Controlled natural languages (CNLs) are subsets of natural languages that are obtained by restricting the grammar and vocabulary in order to reduce or eliminate ambiguity and complexity. Traditionally, controlled languages fall into two major types: those that improve readability for human readers (e.g. non-native speakers), and those that enable reliable automatic semantic analysis of the language. The first type of languages (often called "simplified" or "technical" languages), for example ASD Simplified Technical English, Caterpillar Technical English, IBM's Easy English, are used in the industry to increase the quality of technical documentation, and possibly simplify the semi-automatic translation of the documentation. These languages restrict the writer by general rules such as "Keep sentences short", "Avoid the use of pronouns", "Only use dictionary-approved words", and "Use only the active voice". The second type of languages have a formal syntax and formal semantics, and can be mapped to an existing formal language, such as first-order logic. Thus, those languages can be used as knowledge representation languages, and writing of those languages is supported by fully automatic consistency and redundancy checks, query answering, etc. == Languages == Existing controlled natural languages include: == Encoding == IETF has reserved simple as a BCP 47 variant subtag for simplified versions of languages.
POP-11
POP-11 is a reflective, incrementally compiled programming language with many of the features of an interpreted language. It is the core language of the Poplog programming environment developed originally by the University of Sussex, and recently in the School of Computer Science at the University of Birmingham, which hosts the main Poplog website. POP-11 is an evolution of the language POP-2, developed in Edinburgh University, and features an open stack model (like Forth, among others). It is mainly procedural, but supports declarative language constructs, including a pattern matcher, and is mostly used for research and teaching in artificial intelligence, although it has features sufficient for many other classes of problems. It is often used to introduce symbolic programming techniques to programmers of more conventional languages like Pascal, who find POP syntax more familiar than that of Lisp. One of POP-11's features is that it supports first-class functions. POP-11 is the core language of the Poplog system. The availability of the compiler and compiler subroutines at run-time (a requirement for incremental compiling) gives it the ability to support a far wider range of extensions (including run-time extensions, such as adding new data-types) than would be possible using only a macro facility. This made it possible for (optional) incremental compilers to be added for Prolog, Common Lisp and Standard ML, which could be added as required to support either mixed language development or development in the second language without using any POP-11 constructs. This made it possible for Poplog to be used by teachers, researchers, and developers who were interested in only one of the languages. The most successful product developed in POP-11 was the Clementine data mining system, developed by ISL. After SPSS bought ISL, they renamed Clementine to SPSS Modeler and decided to port it to C++ and Java, and eventually succeeded with great effort, and perhaps some loss of the flexibility provided by the use of an AI language. POP-11 was for a time available only as part of an expensive commercial package (Poplog), but since about 1999 it has been freely available as part of the open-source software version of Poplog, including various added packages and teaching libraries. An online version of ELIZA using POP-11 is available at Birmingham. At the University of Sussex, David Young used POP-11 in combination with C and Fortran to develop a suite of teaching and interactive development tools for image processing and vision, and has made them available in the Popvision extension to Poplog. == Simple code examples == Here is an example of a simple POP-11 program: define Double(Source) -> Result; Source2 -> Result; enddefine; Double(123) => That prints out: 246 This one includes some list processing: define RemoveElementsMatching(Element, Source) -> Result; lvars Index; [[% for Index in Source do unless Index = Element or Index matches Element then Index; endunless; endfor; %]] -> Result; enddefine; RemoveElementsMatching("the", [[the cat sat on the mat]]) => ;;; outputs [[cat sat on mat]] RemoveElementsMatching("the", [[the cat] [sat on] the mat]) => ;;; outputs [[the cat] [sat on] mat] RemoveElementsMatching([[= cat]], [[the cat]] is a [[big cat]]) => ;;; outputs [[is a]] Examples using the POP-11 pattern matcher, which makes it relatively easy for students to learn to develop sophisticated list-processing programs without having to treat patterns as tree structures accessed by 'head' and 'tail' functions (CAR and CDR in Lisp), can be found in the online introductory tutorial. The matcher is at the heart of the SimAgent (sim_agent) toolkit. Some of the powerful features of the toolkit, such as linking pattern variables to inline code variables, would have been very difficult to implement without the incremental compiler facilities.
Master/Session
In cryptography, Master/Session is a key management scheme in which a pre-shared Key Encrypting Key (called the "Master" key) is used to encrypt a randomly generated and insecurely communicated Working Key (called the "Session" key). The Working Key is then used for encrypting the data to be exchanged. Its advantage is simplicity, but it suffers the disadvantage of having to communicate the pre-shared Key Exchange Key, which can be difficult to update in the event of compromise. The Master/Session technique was created in the days before asymmetric techniques, such as Diffie-Hellman, were invented. This technique still finds widespread use in the financial industry, and is routinely used between corporate parties such as issuers, acquirers, switches. Its use in device communications (such as PIN pads), however, is in decline given the advantages of techniques such as DUKPT.
Social television
Social television is the union of television and social media. Millions of people now share their TV experience with other viewers on social media such as Twitter and Facebook using smartphones and tablets. TV networks and rights holders are increasingly sharing video clips on social platforms to monetise engagement and drive tune-in. The social TV market covers the technologies that support communication and social interaction around TV as well as companies that study television-related social behavior and measure social media activities tied to specific TV broadcasts – many of which have attracted significant investment from established media and technology companies. The market is also seeing numerous tie-ups between broadcasters and social networking players such as Twitter and Facebook. The market is expected to be worth $256bn by 2017. Social TV was named one of the 10 most important emerging technologies by the MIT Technology Review on Social TV in 2010. And in 2011, David Rowan, the editor of Wired magazine, named Social TV at number three of six in his peek into 2011 and what tech trends to expect to get traction. Ynon Kreiz, CEO of the Endemol Group told the audience at the Digital Life Design (DLD) conference in January 2011: "Everyone says that social television will be big. I think it's not going to be big—it's going to be huge". Much of the investment in the earlier years of social TV went into standalone social TV apps. The industry believed these apps would provide an appealing and complimentary consumer experience which could then be monetized with ads. These apps featured TV listings, check-ins, stickers and synchronised second-screen content but struggled to attract users away from Twitter and Facebook. Most of these companies have since gone out of business or been acquired amid a wave of consolidation and the market has instead focused on the activities of the social media channels themselves – such as Twitter Amplify, Facebook Suggested Videos and Snapchat Discover – and the technologies that support them. == Twitter == Twitter and Facebook are both helping users connect around media, which can provoke strong debate and engagement. Both social platforms want to be the 'digital watercooler' and host conversation around TV because the engagement and data about what media people consume can then be used to generate advertising revenue. As an open platform, conversation on Twitter is closely aligned with real-time events. In May 2013, it launched Twitter Amplify – an advertising product for media and consumer brands. With Amplify, Twitter runs video highlights from major live broadcasts, with advertisers' names and messages playing before the clip. By February 2014, all four major U.S. TV networks had signed up to the Amplify program, bringing a variety of premium TV content onto the social platform in the form of in-tweet real-time video clips. In June 2014, Twitter acquired its Twitter Amplify partner in the U.S. SnappyTV, a company that was helping broadcasters and rights holders to share video content both organically across social and via Twitter's Amplify program. Twitter continues to rely on Grabyo, which has also struck numerous deals with some of the largest broadcasters and rights holders in Europe and North America to share video content across Facebook and Twitter. == Facebook == Facebook made significant changes to its platform in 2014 including updates to its algorithm to enhance how it serves video in users' feeds. It also launched video autoplay to get users to watch the videos in their feeds. It rapidly surpassed Twitter and by the end of 2014 it was enjoying three billion video views a day on its platform and had announced a partnership with the NFL, one of Twitter's most active Twitter Amplify partners. In April 2015, at its F8 Developer Conference, it revealed it was working with Grabyo among other technology partners to bring video onto its platform. Then in July it announced it would be launching Facebook Suggested Videos, bringing related videos and ads to anyone that clicks on a video – a move that not only competed with Twitter's commercial video offering but also put it in direct competition with YouTube. == TV Time == TV Time is a television dedicated social network that allows users to keep track of the television series they watch, as well as films. It also allows them to express their reaction to the media they have seen with episode specific voting for favorite characters and emotional reaction to episodes, as well as commenting in episode restrictive pages. This way users are able to avoid spoilers while also finding a precise audience and community for each of their interactions, as opposed to bigger, non-television dedicated social medias such as Facebook and Twitter where the likelihood of unintentionally reading spoilers is much higher. TV Time offers an analytics service called "TVLytics" where the votes and reactions collected from users can be studied for research and television production purposes. == Advertising == According to Businessinsider.com, there are variety of applications for social TV, including support for TV ad sales, optimizing TV ad buys, making ad buys more efficient, as a complement to audience measurement, and eventually, audience forecasting and real-time optimization. Social TV data can ease access to focus groups and may create a positive feedback loop for generating ultra-sticky TV programming and multi-screen ad campaigns. == In numbers == Viewers share their TV experience on social media in real-time as events unfold: between 88-100m Facebook users login to the platform during the primetime hours of 8pm – 11pm in the US. The volume of social media engagement in TV is also rising – according to Nielsen SocialGuide, there was a 38% increase in tweets about TV in 2013 to 263m. For the 2014 Super Bowl, Twitter reported that a record 24.9 million tweets about the game were sent during the telecast, peaking at 381,605 tweets per minute. Facebook reported that 50 million people discussed the Super Bowl, generating 185 million interactions. The 2014 Oscars generated 5m tweets, viewed by an audience of 37m unique Twitter users and delivering 3.3bn impressions globally as conversation and key moments were shared virally across the platform. In 2014 the All England Lawn Tennis Club (AELTC), hosts of Wimbledon, used Grabyo to share video content across social. The videos were viewed 3.5 million times across Facebook and Twitter. In partnered with Grabyo again in 2015 and the videos generated over 48 million views across Facebook and Twitter. == Television shows with social integration == Here are some examples of how TV executives are integrating social elements with TV shows: C-SPAN streamed tweets from US Senators and Representatives during the quorum call The Voice had the judges of the program tweet during the show and the posts scrolls on the bottom of the screen. The use of Twitter also led to an increase in viewers. "Glee" Entertainment Weekly created a second screen viewing platform for the Glee season 3 premiere. == Related publications == Erika Jonietz. "Making TV Social, Virtually" MIT Technology Review. (January 11, 2010) AmigoTV (Alcatel-Lucent; Coppens et al.) – 2004 www.ist-ipmedianet.org/Alcatel_EuroiTV2004_AmigoTV_short_paper_S4-2.pdf Nextream (MIT Media Lab, Martin et al.) – 2010 Social Interactive Television: Immersive Shared Experiences and Perspectives (P. Cesar, D. Geerts, and K. Chorianopoulos (eds.)) – 2009 Social TV and the Emergence of Interactive TV – Multimedia Research Group – November 2010 Interactive Social TV on Service Oriented Environments: Challenges and Enablers (May 2011) == Systems == Boxee – acquired by Samsung GetGlue – acquired by i.TV Grabyo KIT digital Miso TV Tank Top TV WiO Xbox Live
List of broadband over power line deployments
This is a list of broadband over power line deployments. In this sense, "broadband" usually refers to Internet access using power line communication technology. == BPL pilot projects - 1st Gen (UPA) == === Inactive pilot projects === North America: United States: The United Telecom Council publishes the Federal Communications Commission (FCC)-mandated BPL Interference Resolution website, which provides a list of all BPL deployments in the US. Canada: Quebec: As of 2005, PLC communication technology developed by Ariane Controls is being installed inside and outside existing buildings to control lights and other energy-hungry devices. The cheap devices allow energy consumption to be better managed, and so save much energy and bring a clear return on investment. Western Europe: Sweden: Vattenfall is using PLC technology at 1200 baud for automatic meter reading based on an Iskraemeco product. Central and Eastern Europe, and Eurasia: Russian Federation: Electro-com has deployed widely BPL/PLC technology and offers internet access service in Moscow, Nizhny Novgorod, Ryazan, Kaluga and Rostov-on-Don, planning to extend coverage to main Russian cities. Currently the company does not provide other services, though plans to start providing telephone, and television services someday. Base equipment is a DefiDev modem with a DS2 chipset. The company had 35,000 subscribers and an annual growth of 15-20%. The company has, however, halted operations in Moscow in September, 2008, having sold its client network to an IDSL internet provider. Romania: In January, 2006, the Ministry of Communications and Information Technology introduced a PLC trial in the rural locality of Band, Mureș County, offering phone and broadband internet access for €7 per month. The technology was introduced to 50 households. Montenegro: In March, 2002, the Internet Crna Gora biggest internet provider in Montenegro launched a pilot project in town of Cetinje. Serbia: In August 2002, the Star Engineering from Niš launched a pilot project to show a completely new way to access the Internet, which is a new in that time in most countries around the world. Hungary: The first powerline service in Hungary was realized in September, 2003, in the Riverside apartment house in Budapest by 23Vnet Ltd. The PLC equipment was supplied by ASCOM Powerline. After four months the service was counting 100 users from 450 apartment owners. The bandwidth is 4.5 Mbit/s. Asia, Pacific, and Oceania: Indonesia: PT Kejora Gemilang Internusa "KEJORA", under their banner PLANET BROADBAND, is currently rolling out broadband over power line, with over 300,000 homes expected to be enabled by August 2010. PT. Kejora Gemilang Internusa signed an 8-year Joint Venture concession agreement with ICON+ a division of PT. Perusahaan Listrik Negara (Indonesia electricity company). Under the terms of the agreement PLAnet Broadband are to supply BPL/PLC to Jakarta West and West Java. Another company, PT. Broadband Powerline Indonesia, has been developing broadband over power line in apartment buildings since 2006. PT. BPI also produces data couplers to make broadband over powerline possible in three phases (R, S, T) with a single master. India : In India IIIT Allahabad has completed a project in co-operation with Corinex Communications Canada to implement a prototype of BPL for University campus and nearby villages. Africa and the Middle East: Egypt: The Engineering Office for Integrated Projects (EOIP) has deployed PLC technology widely in Alexandria, Fayed, and Tanta. Based on a locally developed system, the company provides AMR for electricity utilities. Currently, the company has about 70,000 subscribers. South Africa: Goal Technology Solutions (GTS) trialled the technology and is offering service in the suburbs of Pretoria, and plans to extend it to other areas. The tests were done with Mitsubishi equipment using a DS2 chipset, and the company claims a maximum throughput of 90 Mbit/s although initially only "512 Kbits/s ADSL equivalent speeds" are available. Now it uses DefiDev's equipment, and according to GTS's website, it will expand available bandwidth up to 5-20 Mbit/s. Ghana: Cactel Communications, Ltd. successfully deployed an MV solution pilot project in the Graphic Communications Group in Accra in June, 2005. A Cactel Remote Energy Management System (REMS) pilot project for the Electricity Company of Ghana (ECG) is running a 40-user pilot project at the University of Ghana in Legon. The current project combines fiber, radio link, Wi-Fi and PLC to provide broadband internet access and telephony. It showcases the interoperability of PLC technology and the company's expertise in emerging market design and deployment. Cactel hopes to deploy nationally, and is in deliberations with the national stakeholders and with Ghana's Ministry of Communications (MoC). AllTerra Communications successfully implemented a pilot test of broadband over power lines in Akosombo. In partnership with VRA, this test involves demonstrating transmission of broadband from medium to low voltage signals. AllTerra is working with VRA to expand the pilot project to include essential grid management utilities that will help balance and manage the current electricity transmission throughout their various substations. Using IT as a catalyst for economic development, AllTerra is expanding into numerous areas throughout Ghana. Vobiss Solutions Ltd successfully implemented a Hybrid Fibre BPL pilot network within EMEFS Hillview Estate in collaboration with ECG. Saudi Arabia: ElectroNet has been working with the Saudi Electric Company since 2005 on a pilot project using broadband over power lines over medium voltage cables and linking into low voltage distribution within a shopping mall. The pilot project also integrates automatic meter readers. Powerlines Communications Co. Ltd. implemented an AMR pilot project for Saudi Electricity Company in 2006. The project was located in the city of Jeddah on the west coast of Saudi Arabia. Digital KWh meters were installed in parallel with analog KWh meters. Readings taken by the Saudi Electricity Company showed variations of less than 1%. A BPL pilot project was included. Saudi Arabian Computer Management Consultants (SACMAC) has signed a deal to become an official system integrator and distributor for Mitsubishi PLC. It is expected to become a great success, because the existing broadband service, monopolized by the Saudi Telecom Company, is expensive and has poor customer service (some clients report that company techs arrive months after ordering). SACMAC has declined to talk about specifics of availability and price but says it will start rolling out the service in a few months (as of May 2006) and its price will be lower than current broadband providers. === Concluded pilot projects === The following pilot projects have ended: Australia, Tasmania: In November 2007, electricity retailer Aurora Energy ended its involvement with BPL and announced it was switching to Optical Fiber. This ended their commercial trial begun in September 2005, offering BPL services to 500 homes in the suburb of Tolmans Hill near Hobart, which had followed a successful technological trial earlier that year. Portugal ended BPL/PLC deployments in the country in October 2006, reportedly for economic reasons., Russian Federation: In September 2008, Russia's only BPL provider Electro-com ended deployments in Moscow for economic reasons. Spain: In May 2007 Iberdrola and Endesa (the main power companies in Spain) ended their projects to deploy PLC. United States: As of July 2010, the City of Manassas, VA has shut down their BPL deployment, which was the largest in the country. As of April 2007, Motorola has shuttered its Powerline LV Access BPL and reportedly plans to re-purpose the technology to a new system called Powerline MU, which is for use within multiple-unit dwellings. Motorola's system uses only residential-side low-voltage power lines for transmission to reduce the antenna effect, and successfully demonstrated frequency-notching for reduced potential for interference over the Amperion Inc. and Current Technologies LLC systems. Motorola invited the American Radio Relay League to participate with these tests, and even installed the Motorola system at their headquarters. Preliminary results were very positive with regard to interference, because the Motorola system does not use BPL on the powerlines leading up to the neighborhood. The BPL carrier is only used for the last leg of the trip from the pole to the house, and gets the signal to the pole via radio. This limits the interference to the area surrounding the last leg to the house. === Dismantled pilot projects === The following other BPL trials in the US are dismantled as of May 2008:
QANDA
QANDA (stands for 'Q and A') is an AI-based learning platform developed by Mathpresso Inc., a South Korea-based education technology company. Its best known feature is a solution search, which uses optical character recognition technology to scan problems and provide step-by-step solutions and learning content. As of March 2024, QANDA solved over 6.3 billion questions. QANDA has 90 million total registered users and has reached 8 million monthly active users (MAU) in 50 countries. 90% of the cumulative users are from overseas such as Vietnam and Indonesia. In January 2024, its MathGPT, a math-specific small large language model set a new world record, surpassed Microsoft's 'ToRA 13B', the previous record holder in benchmarks assessing mathematical performance such as 'MATH' (high school math) and 'GSM8K' (grade school math). 'MathGPT' was co-developed with Upstage and KT. In March 2024, Mathpresso launched 'Cramify' (formerly known as Prep.Pie), an AI-powered study material generator designed to create personalized exam prep materials for U.S. college students. It uses generative AI to create customized study materials uploaded by students. Its features include a range of tools including study summarizer and question solver. == History == Co-founder Jongheun ‘Ray’ Lee first came up with the idea of QANDA during his freshman year in college. While he was tutoring to earn money, Lee realized that the quality of education a student receives is greatly based on their location. Lee saw his K-12 students were regularly asking similar questions and realized that these questions were from a pre-selected number of textbooks currently being used in schools. He decided to team up with his high school friend, Yongjae ‘Jake’ Lee to build a platform whereby, one uses a mobile app to scan and submit questions, and students can ask and receive detailed responses. Lee's school friends, Wonguk Jung and Hojae Jeong, joined the team. In June 2015, Mathpresso, Inc. was founded in Seoul, South Korea. In January 2016, Mathpresso's first product QANDA was launched. It supported a Q&A feature between students and tutors. In October 2017, QANDA introduced an AI-based search capability that permitted users to search for answers in seconds. In April 2020, Jake Yongjae Lee(CEO & co-founder) and Ray Jongheun Lee (co-founder) were selected as Forbes 30 under 30 Asia. In June 2021, QANDA raised $50 million in series C funding. Jake Yongjae Lee was recognized as an Innovator Under 35 by MIT Technology Review. In November 2021, QANDA secured a strategic investment from Google. Since its inception, it has received backing in Series C funding from investors namely Google, Yellowdog, GGV Capital, Goodwater Capital, KDB, and SKS Private Equity with participation from SoftBank Ventures Asia, Legend Capital, Mirae Asset Venture Investment, and Smilegate Investment. In September 2023, Mathpresso has raised $8 million (10 billion KRW) from Korea's telecom giant, KT. The total cumulative investment is about 130 million US dollars. The partnership aims to accelerate the development of an education-specific Large Language Model. The company intends to incorporate the LLM model to fortify its AI tutor, which later will be integrated into the existing services: QANDA App, B2B & B2G Saas, and 1:1 online tutoring (QANDA Tutor). == Features == QANDA features OCR-based solution search, one-on-one Q&A tutoring, a study timer. In 2021, QANDA launched additional features, including the premium subscription model that offers unlimited “byte-sized” micro-video lectures and the community feature that enhances collaborative learning. In 2021, QANDA launched QANDA Tutor, a tablet-based 1:1 tutoring service and QANDA Study, a 1:N online school in Vietnam. In 2022, QANDA launched an exam prep feature that offers past exam materials from school via online. This feature is currently available in South Korea. In August 2023, QANDA launched a beta version of an LLM-powered AI Tutor. == Awards and recognition == Best Hidden Gems of 2017 by Google Playstore 2018 AWS AI Startup Challenge Award National representative for the Google AI for Social Good APAC, 2018 Best Self-Improvement Apps of 2018 by Google Playstore GSV Edtech 150 — the Most Transformational Growth Companies in Digital Learning Speaker at the Google App Summit, 2021 Selected as a prospect unicorn company by Korea Technology Finance Corporation in 2023 Winner of G20-DIA Global Pitching in 2023 2021, 2022, 2023 East Asia EdTech 150 by HolonIQ
Knapsack problem
The knapsack problem is the following problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine which items to include in the collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items. The problem often arises in resource allocation where the decision-makers have to choose from a set of non-divisible projects or tasks under a fixed budget or time constraint, respectively. The knapsack problem has been studied for more than a century, with early works dating back to 1897. The subset sum problem is a special case of the decision and 0-1 problems where for each kind of item, the weight equals the value: w i = v i {\displaystyle w_{i}=v_{i}} . In the field of cryptography, the term knapsack problem is often used to refer specifically to the subset sum problem. The subset sum problem is one of Karp's 21 NP-complete problems. == Applications == Knapsack problems appear in real-world decision-making processes in a wide variety of fields, such as finding the least wasteful way to cut raw materials, selection of investments and portfolios, selection of assets for asset-backed securitization, and generating keys for the Merkle–Hellman and other knapsack cryptosystems. One early application of knapsack algorithms was in the construction and scoring of tests in which the test-takers have a choice as to which questions they answer. For small examples, it is a fairly simple process to provide the test-takers with such a choice. For example, if an exam contains 12 questions each worth 10 points, the test-taker need only answer 10 questions to achieve a maximum possible score of 100 points. However, on tests with a heterogeneous distribution of point values, it is more difficult to provide choices. Feuerman and Weiss proposed a system in which students are given a heterogeneous test with a total of 125 possible points. The students are asked to answer all of the questions to the best of their abilities. Of the possible subsets of problems whose total point values add up to 100, a knapsack algorithm would determine which subset gives each student the highest possible score. A 1999 study of the Stony Brook University Algorithm Repository showed that, out of 75 algorithmic problems related to the field of combinatorial algorithms and algorithm engineering, the knapsack problem was the 19th most popular and the third most needed after suffix trees and the bin packing problem. == Definition == The most common problem being solved is the 0-1 knapsack problem, which restricts the number x i {\displaystyle x_{i}} of copies of each kind of item to zero or one. Given a set of n {\displaystyle n} items numbered from 1 up to n {\displaystyle n} , each with a weight w i {\displaystyle w_{i}} and a value v i {\displaystyle v_{i}} , along with a maximum weight capacity W {\displaystyle W} , maximize ∑ i = 1 n v i x i {\displaystyle \sum _{i=1}^{n}v_{i}x_{i}} subject to ∑ i = 1 n w i x i ≤ W {\displaystyle \sum _{i=1}^{n}w_{i}x_{i}\leq W} and x i ∈ { 0 , 1 } {\displaystyle x_{i}\in \{0,1\}} . Here x i {\displaystyle x_{i}} represents the number of instances of item i {\displaystyle i} to include in the knapsack. Informally, the problem is to maximize the sum of the values of the items in the knapsack so that the sum of the weights is less than or equal to the knapsack's capacity. The bounded knapsack problem (BKP) removes the restriction that there is only one of each item, but restricts the number x i {\displaystyle x_{i}} of copies of each kind of item to a maximum non-negative integer value c {\displaystyle c} : maximize ∑ i = 1 n v i x i {\displaystyle \sum _{i=1}^{n}v_{i}x_{i}} subject to ∑ i = 1 n w i x i ≤ W {\displaystyle \sum _{i=1}^{n}w_{i}x_{i}\leq W} and x i ∈ { 0 , 1 , 2 , … , c } . {\displaystyle x_{i}\in \{0,1,2,\dots ,c\}.} The unbounded knapsack problem (UKP) places no upper bound on the number of copies of each kind of item and can be formulated as above except that the only restriction on x i {\displaystyle x_{i}} is that it is a non-negative integer. maximize ∑ i = 1 n v i x i {\displaystyle \sum _{i=1}^{n}v_{i}x_{i}} subject to ∑ i = 1 n w i x i ≤ W {\displaystyle \sum _{i=1}^{n}w_{i}x_{i}\leq W} and x i ∈ N . {\displaystyle x_{i}\in \mathbb {N} .} One example of the unbounded knapsack problem is given using the figure shown at the beginning of this article and the text "if any number of each book is available" in the caption of that figure. == Computational complexity == The knapsack problem is interesting from the perspective of computer science for many reasons: The decision problem form of the knapsack problem (Can a value of at least V be achieved without exceeding the weight W?) is NP-complete, thus there is no known algorithm that is both correct and fast (polynomial-time) in all cases. There is no known polynomial algorithm which can tell, given a solution, whether it is optimal (which would mean that there is no solution with a larger V). This problem is co-NP-complete. There is a pseudo-polynomial time algorithm using dynamic programming. There is a fully polynomial-time approximation scheme, which uses the pseudo-polynomial time algorithm as a subroutine, described below. Many cases that arise in practice, and "random instances" from some distributions, can nonetheless be solved exactly. There is a link between the "decision" and "optimization" problems in that if there exists a polynomial algorithm that solves the "decision" problem, then one can find the maximum value for the optimization problem in polynomial time by applying this algorithm iteratively while increasing the value of k. On the other hand, if an algorithm finds the optimal value of the optimization problem in polynomial time, then the decision problem can be solved in polynomial time by comparing the value of the solution output by this algorithm with the value of k. Thus, both versions of the problem are of similar difficulty. One theme in research literature is to identify what the "hard" instances of the knapsack problem look like, or viewed another way, to identify what properties of instances in practice might make them more amenable than their worst-case NP-complete behaviour suggests. The goal in finding these "hard" instances is for their use in public-key cryptography systems, such as the Merkle–Hellman knapsack cryptosystem. More generally, better understanding of the structure of the space of instances of an optimization problem helps to advance the study of the particular problem and can improve algorithm selection. Furthermore, notable is the fact that the hardness of the knapsack problem depends on the form of the input. If the weights and profits are given as integers, it is weakly NP-complete, while it is strongly NP-complete if the weights and profits are given as rational numbers. However, in the case of rational weights and profits it still admits a fully polynomial-time approximation scheme. === Unit-cost models === The NP-hardness of the Knapsack problem relates to computational models in which the size of integers matters (such as the Turing machine). In contrast, decision trees count each decision as a single step. Dobkin and Lipton show an 1 2 n 2 {\displaystyle {1 \over 2}n^{2}} lower bound on linear decision trees for the knapsack problem, that is, trees where decision nodes test the sign of affine functions. This was generalized to algebraic decision trees by Steele and Yao. If the elements in the problem are real numbers or rationals, the decision-tree lower bound extends to the real random-access machine model with an instruction set that includes addition, subtraction and multiplication of real numbers, as well as comparison and either division or remaindering ("floor"). This model covers more algorithms than the algebraic decision-tree model, as it encompasses algorithms that use indexing into tables. However, in this model all program steps are counted, not just decisions. An upper bound for a decision-tree model was given by Meyer auf der Heide who showed that for every n there exists an O(n4)-deep linear decision tree that solves the subset-sum problem with n items. Note that this does not imply any upper bound for an algorithm that should solve the problem for any given n. == Solving == Several algorithms are available to solve knapsack problems, based on the dynamic programming approach, the branch and bound approach or hybridizations of both approaches. === Dynamic programming in-advance algorithm === The unbounded knapsack problem (UKP) places no restriction on the number of copies of each kind of item. Besides, here we assume that x i > 0 {\displaystyle x_{i}>0} m [ w ′ ] = max ( ∑ i = 1 n v i x i ) {\displaystyle m[w']=\max \left(\sum _{i=1}^{n}v_{i}x_{i}\right)} subject to ∑