Cobham's theorem

Cobham's theorem is a theorem in combinatorics on words that has important connections with number theory, notably transcendental numbers, and automata theory. Informally, the theorem gives the condition for the members of a set S of natural numbers written in bases b1 and base b2 to be recognised by finite automata. Specifically, consider bases b1 and b2 such that they are not powers of the same integer. Cobham's theorem states that S written in bases b1 and b2 is recognised by finite automata if and only if S differs by a finite set from a finite union of arithmetic progressions. The theorem was proved by Alan Cobham in 1969 and has since given rise to many extensions and generalisations. == Definitions == Let n > 0 {\displaystyle n>0} be an integer. The representation of a natural number n {\textstyle n} in base b {\textstyle b} is the sequence of digits n 0 n 1 ⋯ n h {\displaystyle n_{0}n_{1}\cdots n_{h}} such that n = n 0 + n 1 b + ⋯ + n h b h {\displaystyle n=n_{0}+n_{1}b+\cdots +n_{h}b^{h}} where 0 ≤ n 0 , n 1 , … , n h < b {\displaystyle 0\leq n_{0},n_{1},\ldots ,n_{h} 0 {\displaystyle n_{h}>0} . The word n 0 n 1 ⋯ n h {\displaystyle n_{0}n_{1}\cdots n_{h}} is often denoted ⟨ n ⟩ b {\displaystyle \langle n\rangle _{b}} , or more simply, n b {\displaystyle n_{b}} . A set of natural numbers S is recognisable in base b {\textstyle b} or more simply b {\textstyle b} -recognisable or b {\textstyle b} -automatic if the set { n b ∣ n ∈ S } {\displaystyle \{n_{b}\mid n\in S\}} of the representations of its elements in base b {\displaystyle b} is a language recognisable by a finite automaton on the alphabet { 0 , 1 , … , b − 1 } {\displaystyle \{0,1,\ldots ,b-1\}} . Two positive integers k {\displaystyle k} and ℓ {\displaystyle \ell } are multiplicatively independent if there are no non-negative integers p {\displaystyle p} and q {\displaystyle q} such that k p = ℓ q {\displaystyle k^{p}=\ell ^{q}} . For example, 2 and 3 are multiplicatively independent, but 8 and 16 are not since 8 4 = 16 3 {\displaystyle 8^{4}=16^{3}} . Two integers are multiplicatively dependent if and only if they are powers of a same third integer. == Problem statements == === Original problem statement === More equivalent statements of the theorem have been given. The original version by Cobham is the following: Another way to state the theorem is by using automatic sequences. Cobham himself calls them "uniform tag sequences." The following form is found in Allouche and Shallit's book:We can show that the characteristic sequence of a set of natural numbers S recognisable by finite automata in base k is a k-automatic sequence and that conversely, for all k-automatic sequences u {\displaystyle u} and all integers 0 ≤ i < k {\displaystyle 0\leq i 1 {\displaystyle \alpha >1} is the dominant eigenvalue of the matrix of morphism f {\displaystyle f} , namely, the matrix M ( f ) = ( m x , y ) x ∈ B , y ∈ A {\displaystyle M(f)=(m_{x,y})_{x\in B,y\in A}} , where m x , y {\displaystyle m_{x,y}} is the number of occurrences of the letter x {\displaystyle x} in the word f ( y ) {\displaystyle f(y)} . A set S of natural numbers is α {\displaystyle \alpha } -recognisable if its characteristic sequence s {\displaystyle s} is α {\displaystyle \alpha } -substitutive. A last definition: a Perron number is an algebraic number z > 1 {\displaystyle z>1} such that all its conjugates belong to the disc { z ′ ∈ C , | z ′ | < z } {\displaystyle \{z'\in \mathbb {C} ,|z'|

Image tracing

In computer graphics, image tracing, raster-to-vector conversion or raster vectorization is the conversion of raster graphics into vector graphics. == Background == An image does not have any structure: it is just a collection of marks on paper, grains in film, or pixels in a bitmap. While such an image is useful, it has some limits. If the image is magnified enough, its artifacts appear. The halftone dots, film grains, and pixels become apparent. Images of sharp edges become fuzzy or jagged. See, for example, pixelation. Ideally, a vector image does not have the same problem. Edges and filled areas are represented as mathematical curves or gradients, and they can be magnified arbitrarily (though of course the final image must also be rasterized in to be rendered, and its quality depends on the quality of the rasterization algorithm for the given inputs). The task in vectorization is to convert a two-dimensional image into a two-dimensional vector representation of the image. It is not examining the image and attempting to recognize or extract a three-dimensional model that may be depicted; i.e. it is not a vision system. For most applications, vectorization also does not involve optical character recognition; characters are treated as lines, curves, or filled objects without attaching any significance to them. In vectorization, the shape of the character is preserved, so artistic embellishments remain. Vectorization is the inverse operation corresponding to rasterization, as integration is to differentiation. And, just as with these other operations, while rasterization is fairly straightforward and algorithmic, vectorization involves the reconstruction of lost information and therefore requires heuristic methods. Synthetic images such as maps, cartoons, logos, clip art, and technical drawings are suitable for vectorization. Those images could have been originally made as vector images because they are based on geometric shapes or drawn with simple curves. Continuous tone photographs (such as live portraits) are not good candidates for vectorization. The input to vectorization is an image, but an image may come in many forms such as a photograph, a drawing on paper, or one of several raster file formats. Programs that do raster-to-vector conversion may accept bitmap formats such as TIFF, BMP and PNG. The output is a vector file format. Common vector formats are SVG, DXF, EPS, EMF and AI. Vectorization can be used to update images or recover work. Personal computers often come with a simple paint program that produces a bitmap output file. These programs allow users to make simple illustrations by adding text, drawing outlines, and filling outlines with a specific color. Only the results of these operations (the pixels) are saved in the resulting bitmap; the drawing and filling operations are discarded. Vectorization can be used to recapture some of the information that was lost. Vectorization is also used to recover information that was originally in a vector format but has been lost or has become unavailable. A company may have commissioned a logo from a graphic arts firm. Although the graphics firm used a vector format, the client company may not have received a copy of that format. The company may then acquire a vector format by scanning and vectorizing a paper copy of the logo. == Process == Vectorization starts with an image. === Manual === The image can be vectorized manually. A person could look at the image, make some measurements, and then write the output file by hand. That was the case for the vectorization of a technical illustration about neutrinos. The illustration has a few geometric shapes and a lot of text; it was relatively easy to convert the shapes, and the SVG vector format allows the text (even subscripts and superscripts) to be entered easily. The original image did not have any curves (except for the text), so the conversion is straightforward. Curves make the conversion more complicated. Manual vectorization of complicated shapes can be facilitated by the tracing function built into some vector graphics editing programs. If the image is not yet in machine readable form, then it has to be scanned into a usable file format. Once there is a machine-readable bitmap, the image can be imported into a graphics editing program (such as Adobe Illustrator, CorelDRAW, or Inkscape). Then a person can manually trace the elements of the image using the program's editing features. Curves in the original image can be approximated with lines, arcs, and Bézier curves. An illustration program allows spline knots to be adjusted for a close fit. Manual vectorization is possible, but it can be tedious. Although graphics drawing programs have been around for a long time, artists may find the freehand drawing facilities awkward even when a drawing tablet is used. Instead of using a program, Pepper recommends making an initial sketch on paper. Instead of scanning the sketch and tracing it freehand in the computer, Pepper states: "Those proficient with a graphic tablet and stylus could make the following changes directly in CorelDRAW by using a scan of the sketch as an underlay and drawing over it. I prefer to use pen and ink, and a light table"; most of the final image was traced by hand in ink. Later the line-drawing image was scanned at 600 dpi, cleaned up in a paint program, and then automatically traced with a program. Once the black and white image was in the graphics program, some other elements were added and the figure was colored. Similarly, Ploch recreated a design from a digital photograph. The JPEG was imported and some "basic shapes" were traced by hand and colored in the graphics drawing program; more complex shapes were handled differently. Ploch used a bitmap editor to remove the background and crop the more complex image components. He then printed the image and traced it by hand onto tracing paper to get a clean black and white line drawing. That drawing was scanned and then vectorized with a program. === Automatic === Some programs automate the vectorization process. Example programs are Adobe Illustrator, Inkscape, Corel's PowerTRACE, and Potrace. Some of these programs have a command line interface while others are interactive that allow the user to adjust the conversion settings and view the result. Adobe Streamline is not only an interactive program, but it also allows a user to manually edit the input bitmap and the output curves. Corel's PowerTRACE is accessed through CorelDRAW; CorelDRAW can be used to modify the input bitmap and edit the output curves. Adobe Illustrator has a facility to trace individual curves. Automated programs can have mixed results. A program (PowerTRACE) was used to convert a PNG map to SVG. The program did a good job on the map boundaries (the most tedious task in the tracing) and the settings dropped out all the text (small objects). The text was manually re-inserted. Other conversions may not go as well. The results depend on having high-quality scans, reasonable settings, and good algorithms. Scanned images often have a lot of noise, which can require additional work to clean up. == Options == There are many different image styles and possibilities, and no single vectorization method works well on all images. Consequently, vectorization programs have many options that influence the result. One issue is what the predominant shapes are. If the image is of a fill-in form, then it will probably have just vertical and horizontal lines of a constant width. The program's vectorization should take that into account. On the other hand, a CAD drawing may have lines at any angle, there may be curved lines, and there may be several line weights (thick for objects and thin for dimension lines). Instead of (or in addition to) curves, the image may contain outlines filled with the same color. Adobe Streamline allows users to select a combination of line recognition (horizontal and vertical lines), centerline recognition, or outline recognition. Streamline also allows small outline shapes to be thrown out; the notion is such small shapes are noise. The user may set the noise level between 0 and 1000; an outline that has fewer pixels than that setting is discarded. Another issue is the number of colors in the image. Even images that were created as black on white drawings may end up with many shades of gray. Some line-drawing routines employ anti-aliasing; a pixel completely covered by the line will be black, but a pixel that is only partially covered will be gray. If the original image is on paper and is scanned, there is a similar result: edge pixels will be gray. Sometimes images are compressed (e.g., JPEG images), and the compression will introduce gray levels. Many of the vectorization programs will group same-color pixels into lines, curves, or outlined shapes. If each possible color is grouped into its object, there can be an enormous number of objects. Instead, the user is asked to s

Enumeration algorithm

In computer science, an enumeration algorithm is an algorithm that enumerates the answers to a computational problem. Formally, such an algorithm applies to problems that take an input and produce a list of solutions, similarly to function problems. For each input, the enumeration algorithm must produce the list of all solutions, without duplicates, and then halt. The performance of an enumeration algorithm is measured in terms of the time required to produce the solutions, either in terms of the total time required to produce all solutions, or in terms of the maximal delay between two consecutive solutions and in terms of a preprocessing time, counted as the time before outputting the first solution. This complexity can be expressed in terms of the size of the input, the size of each individual output, or the total size of the set of all outputs, similarly to what is done with output-sensitive algorithms. == Formal definitions == An enumeration problem P {\displaystyle P} is defined as a relation R {\displaystyle R} over strings of an arbitrary alphabet Σ {\displaystyle \Sigma } : R ⊆ Σ ∗ × Σ ∗ {\displaystyle R\subseteq \Sigma ^{}\times \Sigma ^{}} An algorithm solves P {\displaystyle P} if for every input x {\displaystyle x} the algorithm produces the (possibly infinite) sequence y {\displaystyle y} such that y {\displaystyle y} has no duplicate and z ∈ y {\displaystyle z\in y} if and only if ( x , z ) ∈ R {\displaystyle (x,z)\in R} . The algorithm should halt if the sequence y {\displaystyle y} is finite. == Common complexity classes == Enumeration problems have been studied in the context of computational complexity theory, and several complexity classes have been introduced for such problems. A very general such class is EnumP, the class of problems for which the correctness of a possible output can be checked in polynomial time in the input and output. Formally, for such a problem, there must exist an algorithm A which takes as input the problem input x, the candidate output y, and solves the decision problem of whether y is a correct output for the input x, in polynomial time in x and y. For instance, this class contains all problems that amount to enumerating the witnesses of a problem in the class NP. Other classes that have been defined include the following. In the case of problems that are also in EnumP, these problems are ordered from least to most specific: Output polynomial, the class of problems whose complete output can be computed in polynomial time. Incremental polynomial time, the class of problems where, for all i, the i-th output can be produced in polynomial time in the input size and in the number i. Polynomial delay, the class of problems where the delay between two consecutive outputs is polynomial in the input (and independent from the output). Strongly polynomial delay, the class of problems where the delay before each output is polynomial in the size of this specific output (and independent from the input or from the other outputs). The preprocessing is generally assumed to be polynomial. Constant delay, the class of problems where the delay before each output is constant, i.e., independent from the input and output. The preprocessing phase is generally assumed to be polynomial in the input. == Common techniques == Backtracking: The simplest way to enumerate all solutions is by systematically exploring the space of possible results (partitioning it at each successive step). However, performing this may not give good guarantees on the delay, i.e., a backtracking algorithm may spend a long time exploring parts of the space of possible results that do not give rise to a full solution. Flashlight search: This technique improves on backtracking by exploring the space of all possible solutions but solving at each step the problem of whether the current partial solution can be extended to a partial solution. If the answer is no, then the algorithm can immediately backtrack and avoid wasting time, which makes it easier to show guarantees on the delay between any two complete solutions. In particular, this technique applies well to self-reducible problems. Closure under set operations: If we wish to enumerate the disjoint union of two sets, then we can solve the problem by enumerating the first set and then the second set. If the union is non disjoint but the sets can be enumerated in sorted order, then the enumeration can be performed in parallel on both sets while eliminating duplicates on the fly. If the union is not disjoint and both sets are not sorted then duplicates can be eliminated at the expense of a higher memory usage, e.g., using a hash table. Likewise, the cartesian product of two sets can be enumerated efficiently by enumerating one set and joining each result with all results obtained when enumerating the second step. == Examples of enumeration problems == The vertex enumeration problem, where we are given a polytope described as a system of linear inequalities and we must enumerate the vertices of the polytope. Enumerating the minimal transversals of a hypergraph. This problem is related to monotone dualization and is connected to many applications in database theory and graph theory. Enumerating the answers to a database query, for instance a conjunctive query or a query expressed in monadic second-order. There have been characterizations in database theory of which conjunctive queries could be enumerated with linear preprocessing and constant delay. The problem of enumerating maximal cliques in an input graph, e.g., with the Bron–Kerbosch algorithm Listing all elements of structures such as matroids and greedoids Several problems on graphs, e.g., enumerating independent sets, paths, cuts, etc. Enumerating the satisfying assignments of representations of Boolean functions, e.g., a Boolean formula written in conjunctive normal form or disjunctive normal form, a binary decision diagram such as an OBDD, or a Boolean circuit in restricted classes studied in knowledge compilation, e.g., NNF. == Connection to computability theory == The notion of enumeration algorithms is also used in the field of computability theory to define some high complexity classes such as RE, the class of all recursively enumerable problems. This is the class of sets for which there exist an enumeration algorithm that will produce all elements of the set: the algorithm may run forever if the set is infinite, but each solution must be produced by the algorithm after a finite time.

Adaptive algorithm

An adaptive algorithm is an algorithm that changes its behavior at the time it is run, based on information available and on a priori defined reward mechanism (or criterion). Such information could be the story of recently received data, information on the available computational resources, or other run-time acquired (or a priori known) information related to the environment in which it operates. Among the most used adaptive algorithms is the Widrow-Hoff’s least mean squares (LMS), which represents a class of stochastic gradient-descent algorithms used in adaptive filtering and machine learning. In adaptive filtering the LMS is used to mimic a desired filter by finding the filter coefficients that relate to producing the least mean square of the error signal (difference between the desired and the actual signal). For example, stable partition, using no additional memory is O(n lg n) but given O(n) memory, it can be O(n) in time. As implemented by the C++ Standard Library, stable_partition is adaptive and so it acquires as much memory as it can get (up to what it would need at most) and applies the algorithm using that available memory. Another example is adaptive sort, whose behavior changes upon the presortedness of its input. An example of an adaptive algorithm in radar systems is the constant false alarm rate (CFAR) detector. In machine learning and optimization, many algorithms are adaptive or have adaptive variants, which usually means that the algorithm parameters such as learning rate are automatically adjusted according to statistics about the optimisation thus far (e.g. the rate of convergence). Examples include adaptive simulated annealing, adaptive coordinate descent, adaptive quadrature, AdaBoost, Adagrad, Adadelta, RMSprop, and Adam. In data compression, adaptive coding algorithms such as Adaptive Huffman coding or Prediction by partial matching can take a stream of data as input, and adapt their compression technique based on the symbols that they have already encountered. In signal processing, the Adaptive Transform Acoustic Coding (ATRAC) codec used in MiniDisc recorders is called "adaptive" because the window length (the size of an audio "chunk") can change according to the nature of the sound being compressed, to try to achieve the best-sounding compression strategy.

SIGMOD Edgar F. Codd Innovations Award

The ACM SIGMOD Edgar F. Codd Innovations Award is a lifetime research achievement award given by the ACM Special Interest Group on Management of Data, at its yearly flagship conference (also called SIGMOD). According to its homepage, it is given "for innovative and highly significant contributions of enduring value to the development, understanding, or use of database systems and databases". The award has been given since 1992. Until 2003, this award was known as the “SIGMOD Innovations Award.” In 2004, SIGMOD, with the unanimous approval of ACM Council, decided to rename the award to honor Dr. E.F. (Ted) Codd (1923 – 2003) who invented the relational data model and was responsible for the significant development of the database field as a scientific discipline. == Recipients ==

Kindara

Kindara is a femtech company headquartered in Colorado that develops apps that help women identify their fertile window. The products are used for women trying to get pregnant, or women who want to track their menstrual cycle for overall health. Their latest product, Priya Fertility and Ovulation Monitor, maximizes a woman's chance of getting pregnancy by identifying her most fertile days. == Overview == Kindara was founded in 2011 by husband-and-wife team Will Sacks and Kati Bicknell. The company launched its free mobile application in 2012. Kindara's mobile application allows women to track signs of fertility, such as basal body temperature, cervical fluid, and the position of the cervix to determine when ovulation is occurring. Kindara also sells a thermometer, Wink, which records basal body temperature and syncs automatically to the Kindara fertility application. In 2018, Kindara was acquired by the company Prima-Temp.

Paper data storage

Paper data storage refers to the use of paper as a data storage device. This includes writing, illustrating, and the use of data that can be interpreted by a machine or is the result of the functioning of a machine. A defining feature of paper data storage is the ability of humans to produce it with only simple tools and interpret it visually. Though now mostly obsolete, paper was once an important form of computer data storage as both paper tape and punch cards were a common staple of working with computers before the 1980s. == History == Before paper was used for storing data, it had been used in several applications for storing instructions to specify a machine's operation. The earliest use of paper to store instructions for a machine was the work of Basile Bouchon who, in 1725, used punched paper rolls to control textile looms. This technology was later developed into the wildly successful Jacquard loom. The 19th century saw several other uses of paper for controlling machines. In 1846, telegrams could be prerecorded on punched tape and rapidly transmitted using Alexander Bain's automatic telegraph. Several inventors took the concept of a mechanical organ and used paper to represent the music. In the late 1880s Herman Hollerith invented the recording of data on a medium that could then be read by a machine. Prior uses of machine readable media, above, had been for control (automatons, piano rolls, looms, ...), not data. "After some initial trials with paper tape, he settled on punched cards..." Hollerith's method was used in the 1890 census. Hollerith's company eventually became the core of IBM. Other technologies were also developed that allowed machines to work with marks on paper instead of punched holes. This technology was widely used for tabulating votes and grading standardized tests. Banks used magnetic ink on checks, supporting MICR scanning. In an early electronic computing device, the Atanasoff–Berry Computer, electric sparks were used to singe small holes in paper cards to represent binary data. The altered dielectric constant of the paper at the location of the holes could then be used to read the binary data back into the machine by means of electric sparks of lower voltage than the sparks used to create the holes. This form of paper data storage was never made reliable and was not used in any subsequent machine. == Modern techniques == === 1D barcodes === Barcodes make it possible for any object that was to be sold or transported to have some computer readable information securely attached to it. Universal Product Code barcodes, first used in 1974, are ubiquitous today. Some people recommend a width of at least 3 pixels for each minimum-width gap and each minimum-width bar for 1D barcodes. The density is about 50 bits per linear inch (about 2 bit/mm). === 2D barcodes === 2D barcodes allow to store much more data on paper, up to 2.9 kbyte per barcode. It is recommended to have a width of at least 4 pixels—e.g., a 4 × 4 pixel = 16 pixel module. == Limits == The limits of data storage depend on the technology to write and read such data. The theoretical limits assume a scanner that can perfectly reproduce the printed image at its printing resolution, and a program which can accurately interpret such an image. For example, an 8 in × 10 in (200 mm × 250 mm) 600 dpi black-and-white image contains 3.43 MiB of data, as does a 300 dpi CMYK printed image. A 2,400 ppi True color (24-bit) image contains about 1.29 GiB of information; printing an image maintaining this data would require a printing resolution of about 120,000 dpi in black and white, or 60,000 dpi with CMYK dots.