In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y that admits operations x\z and z/y, loosely analogous to division or implication, when x•y is viewed as multiplication or conjunction, respectively. Called respectively right and left residuals, these operations coincide when the monoid is commutative. The general concept was introduced by Morgan Ward and Robert P. Dilworth in 1939. Examples, some of which existed prior to the general concept, include Boolean algebras, Heyting algebras, residuated Boolean algebras, relation algebras, and MV-algebras. Residuated semilattices omit the meet operation ∧, for example Kleene algebras and action algebras. == Definition == In mathematics, a residuated lattice is an algebraic structure L = (L, ≤, •, I) such that (i) (L, ≤) is a lattice. (ii) (L, •, I) is a monoid. (iii) For all z there exists for every x a greatest y, and for every y a greatest x, such that x•y ≤ z (the residuation properties). In (iii), the "greatest y", being a function of z and x, is denoted x\z and called the right residual of z by x. Think of it as what remains of z on the right after "dividing" z on the left by x. Dually, the "greatest x" is denoted z/y and called the left residual of z by y. An equivalent, more formal statement of (iii) that uses these operations to name these greatest values is (iii)' for all x, y, z in L, y ≤ x\z ⇔ x•y ≤ z ⇔ x ≤ z/y. As suggested by the notation, the residuals are a form of quotient. More precisely, for a given x in L, the unary operations x• and x\ are respectively the lower and upper adjoints of a Galois connection on L, and dually for the two functions •y and /y. By the same reasoning that applies to any Galois connection, we have yet another definition of the residuals, namely, x•(x\y) ≤ y ≤ x\(x•y), and (y/x)•x ≤ y ≤ (y•x)/x, together with the requirement that x•y be monotone in x and y. (When axiomatized using (iii) or (iii)' monotonicity becomes a theorem and hence not required in the axiomatization.) These give a sense in which the functions x• and x\ are pseudoinverses or adjoints of each other, and likewise for •x and /x. This last definition is purely in terms of inequalities, noting that monotonicity can be axiomatized as x • y ≤ (x∨z) • y and similarly for the other operations and their arguments. Moreover, any inequality x ≤ y can be expressed equivalently as an equation, either x∧y = x or x∨y = y. This along with the equations axiomatizing lattices and monoids then yields a purely equational definition of residuated lattices, provided the requisite operations are adjoined to the signature (L, ≤, •, I) thereby expanding it to (L, ∧, ∨, •, I, /, \). When thus organized, residuated lattices form an equational class or variety, whose homomorphisms respect the residuals as well as the lattice and monoid operations. Note that distributivity x • (y ∨ z) = (x • y) ∨ (x • z) and x•0 = 0 are consequences of these axioms and so do not need to be made part of the definition. This necessary distributivity of • over ∨ does not in general entail distributivity of ∧ over ∨, that is, a residuated lattice need not be a distributive lattice. However distributivity of ∧ over ∨ is entailed when • and ∧ are the same operation, a special case of residuated lattices called a Heyting algebra. Alternative notations for x•y include x◦y, x;y (relation algebra), and x⊗y (linear logic). Alternatives for I include e and 1'. Alternative notations for the residuals are x → y for x\y and y ← x for y/x, suggested by the similarity between residuation and implication in logic, with the multiplication of the monoid understood as a form of conjunction that need not be commutative. When the monoid is commutative the two residuals coincide. When not commutative, the intuitive meaning of the monoid as conjunction and the residuals as implications can be understood as having a temporal quality: x•y means x and then y, x → y means had x (in the past) then y (now), and y ← x means if-ever x (in the future) then y (at that time), as illustrated by the natural language example at the end of the examples. == Examples == One of the original motivations for the study of residuated lattices was the lattice of (two-sided) ideals of a ring. Given a ring R, the ideals of R, denoted Id(R), forms a complete lattice with set intersection acting as the meet operation and "ideal addition" acting as the join operation. The monoid operation • is given by "ideal multiplication", and the element R of Id(R) acts as the identity for this operation. Given two ideals A and B in Id(R), the residuals are given by A / B := { r ∈ R ∣ r B ⊆ A } {\displaystyle A/B:=\{r\in R\mid rB\subseteq A\}} B ∖ A := { r ∈ R ∣ B r ⊆ A } {\displaystyle B\setminus A:=\{r\in R\mid Br\subseteq A\}} It is worth noting that {0}/B and B\{0} are respectively the left and right annihilators of B. This residuation is related to the conductor (or transporter) in commutative algebra written as (A:B)=A/B. One difference in usage is that B need not be an ideal of R: it may just be a subset. Boolean algebras and Heyting algebras are commutative residuated lattices in which x•y = x∧y (whence the unit I is the top element 1 of the algebra) and both residuals x\y and y/x are the same operation, namely implication x → y. The second example is quite general since Heyting algebras include all finite distributive lattices, as well as all chains or total orders, for example the unit interval [0,1] in the real line, or the integers and ± ∞ {\displaystyle \pm \infty } . The structure (Z, min, max, +, 0, −, −) (the integers with subtraction for both residuals) is a commutative residuated lattice such that the unit of the monoid is not the greatest element (indeed there is no least or greatest integer), and the multiplication of the monoid is not the meet operation of the lattice. In this example the inequalities are equalities because − (subtraction) is not merely the adjoint or pseudoinverse of + but the true inverse. Any totally ordered group under addition such as the rationals or the reals can be substituted for the integers in this example. The nonnegative portion of any of these examples is an example provided min and max are interchanged and − is replaced by monus, defined (in this case) so that x-y = 0 when x ≤ y and otherwise is ordinary subtraction. A more general class of examples is given by the Boolean algebra of all binary relations on a set X, namely the power set of X2, made a residuated lattice by taking the monoid multiplication • to be composition of relations and the monoid unit to be the identity relation I on X consisting of all pairs (x,x) for x in X. Given two relations R and S on X, the right residual R\S of S by R is the binary relation such that x(R\S)y holds just when for all z in X, zRx implies zSy (notice the connection with implication). The left residual is the mirror image of this: y(S/R)x holds just when for all z in X, xRz implies ySz. This can be illustrated with the binary relations < and > on {0,1} in which 0 < 1 and 1 > 0 are the only relationships that hold. Then x(>\<)y holds just when x = 1, while x()y holds just when y = 0, showing that residuation of < by > is different depending on whether we residuate on the right or the left. This difference is a consequence of the difference between <•> and >•<, where the only relationships that hold are 0(<•>)0 (since 0<1>0) and 1(>•<)1 (since 1>0<1). Had we chosen ≤ and ≥ instead of < and >, ≥\≤ and ≤/≥ would have been the same because ≤•≥ = ≥•≤, both of which always hold between all x and y (since x≤1≥y and x≥0≤y). The Boolean algebra 2Σ of all formal languages over an alphabet (set) Σ forms a residuated lattice whose monoid multiplication is language concatenation LM and whose monoid unit I is the language {ε} consisting of just the empty string ε. The right residual M\L consists of all words w over Σ such that Mw ⊆ L. The left residual L/M is the same with wM in place of Mw. The residuated lattice of all binary relations on X is finite just when X is finite, and commutative just when X has at most one element. When X is empty the algebra is the degenerate Boolean algebra in which 0 = 1 = I. The residuated lattice of all languages on Σ is commutative just when Σ has at most one letter. It is finite just when Σ is empty, consisting of the two languages 0 (the empty language {}) and the monoid unit I = {ε} = 1. The examples forming a Boolean algebra have special properties treated in the article on residuated Boolean algebras. == Residuated semilattice == A residuated semilattice is defined almost identically for residuated lattices, omitting just the meet operation ∧. Thus it is an algebraic structure L = (L, ∨, •, 1, /, \) satisfying all the residuated lattice equations as specified above except those containing an occurrence of the symbol ∧. The option of defining x ≤ y as x∧y = x is then not available, leaving on
Nice (app)
Nice is a photo-sharing mobile app developed by Nice App Mobile Technology Co., Ltd. (Chinese: 北京极赞科技有限公司) in China. The app allows users to tag specific locations on images, enabling detailed labeling of items such as clothing and accessories. The company received a $36 million investment in C-round funding in 2014. Nice had 30 million registered users and 12 million active users as of late 2015. As of January 2024, it remained a popular app, the 6th most-downloaded in the iOS App Store for China. == Official website == Official website
Convergent encryption
Convergent encryption, also known as content hash keying, is a cryptosystem that produces identical ciphertext from identical plaintext files. This has applications in cloud computing to remove duplicate files from storage without the provider having access to the encryption keys. The combination of deduplication and convergent encryption was described in a backup system patent filed by Stac Electronics in 1995. This combination has been used by Farsite, Permabit, Freenet, MojoNation, GNUnet, flud, and the Tahoe Least-Authority File Store. The system gained additional visibility in 2011 when cloud storage provider Bitcasa announced they were using convergent encryption to enable de-duplication of data in their cloud storage service. == Overview == The system computes a cryptographic hash of the plaintext in question. The system then encrypts the plaintext by using the hash as a key. Finally, the hash itself is stored, encrypted with a key chosen by the user. == Known Attacks == Convergent encryption is open to a "confirmation of a file attack" in which an attacker can effectively confirm whether a target possesses a certain file by encrypting an unencrypted, or plain-text, version and then simply comparing the output with files possessed by the target. This attack poses a problem for a user storing information that is non-unique, i.e. also either publicly available or already held by the adversary - for example: banned books or files that cause copyright infringement. An argument could be made that a confirmation of a file attack is rendered less effective by adding a unique piece of data such as a few random characters to the plain text before encryption; this causes the uploaded file to be unique and therefore results in a unique encrypted file. However, some implementations of convergent encryption where the plain-text is broken down into blocks based on file content, and each block then independently convergently encrypted may inadvertently defeat attempts at making the file unique by adding bytes at the beginning or end. Even more alarming than the confirmation attack is the "learn the remaining information attack" described by Drew Perttula in 2008. This type of attack applies to the encryption of files that are only slight variations of a public document. For example, if the defender encrypts a bank form including a ten digit bank account number, an attacker that is aware of generic bank form format may extract defender's bank account number by producing bank forms for all possible bank account numbers, encrypt them and then by comparing those encryptions with defender's encrypted file deduce the bank account number. Note that this attack can be extended to attack a large number of targets at once (all spelling variations of a target bank customer in the example above, or even all potential bank customers), and the presence of this problem extends to any type of form document: tax returns, financial documents, healthcare forms, employment forms, etc. Also note that there is no known method for decreasing the severity of this attack -- adding a few random bytes to files as they are stored does not help, since those bytes can likewise be attacked with the "learn the remaining information" approach. The only effective approach to mitigating this attack is to encrypt the contents of files with a non-convergent secret before storing (negating any benefit from convergent encryption), or to simply not use convergent encryption in the first place.
NRENum.net
The NRENum.net service is an end-user ENUM service run by TERENA and the participating national research and education networking organisations (NRENs), primarily for academia. NRENum.net is considered as a complementary service and a valid alternative to the Golden ENUM tree. The domain nrenum.net is being populated in order to provide the infrastructure in DNS for storage of E.164 numbers. The NRENum.net service includes the operation of the Tier-0 root Domain Name Server(s) and the delegation of county codes to NRENum.net Registries. NRENum.net is a registered community trademark of TERENA. == Service description == E.164 Telephone Number Mapping (ENUM) is a standard protocol that is the result of work of the Internet Engineering Task Force's Telephone Number Mapping working group. ENUM translates a telephone number into a domain name. This allows users to continue to use the existing phone number formats they are familiar with, while allowing the call to be routed using DNS. This makes ENUM a quick, stable and cheap link between telecommunications systems and the Internet. RFC 3761 discusses the use of the Domain Name System for storage of E.164 numbers. More specifically, how DNS can be used for identifying available services connected to one E.164 number. The RIPE NCC provides DNS operations for e164.arpa (known as Golden ENUM tree) in accordance with the instructions from the Internet Architecture Board. The NRENum.net service is an end-user ENUM service run by TERENA and the participating NRENs primarily for academia. NRENum.net is considered as a complementary service and a valid alternative to the Golden ENUM tree. The domain nrenum.net is being populated in order to provide the infrastructure in DNS for storage of E.164 numbers. The NRENum.net service includes the operation of the Tier-0 root Domain Name Servers and the delegation of county codes to NRENum.net Registries. NRENum.net is a registered community trademark of TERENA. NRENum.net facilitates services such as Voice over IP and videoconferencing. NRENum.net tree refers to the tree structure where: Tier-0 root Domain Name Servers (technically one master and several secondary servers ensuring resilience) are run by the hosting organisations and coordinated by the NRENum.net Operations Team. Tier-1 Domain Name Servers are run by the NRENum.net (national or regional) Registries responsible for the country code(s) delegated. Tier-2 and lower DNS sub-delegations may be implemented, regulated by the national service policies. An NRENum.net Registry is an entity that is authorised by the NRENum.net Operations Team to operate the national or regional Tier-1 Domain Name Server and be responsible for the county code(s) delegated. In many countries there is a National Research and Education Networking organisation (NREN) that acts as the Registry of the country. An NRENum.net Registrar is responsible for the number/block registration in the Tier-1 DNS and a Number Validation Entity is responsible for the validation of the E.164 telephone numbers to be registered. The NREN may at the same time have the role of the NRENum.net Registry, Registrar and Validation Entity for the country code(s) delegated. A Registrant (end user) is an E.164 telephone number holder. Holders of E.164 numbers who want to be listed in the service must contact the appropriate NRENum.net Registrar. Number (block) delegation is the technical process of assigning country codes to national registries, or number blocks under country codes to end users. Number (block) registration is the technical process of configuring DNS and populating it with the appropriate ENUM records (i.e., adding NAPTR records to DNS) via registrars. The ITU-T strictly regulates the number structure of valid E.164 telephone numbers and assigns number blocks to national authorities (telecom regulators) or recently to global entities directly. The national authorities can further delegate the number ranges to local operators within the country or region. A virtual number has either a non-valid E.164 number structure (e.g., longer than 15 digits) or has a valid structure but is not assigned to any national authorities or operators. The number Validation Entity is responsible for checking the numbers to be registered to NRENum.net. == History == The idea for the NRENum.net service was conceived in 2006. NRENum.net became operational in August 2006, and was run by Bernie Höneisen, a staff member of SWITCH, and Kewin Stöckigt, a staff member of AARNet, as a private service, with technical support from SWITCH and the participants in the TERENA Task Force on Enhanced Communication Services (TF-ECS). When that task force completed its activities in 2008, TERENA agreed to take over the coordination of the NRENum.net service. By that time, nine NRENs had joined NRENum.net. The service continued to grow during the next years, and in March 2012 NRENum.net went global when RNP from Brazil joined the service as its 14th partificpant and the first outside Europe. In 2011, the participants decided to migrate the operation of the service's master Domain Name Server to NIIF and the operation of the two secondary DNSs to CARNET and SWITCH. In 2013, Internet2, AARNet and NORDUnet set up additional secondary Domain Name Servers for their regions, thereby completing the global distribution of DNS slaves and bringing the resilience of the NRENum.net infrastructure to a high level. == Governance == TERENA has established a lightweight global governance structure. The Global NRENum.net Governance Committee (GNGC) is the highest-level strategic body responsible for overall NRENum.net service definition, sustainability and long-term strategy. This includes formulating and recommending service governance principles and policies. Its members are nominated by the NRENum.net Registries in the various world regions, and are appointed by TERENA. The GNGC is composed of two members representing Europe, two representing the Asia-Pacific region, and two representing the Americas. The NRENum.net Operations Team is responsible for the day-to-day operations of the Tier-0 root DNSs and the handling of country code delegation requests. It may escalate technical or policy issues to the GNGC for discussion. TERENA is responsible for ensuring the correct and secure operations of the NRENum.net service performed by the NRENum.net Operations Team and governance by the GNGC. TERENA also supports the development of technical improvements to the NRENum.net service and promotes the deployment of NRENum.net worldwide. == Geographical deployment == Thirty-two county codes are delegated in the NRENum.net service. Below these are listed per world region. === Europe === === Asia-Pacific === === North America === +1 United States (Internet2) === Latin America === === Caribbean === === Africa === +262 Réunion, Mayotte (RENATER)
Kruskal count
The Kruskal count (also known as Kruskal's principle, Dynkin–Kruskal count, Dynkin's counting trick, Dynkin's card trick, coupling card trick or shift coupling) is a probabilistic concept originally demonstrated by the Russian mathematician Evgenii Borisovich Dynkin in the 1950s or 1960s discussing coupling effects and rediscovered as a card trick by the American mathematician Martin David Kruskal in the early 1970s as a side-product while working on another problem. It was published by Kruskal's friend Martin Gardner and magician Karl Fulves in 1975. This is related to a similar trick published by magician Alexander F. Kraus in 1957 as Sum total and later called Kraus principle. Besides uses as a card trick, the underlying phenomenon has applications in cryptography, code breaking, software tamper protection, code self-synchronization, control-flow resynchronization, design of variable-length codes and variable-length instruction sets, web navigation, object alignment, and others. == Card trick == The trick is performed with cards, but is more a magical-looking effect than a conventional magic trick. The magician has no access to the cards, which are manipulated by members of the audience. Thus sleight of hand is not possible. Rather the effect is based on the mathematical fact that the output of a Markov chain, under certain conditions, is typically independent of the input. A simplified version using the hands of a clock performed by David Copperfield is as follows. A volunteer picks a number from one to twelve and does not reveal it to the magician. The volunteer is instructed to start from 12 on the clock and move clockwise by a number of spaces equal to the number of letters that the chosen number has when spelled out. This is then repeated, moving by the number of letters in the new number. The output after three or more moves does not depend on the initially chosen number and therefore the magician can predict it.
AARON
AARON is the collective name for a series of computer programs written by artist Harold Cohen that create original artistic images autonomously, which set it apart from previous programs. Proceeding from Cohen's initial question "What are the minimum conditions under which a set of marks functions as an image?", AARON was in development between 1972 and the 2010s. As the software is not open source, its development effectively ended with Cohen's death in 2016. The name "AARON" does not seem to be an acronym; rather, it was a name chosen to start with the letter "A" so that the names of successive programs could follow it alphabetically. However, Cohen did not create any other major programs. Initial versions of AARON created abstract drawings that grew more complex through the 1970s. More representational imagery was added in the 1980s; first rocks, then plants, then people. In the 1990s more representational figures set in interior scenes were added, along with color. AARON returned to more abstract imagery, this time in color, in the early 2000s. Cohen used machines that allowed AARON to produce physical artwork. The first machines drew in black and white using a succession of custom-built "turtle" and flatbed plotter devices. Cohen would sometimes color these images by hand in fabric dye (Procion), or scale them up to make larger paintings and murals. In the 1990s Cohen built a series of digital painting machines to output AARON's images in ink and fabric dye. His later work used a large-scale inkjet printer on canvas. Development of AARON began in the C programming language then switched to Lisp in the early 1990s. Cohen credits Lisp with helping him solve the challenges he faced in adding color capabilities to AARON. An article about Cohen appeared in Computer Answers that describes AARON and shows two line drawings that were exhibited at the Tate gallery. The article goes on to describe the workings of AARON, then running on a DEC VAX 750 minicomputer. Raymond Kurzweil's company has produced a downloadable screensaver of AARON for Microsoft Windows PCs. This version of AARON can also produce printable images. AARON's source code is not publicly available, but Cohen has described AARON's operations in various essays and it is discussed in abstract in Pamela McCorduck's book. AARON cannot learn new styles or imagery on its own; each new capability must be hand-coded by Cohen. It is capable of producing a practically infinite supply of distinct images in its own style. Examples of these images have been exhibited in galleries worldwide. AARON's artwork has been used as an artistic equivalent of the Turing test. It does seem however that AARON's output follows a noticeable formula (figures standing next to a potted plant, framed within a colored square is a common theme). Cohen is very careful not to claim that AARON is creative. But he does ask "If what AARON is making is not art, what is it exactly, and in what ways, other than its origin, does it differ from the 'real thing?' If it is not thinking, what exactly is it doing?" — The further exploits of AARON, Painter. The Whitney Museum featured AARON in 2024, showcasing the evolution of AARON as the earliest artificial intelligence (AI) program for artmaking.
Data recovery
In computing, data recovery is a process of retrieving deleted, inaccessible, lost, corrupted, damaged, or overwritten data from secondary storage, removable media or files, when the data stored in them cannot be accessed in a usual way. The data is most often salvaged from storage media such as internal or external hard disk drives (HDDs), solid-state drives (SSDs), USB flash drives, magnetic tapes, CDs, DVDs, RAID subsystems, and other electronic devices. Recovery may be required due to physical damage to the storage devices or logical damage to the file system that prevents it from being mounted by the host operating system (OS). Logical failures occur when the hard drive devices are functional but the user or automated-OS cannot retrieve or access data stored on them. Logical failures can occur due to corruption of the engineering chip, lost partitions, firmware failure, or failures during formatting/re-installation. Data recovery can be a very simple or technical challenge. This is why there are specific software companies specialized in this field that help to get back data on your system. == About == The most common data recovery scenarios involve an operating system failure, malfunction of a storage device, logical failure of storage devices, accidental damage or deletion, etc. (typically, on a single-drive, single-partition, single-OS system), in which case the ultimate goal is simply to copy all important files from the damaged media to another new drive. This can be accomplished using a Live CD, or DVD by booting directly from a ROM or a USB drive instead of the corrupted drive in question. Many Live CDs or DVDs provide a means to mount the system drive and backup drives or removable media, and to move the files from the system drive to the backup media with a file manager or optical disc authoring software. Such cases can often be mitigated by disk partitioning and consistently storing valuable data files (or copies of them) on a different partition from the replaceable OS system files. Another scenario involves a drive-level failure, such as a compromised file system or drive partition, or a hard disk drive failure. In any of these cases, the data is not easily read from the media devices. Depending on the situation, solutions involve repairing the logical file system, partition table, or master boot record, or updating the firmware or drive recovery techniques ranging from software-based recovery of corrupted data, to hardware- and software-based recovery of damaged service areas (also known as the hard disk drive's "firmware"), to hardware replacement on a physically damaged drive which allows for the extraction of data to a new drive. If a drive recovery is necessary, the drive itself has typically failed permanently, and the focus is rather on a one-time recovery, salvaging whatever data can be read. In a third scenario, files have been accidentally "deleted" from a storage medium by the users. Typically, the contents of deleted files are not removed immediately from the physical drive; instead, references to them in the directory structure are removed, and thereafter space the deleted data occupy is made available for later data overwriting. In the mind of end users, deleted files cannot be discoverable through a standard file manager, but the deleted data still technically exists on the physical drive. In the meantime, the original file contents remain, often several disconnected fragments, and may be recoverable if not overwritten by other data files. The term "data recovery" is also used in the context of forensic applications or espionage, where data which have been encrypted, hidden, or deleted, rather than damaged, are recovered. Sometimes data present in the computer gets encrypted or hidden due to reasons like virus attacks which can only be recovered by some computer forensic experts. == Physical damage == A wide variety of failures can cause physical damage to storage media, which may result from human errors and natural disasters. CD-ROMs can have their metallic substrate or dye layer scratched off; hard disks can suffer from a multitude of mechanical failures, such as head crashes, PCB failure, and failed motors; tapes can simply break. Physical damage to a hard drive, even in cases where a head crash has occurred, does not necessarily mean permanent data loss. However, in extreme cases, such as prolonged exposure to moisture and corrosion —like the lost Bitcoin hard drive of James Howells, buried in the Newport landfill for over a decade — recovery is usually impossible. In rare cases, forensic techniques such as magnetic force microscopy (MFM) have been explored to detect residual magnetic traces when data holds exceptional value. Other techniques employed by many professional data recovery companies can typically salvage most, if not all, of the data that had been lost when the failure occurred. Of course, there are exceptions to this, such as cases where severe damage to the hard drive platters may have occurred. However, if the hard drive can be repaired and a full image or clone created, then the logical file structure can be rebuilt in most instances. Most physical damage cannot be repaired by end users. For example, opening a hard disk drive in a normal environment can allow airborne dust to settle on the platter and become caught between the platter and the read/write head. During normal operation, read/write heads float 3 to 6 nanometers above the platter surface, and the average dust particles found in a normal environment are typically around 30,000 nanometers in diameter. When these dust particles get caught between the read/write heads and the platter, they can cause new head crashes that further damage the platter and thus compromise the recovery process. Furthermore, end users generally do not have the hardware or technical expertise required to make these repairs. Consequently, data recovery companies are often employed to salvage important data with the more reputable ones using class 100 dust- and static-free cleanrooms. === Recovery techniques === Recovering data from physically damaged hardware can involve multiple techniques. Some damage can be repaired by replacing parts in the hard disk. This alone may make the disk usable, but there may still be logical damage. A specialized disk-imaging procedure is used to recover every readable bit from the surface. Once this image is acquired and saved on a reliable medium, the image can be safely analyzed for logical damage and will possibly allow much of the original file system to be reconstructed. ==== Hardware repair ==== A common misconception is that a damaged printed circuit board (PCB) may be simply replaced during recovery procedures by an identical PCB from a healthy drive. While this may work in rare circumstances on hard disk drives manufactured before 2003, it will not work on newer drives. Electronics boards of modern drives usually contain drive-specific adaptation data (generally a map of bad sectors and tuning parameters) and other information required to properly access data on the drive. Replacement boards often need this information to effectively recover all of the data. The replacement board may need to be reprogrammed. Some manufacturers (Seagate, for example) store this information on a serial EEPROM chip, which can be removed and transferred to the replacement board. Each hard disk drive has what is called a system area or service area; this portion of the drive, which is not directly accessible to the end user, usually contains drive's firmware and adaptive data that helps the drive operate within normal parameters. One function of the system area is to log defective sectors within the drive; essentially telling the drive where it can and cannot write data. The sector lists are also stored on various chips attached to the PCB, and they are unique to each hard disk drive. If the data on the PCB do not match what is stored on the platter, then the drive will not calibrate properly. In most cases the drive heads will click because they are unable to find the data matching what is stored on the PCB. == Logical damage == The term "logical damage" refers to situations in which the error is not a problem in the hardware and requires software-level solutions. === Corrupt partitions and file systems, media errors === In some cases, data on a hard disk drive can be unreadable due to damage to the partition table or file system, or to (intermittent) media errors. In the majority of these cases, at least a portion of the original data can be recovered by repairing the damaged partition table or file system using specialized data recovery software such as TestDisk; software like ddrescue can image media despite intermittent errors, and image raw data when there is partition table or file system damage. This type of data recovery can be performed by people without expertise in drive hardware as it requires no special physica