∑ τ = 0 T ∑ j = 1 n w i j ( t ) N j ( t − τ ) {\displaystyle \sum _{\tau =0}^{T}\sum _{j=1}^{n}w_{ij}(t)N_{j}(t-\tau )} for some T ≥ 1 {\displaystyle T\geq 1} . By "facilitation" they meant the same as extinction, except that b j ≤ 0 {\displaystyle b_{j}\leq 0} . Theorem 6 states that these are equivalent. They considered neural networks that do not change, and those that change by Hebbian learning. That is, they assume that at t = 0 {\displaystyle t=0} , some excitatory synaptic connections are not active. If at any t {\displaystyle t} , both N i ( t ) = 1 , N j ( t ) = 1 {\displaystyle N_{i}(t)=1,N_{j}(t)=1} , then any latent excitatory synapse between i , j {\displaystyle i,j} becomes active. Theorem 7 states that these are equivalent. === Logical expressivity === They considered "temporal propositional expressions" (TPE), which are propositional formulas with one free variable t {\displaystyle t} . For example, N 1 ( t ) ∨ N 2 ( t ) ∧ ¬ N 3 ( t ) {\displaystyle N_{1}(t)\vee N_{2}(t)\wedge \neg N_{3}(t)} is such an expression. Theorem 1 and 2 together showed that neural nets without circles are equivalent to TPE. For neural nets with loops, they noted that "realizable P r {\displaystyle Pr} may involve reference to past events of an indefinite degree of remoteness". These then encodes for sentences like "There was some x such that x was a ψ" or ( ∃ x ) ( ψ x ) {\displaystyle (\exists x)(\psi x)} . Theorems 8 to 10 showed that neural nets with loops can encode all first-order logic with equality and conversely, any looped neural networks is equivalent to a sentence in first-order logic with equality, thus showing that they are equivalent in logical expressiveness. As a remark, they noted that a neural network, if furnished with a tape, scanners, and write-heads, is equivalent to a Turing machine, and conversely, every Turing machine is equivalent to some such neural network. Thus, these neural networks are equivalent to Turing computability and Church's lambda-definability. == Context == === Previous work === The paper built upon several previous strands of work. In the symbolic logic side, it built on the previous work by Carnap, Whitehead, and Russell. This was contributed by Walter Pitts, who had a strong proficiency with symbolic logic. Pitts provided mathematical and logical rigor to McCulloch’s vague ideas on psychons (atoms of psychological events) and circular causality. In the neuroscience side, it built on previous work by the mathematical biology research group centered around Nicolas Rashevsky, of which McCulloch was a member. The paper was published in the Bulletin of Mathematical Biophysics, which was founded by Rashevsky in 1939. During the late 1930s, Rashevsky's research group was producing papers that had difficulty publishing in other journals at the time, so Rashevsky decided to found a new journal exclusively devoted to mathematical biophysics. Also in the Rashevsky's group was Alston Scott Householder, who in 1941 published an abstract model

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  Top 10 AI Writing Assistants Compared (2026)

  Trying to pick the best AI writing assistant? An AI writing assistant is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI writing assistant slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.

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  Tree transducer

  In theoretical computer science and formal language theory, a tree transducer (TT) is an abstract machine taking as input a tree, and generating output – generally other trees, but models producing words or other structures exist. Roughly speaking, tree transducers extend tree automata in the same way that word transducers extend word automata. Manipulating tree structures instead of words enable TT to model syntax-directed transformations of formal or natural languages. However, TT are not as well-behaved as their word counterparts in terms of algorithmic complexity, closure properties, etcetera. In particular, most of the main classes are not closed under composition. The main classes of tree transducers are: == Top-Down Tree Transducers (TOP) == A TOP T is a tuple (Q, Σ, Γ, I, δ) such that: Q is a finite set, the set of states; Σ is a finite ranked alphabet, called the input alphabet; Γ is a finite ranked alphabet, called the output alphabet; I is a subset of Q, the set of initial states; and δ is a set of rules of the form q ( f ( x 1 , … , x n ) ) → u {\displaystyle q(f(x_{1},\dots ,x_{n}))\to u} , where f is a symbol of Σ, n is the arity of f, q is a state, and u is a tree on Γ and Q × 1.. n {\displaystyle Q\times 1..n} , such pairs being nullary. === Examples of rules and intuitions on semantics === For instance, q ( f ( x 1 , … , x 3 ) ) → g ( a , q ′ ( x 1 ) , h ( q ″ ( x 3 ) ) ) {\displaystyle q(f(x_{1},\dots ,x_{3}))\to g(a,q'(x_{1}),h(q''(x_{3})))} is a rule – one customarily writes q ( x i ) {\displaystyle q(x_{i})} instead of the pair ( q , x i ) {\displaystyle (q,x_{i})} – and its intuitive semantics is that, under the action of q, a tree with f at the root and three children is transformed into g ( a , q ′ ( x 1 ) , h ( q ″ ( x 3 ) ) ) {\displaystyle g(a,q'(x_{1}),h(q''(x_{3})))} where, recursively, q ′ ( x 1 ) {\displaystyle q'(x_{1})} and q ″ ( x 3 ) {\displaystyle q''(x_{3})} are replaced, respectively, with the application of q ′ {\displaystyle q'} on the first child and with the application of q ″ {\displaystyle q''} on the third. === Semantics as term rewriting === The semantics of each state of the transducer T, and of T itself, is a binary relation between input trees (on Σ) and output trees (on Γ). A way of defining the semantics formally is to see δ {\displaystyle \delta } as a term rewriting system, provided that in the right-hand sides the calls are written in the form q ( x i ) {\displaystyle q(x_{i})} , where states q are unary symbols. Then the semantics [ [ q ] ] {\displaystyle [\![q]\!]} of a state q is given by [ [ q ] ] = { u ↦ v ∣ u is a tree on Σ , v is a tree on Γ , and q ( u ) → δ ∗ v } . {\displaystyle [\![q]\!]=\{u\mapsto v\mid u{\text{ is a tree on }}\Sigma ,\ v{\text{ is a tree on }}\Gamma {\text{, and }}q(u)\to _{\delta }^{}v\}.} The semantics of T is then defined as the union of the semantics of its initial states: [ [ T ] ] = ⋃ q ∈ I [ [ q ] ] . {\displaystyle [\![T]\!]=\bigcup _{q\in I}[\![q]\!].} === Determinism and domain === As with tree automata, a TOP is said to be deterministic (abbreviated DTOP) if no two rules of δ share the same left-hand side, and there is at most one initial state. In that case, the semantics of the DTOP is a partial function from input trees (on Σ) to output trees (on Γ), as are the semantics of each of the DTOP's states. The domain of a transducer is the domain of its semantics. Likewise, the image of a transducer is the image of its semantics. === Properties of DTOP === DTOP are not closed under union: this is already the case for deterministic word transducers. The domain of a DTOP is a regular tree language. Furthermore, the domain is recognisable by a deterministic top-down tree automaton (DTTA) of size at most exponential in that of the initial DTOP. That the domain is DTTA-recognizable is not surprising, considering that the left-hand sides of DTOP rules are the same as for DTTA. As for the reason for the exponential explosion in the worst case (that does not exist in the word case), consider the rule q ( f ( x 1 , x 2 ) ) → g ( p 1 ( x 1 ) , p 2 ( x 1 ) , p 3 ( x 2 ) ) {\displaystyle q(f(x_{1},x_{2}))\to g(p_{1}(x_{1}),p_{2}(x_{1}),p_{3}(x_{2}))} . In order for the computation to succeed, it must succeed for both children. That means that the right child must be in the domain of p 3 {\displaystyle p_{3}} . As for the left child, it must be in the domain of both p 1 {\displaystyle p_{1}} and p 2 {\displaystyle p_{2}} . Generally, since subtrees can be copied, a single subtree can be evaluated by multiple states during a run, despite the determinism, and unlike DTTA. Thus the construction of the DTTA recognising the domain of a DTOP must account for sets of states and compute the intersections of their domains, hence the exponential. In the special case of linear DTOP, that is to say DTOP where each x i {\displaystyle x_{i}} appears at most once in the right-hand side of each rule, the construction is linear in time and space. The image of a DTOP is not a regular tree language. Consider the transducer coding the transformation f ( x ) → g ( x , x ) {\displaystyle f(x)\to g(x,x)} ; that is, duplicate the child of the input. This is easily done by a rule q ( f ( x 1 ) ) → g ( p ( x 1 ) , p ( x 1 ) ) {\displaystyle q(f(x_{1}))\to g(p(x_{1}),p(x_{1}))} , where p encodes the identity. Then, absent any restrictions on the first child of the input, the image is a classical non-regular tree language. However, the domain of a DTOP cannot be restricted to a regular tree language. That is to say, given a DTOP T and a language L, one cannot in general build a DTOP T ′ {\displaystyle T'} such that the semantics of T ′ {\displaystyle T'} is that of T, restricted to L. This property is linked to the reason deterministic top-down tree automata are less expressive than bottom-up automata: once you go down a given path, information from other paths is inaccessible. Consider the transducer coding the transformation f ( x , y ) → y {\displaystyle f(x,y)\to y} ; that is, output the right child of the input. This is easily done by a rule q ( f ( x 1 , x 2 ) ) → p ( x 2 ) {\displaystyle q(f(x_{1},x_{2}))\to p(x_{2})} , where p encodes the identity. Now let's say we want to restrict this transducer to the finite (and thus, in particular, regular) domain { f ( c , a ) , f ( c , b ) } {\displaystyle \{f(c,a),\ f(c,b)\}} . We must use the rules q ( f ( x 1 , x 2 ) ) → p ( x 2 ) , p ( a ) → a , p ( b ) → b {\displaystyle q(f(x_{1},x_{2}))\to p(x_{2}),\ p(a)\to a,\ p(b)\to b} . But in the first rule, x 1 {\displaystyle x_{1}} does not appear at all, since nothing is produced from the left child. Thus, it is not possible to test that the left child is c. In contrast, since we produce from the right child, we can test that it is a or b. In general, the criterion is that DTOP cannot test properties of subtrees from which they do not produce output. DTOP are not closed under composition. However this problem can be solved by the addition of a lookahead: a tree automaton, coupled to the transducer, that can perform tests on the domain which the transducer is incapable of. This follows from the point about domain restriction: composing the DTOP encoding identity on { f ( c , a ) , f ( c , b ) } {\displaystyle \{f(c,a),\ f(c,b)\}} with the one encoding f ( x , y ) → y {\displaystyle f(x,y)\to y} must yield a transducer with the semantics { f ( c , a ) ↦ a , f ( c , b ) ↦ b } {\displaystyle \{f(c,a)\mapsto a,\ f(c,b)\mapsto b\}} , which we know is not expressible by a DTOP. The typechecking problem—testing whether the image of a regular tree language is included in another regular tree language—is decidable. The equivalence problem—testing whether two DTOP define the same functions—is decidable. == Bottom-Up Tree Transducers (BOT) == As in the simpler case of tree automata, bottom-up tree transducers are defined similarly to their top-down counterparts, but proceed from the leaves of the tree to the root, instead of from the root to the leaves. Thus the main difference is in the form of the rules, which are of the form f ( q 1 ( x 1 ) , … , q n ( x n ) ) → q ( u ) {\displaystyle f(q_{1}(x_{1}),\dots ,q_{n}(x_{n}))\to q(u)} .

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  AI Website Builders Reviews: What Actually Works in 2026

  Trying to pick the best AI website builder? An AI website builder is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI website builder slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.

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  Hallucination (artificial intelligence)

  In the field of artificial intelligence (AI), a hallucination or artificial hallucination (also called bullshitting, confabulation, or delusion) is a response generated by AI that contains false or misleading information presented as fact. This term draws a loose analogy with human psychology, where a hallucination typically involves false percepts. For example, a chatbot powered by large language models (LLMs), like ChatGPT, may embed plausible-sounding random falsehoods within its generated content. Detecting and mitigating errors and hallucinations pose significant challenges for practical deployment and reliability of LLMs in high-stakes scenarios, such as chip design, supply chain logistics, and medical diagnostics. Some software engineers and statisticians have criticized the specific term "AI hallucination" for unreasonably anthropomorphizing computers. Symbolic artificial intelligence models generally do not produce hallucinations, unlike large language models. == Term == === Origin === Since the 1980s, the term "hallucination" has been used in computer vision with a positive connotation to describe the process of adding detail to an image. For example, the task of generating high-resolution face images from low-resolution inputs is called face hallucination. The first documented use of the term "hallucination" in this sense is in the PhD thesis of Eric Mjolsness in 1986. A notable work is the face hallucination algorithm by Simon Baker and Takeo Kanade published in 1999. In the 2000s, hallucinations were described in statistical machine translation as a failure mode. Since the 2010s, the term has undergone a semantic shift to signify the generation of factually incorrect or misleading outputs by AI systems in tasks like machine translation and object detection. In 2015, hallucinations were identified in visual semantic role labeling tasks by Saurabh Gupta and Jitendra Malik. In 2015, computer scientist Andrej Karpathy used the term "hallucinated" in a blog post to describe his recurrent neural network (RNN) language model generating an incorrect citation link. In 2017, Google researchers used the term to describe the responses generated by neural machine translation (NMT) models when they are not related to the source text, and in 2018, the term was used in computer vision to describe instances where non-existent objects are erroneously detected because of adversarial attacks. In July 2021, Meta warned during its release of BlenderBot 2 that the system is prone to "hallucinations", which Meta defined as "confident statements that are not true". Following OpenAI's ChatGPT release in beta version in November 2022, some users complained that such chatbots often seem to pointlessly embed plausible-sounding random falsehoods within their generated content. Many news outlets, including The New York Times, started to use the term "hallucinations" to describe these models' frequently incorrect or inconsistent responses. In 2023, the Cambridge dictionary updated its definition of hallucination to include this new sense specific to the field of AI. Some researchers have highlighted a lack of consistency in how the term is used, but also identified several alternative terms in the literature, such as confabulations, fabrications, and factual errors. === Definitions and alternatives === Uses, definitions and characterizations of the term "hallucination" in the context of LLMs include: "a tendency to invent facts in moments of uncertainty" (OpenAI, May 2023) "a model's logical mistakes" (OpenAI, May 2023) "fabricating information entirely, but behaving as if spouting facts" (CNBC, May 2023) "making up information" (The Verge, February 2023) "probability distributions" (in scientific contexts) Journalist Benj Edwards, in Ars Technica, writes that the term "hallucination" is controversial, but that some form of metaphor remains necessary; Edwards suggests "confabulation" as an analogy for processes that involve "creative gap-filling". In July 2024, a White House report on fostering public trust in AI research mentioned hallucinations only in the context of reducing them. Notably, when acknowledging David Baker's Nobel Prize-winning work with AI-generated proteins, the Nobel committee avoided the term entirely, instead referring to "imaginative protein creation". Hicks, Humphries, and Slater, in their article in Ethics and Information Technology, argue that the output of LLMs is "bullshit" under Harry Frankfurt's definition of the term, and that the models are "in an important way indifferent to the truth of their outputs", with true statements only accidentally true, and false ones accidentally false. Some researchers also use the derogatory term "botshit", often referring to uncritical use of AI. === Criticism === In the scientific community, some researchers avoid the term "hallucination", seeing it as potentially misleading. It has been criticized by Usama Fayyad, executive director of the Institute for Experimental Artificial Intelligence at Northeastern University, on the grounds that it misleadingly personifies large language models and is vague. Mary Shaw said, "The current fashion for calling generative AI's errors 'hallucinations' is appalling. It anthropomorphizes the software, and it spins actual errors as somehow being idiosyncratic quirks of the system even when they're objectively incorrect." In Salon, statistician Gary Smith argues that LLMs "do not understand what words mean" and consequently that the term "hallucination" unreasonably anthropomorphizes the machine. Murray Shanahan argues that anthropomorphic framing of LLM capabilities, including terms like "hallucination", encourages users and researchers to attribute cognitive processes to systems that operate through statistical pattern completion, and advocates for more careful linguistic practices when discussing LLM behavior. Kristina Šekrst argues that applying psychological vocabulary to LLM outputs obscures the difference between the appearance of mental properties and their genuine presence. Förster & Skop assert that tech companies use the hallucination metaphor to anthropomorphize models and deflect responsibility for non-factual outputs. Some see the AI outputs not as illusory but as prospective—that is, having some chance of being true, similar to early-stage scientific conjectures. The term has also been criticized for its association with psychedelic drug experiences. == In natural language generation == In natural language generation, there are several reasons why natural language models hallucinate: === Hallucination from data === Hallucinations can stem from incomplete, inaccurate or unrepresentative data sets. === Modeling-related causes === The pre-training of generative pretrained transformers (GPT) involves predicting the next word. It incentivizes GPT models to "give a guess" about what the next word is, even when they lack information. Some researchers take an anthropomorphic perspective and posit that hallucinations arise from a tension between novelty and usefulness. For instance, Amabile and Pratt define human creativity as the production of novel and useful ideas. By extension, a focus on novelty in machine creativity can lead to the production of original but inaccurate responses—that is, falsehoods—whereas a focus on usefulness may result in memorized content lacking originality. By 2022, newspapers such as The New York Times expressed concern that, as the adoption of bots based on large language models continued to grow, unwarranted user confidence in bot output could lead to problems. === Interpretability research === In 2025, interpretability research by Anthropic on the LLM Claude identified internal circuits that cause it to decline to answer questions unless it knows the answer. By default, the circuit is active and the LLM doesn't answer. When the LLM has sufficient information, these circuits are inhibited and the LLM answers the question. Hallucinations were found to occur when this inhibition happens incorrectly, such as when Claude recognizes a name but lacks sufficient information about that person, causing it to generate plausible but untrue responses. === Examples === On 15 November 2022, researchers from Meta AI published Galactica, designed to "store, combine and reason about scientific knowledge". Content generated by Galactica came with the warning: "Outputs may be unreliable! Language Models are prone to hallucinate text." In one case, when asked to draft a paper on creating avatars, Galactica cited a fictitious paper from a real author who works in the relevant area. Meta withdrew Galactica on 17 November due to offensiveness and inaccuracy. OpenAI's ChatGPT, released in beta version to the public on November 30, 2022, was based on the foundation model GPT-3.5 (a revision of GPT-3). Professor Ethan Mollick of Wharton called it an "omniscient, eager-to-please intern who sometimes lies to you". Data scientist Teresa Kuba

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  AI Humanizers: Free vs Paid (2026)

  Trying to pick the best AI humanizer? An AI humanizer is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI humanizer slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.

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  Postediting

  Post-editing (or postediting) is the process whereby humans amend machine-generated translation to achieve an acceptable final product. A person who post-edits is called a post-editor. The concept of post-editing is linked to that of pre-editing. In the process of translating a text via machine translation, best results may be gained by pre-editing the source text – for example by applying the principles of controlled language – and then post-editing the machine output. It is distinct from editing, which refers to the process of improving human generated text (a process which is often known as revision in the field of translation). Post-edited text may afterwards be revised to ensure the quality of the language choices are proofread to correct simple mistakes. Post-editing involves the correction of machine translation output to ensure that it meets a level of quality negotiated in advance between the client and the post-editor. Light post-editing aims at making the output simply understandable; full post-editing at making it also stylistically appropriate. With advances in machine translation full post-editing is becoming an alternative to manual translation. Practically all computer-assisted translation (CAT) tools now support post-editing of machine translated output. == Post-editing and machine translation == Machine translation left the labs to start being used for its actual purpose in the late seventies at some big institutions such as the European Commission and the Pan-American Health Organization, and then, later, at some corporations such as Caterpillar and General Motors. First studies on post-editing appeared in the eighties, linked to those implementations. To develop appropriate guidelines and training, members of the Association for Machine Translation in the Americas (AMTA) and the European Association for Machine Translation (EAMT) set a Post-editing Special Interest Group in 1999. After the nineties, advances in computer power and connectivity sped machine translation development and allowed for its deployment through the web browser, including as a free, useful adjunct to the main search engines (Google Translate, Bing Translator, Yahoo! Babel Fish). A wider acceptance of less than perfect machine translation was accompanied also by a wider acceptance of post-editing. With the demand for localisation of goods and services growing at a pace that could not be met by human translation, not even assisted by translation memory and other translation management technologies, industry bodies such as the Translation Automation Users Society (TAUS) expect machine translation and post-editing to play a much bigger role within the next few years. The use of Machine Translation suggests sometimes pre-editing. Human translators possess significantly more sophisticated cognitive abilities than machine translation (MT) systems. They leverage a wealth of life experience, common sense, and multi-sensory input to understand context, identify semantic intent, and add cultural nuances to translations. This remains true even as MT capabilities continue to improve. Unlike MT systems, which primarily focus on literal word-for-word conversion, human translators grasp the underlying meaning and intent, even when information is implicit. They "read between the lines," guided by their understanding of the world. Essentially, MT models excel at text string prediction, not true comprehension. Their success often stems from framing problems as prediction tasks, such as in self-driving cars or fraud detection. Studies have demonstrated that integrating adaptive MT with post-editing interfaces can lead to reductions in technical effort and time, improving overall translation efficiency. These systems are also supported by research that highlights the benefits of adaptive MT in real-world translation scenarios. For example, incremental adaptation in Neural Machine Translation (NMT) for professional post-editors has been shown to improve translation quality and reduce time spent on edits, showcasing how human expertise and machine assistance can complement each other effectively. == Light and full post-editing == For many years, no widely accepted, standardized post-editing guidelines existed; however, in 2017, ISO standard 18587:2017: Translation services — Post-editing of machine translation output — Requirements was published. Studies in the eighties distinguished between degrees of post-editing which, in the context of the European Commission Translation Service, were first defined as conventional and rapid or full and rapid. Light and full post-editing seems the wording most used today. Light post-editing implies minimal intervention by the post-editor, with the aim of ensuring quality is "good enough" or "understandable"; the expectation is that the client will use it for inbound purposes only, often when the text is needed urgently, or has a short time span. Full post-editing involves a greater level of intervention to achieve a degree of quality to be negotiated between client and post-editor; the expectation is that the outcome will be a text that is not only understandable but presented in some stylistically appropriate way, so it can be used for assimilation and even for dissemination, for inbound and for outbound purposes. The quality is expected to be publishable and equivalent to that of a human translation. The assumption, however, has been that it takes less effort for translators to work directly from the source text than to post-edit the machine generated version. With advances in machine translation, this may be changing. For some language pairs and for some tasks, and with engines that have been customised with domain specific good quality data, some clients are already requesting translators to post-edit instead of translating from scratch, in the belief that they will attain similar quality at a lower cost. The light/full classification, developed in the nineties when machine translation still came on a CD-ROM, may not suit advances in machine translation at the light post-editing end either. For some language pairs and some tasks, particularly if the source has been pre-edited, raw machine output may be good enough for gisting purposes without requiring subsequent human intervention. == Post-editing efficiency == Post-editing is used when raw machine translation is not good enough and human translation not required. Industry advises post-editing to be used when it can at least double the productivity of manual translation, even fourfold it in the case of light post-editing (1000 words per hour vs. 250 wph). However, post-editing efficiency is difficult to predict. Various studies from both academia and industry have claimed that post-editing is generally faster than translating from scratch, regardless of language pairs or translators' experience. There is, however, no agreement about how much time can be saved through post-editing in practice (if any at all): While the industry reports on time savings around 40%, some academic studies suggest that time savings under actual working conditions are more likely to be between 0–20%, or that it may depend on the terminological proximity between the source and target languages. Professionals have also reported negative productivity gains where corrections require more time than to translate from scratch. == Post-editing and the language industry == After some thirty years, post-editing is still "a nascent profession". What the right profile of the post-editor is, has not yet been fully studied. Post-editing overlaps with translating and editing, but only partially. Most think the ideal post-editor will be a translator keen to be trained on the specific skills required, but there are some who think a bilingual without a background in translation may be easier to train. Not much is known either on who the actual post-editors are, whether they tend to be professional translators, whether they work mostly as in-house employees or self-employed, and on which conditions. Many professional translators dislike post-editing, among other reasons because it tends to be paid at lower rates than conventional translations, with the International Association of Professional Translators and Interpreters (IAPTI) having been particularly vocal about it.

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  Best AI Sales Assistants in 2026

  Shopping for the best AI sales assistant? An AI sales assistant is software that uses machine learning to help you get more done — it keeps getting smarter as the underlying models improve. Pricing, accuracy, and the size of the model behind the tool are the three factors that most affect daily usefulness. Whether you are a beginner or a pro, the right AI sales assistant slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.

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  Elastix (image registration)

  Elastix is an image registration toolbox built upon the Insight Segmentation and Registration Toolkit (ITK). It is entirely open-source and provides a wide range of algorithms employed in image registration problems. Its components are designed to be modular to ease a fast and reliable creation of various registration pipelines tailored for case-specific applications. It was first developed by Stefan Klein and Marius Staring under the supervision of Josien P.W. Pluim at Image Sciences Institute (ISI). Its first version was command-line based, allowing the final user to employ scripts to automatically process big data-sets and deploy multiple registration pipelines with few lines of code. Nowadays, to further widen its audience, a version called SimpleElastix is also available, developed by Kasper Marstal, which allows the integration of elastix with high level languages, such as Python, Java, and R. == Image registration fundamentals == Image registration is a well-known technique in digital image processing that searches for the geometric transformation that, applied to a moving image, obtains a one-to-one map with a target image. Generally, the images acquired from different sensors (multimodal), time instants (multitemporal), and points of view (multiview) should be correctly aligned to proceed with further processing and feature extraction. Even though there are a plethora of different approaches to image registration, the majority is composed of the same macro building blocks, namely the transformation, the interpolator, the metric, and the optimizer. Registering two or more images can be framed as an optimization problem that requires multiple iterations to converge to the best solution. Starting from an initial transformation computed from the image moments the optimization process searches for the best transformation parameters based on the value of the selected similarity metric. The figure on the right shows the high-level representation of the registration of two images, where the reference remains constant during the entire process, while the moving one will be transformed according to the transformation parameters. In other words, the registration ends when the similarity metric, which is a mathematical function with a certain number of parameters to be optimized, reaches the optimal value which is highly dependent on the specific application. == Main building blocks == Following the structure of the image registration workflow, the elastix toolbox proposes a modular solution that implements for each of the building blocks different algorithms, highly employed in medical image registration, and helps the final users to build their specific pipeline by selecting the most suitable algorithm for each of the main building blocks. Each block is easily configurable both by selecting pre-defined initialization values or by trying multiple sets of parameters and then choosing the most performing one. The registration is performed on images, and the elastix toolbox supports all the data formats supported by ITK, ranging from JPEG and PNG to medical standard formats such as DICOM and NIFTI. It also stores physical pixel spacing, the origin and the relative position to an external world reference system, when provided in the metadata, to facilitate the registration process, especially in medical field applications. === Transformation === The transformation is an essential building block, since it defines the allowable transformations. In image registration, the main distinction can be done between parallel-to-parallel and parallel-to-non parallel (deformable) line mapping transformations. In the elastix toolbox, the final users can select one transformation or compose more transformations either through addition or via composition. Below are reported the different transformation models in order of increasing flexibility, along with the corresponding elastix class names between brackets. Translation (TranslationTransform) allows only translations Rigid (EulerTransform) expands the translation adding rotations and the object is seen as a rigid body Similarity (SimilarityTransform) expands the rigid transformation by introducing isotropic scaling Affine (AffineTransform) expands the rigid transformation allowing both scaling and shear B-splines (BSplineTransform) is a deformable transformation usually preceded by a rigid or affine one Thin-plate splines (SplineKernelTransform) is a deformable transformation belonging to the class of kernel-based transformations that is a composition of and affine and a non-rigid part === Metric === The similarity metric is the mathematical function whose parameters should be optimized to reach the desired registration, and, during the process, it is computed multiple times. Below are reported the available metrics computed employing the reference and the transformed images and the corresponding elastix class names between brackets. Mean squared difference (AdvancedMeanSquares) to be used for mono-modal applications Normalized correlation coefficient (AdvancedNormalizedCorrelation) to be used for images that have an intensity linear relationship Mutual information (AdvancedMattesMutualInformation) to be used for both mono- and multi-modal applications and optimized to reach better performance compared to the normalized version Normalized mutual information (NormalizedMutualInformation) for both mono- and multi-modal applications Kappa statistic (AdvancedKappaStatistic) to be used only for binary images === Sampler === For the computation of the similarity metrics, it is not always necessary to consider all the voxels and, sometimes, it can be useful to use only a fraction of the voxels of the images, i.e. to reduce the execution time for big input images. Below are reported the available criteria for selecting a fraction of the voxels for the similarity metric computation and the corresponding elastix class names between brackets. Full (Full) to employ all the voxels Grid (Grid) to employ a regular grid defined by the user to downsample the image Random (Random) to randomly select a percentage of voxels defined by the users (all voxels have equal probability to be selected) Random coordinate (RandomCoordinate) like the random criterion, but in this case also off-grid positions can be selected to simplify the optimization process === Interpolator === After the application of the transformation, it may occur that the voxels used for the similarity metric computation are at non-voxel positions, so intensity interpolation should be performed to ensure the correctness of the computed values. Below are reported the implemented interpolators and the corresponding elastix class names between brackets. Nearest neighbor (NearestNeighborInterpolator) exploits little resources, but gives low quality results Linear (LinearInterpolator) is sufficient in general applications N-th order B-spline (BSplineInterpolator) can be used to increase the order N, increasing quality and computation time. N=0 and N=1 indicate the nearest neighbor and linear cases respectively. === Optimizer === The optimizer defines the strategy employed for searching the best transformation parameter to reach the correct registration, and it is commonly an iterative strategy. Below are reported some of the implemented optimization strategies. Gradient descent Robbins-Monro, similar to the gradient descent, but employing an approximation of the cost function derivatives A wider range of optimizers is also available, such as Quasi-Newton or evolutionary strategies. === Other features === The elastix software also offers other features that can be employed to speed up the registration procedure and to provide more advanced algorithms to the end-users. Some examples are the introduction of blur and Gaussian pyramid to reduce data complexity, and multi-image and multi-metric framework to deal with more complex applications. == Applications == Elastix has applications mainly in the medical field, where image registration is fundamental to get comprehensive information regarding the analysed anatomical region. It is widely employed in image-guided surgery, tumour monitoring, and treatment assessment. For example, in radiotherapy planning, image registration allows to correctly deliver the treatment and evaluate the obtained results. Thanks to the wide range of implemented algorithms, the use of the elastix software allows physicians and researchers to test different registration pipelines from the simplest to more complex ones, and to save the best one as a configuration file. This file and the fact that the software is completely open-source makes it easy to reproduce the work, that can help supporting the open science paradigm, and allows fast reuse on different patients data. In image-guided surgery, registration time and accuracy are critical points, considering that, during the registration, the patient is on the operating table, and the imag

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  Deborah Raji

  Inioluwa Deborah Raji (born 1995/1996) is a Nigerian-Canadian computer scientist and socio-tech leader who works on algorithmic bias, AI accountability, and algorithmic auditing. A current Mozilla fellow, she has been recognized by MIT Technology Review and Forbes as one of the world's top young innovators. Raji started her work with racial bias in technology during her internship with Clarifai when she recognized that people of color were more often tagged for NSFW compared to white people. Raji has previously worked with Joy Buolamwini, Timnit Gebru, and the Algorithmic Justice League on researching gender and racial bias in facial recognition technology. Her work on racial bias in facial recognition has forced companies to ultimately change their practices. She has also worked with Google’s Ethical AI team and been a research fellow at the Partnership on AI and AI Now Institute at New York University working on how to operationalize ethical considerations in machine learning engineering practice. She was working on a computer vision model that would help clients flag inappropriate images as NSFW. == Early life and education == Raji was born in Port Harcourt, Nigeria, and moved to Mississauga, Ontario, Canada, when she was four years old. Eventually her family moved to Ottawa. She attended Colonel By Secondary School and completed the International Baccalaureate programme. She studied Engineering Science at the University of Toronto, graduating in 2019. In 2015, she founded Project Include, a nonprofit providing increased student access to engineering education, mentorship, and resources in low income and immigrant communities in the Greater Toronto Area. She started a Doctor of Philosophy - PhD, in Computer Science from the University of California, Berkeley in Aug 2021. == Career and research == Raji worked with Joy Buolamwini at the MIT Media Lab and Algorithmic Justice League, where she audited commercial facial recognition technologies from Microsoft, Amazon, IBM, Face++, and Kairos. They found that these technologies were significantly less accurate for darker-skinned women than for white men. With support from other top AI researchers and increased public pressure and campaigning, their work led IBM and Amazon to agree to support facial recognition regulation and later halt the sale of their product to police for at least a year. Raji also interned at machine learning startup Clarifai, where she worked on a computer vision model for flagging images. She participated in a research mentorship program at Google and worked with their Ethical AI team on creating model cards, a documentation framework for more transparent machine learning model reporting. She also co-led the development of internal auditing practices at Google. Her contributions at Google were separately presented and published at the AAAI conference and ACM Conference on Fairness, Accountability, and Transparency. In 2019, Raji was a summer research fellow at The Partnership on AI working on setting industry machine learning transparency standards and benchmarking norms. Raji was a Tech Fellow at the AI Now Institute worked on algorithmic and AI auditing. Currently, she is a fellow at the Mozilla Foundation researching algorithmic auditing and evaluation. Raji's work on bias in facial recognition systems has been highlighted in the 2020 documentary Coded Bias directed by Shalini Kantayya. She also took part in the 2026 documentary The AI Doc: Or How I Became an Apocaloptimist directed by Daniel Roher. == Awards == 2019 Venture Beat AI Innovations Award in category AI for Good (received with Joy Buolamwini and Timnit Gebru) 2020 MIT Technology Review 35 Under 35 Innovator Award 2020 EFF Pioneer Award (received with Buolamwini and Gebru) 2021 Forbes 30 Under 30 Award in Enterprise Technology 2021 100 Brilliant Women in AI Ethics Hall of Fame Honoree 2023 Time magazine 100 Most Influential People in AI

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  Muller automaton

  In automata theory, a Muller automaton is a type of an ω-automaton. The acceptance condition separates a Muller automaton from other ω-automata. The Muller automaton is defined using a Muller acceptance condition, i.e. the set of all states visited infinitely often must be an element of the acceptance set. Both deterministic and non-deterministic Muller automata recognize the ω-regular languages. They are named after David E. Muller, an American mathematician and computer scientist, who invented them in 1963. == Formal definition == Formally, a deterministic Muller-automaton is a tuple A = (Q,Σ,δ,q0,F) that consists of the following information: Q is a finite set. The elements of Q are called the states of A. Σ is a finite set called the alphabet of A. δ: Q × Σ → Q is a function, called the transition function of A. q0 is an element of Q, called the initial state. F is a set of sets of states. Formally, F ⊆ P(Q) where P(Q) is powerset of Q. F defines the acceptance condition. A accepts exactly those runs in which the set of infinitely often occurring states is an element of F In a non-deterministic Muller automaton, the transition function δ is replaced with a transition relation Δ that returns a set of states and the initial state q0 is replaced by a set of initial states Q0. Generally, 'Muller automaton' refers to a non-deterministic Muller automaton. For more comprehensive formalisation look at ω-automaton. == Equivalence with other ω-automata == The Muller automata are equally expressive as parity automata, Rabin automata, Streett automata, and non-deterministic Büchi automata, to mention some, and strictly more expressive than the deterministic Büchi automata. The equivalence of the above automata and non-deterministic Muller automata can be shown very easily as the accepting conditions of these automata can be emulated using the acceptance condition of Muller automata and vice versa. McNaughton's theorem demonstrates the equivalence of non-deterministic Büchi automaton and deterministic Muller automaton. Thus, deterministic and non-deterministic Muller automata are equivalent in terms of the languages they can accept. == Transformation to non-deterministic Muller automata == Following is a list of automata constructions that each transforms a type of ω-automata to a non-deterministic Muller automaton. From Büchi automata If B is the set of final states in a Büchi automaton with the set of states Q, we can construct a Muller automaton with same set of states, transition function and initial state with the Muller accepting condition as F = { X | X ∈ P(Q) ∧ X ∩ B ≠ ∅}. From Rabin automata/parity automata Similarly, the Rabin conditions ( E j , F j ) {\displaystyle (E_{j},F_{j})} can be emulated by constructing the acceptance set in the Muller automaton as all sets F ⊆ Q {\displaystyle F\subseteq Q} that satisfy F ∩ E j = ∅ {\displaystyle F\cap E_{j}=\emptyset } and F ∩ F j ≠ ∅ {\displaystyle F\cap F_{j}\neq \emptyset } , for some j. Note that this covers the case of parity automata too, as the parity acceptance condition can be expressed as a Rabin acceptance condition easily. From Streett automata The Streett conditions ( E j , F j ) {\displaystyle (E_{j},F_{j})} can be emulated by constructing the acceptance set in the Muller automaton as all sets F ⊆ Q {\displaystyle F\subseteq Q} that satisfy F ∩ F j = ∅ ⟹ F ∩ E j = ∅ {\displaystyle F\cap F_{j}=\emptyset \implies F\cap E_{j}=\emptyset } , for all j. == Transformation to deterministic Muller automata == From Büchi automaton McNaughton's theorem provides a procedure to transform any non-deterministic Büchi automaton into a deterministic Muller automaton.

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