Allen's interval algebra is a calculus for temporal reasoning that was introduced by James F. Allen in 1983. The calculus defines possible relations between time intervals and provides a composition table that can be used as a basis for reasoning about temporal descriptions of events. == Formal description == === Relations === The following 13 base relations capture the possible relations between two intervals. To see that the 13 relations are exhaustive, note that each point of X {\displaystyle X} can be at 5 possible locations relative to Y {\displaystyle Y} : before, at the start, within, at the end, after. These give 5 + 4 + 3 + 2 + 1 = 15 {\displaystyle 5+4+3+2+1=15} possible relative positions for the start and the end of X {\displaystyle X} . Of these, we cannot have X 0 = X 1 = Y 0 {\displaystyle X_{0}=X_{1}=Y_{0}} since X 0 < X 1 {\displaystyle X_{0} In machine learning, the kernel embedding of distributions (also called the kernel mean or mean map) comprises a class of nonparametric methods in which a probability distribution is represented as an element of a reproducing kernel Hilbert space (RKHS). A generalization of the individual data-point feature mapping done in classical kernel methods, the embedding of distributions into infinite-dimensional feature spaces can preserve all of the statistical features of arbitrary distributions, while allowing one to compare and manipulate distributions using Hilbert space operations such as inner products, distances, projections, linear transformations, and spectral analysis. This learning framework is very general and can be applied to distributions over any space Ω {\displaystyle \Omega } on which a sensible kernel function (measuring similarity between elements of Ω {\displaystyle \Omega } ) may be defined. For example, various kernels have been proposed for learning from data which are: vectors in R d {\displaystyle \mathbb {R} ^{d}} , discrete classes/categories, strings, graphs/networks, images, time series, manifolds, dynamical systems, and other structured objects. The theory behind kernel embeddings of distributions has been primarily developed by Alex Smola, Le Song, Arthur Gretton, and Bernhard Schölkopf. A review of recent works on kernel embedding of distributions can be found in. The analysis of distributions is fundamental in machine learning and statistics, and many algorithms in these fields rely on information theoretic approaches such as entropy, mutual information, or Kullback–Leibler divergence. However, to estimate these quantities, one must first either perform density estimation, or employ sophisticated space-partitioning/bias-correction strategies which are typically infeasible for high-dimensional data. Commonly, methods for modeling complex distributions rely on parametric assumptions that may be unfounded or computationally challenging (e.g. Gaussian mixture models), while nonparametric methods like kernel density estimation (Note: the smoothing kernels in this context have a different interpretation than the kernels discussed here) or characteristic function representation (via the Fourier transform of the distribution) break down in high-dimensional settings. Methods based on the kernel embedding of distributions sidestep these problems and also possess the following advantages: Data may be modeled without restrictive assumptions about the form of the distributions and relationships between variables Intermediate density estimation is not needed Practitioners may specify the properties of a distribution most relevant for their problem (incorporating prior knowledge via choice of the kernel) If a characteristic kernel is used, then the embedding can uniquely preserve all information about a distribution, while thanks to the kernel trick, computations on the potentially infinite-dimensional RKHS can be implemented in practice as simple Gram matrix operations Dimensionality-independent rates of convergence for the empirical kernel mean (estimated using samples from the distribution) to the kernel embedding of the true underlying distribution can be proven. Learning algorithms based on this framework exhibit good generalization ability and finite sample convergence, while often being simpler and more effective than information theoretic methods Thus, learning via the kernel embedding of distributions offers a principled drop-in replacement for information theoretic approaches and is a framework which not only subsumes many popular methods in machine learning and statistics as special cases, but also can lead to entirely new learning algorithms. == Definitions == Let X {\displaystyle X} denote a random variable with domain Ω {\displaystyle \Omega } and distribution P {\displaystyle P} . Given a symmetric, positive-definite kernel k : Ω × Ω → R {\displaystyle k:\Omega \times \Omega \rightarrow \mathbb {R} } the Moore–Aronszajn theorem asserts the existence of a unique RKHS H {\displaystyle {\mathcal {H}}} on Ω {\displaystyle \Omega } (a Hilbert space of functions f : Ω → R {\displaystyle f:\Omega \to \mathbb {R} } equipped with an inner product ⟨ ⋅ , ⋅ ⟩ H {\displaystyle \langle \cdot ,\cdot \rangle _{\mathcal {H}}} and a norm ‖ ⋅ ‖ H {\displaystyle \|\cdot \|_{\mathcal {H}}} ) for which k {\displaystyle k} is a reproducing kernel, i.e., in which the element k ( x , ⋅ ) {\displaystyle k(x,\cdot )} satisfies the reproducing property ⟨ f , k ( x , ⋅ ) ⟩ H = f ( x ) ∀ f ∈ H , ∀ x ∈ Ω . {\displaystyle \langle f,k(x,\cdot )\rangle _{\mathcal {H}}=f(x)\qquad \forall f\in {\mathcal {H}},\quad \forall x\in \Omega .} One may alternatively consider x ↦ k ( x , ⋅ ) {\displaystyle x\mapsto k(x,\cdot )} as an implicit feature mapping φ : Ω → H {\displaystyle \varphi :\Omega \rightarrow {\mathcal {H}}} (which is therefore also called the feature space), so that k ( x , x ′ ) = ⟨ φ ( x ) , φ ( x ′ ) ⟩ H {\displaystyle k(x,x')=\langle \varphi (x),\varphi (x')\rangle _{\mathcal {H}}} can be viewed as a measure of similarity between points x , x ′ ∈ Ω . {\displaystyle x,x'\in \Omega .} While the similarity measure is linear in the feature space, it may be highly nonlinear in the original space depending on the choice of kernel. === Kernel embedding === The kernel embedding of the distribution P {\displaystyle P} in H {\displaystyle {\mathcal {H}}} (also called the kernel mean or mean map) is given by: μ X := E [ k ( X , ⋅ ) ] = E [ φ ( X ) ] = ∫ Ω φ ( x ) d P ( x ) {\displaystyle \mu _{X}:=\mathbb {E} [k(X,\cdot )]=\mathbb {E} [\varphi (X)]=\int _{\Omega }\varphi (x)\ \mathrm {d} P(x)} If P {\displaystyle P} allows a square integrable density p {\displaystyle p} , then μ X = E k p {\displaystyle \mu _{X}={\mathcal {E}}_{k}p} , where E k {\displaystyle {\mathcal {E}}_{k}} is the Hilbert–Schmidt integral operator. A kernel is characteristic if the mean embedding μ : { family of distributions over Ω } → H {\displaystyle \mu :\{{\text{family of distributions over }}\Omega \}\to {\mathcal {H}}} is injective. Each distribution can thus be uniquely represented in the RKHS and all statistical features of distributions are preserved by the kernel embedding if a characteristic kernel is used. === Empirical kernel embedding === Given n {\displaystyle n} training examples { x 1 , … , x n } {\displaystyle \{x_{1},\ldots ,x_{n}\}} drawn independently and identically distributed (i.i.d.) from P , {\displaystyle P,} the kernel embedding of P {\displaystyle P} can be empirically estimated as μ ^ X = 1 n ∑ i = 1 n φ ( x i ) {\displaystyle {\widehat {\mu }}_{X}={\frac {1}{n}}\sum _{i=1}^{n}\varphi (x_{i})} === Joint distribution embedding === If Y {\displaystyle Y} denotes another random variable (for simplicity, assume the co-domain of Y {\displaystyle Y} is also Ω {\displaystyle \Omega } with the same kernel k {\displaystyle k} which satisfies ⟨ φ ( x ) ⊗ φ ( y ) , φ ( x ′ ) ⊗ φ ( y ′ ) ⟩ = k ( x , x ′ ) k ( y , y ′ ) {\displaystyle \langle \varphi (x)\otimes \varphi (y),\varphi (x')\otimes \varphi (y')\rangle =k(x,x')k(y,y')} ), then the joint distribution P ( x , y ) ) {\displaystyle P(x,y))} can be mapped into a tensor product feature space H ⊗ H {\displaystyle {\mathcal {H}}\otimes {\mathcal {H}}} via C X Y = E [ φ ( X ) ⊗ φ ( Y ) ] = ∫ Ω × Ω φ ( x ) ⊗ φ ( y ) d P ( x , y ) {\displaystyle {\mathcal {C}}_{XY}=\mathbb {E} [\varphi (X)\otimes \varphi (Y)]=\int _{\Omega \times \Omega }\varphi (x)\otimes \varphi (y)\ \mathrm {d} P(x,y)} By the equivalence between a tensor and a linear map, this joint embedding may be interpreted as an uncentered cross-covariance operator C X Y : H → H {\displaystyle {\mathcal {C}}_{XY}:{\mathcal {H}}\to {\mathcal {H}}} from which the cross-covariance of functions f , g ∈ H {\displaystyle f,g\in {\mathcal {H}}} can be computed as Cov ( f ( X ) , g ( Y ) ) := E [ f ( X ) g ( Y ) ] − E [ f ( X ) ] E [ g ( Y ) ] = ⟨ f , C X Y g ⟩ H = ⟨ f ⊗ g , C X Y ⟩ H ⊗ H {\displaystyle \operatorname {Cov} (f(X),g(Y)):=\mathbb {E} [f(X)g(Y)]-\mathbb {E} [f(X)]\mathbb {E} [g(Y)]=\langle f,{\mathcal {C}}_{XY}g\rangle _{\mathcal {H}}=\langle f\otimes g,{\mathcal {C}}_{XY}\rangle _{{\mathcal {H}}\otimes {\mathcal {H}}}} Given n {\displaystyle n} pairs of training examples { ( x 1 , y 1 ) , … , ( x n , y n ) } {\displaystyle \{(x_{1},y_{1}),\dots ,(x_{n},y_{n})\}} drawn i.i.d. from P {\displaystyle P} , we can also empirically estimate the joint distribution kernel embedding via C ^ X Y = 1 n ∑ i = 1 n φ ( x i ) ⊗ φ ( y i ) {\displaystyle {\widehat {\mathcal {C}}}_{XY}={\frac {1}{n}}\sum _{i=1}^{n}\varphi (x_{i})\otimes \varphi (y_{i})} === Conditional distribution embedding === Given a conditional distribution P ( y ∣ x ) , {\displaystyle P(y\mid x),} one can define the corresponding RKHS embedding as μ Y ∣ x = E [ φ ( Y ) ∣ X ] = ∫ Ω φ ( y ) d P ( y ∣ x ) {\displaystyle \mu _{Y\mid x}=\mathbb {E} [\varphi (Y)\mid X]=\int _{\Omega Mojo is an in-development proprietary programming language based on Python available for Linux and macOS. Mojo aims to combine the usability of a high-level programming language, specifically Python, with the performance of a system programming language such as C++, Rust, and Zig. As of October 2025, the Mojo compiler is closed source with an open source standard library. Modular, the company behind Mojo, has stated an intent to open source the Mojo language, committing to open-source Mojo in "fall 2026". Mojo builds on the Multi-Level Intermediate Representation (MLIR) compiler software framework, instead of directly on the lower level LLVM compiler framework like many languages such as Julia, Swift, C++, and Rust. MLIR is a newer compiler framework that allows Mojo to exploit higher level compiler passes unavailable in LLVM alone, and allows Mojo to compile down and target more than only central processing units (CPUs), including producing code that can run on graphics processing units (GPUs), Tensor Processing Units (TPUs), application-specific integrated circuits (ASICs) and other accelerators. It can also often more effectively use certain types of CPU optimizations directly, like single instruction, multiple data (SIMD) with minor intervention by a developer, as occurs in many other languages. According to Jeremy Howard of fast.ai, Mojo can be seen as "syntax sugar for MLIR" and for that reason Mojo is well optimized for applications like artificial intelligence (AI). == Origin and development history == The Mojo programming language was created by Modular Inc, which was founded by Chris Lattner, the original architect of the Swift programming language and LLVM, and Tim Davis, a former Google employee. The intention behind Mojo is to bridge the gap between Python’s ease of use and the fast performance required for cutting-edge AI applications. According to public change logs, Mojo development goes back to 2022. In May 2023, the first publicly testable version was made available online via a hosted playground. By September 2023 Mojo was available for local download for Linux and by October 2023 it was also made available for download on Apple's macOS. In March 2024, Modular open sourced the Mojo standard library and started accepting community contributions under the Apache 2.0 license. == Features == Mojo was created for an easy transition from Python. The language has syntax similar to Python's, with inferred static typing, and allows users to import Python modules. It uses LLVM and MLIR as its compilation backend. The language also intends to add a foreign function interface to call C/C++ and Python code. The language is not source-compatible with Python 3, only providing a subset of its syntax, e.g. missing the global keyword, list and dictionary comprehensions, and support for classes. Further, Mojo also adds features that enable performant low-level programming: fn for creating typed, compiled functions and "struct" for memory-optimized alternatives to classes. Mojo structs support methods, fields, operator overloading, and decorators. The language also provides a borrow checker, an influence from Rust. Mojo def functions use value semantics by default (functions receive a copy of all arguments and any modifications are not visible outside the function), while Python functions use reference semantics (functions receive a reference on their arguments and any modification of a mutable argument inside the function is visible outside). The language is not currently open source, but it is planned to be made open source in the future. Modular has since committed to open-sourcing the Mojo language in "fall 2026". == Programming examples == In Mojo, functions can be declared using both fn (for performant functions) or def (for Python compatibility). Basic arithmetic operations in Mojo with a def function: and with an fn function: The manner in which Mojo employs var and let for mutable and immutable variable declarations respectively mirrors the syntax found in Swift. In Swift, var is used for mutable variables, while let is designated for constants or immutable variables. Variable declaration and usage in Mojo: STUDENT is an early artificial intelligence program that solves algebra word problems. It is written in Lisp by Daniel G. Bobrow as his PhD thesis in 1964 (Bobrow 1964). It was designed to read and solve the kind of word problems found in high school algebra books. The program is often cited as an early accomplishment of AI in natural language processing. == Technical description == Within Project MAC at MIT, the STUDENT system was an early example of a question answering software, which uniquely involved natural language processing and symbolic programming. Other early attempts for solving algebra story problems were realized with 1960s hardware and software as well: for example, the Philips, Baseball and Synthex systems. STUDENT accepts an algebra story written in the English language as input, and generates a number as output. This is realized with a layered pipeline that consists of heuristics for pattern transformation. At first, sentences in English are converted into kernel sentences, which each contain a single piece of information. Next, the kernel sentences are converted into mathematical expressions. The knowledge base that supports the transformation contains 52 facts. STUDENT uses a rule-based system with logic inference. The rules are pre-programmed by the software developer and are able to parse natural language. More powerful techniques for natural language processing, such as machine learning, came into use later as hardware grew more capable, and gained popularity over simpler rule-based systems. In computer science, semantic knowledge management is a set of practices that seeks to classify content so that the knowledge it contains may be immediately accessed and transformed for delivery to the desired audience, in the required format. This classification of content is semantic in its nature – identifying content by its type or meaning within the content itself and via external, descriptive metadata – and is achieved by employing XML technologies. The specific outcomes of these practices are: Maintain content for multiple audiences together in a single document Transform content into various delivery formats without re-authoring Search for content more effectively Involve more subject-matter experts in the creation of content without reducing quality Reduce production costs for delivery formats Reduce the manual administration of getting the right knowledge to the right people Reduce the cost and time to localize content == Notable semantic knowledge management systems == Learn eXact Thinking Cap LCMS Thinking Cap LMS Xyleme LCMS iMapping Computer Dreams is a 1988 film created by Digital Vision Entertainment and released by MPI Home Video. Written, produced and directed by Geoffrey de Valois and hosted by Amanda Pays, it consists primarily of clips and behind-the-scenes work of early computer graphics animation. Notably included are Luxo Jr. and Red's Dream, the first two short films from Pixar. The film is an hour long and features an electronic score by Music Fantastic. It was revised and re-released on DVD as The History of Computer Animation, Volume 2. It won the Winner Gold Special Jury Award at the 1989 Houston International Film Festival, and the 1989 Golden Decade Award from the US Film & Video Festival. Music used includes: Gail Lennon - Desire, Gail Lennon - Like A Dream, Shandi Sinnamon - Making It, The Catholic Church views artificial intelligence as a significant technological development that must be governed by strict ethical principles rooted in human dignity and the common good. In January 2025, the Church issued the doctrinal note Antiqua et nova co-issued by the Dicastery for the Doctrine of the Faith and the Dicastery for Culture and Education. It addresses the "relationship between artificial intelligence and human intelligence" and offers reflections on the "anthropological and ethical challenges raised by AI". In August 2025, Time magazine included Pope Leo XIV in its 2025 list of the World’s Most Influential People in Artificial Intelligence. In May 2026, Pope Leo XIV approved the creation of a new Vatican commission on artificial intelligence. He released his first papal encyclical, titled Magnifica humanitas, on the topic later in the month.Kernel embedding of distributions
Mojo (programming language)
STUDENT
Semantic knowledge management
Computer Dreams
Catholic Church and artificial intelligence