T-norm

In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersection in a lattice and conjunction in logic. The name triangular norm refers to the fact that in the framework of probabilistic metric spaces t-norms are used to generalize the triangle inequality of ordinary metric spaces. == Definition == A t-norm is a function T: [0, 1] × [0, 1] → [0, 1] that satisfies the following properties: Commutativity: T(a, b) = T(b, a) Monotonicity: T(a, b) ≤ T(c, d) if a ≤ c and b ≤ d Associativity: T(a, T(b, c)) = T(T(a, b), c) The number 1 acts as identity element: T(a, 1) = a Since a t-norm is a binary algebraic operation on the interval [0, 1], infix algebraic notation is also common, with the t-norm usually denoted by ∗ {\displaystyle } . The defining conditions of the t-norm are exactly those of a partially ordered abelian monoid on the real unit interval [0, 1]. (Cf. ordered group.) The monoidal operation of any partially ordered abelian monoid L is therefore by some authors called a triangular norm on L. === Classification of t-norms === A t-norm is called continuous if it is continuous as a function, in the usual interval topology on [0, 1]2. (Similarly for left- and right-continuity.) A t-norm is called strict if it is continuous and strictly monotone. A t-norm is called nilpotent if it is continuous and each x in the open interval (0, 1) is nilpotent, that is, there is a natural number n such that x ∗ {\displaystyle } ... ∗ {\displaystyle } x (n times) equals 0. A t-norm ∗ {\displaystyle } is called Archimedean if it has the Archimedean property, that is, if for each x, y in the open interval (0, 1) there is a natural number n such that x ∗ {\displaystyle } ... ∗ {\displaystyle } x (n times) is less than or equal to y. The usual partial ordering of t-norms is pointwise, that is, T1 ≤ T2 if T1(a, b) ≤ T2(a, b) for all a, b in [0, 1]. As functions, pointwise larger t-norms are sometimes called stronger than those pointwise smaller. In the semantics of t-norm fuzzy logics, however, the larger a t-norm, the weaker (in terms of logical strength) conjunction it represents. == Prominent examples == Minimum t-norm ⊤ m i n ( a , b ) = min { a , b } , {\displaystyle \top _{\mathrm {min} }(a,b)=\min\{a,b\},} also called the Gödel t-norm, as it is the standard semantics for conjunction in Gödel fuzzy logic. Besides that, it occurs in most t-norm based fuzzy logics as the standard semantics for weak conjunction. It is the pointwise largest t-norm (see the properties of t-norms below). Product t-norm ⊤ p r o d ( a , b ) = a ⋅ b {\displaystyle \top _{\mathrm {prod} }(a,b)=a\cdot b} (the ordinary product of real numbers). Besides other uses, the product t-norm is the standard semantics for strong conjunction in product fuzzy logic. It is a strict Archimedean t-norm. Łukasiewicz t-norm ⊤ L u k ( a , b ) = max { 0 , a + b − 1 } . {\displaystyle \top _{\mathrm {Luk} }(a,b)=\max\{0,a+b-1\}.} The name comes from the fact that the t-norm is the standard semantics for strong conjunction in Łukasiewicz fuzzy logic. It is a nilpotent Archimedean t-norm, pointwise smaller than the product t-norm. Drastic t-norm ⊤ D ( a , b ) = { b if a = 1 a if b = 1 0 otherwise. {\displaystyle \top _{\mathrm {D} }(a,b)={\begin{cases}b&{\mbox{if }}a=1\\a&{\mbox{if }}b=1\\0&{\mbox{otherwise.}}\end{cases}}} The name reflects the fact that the drastic t-norm is the pointwise smallest t-norm (see the properties of t-norms below). It is a right-continuous Archimedean t-norm. Nilpotent minimum ⊤ n M ( a , b ) = { min ( a , b ) if a + b > 1 0 otherwise {\displaystyle \top _{\mathrm {nM} }(a,b)={\begin{cases}\min(a,b)&{\mbox{if }}a+b>1\\0&{\mbox{otherwise}}\end{cases}}} is a standard example of a t-norm that is left-continuous, but not continuous. Despite its name, the nilpotent minimum is not a nilpotent t-norm. Hamacher product ⊤ H 0 ( a , b ) = { 0 if a = b = 0 a b a + b − a b otherwise {\displaystyle \top _{\mathrm {H} _{0}}(a,b)={\begin{cases}0&{\mbox{if }}a=b=0\\{\frac {ab}{a+b-ab}}&{\mbox{otherwise}}\end{cases}}} is a strict Archimedean t-norm, and an important representative of the parametric classes of Hamacher t-norms and Schweizer–Sklar t-norms. == Properties of t-norms == The drastic t-norm is the pointwise smallest t-norm and the minimum is the pointwise largest t-norm: ⊤ D ( a , b ) ≤ ⊤ ( a , b ) ≤ ⊤ m i n ( a , b ) , {\displaystyle \top _{\mathrm {D} }(a,b)\leq \top (a,b)\leq \mathrm {\top _{min}} (a,b),} for any t-norm ⊤ {\displaystyle \top } and all a, b in [0, 1]. In particular, we have that: ⊤ D ( a , b ) ≤ ⊤ L u k ( a , b ) ≤ ⊤ p r o d ( a , b ) ≤ ⊤ m i n ( a , b ) , {\displaystyle \top _{\mathrm {D} }(a,b)\leq \top _{\mathrm {Luk} }(a,b)\leq \top _{\mathrm {prod} }(a,b)\leq \mathrm {\top _{min}} (a,b),} for all a, b in [0, 1]. For every t-norm T, the number 0 acts as null element: T(a, 0) = 0 for all a in [0, 1]. A t-norm T has zero divisors if and only if it has nilpotent elements; each nilpotent element of T is also a zero divisor of T. The set of all nilpotent elements is an interval [0, a] or [0, a), for some a in [0, 1]. === Properties of continuous t-norms === Although real functions of two variables can be continuous in each variable without being continuous on [0, 1]2, this is not the case with t-norms: a t-norm T is continuous if and only if it is continuous in one variable, i.e., if and only if the functions fy(x) = T(x, y) are continuous for each y in [0, 1]. Analogous theorems hold for left- and right-continuity of a t-norm. A continuous t-norm is Archimedean if and only if 0 and 1 are its only idempotents. A continuous Archimedean t-norm is strict if 0 is its only nilpotent element; otherwise it is nilpotent. By definition, moreover, a continuous Archimedean t-norm T is nilpotent if and only if each x < 1 is a nilpotent element of T. Thus with a continuous Archimedean t-norm T, either all or none of the elements of (0, 1) are nilpotent. If it is the case that all elements in (0, 1) are nilpotent, then the t-norm is isomorphic to the Łukasiewicz t-norm; i.e., there is a strictly increasing function f such that ⊤ ( x , y ) = f − 1 ( ⊤ L u k ( f ( x ) , f ( y ) ) ) . {\displaystyle \top (x,y)=f^{-1}(\top _{\mathrm {Luk} }(f(x),f(y))).} If on the other hand it is the case that there are no nilpotent elements of T, the t-norm is isomorphic to the product t-norm. In other words, all nilpotent t-norms are isomorphic, the Łukasiewicz t-norm being their prototypical representative; and all strict t-norms are isomorphic, with the product t-norm as their prototypical example. The Łukasiewicz t-norm is itself isomorphic to the product t-norm undercut at 0.25, i.e., to the function p(x, y) = max(0.25, x ⋅ y) on [0.25, 1]2. For each continuous t-norm, the set of its idempotents is a closed subset of [0, 1]. Its complement—the set of all elements that are not idempotent—is therefore a union of countably many non-overlapping open intervals. The restriction of the t-norm to any of these intervals (including its endpoints) is Archimedean, and thus isomorphic either to the Łukasiewicz t-norm or the product t-norm. For such x, y that do not fall into the same open interval of non-idempotents, the t-norm evaluates to the minimum of x and y. These conditions actually give a characterization of continuous t-norms, called the Mostert–Shields theorem, since every continuous t-norm can in this way be decomposed, and the described construction always yields a continuous t-norm. The theorem can also be formulated as follows: A t-norm is continuous if and only if it is isomorphic to an ordinal sum of the minimum, Łukasiewicz, and product t-norm. A similar characterization theorem for non-continuous t-norms is not known (not even for left-continuous ones), only some non-exhaustive methods for the construction of t-norms have been found. == Residuum == For any left-continuous t-norm ⊤ {\displaystyle \top } , there is a unique binary operation ⇒ {\displaystyle \Rightarrow } on [0, 1] such that ⊤ ( z , x ) ≤ y {\displaystyle \top (z,x)\leq y} if and only if z ≤ ( x ⇒ y ) {\displaystyle z\leq (x\Rightarrow y)} for all x, y, z in [0, 1]. This operation is called the residuum of the t-norm. In prefix notation, the residuum of a t-norm ⊤ {\displaystyle \top } is often denoted by ⊤ → {\displaystyle {\vec {\top }}} or by the letter R. The interval [0, 1] equipped with a t-norm and its residuum forms a residuated lattice. The relation between a t-norm T and its residuum R is an instance of adjunction (specifically, a Galois connection): the residuum forms a right adjoint R(x, –) to the functor T(–, x) for each x in the lattice [0, 1] taken as a poset category. In the standard semantics of t-norm based fuzzy logics, where conjunction is interpreted by a t-norm, the residuum plays the role of implication (often

AI literacy

AI literacy or artificial intelligence literacy is "a set of competencies that enables individuals to critically evaluate AI technologies; communicate and collaborate effectively with AI; and use AI as a tool online, at home, and in the workplace." AI is employed in a variety of applications, including self-driving automobiles, virtual assistants and text generation by generative AI models. Users of these tools should be able to make informed decisions. AI literacy may have an impact on students' future employment prospects. With the rise of generative AI platforms, AI literacy has become a topic of conversation in the field of education. Some think AI literacy is essential for school and college students, while others restrict or prohibit the use of AI in assignments, viewing it as a form of academic dishonesty. However, many researchers and educational institutions promote a more nuanced approach, encouraging critical engagement with AI while developing policies that balance academic integrity with opportunities for learning. == Definitions == Other definitions of AI literacy include the ability to understand, use, monitor, and critically reflect on AI applications. That use of the term usually refers to teaching skills and knowledge to the general public, particularly those who are not adept in AI and the ability to understand, use, evaluate, and ethically navigate AI. As research into AI literacy is still emerging and focused on developing context-specific skills, there is not yet a single, broadly agreed-upon definition. AI literacy is linked to other forms of literacy. AI literacy requires digital literacy, whereas scientific and computational literacy may inform it. Data literacy also significantly overlaps with it. == Categories == AI literacy encompasses multiple categories, including a theoretical understanding of how artificial intelligence works, the usage of artificial intelligence technologies, and the critical appraisal of artificial intelligence, and its ethics. === Know and understand AI === Knowledge and understanding of AI refers to a basic understanding of what artificial intelligence is and how it works. This includes familiarity with machine learning algorithms and the limitations and biases present in AI systems. Users who know and understand AI should be familiar with various technologies that use artificial intelligence, including cognitive systems, robotics and machine learning. This includes recognizing that large language models (LLMs) are machine learning models trained on extensive datasets which generate new text rather than retrieving pre-written responses. === Use and apply AI === Using and applying AI refers to the ability to use AI tools to solve problems and perform tasks such as programming and analyzing big data. Some consider prompt engineering, the practice of designing effective prompts to guide generative AI platforms more effectively, as another competency within AI literacy. === Evaluate and create AI === Evaluation and creation refers to the ability to critically evaluate the quality and reliability of AI systems. It also refers to designing and building fair and ethical AI systems. To evaluate correctly, users should also learn in which areas AI is strong, and in which areas it is weak. === AI ethics === AI ethics refers to understanding the moral implications of AI, and the making informed decisions regarding the use of AI tools. This area includes considerations such as: Accountability: Hold AI actors accountable for the operation of AI systems and adherence to ethical ideals. Accuracy: Identify and report sources of error and uncertainty in algorithms and data. Auditability: Enable other parties to audit and assess algorithm behavior via transparent information sharing. Explainability: Make sure that algorithmic judgments and the underlying data can be presented in simple language. Fairness: Prevent biases and consider varied viewpoints. To do so, increase the diversity of researchers in the field. Human Centricity and Well-being: Prioritize human well-being in AI development and deployment. Human rights Alignment: Ensure that technology do not infringe internationally recognized human rights. Inclusivity: Make AI accessible to everyone. Progress: Choose high value initiatives. Responsibility, accountability, and transparency: Foster trust via responsibility, accountability, and fairness. Robustness and Security: Make AI systems safe, secure, and resistant to manipulation or data breach. Sustainability: Choose implementations that generate long-term, useful benefits. Environmental Implications: How this tool impacts the environment, any restrictions or laws, if this impact is worth the effects or not. === Enabling AI === Support AI by developing associated knowledge and skills such as programming and statistics. == Promoting AI literacy == Several governments have recognized the need to promote AI literacy, including among adults. Such programs have been published in the United States, China, Germany and Finland. Programs intended for the general public usually consist of short and easy to understand online study units. Programs intended for children are usually project-based. Programs for students at colleges and universities often address the specific professional needs of the student, depending on their field of study. Beyond the education system, AI literacy can also be developed in the community, for example in museums. === Schools === Schools use diverse pedagogies to promote AI literacy. These include: Performing a Turing test with an intelligent agent Creating chatbots Building apps using Blockly-based programming Project-based learning Building robots Data visualization Training AI models Artificial intelligence curricula can improve students' understanding of topics such as machine learning, neural networks, and deep learning. === Higher education === Before the second decade of the 21st century, artificial intelligence was studied mainly in STEM courses. Later, projects emerged to increase artificial intelligence education, specifically to promote AI literacy. Most courses start with one or more study units that deal with basic questions such as what artificial intelligence is, where it comes from, what it can do and what it can't do. Most courses also refer to machine learning and deep learning. Some of the courses deal with moral issues in artificial intelligence. In Ireland, the Higher Education Authority published Generative AI in Higher Education Teaching & Learning: Policy Framework in December 2025, which encouraged higher education institutions to embed AI literacy across programmes as a core graduate attribute. ==== Disciplinary policy ==== As a response to the increase of generative AI use in education, several disciplines formed committees or task forces to examine context-specific approaches toward AI literacy. In spring 2025, the Modern Language Association and Conference on College Composition and Communication Joint Task Force finished development of three working papers, a guide on AI literacy for students, and a collection of resources addressing AI use in writing. The task force emphasized the need for "a culture of critical AI literacy" and included guidelines not only for students but also educators and institutions, highlighting the need for modeling ethical AI use in planning processes. Similarly, a committee formed by the American Historical Association Council published "Guiding Principles for Artificial Intelligence in History Education" which encouraged "clear and transparent engagement with generative AI." The guidelines demonstrate the value of criticality when working with generative AI in thinking and research.

A.I. Insight forums

The Artificial Intelligence Insight forums, also known as the A.I. Insight forums, are a series of forums to build consensus on how the United States Congress should craft A.I. legislation. Organized by Senate Majority Leader Charles "Chuck" Schumer, the first of nine closed-door forums convened on September 13, 2023. == Background == Amid a surge in the popularity and advancement of artificial intelligence, senator Chuck Schumer launched an effort to establish a framework for the regulation of A.I. in April 2023. By the end of June, a preliminary framework – dubbed the "SAFE Innovation Framework" – was established and presented to Congress. Schumer also announced a series of forums wherein tech leaders who were well-acquainted with A.I. would help to "educate" Congress on the risks and problems that A.I. poses. Many tech leaders including Sam Altman, Elon Musk, and Sundar Pichai were set to attend the meetings. Many U.S. lawmakers and senators such as Mike Rounds and Todd Young were also set to attend. == September 13 forum == The overarching consensus following the conclusion of the September 13 forum was that there "should be" regulations regarding the use and advancement of A.I., but it should not be made "too fast". Many tech executives who attended the forum also warned senators of the risks and threats that A.I. could pose. Musk, who attended the forum, stated afterwards that there was "overwhelming consensus" on the regulation of A.I. === Invitees === This is a list of people who were invited to attend the September 13 forum. Elon Musk (Tesla, SpaceX, X Corp.) Sam Altman (OpenAI) Bill Gates (ex–Microsoft) Jensen Huang (Nvidia) Alex Karp (Palantir) Satya Nadella (Microsoft) Arvind Krishna (IBM) Sundar Pichai (Alphabet Inc., Google) Eric Schmidt (ex–Google) Mark Zuckerberg (Meta) Charles Rivkin (Motion Picture Association) Liz Shuler (AFL-CIO) Meredith Stiehm (Writers Guild of America) Randi Weingarten (American Federation of Teachers) Maya Wiley (LCCHR) == October 24 forum == The second of nine forums was hosted on October 24, 2023, as federal A.I. regulation drew nearer. According to Schumer's office, the forum was centered mainly on how A.I. could "enable innovation", and the innovation that is needed for the safe progression of A.I. At the forum, Senators Brian Schatz and John Kennedy introduced the "Schatz-Kennedy A.I. Labeling Act", a new piece of A.I. legislation that would provide "more transparency on A.I.-generated content". Following the forum, Senator Rounds stated that in order to fuel the development of A.I., a total estimated $56 billion would be needed for the next three years. Rounds, alongside Senator Young and Schumer, also highlighted the need to outcompete China and workforce initiatives. === Invitees === 21 people were invited to attend the forum, and were composed largely of venture capitalists, academics, civil rights campaigners, and industry figures. Some key figures included venture capitalists Marc Andreessen and John Doerr. == Future == Over the course of fall 2023, there is slated to be a total of nine forums on the topic of A.I., with the first hosted on September 13.

CogX Festival

CogX Festival is a global festival focusing on the impact of artificial intelligence (AI) and emerging technology on industry, government, and society. It takes place annually, usually in September, in London, England. Founded by Charlie Muirhead and Tabitha Goldstaub in 2017, CogX aims to facilitate dialogue and understanding about AI and its implications across various sectors. CogX Festival 2023 was held from September 12 to September 14 across multiple sites in London. == History == The inaugural CogX event took place in 2017, intending to bring together experts from diverse fields to discuss the role and impact of AI and emerging technologies. Since then, it has evolved to include a broader range of topics and attract a diverse audience. In 2018, the first CogX Awards festival was hosted. That year, over 50 awards were shown to 300 guests. In 2021, CogX and Hopin, a video conferencing software, signed an agreement lasting 4 years to make CogX a hybrid conference due to the COVID-19 pandemic. CogX 2021 attracted over 5,000 attendees in-person and over 100,000 virtually. In 2022, they returned to a live event format after two years of hybrid events and controlled physical attendance. They also launched the CogX app, which curated insights from the world's top podcasts. In 2023, after he had delivered the keynote address guest speaker Stephen Fry fell off the stage and subsequently broke his leg, hip, pelvis and a "bunch of ribs". A court filing in 2026 revealed that Fry was seeking £100,000 in damages from CogX Festival Ltd and creative agency Blonstein Events. == Programming == The festival features sessions, discussions, workshops, and exhibitions, encompassing various domains of AI and technology. In recent CogX Festivals, they have featured summits encompassing topics like global leadership and industry transformation.

Someday (short story)

"Someday" is a science fiction short story by American writer Isaac Asimov. It was first published in the August 1956 issue of Infinity Science Fiction and reprinted in the collections Earth Is Room Enough (1957), The Complete Robot (1982), Robot Visions (1990), and The Complete Stories, Volume 1 (1990). == Plot summary == The story is set in a future where computers play a central role in organizing society. Humans are employed as computer operators, but they leave most of the thinking to machines. Indeed, whilst binary programming is taught at school, reading and writing have become obsolete. The story concerns a pair of boys who dismantle and upgrade an old Bard, a child's computer whose sole function is to generate random fairy tales. The boys download a book about computers into the Bard's memory in an attempt to expand its vocabulary, but the Bard simply incorporates computers into its standard fairy tale repertoire. The story ends with the boys excitedly leaving the room after deciding to go to the library to learn "squiggles" (writing) as a means of passing secret messages to one another. As they leave, one of the boys accidentally kicks the Bard's on switch. The Bard begins reciting a new story about a poor mistreated and often ignored robot called the Bard, whose sole purpose is to tell stories, which ends with the words: "the little computer knew then that computers would always grow wiser and more powerful until someday—someday—someday—…"

Condensation algorithm

The condensation algorithm (Conditional Density Propagation) is a computer vision algorithm. The principal application is to detect and track the contour of objects moving in a cluttered environment. Object tracking is one of the more basic and difficult aspects of computer vision and is generally a prerequisite to object recognition. Being able to identify which pixels in an image make up the contour of an object is a non-trivial problem. Condensation is a probabilistic algorithm that attempts to solve this problem. The algorithm itself is described in detail by Isard and Blake in a publication in the International Journal of Computer Vision in 1998. One of the most interesting facets of the algorithm is that it does not compute on every pixel of the image. Rather, pixels to process are chosen at random, and only a subset of the pixels end up being processed. Multiple hypotheses about what is moving are supported naturally by the probabilistic nature of the approach. The evaluation functions come largely from previous work in the area and include many standard statistical approaches. The original part of this work is the application of particle filter estimation techniques. The algorithm's creation was inspired by the inability of Kalman filtering to perform object tracking well in the presence of significant background clutter. The presence of clutter tends to produce probability distributions for the object state which are multi-modal and therefore poorly modeled by the Kalman filter. The condensation algorithm in its most general form requires no assumptions about the probability distributions of the object or measurements. == Algorithm overview == The condensation algorithm seeks to solve the problem of estimating the conformation of an object described by a vector x t {\displaystyle \mathbf {x_{t}} } at time t {\displaystyle t} , given observations z 1 , . . . , z t {\displaystyle \mathbf {z_{1},...,z_{t}} } of the detected features in the images up to and including the current time. The algorithm outputs an estimate to the state conditional probability density p ( x t | z 1 , . . . , z t ) {\displaystyle p(\mathbf {x_{t}} |\mathbf {z_{1},...,z_{t}} )} by applying a nonlinear filter based on factored sampling and can be thought of as a development of a Monte-Carlo method. p ( x t | z 1 , . . . , z t ) {\displaystyle p(\mathbf {x_{t}} |\mathbf {z_{1},...,z_{t}} )} is a representation of the probability of possible conformations for the objects based on previous conformations and measurements. The condensation algorithm is a generative model since it models the joint distribution of the object and the observer. The conditional density of the object at the current time p ( x t | z 1 , . . . , z t ) {\displaystyle p(\mathbf {x_{t}} |\mathbf {z_{1},...,z_{t}} )} is estimated as a weighted, time-indexed sample set { s t ( n ) , n = 1 , . . . , N } {\displaystyle \{s_{t}^{(n)},n=1,...,N\}} with weights π t ( n ) {\displaystyle \pi _{t}^{(n)}} . N is a parameter determining the number of sample sets chosen. A realization of p ( x t | z 1 , . . . , z t ) {\displaystyle p(\mathbf {x_{t}} |\mathbf {z_{1},...,z_{t}} )} is obtained by sampling with replacement from the set s t {\displaystyle s_{t}} with probability equal to the corresponding element of π t {\displaystyle \pi _{t}} . The assumptions that object dynamics form a temporal Markov chain and that observations are independent of each other and the dynamics facilitate the implementation of the condensation algorithm. The first assumption allows the dynamics of the object to be entirely determined by the conditional density p ( x t | x t − 1 ) {\displaystyle p(\mathbf {x_{t}} |\mathbf {x_{t-1}} )} . The model of the system dynamics determined by p ( x t | x t − 1 ) {\displaystyle p(\mathbf {x_{t}} |\mathbf {x_{t-1}} )} must also be selected for the algorithm, and generally includes both deterministic and stochastic dynamics. The algorithm can be summarized by initialization at time t = 0 {\displaystyle t=0} and three steps at each time t: === Initialization === Form the initial sample set and weights by sampling according to the prior distribution. For example, specify as Gaussian and set the weights equal to each other. === Iterative procedure === Sample with replacement N {\displaystyle N} times from the set { s 0 ( n ) , n = 1 , . . . , N } {\displaystyle \{s_{0}^{(n)},n=1,...,N\}} with probability { π 0 ( n ) , n = 1 , . . . , N } {\displaystyle \{\pi _{0}^{(n)},n=1,...,N\}} to generate a realization of p ( x t | z 1 , . . . , z t ) {\displaystyle p(\mathbf {x_{t}} |\mathbf {z_{1},...,z_{t}} )} . Apply the learned dynamics p ( x t | x t − 1 ) {\displaystyle p(\mathbf {x_{t}} |\mathbf {x_{t-1}} )} to each element of this new set, to generate a new set { s t ( n ) } {\displaystyle \{s_{t}^{(n)}\}} . To take into account the current observation z t {\displaystyle \mathbf {z_{t}} } , set π t ( n ) = p ( z t | s ( n ) ) ∑ j = 1 N p ( z t | s ( j ) ) {\displaystyle \pi _{t}^{(n)}={\frac {p(\mathbf {z_{t}} |s^{(n)})}{\sum _{j=1}^{N}p(\mathbf {z_{t}} |s^{(j)})}}} for each element { s t ( n ) } {\displaystyle \{s_{t}^{(n)}\}} . This algorithm outputs the probability distribution p ( x t | z 1 , . . . , z t ) {\displaystyle p(\mathbf {x_{t}} |\mathbf {z_{1},...,z_{t}} )} which can be directly used to calculate the mean position of the tracked object, as well as the other moments of the tracked object. Cumulative weights can instead be used to achieve a more efficient sampling. == Implementation considerations == Since object-tracking can be a real-time objective, consideration of algorithm efficiency becomes important. The condensation algorithm is relatively simple when compared to the computational intensity of the Ricatti equation required for Kalman filtering. The parameter N {\displaystyle N} , which determines the number of samples in the sample set, will clearly hold a trade-off in efficiency versus performance. One way to increase efficiency of the algorithm is by selecting a low degree of freedom model for representing the shape of the object. The model used by Isard 1998 is a linear parameterization of B-splines in which the splines are limited to certain configurations. Suitable configurations were found by analytically determining combinations of contours from multiple views, of the object in different poses, and through principal component analysis (PCA) on the deforming object. Isard and Blake model the object dynamics p ( x t | x t − 1 ) {\displaystyle p(\mathbf {x_{t}} |\mathbf {x_{t-1}} )} as a second order difference equation with deterministic and stochastic components: p ( x t | x t − 1 ) ∝ e − 1 2 | | B − 1 ( ( x t − x ¯ ) − A ( x t − 1 − x ¯ ) ) | | 2 ) {\displaystyle p(\mathbf {x_{t}} |\mathbf {x_{t-1}} )\propto e^{-{\frac {1}{2}}||B^{-1}((\mathbf {x_{t}} -\mathbf {\bar {x}} )-A(\mathbf {x_{t-1}} -\mathbf {\bar {x}} ))||^{2})}} where x ¯ {\displaystyle \mathbf {\bar {x}} } is the mean value of the state, and A {\displaystyle A} , B {\displaystyle B} are matrices representing the deterministic and stochastic components of the dynamical model respectively. A {\displaystyle A} , B {\displaystyle B} , and x ¯ {\displaystyle \mathbf {\bar {x}} } are estimated via Maximum Likelihood Estimation while the object performs typical movements. The observation model p ( z | x ) {\displaystyle p(\mathbf {z} |\mathbf {x} )} cannot be directly estimated from the data, requiring assumptions to be made in order to estimate it. Isard 1998 assumes that the clutter which may make the object not visible is a Poisson random process with spatial density λ {\displaystyle \lambda } and that any true target measurement is unbiased and normally distributed with standard deviation σ {\displaystyle \sigma } . The basic condensation algorithm is used to track a single object in time. It is possible to extend the condensation algorithm using a single probability distribution to describe the likely states of multiple objects to track multiple objects in a scene at the same time. Since clutter can cause the object probability distribution to split into multiple peaks, each peak represents a hypothesis about the object configuration. Smoothing is a statistical technique of conditioning the distribution based on both past and future measurements once the tracking is complete in order to reduce the effects of multiple peaks. Smoothing cannot be directly done in real-time since it requires information of future measurements. == Applications == The algorithm can be used for vision-based robot localization of mobile robots. Instead of tracking the position of an object in the scene, however, the position of the camera platform is tracked. This allows the camera platform to be globally localized given a visual map of the environment. Extensions of the condensation algorithm have also been used to recognize human gestures in image sequences. This application of the condensation algorithm impacts the ran

The MANIAC

The MANIAC is a 2023 novel by Chilean author Benjamín Labatut, written in English. It is a fictionalised biography of polymath John von Neumann, whom Labatut calls "the smartest human being of the 20th century". The book focuses on von Neumann, but is also about physicist Paul Ehrenfest, the history of artificial intelligence, and Lee Sedol's Go match against AlphaGo. The book received mostly positive reviews from critics. == Background == John von Neumann was a Jewish Hungarian-born polymath who was a prodigy from an early childhood. Von Neumann worked in multiple fields of science, theoretical (mathematical foundations of quantum mechanics, game theory, cellular automata) and applied (nuclear weapons research during the Manhattan Project in World War II, computer architecture later named after him, and many other subjects). Labatut calls him "the smartest human being of the 20th century". The title of the book is derived from an early computer based on von Neumann architecture, built after the war at Los Alamos laboratory, called MANIAC I. Benjamín Labatut is a Chilean author known for his 2020 book When We Cease to Understand the World, a collection of fictionalised stories about famous scientists that received positive reviews and was translated into multiple languages from Spanish. The MANIAC is Labatut's first book written in English. In an interview, Labatut said he prefers to write in English: English is my preferred form of thought. ... English is the language I do most if not all my reading it. And it is a far better language than Spanish, in so many ways. Writing "clean" prose in Spanish is almost impossible, because so many of its sounds clash. Borges said that he found English "a far finer language than Spanish" because it's both Germanic and Latin; because of its wonderful vocabulary ("Regal is not exactly the same thing as saying kingly," he explained); because of its physicality; and because you can do almost anything with verbs and prepositions. Labatut was inspired to write The MANIAC by George Dyson's book Turing's Cathedral. == Synopsis == The book has three chapters. The first chapter, "Paul or the Discovery of the Irrational", written in the third person, is about physicist Paul Ehrenfest. The chapter opens with Ehrenfest shooting dead his son Vassily, who suffered from Down syndrome, and then himself. It then recounts Ehrenfest's life story, describing his relationships with his wife Tatyana, his mistress Nelly Meyjes, and his eminent physicist colleagues. It chronicles his descent into despair and depression over his marriage's disintegration, the advent of quantum mechanics, and the direction Europe was heading in with the Nazi Party's rise to power in Germany, looping back to the initial scene of the chapter. The second chapter, "John or the Mad Dreams of Reason", is about John von Neumann, and is written as a series of interviews of his family members, wives, friends, and colleagues, each in a distinctive voice. It is divided into three parts. Part I, "The Limits of Logic", is about his early life, as told by von Neumann's childhood friend Eugene Wigner, mother Margrit Kann, brother Nicholas von Neumann, first wife Mariette Kövesi, and scientists Theodore von Karman, George Polya, and Gábor Szegő. It climaxes with von Neumann's participation in David Hilbert's program to create a logical basis for mathematics based on a consistent set of axioms, a quest ultimately scuppered by Kurt Gödel. Part II, "The Delicate Balance of Terror", discusses von Neumann's role in the Manhattan Project (as told by Richard Feynman); his development of game theory and the doctrine of mutual assured destruction (MAD) (as told by Oskar Morgenstern); and his creation of the MANIAC I computer and the von Neumann architecture (as told by Julian Bigelow). In Part III, "Ghosts in the Machine", Sydney Brenner discusses von Neumann's contributions to biology, his theoretical work on self-replicating and self-repairing machines, and his vision of Von Neumann probes exploring the universe. Nils Aall Barricelli talks about his ideas of digital life and his disagreements with von Neumann. Von Neumann's wife Klára Dán, daughter Marina, and Wigner talk about his final years, personal life, and death. The third chapter, "Lee or The Delusions of Artificial Intelligence", is about Lee Sedol's Go match against AlphaGo. The narrative reverts to the third person. The chapter also tells the story of Demis Hassabis, a chess prodigy in childhood who decided to work on artificial intelligence and founded DeepMind, the company behind AlphaGo. The way is pointed to the future, as artificial intelligence's growing capabilities outpace the human mind. The book ends with Lee Sedol's retirement from Go, and new version of DeepMind's program, AlphaZero, that did not train on human games but nevertheless became the strongest player in Go, chess, and Shogi. == Reception == The book received mostly positive reviews. In his review for The New York Times Tom McCarthy noted the ambiguity of genre: "At its best, as in the stunning opening sequence reconstructing the murder-suicide of the physicist Paul Ehrenfest and his disabled son, or in the final section's gripping account of a computer defeating the world's best human Go player, you just throw up your hands and think, Who cares what discourse label we assign this stuff? It's great." Becca Rothfeld of the Washington Post praised the book, writing that it is "Labatut's latest virtuosic effort, at once a historical novel and a philosophical foray": "The MANIAC is a work of dark, eerie and singular beauty." She noted that the book "can also be difficult to read" because of its unusual narrative structure: "The book is narrated by a cluttered polyphony of characters, among them both of von Neumann's wives and a number of his teachers and colleagues. ... Like von Neumann, The MANIAC strives to adopt the impartial standpoint of the universe." Killian Fox of The Guardian sees the book as "darkly fascinating novel", and notes Labatut's "impressive dexterity, unpicking complex ideas in long, elegant sentences that propel us forward at speed (this is his first book written in English). Even in the more feverish passages, when yet another great mind succumbs to madness, haunted by the spectres they've helped unleash on the world, he feels in full control of his material." Sam Byers of The Guardian praises the book and the author's style: "The opening chapter of Benjamín Labatut's second novel is such a perfect distillation of his technique that it could serve as a manifesto." and "Readers ... will recognise the sense of breathlessness his best writing can evoke. Seemingly loosened from the laws of physics they describe, his sentences range freely through time and space, connecting not only characters and events, but the delicate tissue of intellectual history, often with a lightness of touch that belies their underlying complexity." He writes on the narrative structure: "Through a cascade of staccato chapters, an ensemble of narrators offer their piecemeal insights." Byers adds that "a brilliant novel is not quite what we end up with" and sees the problem in the "diffusion": "Labatut simply spreads himself too thin. Too many years in too few pages; too many voices with far too little to distinguish them. Initially intriguing, the bite-size monologues quickly come to feel inadequate." Some reviewers did not see the book as a biography. In an essay for the Cleveland Review of Books, Ben Cosman juxtaposes the book with Christopher Nolan's biopic Oppenheimer, and writes that it "follows the development of artificial intelligence—first as an idea at the beginning of the twentieth century, and then as a practicality at the beginning of the twenty-first—through the lives of three men who faced it." He also compared the book's structure to "witness testimony". Another reviewer called the book "perfect for anyone thirsting for more nuclear anxiety after watching Oppenheimer". Garrett Biggs of the Chicago Review of Books writes of the book's style: "Labatut writes about scientists the way Roberto Bolaño writes about poets. They are near mythical figures, captured at the corner of the novel's eye. They become historical in the most fraught sense of the term: subject to rumor and speculation and, eventually, the novel's form inflates their personas into something so large they can only be understood as narrative, never known in any objective capacity." Biggs criticises the last chapter: "the story of artificial intelligence has yet to be written. And so when Labatut's narration editorializes about artificial intelligence as 'a future that inspires hope and horror,' The MANIAC disassembles as a novel and starts to sound like a stale thinkpiece. AlphaGo might represent the first glimmer of a true artificial intelligence, as Labatut suggests. It also could one day be considered nothing more than a souped-up cousin to IBM's DeepBlue.