Iteration means repeating a process to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is the starting point of the next iteration. In mathematics and computer science, iteration (along with the related technique of recursion) is a standard element of algorithms. == Mathematics == In mathematics, iteration may refer to the process of iterating a function, i.e. applying a function repeatedly, using the output from one iteration as the input to the next. Iteration of apparently simple functions can produce complex behaviors and difficult problems – for examples, see the Collatz conjecture and juggler sequences. Another use of iteration in mathematics is in iterative methods which are used to produce approximate numerical solutions to certain mathematical problems. Newton's method is an example of an iterative method. Manual calculation of a number's square root is a common use and a well-known example. == Computing == In computing, iteration is a technique that marks out of a block of statements within a computer program for a defined number of repetitions. That block of statements is said to be iterated. A computer programmer might also refer to that block of statements as an iteration. === Implementations === Loops constitute the most common language constructs for performing iterations. The following pseudocode "iterates" three times the line of code between begin & end through a for loop, and uses the values of i as increments. It is permissible, and often necessary, to use values from other parts of the program outside the bracketed block of statements, to perform the desired function. Iterators constitute alternative language constructs to loops, which ensure consistent iterations over specific data structures. They can eventually save time and effort in later coding attempts. In particular, an iterator allows one to repeat the same kind of operation at each node of such a data structure, often in some pre-defined order. Iteratees are purely functional language constructs, which accept or reject data during the iterations. === Relation with recursion === Recursions and iterations have different algorithmic definitions, even though they can generate identical results. The primary difference is that recursion can be a solution without prior knowledge as to how many times the action must repeat, while a successful iteration requires that foreknowledge. Some types of programming languages, known as functional programming languages, are designed such that they do not set up a block of statements for explicit repetition, as with the for loop. Instead, those programming languages exclusively use recursion. Rather than call out a block of code to repeate a pre-defined number of times, the executing code block instead "divides" the work into a number of separate pieces, after which the code block executes itself on each individual piece. Each piece of work is divided repeatedly until the "amount" of work is as small as possible, at which point the algorithm does that work very quickly. The algorithm then "reverses" and reassembles the pieces into a complete whole. The classic example of recursion is in list-sorting algorithms, such as merge sort. The merge sort recursive algorithm first repeatedly divides the list into consecutive pairs. Each pair is then ordered, then each consecutive pair of pairs, and so forth until the elements of the list are in the desired order. The code below is an example of a recursive algorithm in the Scheme programming language that outputs the same result as the pseudocode under the previous heading. == Education == In some schools of pedagogy, iterations are used to describe the process of teaching or guiding students to repeat experiments, assessments, or projects, until more accurate results are found, or the student has mastered the technical skill. This idea is found in the old adage, "Practice makes perfect." In particular, "iterative" is defined as the "process of learning and development that involves cyclical inquiry, enabling multiple opportunities for people to revisit ideas and critically reflect on their implication." Unlike computing and math, educational iterations are not predetermined; instead, the task is repeated until success according to some external criteria (often a test) is achieved.
Speculative decoding
Speculative decoding is an inference-time optimization for autoregressive large language models (LLMs) that generates multiple tokens per decoding step instead of one. A smaller draft model proposes a sequence of candidate tokens, and the larger target model verifies them in a single forward pass through a modified rejection sampling scheme. The verification preserves the target model's original output distribution, so the technique produces the same results as standard decoding while cutting latency by roughly two to three times. The name is an analogy to speculative execution in CPU design, where a processor runs instructions along a predicted branch before the outcome is known. == Background == Standard autoregressive decoding in large language models generates one token at a time. The model computes a probability distribution over its vocabulary, samples the next token, and feeds that token back as input. For large models, this process is bottlenecked by memory bandwidth rather than arithmetic throughput: loading the model's parameters from high-bandwidth memory (HBM) to the processor takes up most of the wall-clock time at each step. Because of this, a forward pass over one token and a forward pass over several tokens in a batch take roughly the same time. Speculative decoding relies on this property. == Mechanism == The technique alternates between two phases: drafting and verification. During drafting, a fast approximation model generates a short run of K candidate tokens, typically between 3 and 12. The draft model is usually a much smaller version of the target model or a lightweight auxiliary network. During verification, the target model scores the entire draft sequence in one batched forward pass. A modified rejection sampling algorithm compares the draft and target probabilities at each position. If the target model would have been at least as likely to produce a given token, that token is accepted; the first token that fails is resampled from a corrected distribution, and everything after it is thrown out. The result is that the output distribution is the same as if each token had been generated one at a time. How many tokens get accepted per cycle depends on how well the draft model matches the target. For common words and predictable continuations the match tends to be good, so the target model can confirm several tokens at once. == History == An early precursor was blockwise parallel decoding, proposed in 2018 by Stern, Shazeer, and Uszkoreit. Their method predicted multiple future tokens through auxiliary prediction heads and validated them against the autoregressive model, but it only worked with greedy decoding and did not preserve the full sampling distribution. The modern form of the technique came from Yaniv Leviathan, Matan Kalman, and Yossi Matias at Google Research, who posted "Fast Inference from Transformers via Speculative Decoding" on arXiv in November 2022. Separately and at about the same time, Charlie Chen and colleagues at DeepMind arrived at a closely related method they called speculative sampling, published in February 2023. Both papers introduced the use of rejection sampling to guarantee that the output distribution is unchanged. Leviathan et al. showed roughly 2–3x speedup on T5-XXL (11 billion parameters); Chen et al. reported 2–2.5x on the Chinchilla model (70 billion parameters). The Leviathan et al. paper was presented as an oral at the International Conference on Machine Learning in July 2023. == Variants == SpecInfer (Miao et al., 2024) uses multiple small language models to jointly build a tree of candidate continuations rather than a single chain. The target model verifies the whole tree in parallel and keeps the longest valid path, with reported speedups of 1.5–3.5x. Medusa (Cai et al., 2024) takes a different approach by not using a separate draft model at all. Extra lightweight decoding heads are attached to the target model itself, and each one predicts a token at a different future position. The candidates are evaluated through a tree-structured attention mechanism. The authors measured 2.2–3.6x speedup. EAGLE (Li et al., 2024) performs autoregression on the target model's internal feature representations (specifically the second-to-top layer) rather than on tokens directly. On LLaMA 2 Chat 70B, this gave a 2.7–3.5x latency reduction. Later versions added dynamic draft trees (EAGLE-2) and further optimizations (EAGLE-3), reaching 3–6.5x speedup. == Adoption == By 2024, speculative decoding had become a standard part of production LLM serving. Google uses it in the AI Overviews feature of Google Search. Open-source inference frameworks such as vLLM, NVIDIA's TensorRT-LLM, and SGLang all include built-in support for speculative decoding and its variants. Apple, AWS, and Meta have also published research extending the method or deploying it at scale.
Oblivion (2013 film)
Oblivion is a 2013 American epic post-apocalyptic science fiction action film produced and directed by Joseph Kosinski from a screenplay by Karl Gajdusek and Michael deBruyn, starring Tom Cruise in the main role alongside Morgan Freeman, Olga Kurylenko, Andrea Riseborough, Nikolaj Coster-Waldau, and Melissa Leo in supporting roles. Based on Kosinski's unpublished Radical Comics graphic novel of the same name, the film pays homage to 1970s sci-fi, and is a "love story" set in 2077 on an Earth desolated by an alien war; a maintenance technician on the verge of completing his mission finds a woman who survived from a space ship crash, leading him to question his purpose and discover the truth about the war. Oblivion premiered in Buenos Aires on March 26, 2013, and was released in theaters by Universal Pictures on April 19. The film grossed $286 million worldwide on a production budget of $120 million and received mixed reviews from critics. == Plot == In 2017, aliens known as Scavengers attack Earth and destroy the Moon, triggering global natural disasters. Although humanity wins the war using nuclear weapons, Earth is left uninhabitable. Sixty years later, the remnants of humanity have relocated to a colony on Saturn's moon Titan, except for Unit 49—technician Jack and his communications officer Victoria—who are scheduled to join them in two weeks. The pair oversee hydro rigs that convert seawater into fusion energy for the Tet, the last remaining human colony ship in orbit. Though Jack and Victoria are romantically involved and have had their memories erased for security reasons, Jack experiences recurring dreams of an unknown woman. He also secretly visits a hidden, verdant valley where he has built a lakeside cabin and collects relics of Earth's past. While investigating a missing drone—autonomous, highly advanced, and heavily armed machines—Jack is nearly captured by Scavengers. Later, he discovers the Scavengers are transmitting a signal into space. A NASA pod crash-lands at the signal's coordinates, carrying five humans in suspended animation, including the woman from Jack's dreams. A drone arrives and destroys four of the pods, but Jack rescues the remaining one and brings the unconscious woman to Unit 49's base. After reviving her, Jack and Victoria learn that the woman, Julia, has been in stasis aboard the Odyssey spaceship since 2017. Julia insists on recovering the ship's flight recorder. However, she and Jack are captured by Scavengers and brought to the Raven Rock Mountain Complex. Their leader, Malcolm, reveals that the Scavengers are actually surviving humans. Malcolm needs Jack to reprogram a captured drone to deliver a nuclear bomb, built from Odyssey's reactor, to the Tet. Jack refuses, so Malcolm releases him and Julia, urging him to seek the truth in the radiation zone, which is supposedly deadly and off-limits. Julia helps Jack recall that she is his wife, and fragments of his memories begin to return. When they arrive back at Unit 49, a devastated Victoria informs Sally, the Tet's mission controller, that she and Jack are no longer an "effective team." A drone activates and kills Victoria. Jack and Julia destroy the drone, but crash their aircraft inside the radiation zone. There, they encounter another version of Jack—"Jack-52"—who arrives to repair the drone. Jack subdues him, but Julia is seriously injured in the fight. Jack impersonates his clone to infiltrate Unit 52, meets Victoria-52, and steals medical supplies for Julia. They rest at his cabin. At Raven Rock, Malcolm reveals the truth: humanity lost the war, and the Tet is an alien machine intelligence harvesting Earth's resources. After the Moon's destruction, the Tet deployed thousands of clones of astronaut Jack Harper—brainwashed into obedience—to exterminate the remaining humans. Malcolm had assumed these clones were inhuman until witnessing Jack show interest in a discarded book, hinting at lingering humanity. Jack reprograms the captured drone, but it is destroyed in a surprise attack by other drones, leaving Malcolm badly wounded. Jack and Julia resolve to deliver the bomb themselves; Julia enters a stasis pod. En route, Jack listens to the Odyssey's flight recorder, which reveals the original Jack Harper and Victoria were astronauts sent to explore Titan before being confronted by the Tet. The pair were captured, but not before Jack ejected the remaining crew—including Julia—in stasis pods to protect them. Jack gains access to the Tet by claiming he is delivering Julia, as previously instructed. However, the stasis pod contains a dying Malcolm. Jack and Malcolm detonate the bomb, destroying the Tet and themselves. Julia later awakens at the cabin. Three years later, Julia lives there and it is revealed she had a daughter with Jack. A group of Raven Rock survivors arrives, alongside Jack-52, who has begun regaining fragments of his own lost identity. == Cast == Tom Cruise as Jack Harper—Tech 49, a technician who works to repair drones on Earth and questions his mission. Originally, he was the American commander of a mission en route to Titan who was captured by the Tet and cloned to fight humanity. Cruise also plays Jack Harper—Tech 52, a clone who seeks out Julia after the destruction of the Tet. Morgan Freeman as Malcolm Beech, an American veteran soldier and leader of a large community of scavengers, the human survivors of the alien Tet's attacks. Olga Kurylenko as Julia Rusakova Harper, Jack's wife and a Russian crew member on the Odyssey, who was sent back towards Earth by her husband to protect her from the initial contact with the Tet. Andrea Riseborough as Victoria "Vika" Olsen, Jack's communications partner and housemate. Originally, she was the British co-pilot of Jack's mission to Titan who was captured and cloned to assist in the Tet's war on humanity. Riseborough also plays a clone of Vika who Jack misleads to obtain medical supplies. Nikolaj Coster-Waldau as Sergeant Sykes, the main military commander of Beech's community of scavengers who is skeptical of Jack at first. Melissa Leo as the Tet, an alien artificial intelligence seeking to acquire Earth's natural resources and wipe out humanity. Leo also plays Sally, the mission director of Jack and Julia's mission to Titan; her likeness was copied by the Tet to serve as its visual and auditory representation. Zoë Bell as Kara, a soldier and member of the scavengers. == Production == === Development === Joseph Kosinski started the movie process by beginning work on a graphic novel called Oblivion featuring his story. While the completion of this would be teased to the public and the concept was used to pitch the movie, it was never finished and Kosinski claims he never intended to, stating it was "just a stage in the project [of film development]". Arvid Nelson was billed as co-writer and Radical Comics was attached as publisher. The novel was never finished; Kosinski explaining: "the partnership with Radical Comics allowed me to continue working on the story by developing a series of images and continuing to refine the story more over a period of years. Then I basically used all that development as a pitch kit to the studio. So even though we really never released it as an illustrated novel the story is being told as a film, which was always the intention." Walt Disney Pictures, which produced Kosinski's previous film Tron: Legacy (2010), acquired the Oblivion film adaptation rights from Radical Comics and Kosinski after a heated auction in August 2010. The film was a directing vehicle for Kosinski, with Barry Levine producing, and Jesse Berger executive producing. Other studios that made bids on the film were Paramount Pictures, 20th Century Fox, and Universal Pictures. Disney subsequently released the rights after realizing the PG-rated film they envisioned, in line with their family-oriented reputation, would require too many story changes. Universal, which had also bid for the original rights, then bought them from Kosinski and Radical and authorized a PG-13 film version. The film's script was originally written by Kosinski and William Monahan and underwent a first rewrite by Karl Gajdusek. When the film passed into Universal's hands, a final rewrite was done by Michael Arndt, under the pen name "Michael deBruyn". Universal was particularly appreciative of the script, saying, "It's one of the most beautiful scripts we've ever come across." The Bubble Ship operated by Cruise's main character, Jack 49, was inspired by the Bell 47 helicopter (often colloquially referred to as a "bubble cockpit" helicopter), a utilitarian 1947 vehicle with a transparent round canopy that Kosinski saw in the lobby of the Museum of Modern Art in Manhattan, and which he likened to a dragonfly. Daniel Simon, who previously worked with Kosinski as the lead vehicle designer on Tron: Legacy, was tasked with creating the Bubble Ship from this basis, incorporating elements evocative of an advanced fighter
WebCrow
The WebCrow is a research project carried out at the Information Engineering Department of the University of Siena with the purpose of automatically solving crosswords. == The Project == The scientific relevance of the project can be understood considering that cracking crosswords requires human-level knowledge. Unlike chess and related games and there is no closed world configuration space. A first nucleus of technology, such as search engines, information retrieval, and machine learning techniques enable computers to enfold with semantics real-life concepts. The project is based on a software system whose major assumption is to attack crosswords making use of the Web as its primary source of knowledge. WebCrow is very fast and often thrashes human challengers in competitions, especially on multi language crossword schemes. A distinct feature of the WebCrow software system is to combine properly natural language processing (NLP) techniques, the Google web search engine, and constraint satisfaction algorithms from artificial intelligence to acquire knowledge and to fill the schema. The most important component of WebCrow is the Web Search Module (WSM), which implements a domain specific web based question answering algorithm. The way WebCrow approaches crosswords solving is quite different with respect to humans: Whereas we tend to first answer clues we are sure of and then proceed filling the schema by exploiting the already answered clues as hints, WebCrow uses two clearly distinct stages. In the first one, it processes all the clues and tries to answer them all: For each clue it finds many possible candidates and sorts them according to complex ranking models mainly based on a probability criteria. In the second stage, WebCrow uses constraint satisfaction algorithms to fill the grid with the overall most likely combination of clue answers. In order to interact with Google, first of all, WebCrow needs to compose queries on the basis of the given clues. This is done by query expansion, whose purpose is to convert the clue into a query expressed by a simplified and more appropriate language for Google. The retrieved documents are parsed so as to extract a list of word candidates that are congruent with the crossword length constraints. Crosswords can hardly be faced by using encyclopedic knowledge only, since many clues are wordplays or are otherwise purposefully very ambiguous. This enigmatic component of crosswords is faced by a massive use of database of solved crosswords, and by automatic reasoning on a properly organized knowledge base of wired rules. Last but not the least, the final constraint satisfaction step is very effective to fill the correct candidate, even though, unlike humans, the system can not rely on very high confidence on the correctness of the answer. == Competitions == WebCrow speed and effectiveness has been tested many times in man-machine competitions on Italian, English and multi-language crosswords The outcome of the tests is that WebCrow can successfully compete with average human players on single language schemes and reaches expert level performance in multi-language crosswords. However, WebCrow has not reached expert level in single-language crosswords, yet. === ECAI-06 Competition === On August 30, 2006, at the European Conference on Artificial Intelligence (ECAI2006), 25 conference attendees and 53 internet connected crosswords lovers, competed with WebCrow in an official challenge organized within the conference program. The challenge consisted in 5 different crosswords (2 in Italian, 2 in English and one multi-language in Italian and English) and 15 minutes were assigned for each crossword. WebCrow ranked 21 out of 74 participants in the Italian competition, and won both the bilingual and English competitions. === Other Competitions === Several competitions have been held in Florence, Italy within the Creativity Festival in December 2006, and another official conference competition took place in Hyderabad, India in January 2007, within the International Conference of Artificial Intelligence, where it ranked second out of 25 participants.
Residuated Boolean algebra
In mathematics, a residuated Boolean algebra is a residuated lattice whose lattice structure is that of a Boolean algebra. Examples include Boolean algebras with the monoid taken to be conjunction, the set of all formal languages over a given alphabet Σ {\displaystyle \Sigma } under concatenation, the set of all binary relations on a given set X {\displaystyle X} under relational composition, and more generally the power set of any equivalence relation, again under relational composition. The original application was to relation algebras as a finitely axiomatized generalization of the binary relation example, but there exist interesting examples of residuated Boolean algebras that are not relation algebras, such as the language example. == Definition == A residuated Boolean algebra is an algebraic structure ( L , ∧ , ∨ , ¬ , 0 , 1 , ∙ , I , / , ∖ ) {\displaystyle (L,\wedge ,\vee ,\neg ,0,1,\bullet ,\mathbf {I} ,/,\backslash )} such that An equivalent signature better suited to the relation algebra application is ( L , ∧ , ∨ , ¬ , 0 , 1 , ∙ , I , ▹ , ◃ ) {\displaystyle (L,\wedge ,\vee ,\neg ,0,1,\bullet ,\mathbf {I} ,\triangleright ,\triangleleft )} where the unary operations x ∖ {\displaystyle x\backslash } and x ▹ {\displaystyle x\triangleright } are intertranslatable in the manner of De Morgan's laws via x ∖ y = ¬ ( x ▹ ¬ y ) {\displaystyle x\backslash y=\neg (x\triangleright \neg y)} , x ▹ y = ¬ ( x ∖ ¬ y ) {\displaystyle x\triangleright y=\neg (x\backslash \neg y)} , and dually / y {\displaystyle /y} and ◃ y {\displaystyle \triangleleft y} as x / y = ¬ ( ¬ x ◃ y ) {\displaystyle x/y=\neg (\neg x\triangleleft y)} , x ◃ y = ¬ ( ¬ x / y ) {\displaystyle x\triangleleft y=\neg (\neg x/y)} , with the residuation axioms in the residuated lattice article reorganized accordingly (replacing z {\displaystyle z} by ¬ z {\displaystyle \neg z} ) to read ( x ▹ z ) ∧ y = 0 ⇔ ( x ∙ y ) ∧ z = 0 ⇔ ( z ◃ y ) ∧ x = 0 {\displaystyle (x\triangleright z)\wedge y=0\ \Leftrightarrow \ (x\bullet y)\wedge z=0\ \Leftrightarrow \ (z\triangleleft y)\wedge x=0} This De Morgan dual reformulation is motivated and discussed in more detail in the section below on conjugacy. Since residuated lattices and Boolean algebras are each definable with finitely many equations, so are residuated Boolean algebras, whence they form a finitely axiomatizable variety. == Examples == Any Boolean algebra, with the monoid multiplication ∙ {\displaystyle \bullet } taken to be conjunction and both residuals taken to be material implication x → y {\displaystyle x\to y} . Of the remaining 15 binary Boolean operations that might be considered in place of conjunction for the monoid multiplication, only five meet the monotonicity requirement, namely 0 , 1 , x , y {\displaystyle 0,1,x,y} and x ∨ y {\displaystyle x\vee y} . Setting y = z = 0 {\displaystyle y=z=0} in the residuation axiom y ≤ x ∖ z ⇔ x ∙ y ≤ z {\displaystyle y\leq x\backslash z\ \Leftrightarrow \ x\bullet y\leq z} , we have 0 ≤ x ∖ 0 ⇔ x ∙ 0 ≤ 0 {\displaystyle 0\leq x\backslash 0\ \Leftrightarrow \ x\bullet 0\leq 0} , which is falsified by taking x = 1 {\displaystyle x=1} when x ∙ y = 1 {\displaystyle x\bullet y=1} , x {\displaystyle x} , or x ∨ y {\displaystyle x\vee y} . The dual argument for z / y {\displaystyle z/y} rules out x ∙ y = y {\displaystyle x\bullet y=y} . This just leaves x ∙ y = 0 {\displaystyle x\bullet y=0} (a constant binary operation independent of x {\displaystyle x} and y {\displaystyle y} ), which satisfies almost all the axioms when the residuals are both taken to be the constant operation x / y = x ∖ y = 1 {\displaystyle x/y=x\backslash y=1} . The axiom it fails is x ∙ I = x = I ∙ x {\displaystyle x\bullet \mathbf {I} =x=\mathbf {I} \bullet x} , for want of a suitable value for I {\displaystyle \mathbf {I} } . Hence conjunction is the only binary Boolean operation making the monoid multiplication that of a residuated Boolean algebra. The power set 2 X 2 {\displaystyle 2^{X^{2}}} made a Boolean algebra as usual with ∩ {\displaystyle \cap } , ∪ {\displaystyle \cup } and complement relative to X 2 {\displaystyle X^{2}} , and made a monoid with relational composition. The monoid unit I {\displaystyle \mathbf {I} } is the identity relation { ( x , x ) | x ∈ X } {\displaystyle \{(x,x)|x\in X\}} . The right residual R ∖ S {\displaystyle R\backslash S} is defined by x ( R ∖ S ) y ⇔ ∀ z ∈ X , z R x ⇒ z S y {\displaystyle x(R\backslash S)y\ \Leftrightarrow \ \forall z\in X,zRx\Rightarrow zSy} . Dually the left residual S / R {\displaystyle S/R} is defined by y ( S / R ) x ⇔ ∀ z ∈ X , x R z ⇒ y S z {\displaystyle y(S/R)x\ \Leftrightarrow \ \forall z\in X,xRz\Rightarrow ySz} . The power set 2 Σ ∗ {\displaystyle 2^{\Sigma ^{}}} made a Boolean algebra as for Example 2, but with language concatenation for the monoid. Here the set Σ {\displaystyle \Sigma } is used as an alphabet while Σ ∗ {\displaystyle \Sigma ^{}} denotes the set of all finite (including empty) words over that alphabet. The concatenation L M {\displaystyle LM} of languages L {\displaystyle L} and M {\displaystyle M} consists of all words u v {\displaystyle uv} such that u ∈ L {\displaystyle u\in L} and v ∈ M {\displaystyle v\in M} . The monoid unit is the language { ε } {\displaystyle \{\varepsilon \}} consisting of just the empty word ε {\displaystyle \varepsilon } . The right residual M ∖ L {\displaystyle M\backslash L} consists of all words w {\displaystyle w} over Σ {\displaystyle \Sigma } such that M w ⊆ L {\displaystyle Mw\subseteq L} . The left residual L / M {\displaystyle L/M} is the same with w M {\displaystyle wM} in place of M w {\displaystyle Mw} . == Conjugacy == The De Morgan duals ▹ {\displaystyle \triangleright } and ◃ {\displaystyle \triangleleft } of residuation arise as follows. Among residuated lattices, Boolean algebras are special by virtue of having a complementation operation ¬ {\displaystyle \neg } . This permits an alternative expression of the three inequalities y ≤ x ∖ z ⇔ x ∙ y ≤ z ⇔ x ≤ z / y {\displaystyle y\leq x\backslash z\ \Leftrightarrow \ x\bullet y\leq z\ \Leftrightarrow \ x\leq z/y} in the axiomatization of the two residuals in terms of disjointness, via the equivalence x ≤ y ⇔ x ∧ ¬ y = 0 {\displaystyle x\leq y\ \Leftrightarrow \ x\wedge \neg y=0} . Abbreviating x ∧ y = 0 {\displaystyle x\wedge y=0} to x # y {\displaystyle x\#y} as the expression of their disjointness, and substituting ¬ z {\displaystyle \neg z} for z {\displaystyle z} in the axioms, they become with a little Boolean manipulation ¬ ( x ∖ ¬ z ) # y ⇔ x ∙ y # z ⇔ ¬ ( ¬ z / y ) # x {\displaystyle \neg (x\backslash \neg z)\#y\ \Leftrightarrow \ x\bullet y\#z\ \Leftrightarrow \ \neg (\neg z/y)\#x} Now ¬ ( x ∖ ¬ z ) {\displaystyle \neg (x\backslash \neg z)} is reminiscent of De Morgan duality, suggesting that x ∖ {\displaystyle x\backslash } be thought of as a unary operation f {\displaystyle f} , defined by f ( y ) = x ∖ y {\displaystyle f(y)=x\backslash y} , that has a De Morgan dual ¬ f ( ¬ y ) {\displaystyle \neg f(\neg y)} , analogous to ∀ x ϕ ( x ) = ¬ ∃ x ¬ ϕ ( x ) {\displaystyle \forall x\phi (x)=\neg \exists x\neg \phi (x)} . Denoting this dual operation as x ▹ {\displaystyle x\triangleright } , we define x ▹ z {\displaystyle x\triangleright z} as ¬ x ∖ ¬ z {\displaystyle \neg x\backslash \neg z} . Similarly we define another operation z ◃ y {\displaystyle z\triangleleft y} as ¬ ( ¬ z / y ) {\displaystyle \neg (\neg z/y)} . By analogy with x ∖ {\displaystyle x\backslash } as the residual operation associated with the operation x ∙ {\displaystyle x\bullet } , we refer to x ▹ {\displaystyle x\triangleright } as the conjugate operation, or simply conjugate, of x ∙ {\displaystyle x\bullet } . Likewise ◃ y {\displaystyle \triangleleft y} is the conjugate of ∙ y {\displaystyle \bullet y} . Unlike residuals, conjugacy is an equivalence relation between operations: if f {\displaystyle f} is the conjugate of g {\displaystyle g} then g {\displaystyle g} is also the conjugate of f {\displaystyle f} , i.e. the conjugate of the conjugate of f {\displaystyle f} is f {\displaystyle f} . Another advantage of conjugacy is that it becomes unnecessary to speak of right and left conjugates, that distinction now being inherited from the difference between x ∙ {\displaystyle x\bullet } and ∙ x {\displaystyle \bullet x} , which have as their respective conjugates x ▹ {\displaystyle x\triangleright } and ◃ x {\displaystyle \triangleleft x} . (But this advantage accrues also to residuals when x ∖ {\displaystyle x\backslash } is taken to be the residual operation to x ∙ {\displaystyle x\bullet } .) All this yields (along with the Boolean algebra and monoid axioms) the following equivalent axiomatization of a residuated Boolean algebra. y # x ▹ z ⇔ x ∙ y # z ⇔ x # z ◃ y {\displaystyle y\#x\triangleright z\ \Leftrightarrow \ x\bullet y\#z\ \Leftrightarrow \ x\#z\triangleleft y} With this signature it remains the case that this axiomatization can be expressed as
N-jet
An N-jet is the set of (partial) derivatives of a function f ( x ) {\displaystyle f(x)} up to order N. Specifically, in the area of computer vision, the N-jet is usually computed from a scale space representation L {\displaystyle L} of the input image f ( x , y ) {\displaystyle f(x,y)} , and the partial derivatives of L {\displaystyle L} are used as a basis for expressing various types of visual modules. For example, algorithms for tasks such as feature detection, feature classification, stereo matching, tracking and object recognition can be expressed in terms of N-jets computed at one or several scales in scale space.
Wayve
Wayve Technologies Ltd is a British autonomous driving technology company focused on developing self-driving vehicle systems through end-to-end deep learning. Founded in 2017 by researchers from the University of Cambridge, Wayve’s approach eschews detailed 3D maps and hand-coded rules, in favor of a self-learning “AI driver” that learns from camera data and driving experience. The London-headquartered startup has garnered significant attention and funding for its visually-based method. == History == Wayve was founded in Cambridge, England, on August 21, 2017, by Amar Shah and Alex Kendall, two machine learning PhD students at the University of Cambridge. Shah initially served as CEO while Kendall was CTO, and the pair set out to develop an unconventional self-driving car system using machine learning at every layer of the driving task. In May 2018, Wayve emerged from stealth mode with backing from early-stage investors. At this time the company had around 10 employees, and its advisory investors included Uber’s Chief Scientist, Zoubin Ghahramani, who shared Wayve’s vision of a learning-centric driving AI. In 2019, Wayve achieved a milestone by training a car to drive autonomously on public roads it had never seen before, using only cameras, a basic GPS map, and end-to-end deep learning control. The company moved its base to London and secured a $20 million Series A funding round in November 2019. This investment enabled Wayve to launch a pilot fleet of autonomous electric vehicles in central London for real-world testing. During these trials, Wayve’s cars (such as retrofitted Jaguar I-Pace SUVs) began navigating the complex, narrow streets of London to prove the system’s ability to adapt to challenging urban scenarios. In 2020, co-founder Amar Shah departed the company, and Alex Kendall assumed the role of CEO. The startup joined the Microsoft for Startups: Autonomous Driving program in 2020, leveraging Microsoft Azure’s cloud computing for training its machine learning models at scale. It also committed to testing exclusively on electric vehicles, and a goal to reduce carbon emissions. In 2021, Wayve entered pilot programs with major UK retailers. It launched a 12-month autonomous delivery trial with supermarket chain Asda, and received a £10 million ($13.6 million) investment from online grocer Ocado Group as part of a partnership to develop self-driving grocery delivery vans. Ocado’s backing gave Wayve access to a fleet of delivery vans for data collection and testing on busy London routes (with human safety drivers present) to train its AI in urban traffic. In 2022, after a successful Series B funding round, the company extended road testing beyond the UK to other regions, and, by 2023, in multiple countries. The company had begun operating in the United States and in continental Europe, in preparation for larger commercial deployments. In 2023, Wayve announced a collaboration with Nissan to integrate Wayve’s AI-driven software into its ProPilot ADAS system, slated to launch in fiscal year 2027. Wayve received strategic investment from Uber, in 2024, to jointly develop autonomous ride-hailing services. The two companies plan to trial a fully driverless robotaxi service in London, supported by a UK government program to accelerate commercial self-driving pilots to as early as 2026. To demonstrate the scalability of its technology, Wayve conducted an “AI-500” roadshow project, driving in dozens of cities across Asia, Europe, and North America using the same AI model. By mid-2025, it had completed autonomous driving demos in 90 cities without prior HD mapping. In April 2025, Wayve opened its first Asian research hub in Japan, with investment by SoftBank, to improve its model’s generalization using local driving data. That year, the company conducted driving tests in over 500 cities in Europe, North America and Japan without city-specific programming. In February 2026, Nissan, Uber and Wayve announced their collaboration on robotaxi development, with the aim of launching a pilot programme in Tokyo by late 2026. Wayve also formed a strategic alliance with Mercedes-Benz and Stellantis on personal vehicle and robotaxi applications. == Financing and investors == Wayve has been backed by a mix of venture capital (VC) firms, corporate investors, and individuals. Its initial seed funding came from funds such as Compound (NYC) and Firstminute Capital (London), as well as Cambridge-based angel investors, in 2018. Academic Pieter Abbeel and Uber’s chief scientist, Zoubin Ghahramani, were early backers. In November 2019, Wayve raised a $20 million Series A led by Eclipse Ventures, with participation from Balderton Capital and other prior investors. The Series A financing was used to fund the company’s first autonomous trials in London, and marked the first time a European self-driving car startup had secured a U.S. VC as lead investor. In October 2021, Ocado Group invested £10 million (approximately $13.6 million) in Wayve as a strategic partner in autonomous grocery delivery. This brought Wayve’s total funding to around $60 million at that time. The Series B round followed in January 2022, when Wayve announced $200 million in new funding led by Eclipse Ventures, with D1 Capital Partners, Moore Strategic Ventures, and Linse Capital. Balderton, Microsoft and Virgin Group joined as strategic backers. Baillie Gifford and Compound also participated; Ocado increased its stake as a strategic investor; and Meta AI head Yann LeCun and Richard Branson also became investors. Wayve’s Series C in May 2024 closed a $1.05 billion, led by Japan’s SoftBank Group. The funding round was the largest-ever for a UK AI company, and included new investor Nvidia, and returning investors Microsoft and Eclipse Ventures, among others. Uber also joined as a stratgic partner and a stakeholder. The Series C round increased Wayve’s total funding raised to about $1.3 billion to date from investors including SoftBank, Microsoft and Nvidia, and lifted Wayve’s valuation into “unicorn” status. In February 2026, Wayve announced a $1.2 billion Series D funding round; later that month, the company reported that $1.5 billion had been raised from, primarily, Mercedes-Benz, Stellantis, Nissan, and existing backers Uber, Microsoft and Nvidia, increasing Wayve's overall valuation to $8.6 billion. == Technology == Wayve’s self-driving approach centers on end-to-end deep learning and a vision-based AI system. Unlike conventional autonomous vehicles that depend on high-definition maps, hand-coded rules, and arrays of expensive lidar sensors, Wayve’s platform learns to drive predominantly using camera data and machine learning algorithms. The company refers to its AI-driven driving software as an “Embodied AI” or AI Driver, emphasizing that the system learns from experience (both real and simulated) to handle complex or novel situations rather than following pre-programmed instructions, not unlike Tesla's approach. The Wayve hardware-agnostic autonomy stack consists of a suite of video cameras, with basic automotive sensors, mounted on the vehicle, and paired with onboard compute units that are powered by GPUs to run the AI models. This vision-only philosophy is similar to Tesla’s Autopilot/FSDB model, but Wayve’s solution is vehicle-agnostic and mapless. Wayve’s strategy is to provide its driving AI as an OEM-ready platform; it plans to license or embed its technology into vehicles made by established automakers rather than build its own cars. Wayve’s development vehicles currently use Nvidia’s Orin system-on-chip as the onboard computer for running the AI model, but CEO Kendall has noted that the software can run on “whatever GPU [an automaker] already has in their vehicles” Wayve has built a cloud infrastructure, largely on Microsoft Azure, to process petabytes of this data, and uses simulation tools (known internally as the “Wayve Infinity” simulator) to synthetically generate and practice rare or dangerous scenarios for the AI to learn from. == Corporate affairs == Wayve is a privately held company headquartered in London, England, with its primary research and development office in the Kings Cross area of London. The company was initially incorporated as Wayve Technologies Ltd in the UK. Wayve has also established a presence in the U.S., in Silicon Valley); in Canada, with a research hub in Vancouver; in Yokohama, Japan; in Leonberg, Germany; and in Herzliya, Israel. The Leadership team includes research scientists and engineers with backgrounds in computer vision, robotics, and automotive systems. President Erez Dagan was hired in 2024, following two decades at Mobileye; chief scientist Jamie Shotton is formerly of Microsoft Research; CEO Alex Kendall, originally from New Zealand with a PhD in computer vision from Cambridge, took over as CEO in 2020 after the departure of his co-founder Amar Shah.