OntoUML is a language for ontology-driven conceptual modeling. OntoUML is built as a UML extension based on the Unified Foundational Ontology. The foundations of UFO and OntoUML can be traced back to Giancarlo Guizzardi's Ph.D. thesis "Ontological foundations for structural conceptual models". In his work, he proposed a novel foundational ontology for conceptual modeling (UFO) and employed it to evaluate and re-design a fragment of the UML 2.0 metamodel for the purposes of conceptual modeling and domain ontology engineering. == Supporting tools == In 2006, Guizzardi co-founded the Ontology & Conceptual Modeling Research Group (NEMO) located at the Federal University of Espírito Santo (UFES) in Vitória city, state of Espírito Santo, Brazil. Since then, NEMO has been responsible for most of the developments in OntoUML. Several papers about ontologies and OntoUML have been authored by members of the NEMO group.
List of robotics journals
List of robotics journals includes notable academic and scientific journals that focus on research in the field of robotics and automation. == Journals == Acta Mechanica et Automatica Advanced Robotics Annual Review of Control, Robotics, and Autonomous Systems IEEE Robotics and Automation Letters IEEE Transactions on Robotics IEEE Transactions on Field Robotics The International Journal of Advanced Manufacturing Technology International Journal of Humanoid Robotics International Journal of Robotics Research Journal of Cognitive Engineering and Decision Making Journal of Field Robotics Journal of Intelligent & Robotic Systems Paladyn Robotics and Autonomous Systems Robotics Science Robotics SLAS Technology
Lion algorithm
Lion algorithm (LA) is one among the bio-inspired (or) nature-inspired optimization algorithms (or) that are mainly based on meta-heuristic principles. It was first introduced by B. R. Rajakumar in 2012 in the name, Lion’s Algorithm. It was further extended in 2014 to solve the system identification problem. This version was referred as LA, which has been applied by many researchers for their optimization problems. == Inspiration from lion’s social behaviour == Lions form a social system called a "pride", which consists of 1–3 pair of lions. A pride of lions shares a common area known as territory in which a dominant lion is called as territorial lion. The territorial lion safeguards its territory from outside attackers, especially nomadic lions. This process is called territorial defense. It protects the cubs till they become sexually matured. The maturity period is about 2–4 years. The pride undergoes survival fights to protect its territory and the cubs from nomadic lions. Upon getting defeated by the nomadic lions, the dominating nomadic lion takes the role of territorial lion by killing or driving out the cubs of the pride. The lioness of the pride give birth to cubs though the new territorial lion. When the cubs of the pride mature and considered to be stronger than the territorial lion, they take over the pride. This process is called territorial take-over. If territorial take-over happens, either the old territorial lion, which is considered to be laggard, is driven out or it leaves the pride. The stronger lions and lioness form the new pride and give birth to their own cubs == Terminology == In the LA, the terms that are associated with lion’s social system are mapped to the terminology of optimization problems. Few of such notable terms are related here. Lion: A potential solution to be generated or determined as optimal (or) near-optimal solution of the problem. The lion can be a territorial lion and lioness, cubs and nomadic lions that represent the solution based on the processing steps of the LA. Territorial lion: The strongest solution of the pride that tends to meet the objective function. Nomadic lion: A random solution, sometimes termed as nomad, to facilitate the exploration principle Laggard lion: Poor solutions that are failed in the survival fight. Pride: A pool of potential solutions i.e. a lion, lioness and their cubs, that are potential solutions of the search problem. Fertility evaluation: A process of evaluating whether the territorial lion and lioness are able to provide potential solutions in the future generations i.e. It ensures that the lion or lioness converge at every generation. Survival fight: It is a greedy selection process, which is often carried out between the pride and nomadic lion. == Algorithm == The steps involved in LA are given below: Pride Generation: Generate X m a l e {\displaystyle X^{male}} , X f e m a l e {\displaystyle X^{female}} and X 1 n o m a d {\displaystyle X_{1}^{nomad}} Determine f ( X m a l e ) {\displaystyle f(X^{male})} , f ( X f e m a l e ) {\displaystyle f(X^{female})} , f ( X 1 n o m a d ) {\displaystyle f(X_{1}^{nomad})} Initialize f r e f {\displaystyle f^{ref}} as f ( X m a l e ) {\displaystyle f(X^{male})} and N g {\displaystyle N_{g}} as 0 Memorize X m a l e {\displaystyle X^{male}} and X f e m a l e {\displaystyle X^{female}} Apply Fertility evaluation Process Generation of cubpool by mating Gender clustering: Define X c u b m a l e {\displaystyle X_{cub}^{male}} and X c u b f e m a l e {\displaystyle X_{cub}^{female}} Initialize a g e c u b {\displaystyle age_{cub}} as zero Apply Cub growth function Territorial defense: If X m a l e {\displaystyle X^{male}} (or pride) fails in the survival fight i.e. X 1 n o m a d {\displaystyle X_{1}^{nomad}} defeats the pride, go to step 4, else continue Increase a g e c u b {\displaystyle age_{cub}} by 1 and check whether cub attains maturity i.e., if a g e c u b > a g e m a x {\displaystyle age_{cub}>age_{max}} , go to Step 9, else continue Territorial takeover: If X c u b m a l e {\displaystyle X_{cub}^{male}} and X c u b f e m a l e {\displaystyle X_{cub}^{female}} are found to be closer to optimal solution, update X m a l e {\displaystyle X^{male}} and X f e m a l e {\displaystyle X^{female}} Increment N g {\displaystyle N_{g}} by 1 Repeat from Step 5, if termination criterion is not violated, else return X m a l e {\displaystyle X^{male}} as the near-optimal solution == Variants == The LA has been further taken forward to adopt in different problem areas. According to the characteristics of the problem area, significant amendment has been done in the processes and the models used in the LA. Accordingly, diverse variants have been developed by the researchers. They can be broadly grouped as hybrid LAs and non-hybrid LAs. Hybrid LAs are the LAs that are amended by the principle of other meta-heuristics, whereas the Non-hybrid LAs take any scientific amendment inside its operation that are felt to be essential to attend the respective problem area. == Applications == LA is applied in diverse engineering applications that range from network security, text mining, image processing, electrical systems, data mining and many more. Few of the notable applications are discussed here. Networking applications: In WSN, LA is used to solve the cluster head selection problem by determining optimal cluster head. Route discovery problem in both the VANET and MANET are also addressed by the LA in the literature. It is also used to detect attacks in advanced networking scenarios such as Software-Defined Networks (SDN) Power Systems: LA has attended generation rescheduling problem in a deregulated environment, optimal localization and sizing of FACTS devices for power quality enhancement and load-frequency controlling problem Cloud computing: LA is used in optimal container-resource allocation problem in cloud environment and cloud security
Data (word)
The word data is most often used as a singular collective mass noun in educated everyday usage. However, due to the history and etymology of the word, considerable controversy has existed on whether it should be considered a mass noun used with verbs conjugated in the singular, or should be treated as the plural of the now-rarely-used datum. == Usage in English == In one sense, data is the plural form of datum. Datum actually can also be a count noun with the plural datums (see usage in datum article) that can be used with cardinal numbers (e.g., "80 datums"); data (originally a Latin plural) is not used like a normal count noun with cardinal numbers and can be plural with plural determiners such as these and many, or it can be used as a mass noun with a verb in the singular form. Even when a very small quantity of data is referenced (one number, for example), the phrase piece of data is often used, as opposed to datum. The debate over appropriate usage continues, but "data" as a singular form is far more common. In English, the word datum is still used in the general sense of "an item given". In cartography, geography, nuclear magnetic resonance and technical drawing, it is often used to refer to a single specific reference datum from which distances to all other data are measured. Any measurement or result is a datum, though data point is now far more common. Data is indeed most often used as a singular mass noun in educated everyday usage. Some major newspapers, such as The New York Times, use it either in the singular or plural. In The New York Times, the phrases "the survey data are still being analyzed" and "the first year for which data is available" have appeared within one day. The Wall Street Journal explicitly allows this usage in its style guide. The Associated Press style guide classifies data as a collective noun that takes the singular when treated as a unit but the plural when referring to individual items (e.g., "The data is sound" and "The data have been carefully collected"). In scientific writing, data is often treated as a plural, as in These data do not support the conclusions, but the word is also used as a singular mass entity like information (e.g., in computing and related disciplines). British usage now widely accepts treating data as singular in standard English, including everyday newspaper usage at least in non-scientific use. UK scientific publishing still prefers treating it as a plural. Some UK university style guides recommend using data for both singular and plural use, and others recommend treating it only as a singular in connection with computers. The IEEE Computer Society allows usage of data as either a mass noun or plural based on author preference, while IEEE in the editorial style manual indicates to always use the plural form. Some professional organizations and style guides require that authors treat data as a plural noun. For example, the Air Force Flight Test Center once stated that the word data is always plural, never singular.
Hindley–Milner type system
A Hindley–Milner (HM) type system is a classical type system for the lambda calculus with parametric polymorphism. It is also known as Damas–Milner or Damas–Hindley–Milner. It was first described by J. Roger Hindley and later rediscovered by Robin Milner. Luis Damas contributed a close formal analysis and proof of the method in his PhD thesis. Among HM's more notable properties are its completeness and its ability to infer the most general type of a given program without programmer-supplied type annotations or other hints. Algorithm W is an efficient type inference method in practice and has been successfully applied on large code bases, although it has a high theoretical complexity. HM is preferably used for functional programming languages. It was first implemented as part of the type system of the programming language ML. Since then, HM has been extended in various ways, most notably with type class constraints like those in Haskell. == Introduction == As a type inference method, Hindley–Milner is able to deduce the types of variables, expressions and functions from programs written in an entirely untyped style. Being scope sensitive, it is not limited to deriving the types only from a small portion of source code, but rather from complete programs or modules. Being able to cope with parametric types, too, it is core to the type systems of many functional programming languages. It was first applied in this manner in the ML programming language. The origin is the type inference algorithm for the simply typed lambda calculus that was devised by Haskell Curry and Robert Feys in 1958. In 1969, J. Roger Hindley extended this work and proved that their algorithm always inferred the most general type. In 1978, Robin Milner, independently of Hindley's work, provided an equivalent algorithm, Algorithm W. In 1982, Luis Damas finally proved that Milner's algorithm is complete and extended it to support systems with polymorphic references. === Monomorphism vs. polymorphism === In the simply typed lambda calculus, types T are either atomic type constants or function types of form T → T {\displaystyle T\rightarrow T} . Such types are monomorphic. Typical examples are the types used in arithmetic values: 3 : N u m b e r a d d 3 4 : N u m b e r a d d : N u m b e r → N u m b e r → N u m b e r {\displaystyle {\begin{array}{ll}3&:{\mathtt {Number}}\\{\mathtt {add}}\ 3\ 4&:{\mathtt {Number}}\\{\mathtt {add}}&:{\mathtt {Number}}\rightarrow {\mathtt {Number}}\rightarrow {\mathtt {Number}}\end{array}}} Contrary to this, the untyped lambda calculus is neutral to typing at all, and many of its functions can be meaningfully applied to all type of arguments. The trivial example is the identity function i d ≡ λ x . x {\displaystyle {\mathtt {id}}\equiv \lambda x.x} which simply returns whatever value it is applied to. Less trivial examples include parametric types like lists. While polymorphism in general means that operations accept values of more than one type, the polymorphism used here is parametric. One finds the notation of type schemes in the literature, too, emphasizing the parametric nature of the polymorphism. Additionally, constants may be typed with (quantified) type variables. For example, the following type schemes quantify universally over α {\displaystyle \alpha } , meaning that they are true for all possible α {\displaystyle \alpha } : c o n s : ∀ α . α → L i s t α → L i s t α n i l : ∀ α . L i s t α i d : ∀ α . α → α {\displaystyle {\begin{array}{ll}{\mathtt {cons}}&:\forall \alpha .\alpha \rightarrow {\mathtt {List}}\ \alpha \rightarrow {\mathtt {List}}\ \alpha \\{\mathtt {nil}}&:\forall \alpha .{\mathtt {List}}\ \alpha \\{\mathtt {id}}&:\forall \alpha .\alpha \rightarrow \alpha \end{array}}} Polymorphic types can become monomorphic by consistent substitution of their variables. Examples of monomorphic instances are: i d ′ : S t r i n g → S t r i n g n i l ′ : L i s t N u m b e r {\displaystyle {\begin{array}{ll}{\mathtt {id}}'&:{\mathtt {String}}\rightarrow {\mathtt {String}}\\{\mathtt {nil}}'&:{\mathtt {List}}\ {\mathtt {Number}}\end{array}}} More generally, types are polymorphic when they contain type variables, while types without them are monomorphic. Contrary to the type systems used for example in Pascal (1970) or C (1972), which only support monomorphic types, HM is designed with emphasis on parametric polymorphism. The successors of the languages mentioned, like C++ (1985), focused on different types of polymorphism, namely subtyping in connection with object-oriented programming and overloading. While subtyping is incompatible with HM, a variant of systematic overloading is available in the HM-based type system of Haskell. === Let-polymorphism === When extending the type inference for the simply-typed lambda calculus towards polymorphism, one has to decide whether assigning a polymorphic type not only as type of an expression, but also as the type of a λ-bound variable is admissible. This would allow the generic identity type to be assigned to the variable 'id' in: (λ id . ... (id 3) ... (id "text") ... ) (λ x . x) Allowing this gives rise to the polymorphic lambda calculus; however, type inference in this system is not decidable. Instead, HM distinguishes variables that are immediately bound to an expression from more general λ-bound variables, calling the former let-bound variables, and allows polymorphic types to be assigned only to these. This leads to let-polymorphism where the above example takes the form let id = λ x . x in ... (id 3) ... (id "text") ... which can be typed with a polymorphic type for 'id'. As indicated, the expression syntax is extended to make the let-bound variables explicit, and by restricting the type system to allow only let-bound variable to have polymorphic types, while the parameters in lambda-abstractions must get a monomorphic type, type inference becomes decidable. == Overview == The remainder of this article proceeds as follows: The HM type system is defined. This is done by describing a deduction system that makes precise what expressions have what type, if any. From there, it works towards an implementation of the type inference method. After introducing a syntax-driven variant of the above deductive system, it sketches an efficient implementation (algorithm J), appealing mostly to the reader's metalogical intuition. Because it remains open whether algorithm J indeed realises the initial deduction system, a less efficient implementation (algorithm W), is introduced and its use in a proof is hinted. Finally, further topics related to the algorithm are discussed. The same description of the deduction system is used throughout, even for the two algorithms, to make the various forms in which the HM method is presented directly comparable. == The Hindley–Milner type system == The type system can be formally described by syntax rules that fix a language for the expressions, types, etc. The presentation here of such a syntax is not too formal, in that it is written down not to study the surface grammar, but rather the depth grammar, and leaves some syntactical details open. This form of presentation is usual. Building on this, typing rules are used to define how expressions and types are related. As before, the form used is a bit liberal. === Syntax === The expressions to be typed are exactly those of the lambda calculus extended with a let-expression as shown in the adjacent table. Parentheses can be used to disambiguate an expression. The application is left-binding and binds stronger than abstraction or the let-in construct. Types are syntactically split into two groups, monotypes and polytypes. ==== Monotypes ==== Monotypes always designate a particular type. Monotypes τ {\displaystyle \tau } are syntactically represented as terms. Examples of monotypes include type constants like i n t {\displaystyle {\mathtt {int}}} or s t r i n g {\displaystyle {\mathtt {string}}} , and parametric types like M a p ( S e t s t r i n g ) i n t {\displaystyle {\mathtt {Map\ (Set\ string)\ int}}} . The latter types are examples of applications of type functions, for example, from the set { M a p 2 , S e t 1 , s t r i n g 0 , i n t 0 , → 2 } {\displaystyle \{{\mathtt {Map^{2},\ Set^{1},\ string^{0},\ int^{0}}},\ \rightarrow ^{2}\}} , where the superscript indicates the number of type parameters. The complete set of type functions C {\displaystyle C} is arbitrary in HM, except that it must contain at least → 2 {\displaystyle \rightarrow ^{2}} , the type of functions. It is often written in infix notation for convenience. For example, a function mapping integers to strings has type i n t → s t r i n g {\displaystyle {\mathtt {int}}\rightarrow {\mathtt {string}}} . Again, parentheses can be used to disambiguate a type expression. The application binds stronger than the infix arrow, which is right-binding. Type variables are admitted as monotypes. Monotypes are not to be confused with monomorphic types, which exc
Touch 'n Go eWallet
Touch 'n Go eWallet is a Malaysian digital wallet and online payment platform, established in Kuala Lumpur, Malaysia, in July 2017 as a joint venture between Touch 'n Go and Ant Financial. It allows users to make payments at over 280,000 merchant touch points via QR code, as well as perform peer-to-peer (P2P) money transfers. Since then, the e-wallet further diversified for users to pay for tolls via RFID or PayDirect, street parking and various online payment spanning e-hailing, car-sharing apps or taxis, various overhead bills; top-up for mobile prepaid or in-game currencies; purchases on e-commerce websites; food delivery; renewing motor insurance and other insurance/takaful plans; and even movie, bus, trains or airline tickets. == Background == Prior to the launch of the e-wallet service, Touch 'n Go provided stored-value physical all-in-one contactless card (namely Touch 'n Go cards or "TnG cards") that users can use to pay for toll fares, public transportation and parking lots as well as purchases in some retail stores. In 1999, Touch 'n Go also markets SmartTag devices that allow road users to pass through certain toll booths without the need to unwind the car window. The high entry cost of the device (around RM 100 each) also meant that only few can enjoy the seamless experience. In 2009, Touch 'n Go partnered with Maxis to launch FastTap, a new mobile payment service that utilised Near-Field Communication (NFC). Maxis customers can make payments by placing the phone near the card readers (that also supports physical bank cards and Touch ’N Go cards). However, the venture featured only one phone model, Nokia 6212, which greatly limited the public reach. In July 2012, Touch 'n Go announced another collaboration with CIMB and Maxis to create similar NFC-based online transaction service that runs on compatible smartphones. Touch 'n Go Wallet was launched in February 2017 as an QR code-based e-wallet application, to compete with Samsung Pay that utilizes NFC modules. In the controlled pilot test in Taman Tun Dr Ismail, the correspondents can experience basic functionalities (prepaid mobile service reload, bills payment, movie tickets and flight tickets purchase, transfer of money with another user, and payments at participating stores and restaurants). While the deployed version of the app was generally well-received, the existing process to transfer the balance to the physical TnG card stored value from the app garnered unanimous backlash. Test groups felt that the need to head to a self-service terminal named "Pick Up Device" in person within 24 hours for completion, along with the failure to do so (the balance would be credited back to the wallet after 24 hours), was not divulged clearly and also defeated the purpose of convenience, not to mention there were only 2 such terminals. The feature was eventually suspended. On 15 November 2017, Touch 'n Go was granted permission by the Central Bank of Malaysia to form a joint venture with Ant Financial, a Chinese-based financial company that operates Alipay. The partnership allowed the local e-wallet to learn from and build upon the operational model pioneered by Alipay. In June 2018, it was reported that Touch 'n Go was pilot testing the uses of the Touch 'n Go eWallet in Rapid Transit, as the ticketing system was enabled on the Kelana Jaya line in the Klang Valley. Pilot testing only applied to stations in Kelana Jaya, KL Gateway–Universiti, Kerinchi, KL Sentral, Dang Wangi, KLCC, and Ampang Park. The test was reported to be successful in February 2020 and was planned to be fully deployed on the LRT and MRT. Due to unforeseen circumstances, this feature did not come into fruition, the app merely adds in-app purchase of monthly concession cards called "My50". In August 2018, Touch 'n Go announced that selected drivers may experience first-hand a new RFID-based payment (later rebranded as "myRFID") that serves to replace SmartTag devices on closed toll roads with during pilot testing phase commencing on 3 September 2018. On 2 November 2018, participation in the ongoing pilot programme was expanded, allowing more drivers to sign up ahead of the public rollout of the RFID system. During the same period, Touch 'n Go has discontinued the sales of SmartTAG devices in favor of the RFID-based payment system. Initially, the installation of the RFID chip onto the car could only be done by Touch 'n Go staff at the RFID fitment centers, at no cost. As the pilot testing concluded on 15 February 2020, a self-installation kit are being offered to the public on Lazada and Shopee. Support for taxi-hailing mobile apps was added in November 2018 when Touch 'n Go partnered with EzCab and Public Cab, allowing users to make payments via QR code. This was later expanded to support MULA on 7 January 2020, and later MyCar on 4 April 2020. Touch 'n Go eWallet was also the first eWallet to convert Kuala Lumpur's most famous Ramadan bazaar in Kampong Bahru into "Kampong Kashless", a venue that can accept cashless QR payments. It welcomed more than 250,000 Malaysians including local celebrities and government officials. On 1 October 2019, some e-commerce websites owned by the Alibaba Group (TMall and Taobao) began to support Touch 'n Go eWallet payments, Lazada joined the list on 29 October 2019. Touch 'n Go eWallet was one of the three e-wallet services in Malaysia (the other being Boost and GrabPay) that was eligible for its users to receive an RM 30 credit in conjunction of E-Tunai Rakyat program under the Budget 2020 plan, that further normalizes adoption of cashless and mobile payment among Malaysians. Unlike Boost and GrabPay, whose P2P transfers were completely disabled until users have exhausted the RM 30 first, Touch 'n Go eWallet did not impose such measures. in 2020, Touch 'n Go eWallet joined DuitNow, an electronic transaction ecosystem in Malaysia which allows the funds from Touch 'n Go eWallet to be transferred to other competing services and vice versa, by implementing a standard DuitNow QR code deisgn. Japan become the first country outside Malaysia to support Touch 'n Go eWallet payment via Alipay Connect. During the COVID-19 pandemic and the enforcement of the movement control order, use of eWallets (including Touch 'n Go eWallet) increased tremendously among citizens due to its contactless nature of the payment and increased take-out orders at home; which in turn helped small and medium-sized enterprises to thrive. Touch 'n Go eWallet launched its loyalty programme – The Goal Hunter – in October 2020 where on monthly basis, users collect stamps by paying with the app in exchange for rewards that include lucky draws and other vouchers. == Services == Touch 'n Go eWallet app is available for download on both Google Play and Apple Appstore. It utilizes QR code technology for local in-store payments. The Touch 'n Go eWallet app also diversifies payment types, including but not limited to Utility bills Purchase of motor insurance policy Pay Later facility Prepaid reload and Postpaid payment to telecommunications companies loan repayments for courts, MBSJ payments, zakat and PTPTN payment for car parking P2P transfer airline ticket bookings; movie tickets from TGV Cinemas RFID refuelling at Shell stations (defunct after Shell launched its own payment app in 2024) User can reload the eWallet credit by setting up auto-reload, purchasing reload pins from convenience stores (such as 7-Eleven, KK Super Mart, MyNews, Family Mart etc.), reloading by FPX and credit/debit card. The PayDirect feature allows users to link their physical Touch 'n Go cards into the eWallet, where the toll fare can be debited from the eWallet balance when flashing the card near the sensor. In the circumstance of insufficient balance in the app, the toll fare will be deducted from the physical card's balance instead. This also conveniently allows users to view the card's remaining balance. Touch 'n Go eWallet is the first and only eWallet to offer a money-back guarantee when an unauthorised transaction is made on the user’s eWallet account, subject to Terms & Conditions. Payment via QR code scanning, including Touch 'n Go eWallet, becomes a norm in most of the shops/restaurants across Malaysia, including roadside hawkers/stall owners and automatic vending machines. The merchants usually display their owner's individual QR or Business account that they can apply for in-app. The popularity attributes to the low merchant onboarding cost (Unlike NFC payment and debit/credit card that requires purchase or rental of a payment terminal device at a yearly fee.) The app is also one of the few ewallet that supports bidirectional liquidity (alongside MAE developed by Maybank), where funds can be transferred two-way with bank accounts. This is not possible with the other major ewallets (GrabPay, Boost, ShopeePay etc.) where the money that is reloaded to the wallet cannot be transferred to another bank account, unless through manual req
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.