Open-source software security is the measure of assurance or guarantee in the freedom from danger and risk inherent to an open-source software system. == Implementation debate == === Benefits === Proprietary software forces the user to accept the level of security that the software vendor is willing to deliver and to accept the rate that patches and updates are released. It is assumed that any compiler that is used creates code that can be trusted, but it has been demonstrated by Ken Thompson that a compiler can be subverted using a compiler backdoor to create faulty executables that are unwittingly produced by a well-intentioned developer. With access to the source code for the compiler, the developer has at least the ability to discover if there is any mal-intention. Kerckhoffs' principle is based on the idea that an enemy can steal a secure military system and not be able to compromise the information. His ideas were the basis for many modern security practices, and followed that security through obscurity is a bad practice. === Drawbacks === Simply making source code available does not guarantee review. An example of this occurring is when Marcus Ranum, an expert on security system design and implementation, released his first public firewall toolkit. At one time, there were over 2,000 sites using his toolkit, but only 10 people gave him any feedback or patches. Having a large amount of eyes reviewing code can "lull a user into a false sense of security". Having many users look at source code does not guarantee that security flaws will be found and fixed. == Metrics and models == There are a variety of models and metrics to measure the security of a system. These are a few methods that can be used to measure the security of software systems. === Number of days between vulnerabilities === It is argued that a system is most vulnerable after a potential vulnerability is discovered, but before a patch is created. By measuring the number of days between the vulnerability and when the vulnerability is fixed, a basis can be determined on the security of the system. There are a few caveats to such an approach: not every vulnerability is equally bad, and fixing a lot of bugs quickly might not be better than only finding a few and taking a little bit longer to fix them, taking into account the operating system, or the effectiveness of the fix. === Poisson process === The Poisson process can be used to measure the rates at which different people find security flaws between open and closed source software. The process can be broken down by the number of volunteers Nv and paid reviewers Np. The rates at which volunteers find a flaw is measured by λv and the rate that paid reviewers find a flaw is measured by λp. The expected time that a volunteer group is expected to find a flaw is 1/(Nv λv) and the expected time that a paid group is expected to find a flaw is 1/(Np λp). === Morningstar model === By comparing a large variety of open source and closed source projects a star system could be used to analyze the security of the project similar to how Morningstar, Inc. rates mutual funds. With a large enough data set, statistics could be used to measure the overall effectiveness of one group over the other. An example of such as system is as follows: 1 Star: Many security vulnerabilities. 2 Stars: Reliability issues. 3 Stars: Follows best security practices. 4 Stars: Documented secure development process. 5 Stars: Passed independent security review. === Coverity scan === Coverity in collaboration with Stanford University has established a new baseline for open-source quality and security. The development is being completed through a contract with the Department of Homeland Security. They are utilizing innovations in automated defect detection to identify critical types of bugs found in software. The level of quality and security is measured in rungs. Rungs do not have a definitive meaning, and can change as Coverity releases new tools. Rungs are based on the progress of fixing issues found by the Coverity Analysis results and the degree of collaboration with Coverity. They start with Rung 0 and currently go up to Rung 2. Rung 0 The project has been analyzed by Coverity's Scan infrastructure, but no representatives from the open-source software have come forward for the results. Rung 1 At rung 1, there is collaboration between Coverity and the development team. The software is analyzed with a subset of the scanning features to prevent the development team from being overwhelmed. Rung 2 There are 11 projects that have been analyzed and upgraded to the status of Rung 2 by reaching zero defects in the first year of the scan. These projects include: AMANDA, ntp, OpenPAM, OpenVPN, Overdose, Perl, PHP, Postfix, Python, Samba, and Tcl.
Tesla Dojo
Tesla Dojo is a series of supercomputers designed and built by Tesla for computer vision video processing and recognition. It was used for training Tesla's machine learning models to improve its Full Self-Driving (FSD) advanced driver-assistance system. It went into production in July 2023. Dojo's goal was to efficiently process millions of terabytes of video data captured from real-life driving situations from Tesla's 4+ million cars. This goal led to a considerably different architecture than conventional supercomputer designs. In August 2025, Bloomberg News reported that the Dojo project had been disbanded, though it was restarted in January 2026. == History == Tesla operates several massively parallel computing clusters for developing its Autopilot advanced driver assistance system. Its primary unnamed cluster using 5,760 Nvidia A100 graphics processing units (GPUs) was touted by Andrej Karpathy in 2021 at the fourth International Joint Conference on Computer Vision and Pattern Recognition (CCVPR 2021) to be "roughly the number five supercomputer in the world" at approximately 81.6 petaflops, based on scaling the performance of the Nvidia Selene supercomputer, which uses similar components. However, the performance of the primary Tesla GPU cluster has been disputed, as it was not clear if this was measured using single-precision or double-precision floating point numbers (FP32 or FP64). Tesla also operates a second 4,032 GPU cluster for training and a third 1,752 GPU cluster for automatic labeling of objects. The primary unnamed Tesla GPU cluster has been used for processing one million video clips, each ten seconds long, taken from Tesla Autopilot cameras operating in Tesla cars in the real world, running at 36 frames per second. Collectively, these video clips contained six billion object labels, with depth and velocity data; the total size of the data set was 1.5 petabytes. This data set was used for training a neural network intended to help Autopilot computers in Tesla cars understand roads. By August 2022, Tesla had upgraded the primary GPU cluster to 7,360 GPUs. Dojo was first mentioned by Elon Musk in April 2019 during Tesla's "Autonomy Investor Day". In August 2020, Musk stated it was "about a year away" due to power and thermal issues. Dojo was officially announced at Tesla's Artificial Intelligence (AI) Day on August 19, 2021. Tesla revealed details of the D1 chip and its plans for "Project Dojo", a datacenter that would house 3,000 D1 chips; the first "Training Tile" had been completed and delivered the week before. In October 2021, Tesla released a "Dojo Technology" whitepaper describing the Configurable Float8 (CFloat8) and Configurable Float16 (CFloat16) floating point formats and arithmetic operations as an extension of Institute of Electrical and Electronics Engineers (IEEE) standard 754. At the follow-up AI Day in September 2022, Tesla announced it had built several System Trays and one Cabinet. During a test, the company stated that Project Dojo drew 2.3 megawatts (MW) of power before tripping a local San Jose, California power substation. At the time, Tesla was assembling one Training Tile per day. In August 2023, Tesla powered on Dojo for production use as well as a new training cluster configured with 10,000 Nvidia H100 GPUs. In January 2024, Musk described Dojo as "a long shot worth taking because the payoff is potentially very high. But it's not something that is a high probability." In June 2024, Musk explained that ongoing construction work at Gigafactory Texas is for a computing cluster claiming that it is planned to comprise an even mix of "Tesla AI" and Nvidia/other hardware with a total thermal design power of at first 130 MW and eventually exceeding 500 MW. In August 2025, Bloomberg News reported that the Dojo project was disbanded, though Musk announced it would be restarted in January 2026 with a new chip iteration. == Technical architecture == The fundamental unit of the Dojo supercomputer is the D1 chip, designed by a team at Tesla led by ex-AMD CPU designer Ganesh Venkataramanan, including Emil Talpes, Debjit Das Sarma, Douglas Williams, Bill Chang, and Rajiv Kurian. The D1 chip is manufactured by the Taiwan Semiconductor Manufacturing Company (TSMC) using 7 nanometer (nm) semiconductor nodes, has 50 billion transistors and a large die size of 645 mm2 (1.0 square inch). Updating at Artificial Intelligence (AI) Day in 2022, Tesla announced that Dojo would scale by deploying multiple ExaPODs, in which there would be: 10 Cabinets per ExaPOD (1,062,000 cores, 3,000 D1 chips) 2 System Trays per Cabinet (106,200 cores, 300 D1 chips) 6 Training Tiles per System Tray (53,100 cores, along with host interface hardware) 25 D1 chips per Training Tile (8,850 cores) 354 computing cores per D1 chip According to Venkataramanan, Tesla's senior director of Autopilot hardware, Dojo will have more than an exaflop (a million teraflops) of computing power. For comparison, according to Nvidia, in August 2021, the (pre-Dojo) Tesla AI-training center used 720 nodes, each with eight Nvidia A100 Tensor Core GPUs for 5,760 GPUs in total, providing up to 1.8 exaflops of performance. === D1 chip === Each node (computing core) of the D1 processing chip is a general purpose 64-bit CPU with a superscalar core. It supports internal instruction-level parallelism, and includes simultaneous multithreading (SMT). It doesn't support virtual memory and uses limited memory protection mechanisms. Dojo software/applications manage chip resources. The D1 instruction set supports both 64-bit scalar and 64-byte single instruction, multiple data (SIMD) vector instructions. The integer unit mixes reduced instruction set computer (RISC-V) and custom instructions, supporting 8, 16, 32, or 64 bit integers. The custom vector math unit is optimized for machine learning kernels and supports multiple data formats, with a mix of precisions and numerical ranges, many of which are compiler composable. Up to 16 vector formats can be used simultaneously. ==== Node ==== Each D1 node uses a 32-byte fetch window holding up to eight instructions. These instructions are fed to an eight-wide decoder which supports two threads per cycle, followed by a four-wide, four-way SMT scalar scheduler that has two integer units, two address units, and one register file per thread. Vector instructions are passed further down the pipeline to a dedicated vector scheduler with two-way SMT, which feeds either a 64-byte SIMD unit or four 8×8×4 matrix multiplication units. The network on-chip (NOC) router links cores into a two-dimensional mesh network. It can send one packet in and one packet out in all four directions to/from each neighbor node, along with one 64-byte read and one 64-byte write to local SRAM per clock cycle. Hardware native operations transfer data, semaphores and barrier constraints across memories and CPUs. System-wide double data rate 4 (DDR4) synchronous dynamic random-access memory (SDRAM) memory works like bulk storage. ==== Memory ==== Each core has a 1.25 megabytes (MB) of SRAM main memory. Load and store speeds reach 400 gigabytes (GB) per second and 270 GB/sec, respectively. The chip has explicit core-to-core data transfer instructions. Each SRAM has a unique list parser that feeds a pair of decoders and a gather engine that feeds the vector register file, which together can directly transfer information across nodes. ==== Die ==== Twelve nodes (cores) are grouped into a local block. Nodes are arranged in an 18×20 array on a single die, of which 354 cores are available for applications. The die runs at 2 gigahertz (GHz) and totals 440 MB of SRAM (360 cores × 1.25 MB/core). It reaches 376 teraflops using 16-bit brain floating point (BF16) numbers or using configurable 8-bit floating point (CFloat8) numbers, which is a Tesla proposal, and 22 teraflops at FP32. Each die comprises 576 bi-directional serializer/deserializer (SerDes) channels along the perimeter to link to other dies, and moves 8 TB/sec across all four die edges. Each D1 chip has a thermal design power of approximately 400 watts. === Training Tile === The water-cooled Training Tile packages 25 D1 chips into a 5×5 array. Each tile supports 36 TB/sec of aggregate bandwidth via 40 input/output (I/O) chips - half the bandwidth of the chip mesh network. Each tile supports 10 TB/sec of on-tile bandwidth. Each tile has 11 GB of SRAM memory (25 D1 chips × 360 cores/D1 × 1.25 MB/core). Each tile achieves 9 petaflops at BF16/CFloat8 precision (25 D1 chips × 376 TFLOP/D1). Each tile consumes 15 kilowatts; 288 amperes at 52 volts. === System Tray === Six tiles are aggregated into a System Tray, which is integrated with a host interface. Each host interface includes 512 x86 cores, providing a Linux-based user environment. Previously, the Dojo System Tray was known as the Training Matrix, which includes six Training Tiles, 20 Dojo Interface Processor cards across four host servers, and Ethernet-l
Harold Borko
Harold Borko (1922-2012) was an American psychologist and researcher working primarily in the field of information science. == Biography == Borko was born in 1922 in New York City, New York. After serving in the US Army from 1942 to 1946 he obtained a BA in Psychology from the University of California, Los Angeles in 1948 and both his MA and PhD from the University of Southern California in Psychology in 1952. He returned to the army as a psychologist until 1956 after which he began a career working in and teaching information science. He died in California in 2012. == Information Science Career == After leaving the military Borko began working at the RAND Corporation as a Systems Training Specialist in 1956 and moved to the Systems Development Corporation a year later working in the Language Processing and Retrieval department. Alongside this work he taught Psychology at USC from 1957-65 and then moved into teaching Library Science at UCLA from 1965. In 1967 Borko left his role at the Systems Development Corporation and continued as a full-time professor at UCLA until his retirement in 1993.. From 1961 to 1995 Borko authored and co-authored over 100 articles on new developments in the field as well as the historiography of information science. He served as an editor of the Journal of Educational Data Processing from 1963-1975 and as President of the American Society for Information Science in 1966 == Partial list of works == Borko, H. (1962, May). The construction of an empirically based mathematically derived classification system. In Proceedings of the May 1-3, 1962, spring joint computer conference (pp. 279-289). Borko, H., & Bernick, M. (1963). Automatic document classification. Journal of the ACM (JACM), 10(2), 151-162. Borko, H. (1964). The Storage and Retrieval of Educational Information. Journal of Teacher Education, 15(4), 449-452. Borko, H. (1964). Measuring the reliability of subject classification by men and machines. American Documentation, 15(4), 268-273. Borko, H. (1965). The conceptual foundations of information systems. Borko, H. (1968), Information science: What is it?†. Amer. Doc., 19: 3-5. https://doi.org/10.1002/asi.5090190103 Borko, H. (1970). Experiments in book indexing by computer. Information storage and retrieval, 6(1), 5-16. Borko, H. (1985). An introduction to computer-based library systems (Lucy A. Tedd). Education for Information, 3(1), 61.
Penril
Penril DataComm Networks, Inc. was a computer telecommunications hardware company that made some acquisitions and was eventually split into two parts: one was acquired by Bay Networks and the other was a newly formed company named Access Beyond. The focus of both company's products was end-to-end data transfer. By the mid-1990s, with the popularization of the internet, this was no longer of wide interest. == History == Penril, whose earnings reports and other financials were followed by The New York Times in the 1990s, made several acquisitions but also grew internally. Following its Datability acquisition it renamed itself Penril Datability Networks. By the time the 1968-founded Penril was acquired by Bay their name was Penril DataComm Networks. The company, which as of 1985 "had made 14 acquisitions in 12 years," also had done extensive work regarding quality control, and leveraged their product line by what The Washington Post called clever packaging: "software, cables, instructions and telephone support" sold to those less technically skilled as "Network in a Box." == Datability == Datability Software Systems Inc. was the initial name of what by 1991 became 'Datability, Inc.', "a manufacturer of hardware that links computer networks." The 1977-founded firm began as a software consulting company, especially in the area of databases. To speed up project development they built a program generator, which they marketed as Control 10/20 (targeted at users of Digital Equipment Corporation's DECsystem-10 and DECSYSTEM-20). After trying their hand at time-sharing they built hardware to enhance bridging these computers to DEC's VAX product line. In particular they focused on Digital's LAT protocol, selling "boxes" that reimplemented the protocol, at a lower price than DEC's. They later expanded into other areas of telecommunications hardware The firm relocated to a larger manufacturing plant in 1991 and was acquired by Penril in 1993. == Access Beyond == Access Beyond was initially housed by Penril, from which it was spun off. A securities analyst noted that Access began operations with no debt. They subsequently merged with Hayes Corporation. Some of the funds brought to the merger came from a sale by Penril of two of its divisions, each bringing about $4 million. == Ron Howard == Ron Howard, founder of Datability, became part of Penril when the latter acquired the former, and was CEO of Access Beyond when it was spun off by Penril. Access merged with Hayes Microcomputer Products and was renamed Hayes Corp, at which time Howard became executive VP of business development and corporate vice chairman of Hayes. == People == In the matter of hiring immigrants, in an industry where recent arrivals came from a culture of six day work weeks, and subcontracting was then common, these assembly line workers at Penril comprised about 25%, compared to double in other firms. Placement was overseen by government agencies. == Controversy == Penril had a joint development agreement, beginning in 1990, with a Standard Microsystems Corporation (SMSC) subsidiary. A dispute arose, and the matter was brought to court. Penril was awarded $3.5 million in 1996.
NoSQL
NoSQL (originally meaning "not only SQL" or "non-relational") refers to a type of database design that stores and retrieves data differently from the traditional table-based structure of relational databases. Unlike relational databases, which organize data into rows and columns like a spreadsheet, NoSQL databases use a single data structure—such as key–value pairs, wide columns, graphs, or documents—to hold information. Since this non-relational design does not require a fixed schema, it scales easily to manage large, often unstructured datasets. NoSQL systems are sometimes called "Not only SQL" because they can support SQL-like query languages or work alongside SQL databases in polyglot-persistent setups, where multiple database types are combined. Non-relational databases date back to the late 1960s, but the term "NoSQL" emerged in the early 2000s, spurred by the needs of Web 2.0 companies like social media platforms. NoSQL databases are popular in big data and real-time web applications due to their simple design, ability to scale across clusters of machines (called horizontal scaling), and precise control over data availability. These structures can speed up certain tasks and are often considered more adaptable than fixed database tables. However, many NoSQL systems prioritize speed and availability over strict consistency (per the CAP theorem), using eventual consistency—where updates reach all nodes eventually, typically within milliseconds, but may cause brief delays in accessing the latest data, known as stale reads. While most lack full ACID transaction support, some, like MongoDB, include it as a key feature. == Barriers to adoption == Barriers to wider NoSQL adoption include their use of low-level query languages instead of SQL, inability to perform ad hoc joins across tables, lack of standardized interfaces, and significant investments already made in relational databases. Some NoSQL systems risk losing data through lost writes or other forms, though features like write-ahead logging—a method to record changes before they’re applied—can help prevent this. For distributed transaction processing across multiple databases, keeping data consistent is a challenge for both NoSQL and relational systems, as relational databases cannot enforce rules linking separate databases, and few systems support both ACID transactions and X/Open XA standards for managing distributed updates. Limitations within the interface environment are overcome using semantic virtualization protocols, such that NoSQL services are accessible to most operating systems. == History == The term NoSQL was used by Carlo Strozzi in 1998 to name his lightweight Strozzi NoSQL open-source relational database that did not expose the standard Structured Query Language (SQL) interface, but was still relational. His NoSQL RDBMS is distinct from the around-2009 general concept of NoSQL databases. Strozzi suggests that, because the current NoSQL movement "departs from the relational model altogether, it should therefore have been called more appropriately 'NoREL'", referring to "not relational". Johan Oskarsson, then a developer at Last.fm, reintroduced the term NoSQL in early 2009 when he organized an event to discuss "open-source distributed, non-relational databases". The name attempted to label the emergence of an increasing number of non-relational, distributed data stores, including open source clones of Google's Bigtable/MapReduce and Amazon's DynamoDB. == Types and examples == There are various ways to classify NoSQL databases, with different categories and subcategories, some of which overlap. What follows is a non-exhaustive classification by data model, with examples: === Key–value store === Key–value (KV) stores use the associative array (also called a map or dictionary) as their fundamental data model. In this model, data is represented as a collection of key–value pairs, such that each possible key appears at most once in the collection. The key–value model is one of the simplest non-trivial data models, and richer data models are often implemented as an extension of it. The key–value model can be extended to a discretely ordered model that maintains keys in lexicographic order. This extension is computationally powerful, in that it can efficiently retrieve selective key ranges. Key–value stores can use consistency models ranging from eventual consistency to serializability. Some databases support ordering of keys. There are various hardware implementations, and some users store data in memory (RAM), while others on solid-state drives (SSD) or rotating disks (aka hard disk drive (HDD)). === Document store === The central concept of a document store is that of a "document". While the details of this definition differ among document-oriented databases, they all assume that documents encapsulate and encode data (or information) in some standard formats or encodings. Encodings in use include XML, YAML, and JSON and binary forms like BSON. Documents are addressed in the database via a unique key that represents that document. Another defining characteristic of a document-oriented database is an API or query language to retrieve documents based on their contents. Different implementations offer different ways of organizing and/or grouping documents: Collections Tags Non-visible metadata Directory hierarchies Compared to relational databases, collections could be considered analogous to tables and documents analogous to records. But they are different – every record in a table has the same sequence of fields, while documents in a collection may have fields that are completely different. === Graph === Graph databases are designed for data whose relations are well represented as a graph consisting of elements connected by a finite number of relations. Examples of data include social relations, public transport links, road maps, network topologies, etc. Graph databases and their query language == Performance == The performance of NoSQL databases is usually evaluated using the metric of throughput, which is measured as operations per second. Performance evaluation must pay attention to the right benchmarks such as production configurations, parameters of the databases, anticipated data volume, and concurrent user workloads. Ben Scofield rated different categories of NoSQL databases as follows: Performance and scalability comparisons are most commonly done using the YCSB benchmark. == Handling relational data == Since most NoSQL databases lack ability for joins in queries, the database schema generally needs to be designed differently. There are three main techniques for handling relational data in a NoSQL database. (See table join and ACID support for NoSQL databases that support joins.) === Multiple queries === Instead of retrieving all the data with one query, it is common to do several queries to get the desired data. NoSQL queries are often faster than traditional SQL queries, so the cost of additional queries may be acceptable. If an excessive number of queries would be necessary, one of the other two approaches is more appropriate. === Caching, replication and non-normalized data === Instead of only storing foreign keys, it is common to store actual foreign values along with the model's data. For example, each blog comment might include the username in addition to a user id, thus providing easy access to the username without requiring another lookup. When a username changes, however, this will now need to be changed in many places in the database. Thus this approach works better when reads are much more common than writes. === Nesting data === With document databases like MongoDB it is common to put more data in a smaller number of collections. For example, in a blogging application, one might choose to store comments within the blog post document, so that with a single retrieval one gets all the comments. Thus in this approach a single document contains all the data needed for a specific task. == ACID and join support == A database is marked as supporting ACID properties (atomicity, consistency, isolation, durability) or join operations if the documentation for the database makes that claim. However, this doesn't necessarily mean that the capability is fully supported in a manner similar to most SQL databases. == Query optimization and indexing in NoSQL databases == Different NoSQL databases, such as DynamoDB, MongoDB, Cassandra, Couchbase, HBase, and Redis, exhibit varying behaviors when querying non-indexed fields. Many perform full-table or collection scans for such queries, applying filtering operations after retrieving data. However, modern NoSQL databases often incorporate advanced features to optimize query performance. For example, MongoDB supports compound indexes and query-optimization strategies, Cassandra offers secondary indexes and materialized views, and Redis employs custom indexing mechanisms tailored to specific use cases. Systems like El
Catie Cuan
Catie Cuan is an artist, entrepeuneur, and innovator in the field of robotic art and human-robot interaction, where she specializes in choreorobotics, an emerging field at the intersection of choreographic dance and robotics. Catie Cuan is currently one of the academic researchers pioneering the field of choreorobotics and currently holds a post-doctoral fellowship at Stanford University. == Career == Catie Cuan earned a bachelor's degree from the University of California, Berkeley. She graduated with a Ph.D. from the Department of Mechanical Engineering at Stanford University, focusing in robotics. Her most cited publication is about how to improve robotic expressive systems using tools from dance theory, such as the Laban/Bartenieff Movement Analysis. In her most recent research projects, she explores a predictive model of imitation learning for robots moving around humans, a project that advances the field of social robotics. Cuan credits her work in robotics to the experience with her father when he had a stroke and was surrounded by many medical machines, which made her think about how people might feel empowered and hopeful rather than afraid. As a ballet dancer and choreographer, she has performed with the Metropolitan Opera Ballet and the Lyric Opera of Chicago. In 2020, she was the dancer and choreographer of the show Output, which was part of a collaboration with ThoughtWorks Arts and the Pratt Institute. In the production, she danced with an ABB IRB 6700 industrial robot. In 2022, she was named as an IF/THEN ambassador for the American Association for the Advancement of Science. The same year, she was appointed Futurist-in-Residence at the Smithsonian Arts and Industries Building, where she performed at the closing ceremonies of the FUTURES exhibit on July 6, 2022. Cuan has also contributed to product designs, working with IDEO and Dutch interior design firm moooi on their Piro project, which launched a dancing scent diffuser robot during Milan Design Week in June 2022. She is a TED speaker with talks about how to teach robots to dance, and what is coming up for dancing robots in the AI era.
Holographic algorithm
In computer science, a holographic algorithm is an algorithm that uses a holographic reduction. A holographic reduction is a constant-time reduction that maps solution fragments many-to-many such that the sum of the solution fragments remains unchanged. These concepts were introduced by Leslie Valiant, who called them holographic because "their effect can be viewed as that of producing interference patterns among the solution fragments". The algorithms are unrelated to laser holography, except metaphorically. Their power comes from the mutual cancellation of many contributions to a sum, analogous to the interference patterns in a hologram. Holographic algorithms have been used to find polynomial-time solutions to problems without such previously known solutions for special cases of satisfiability, vertex cover, and other graph problems. They have received notable coverage due to speculation that they are relevant to the P versus NP problem and their impact on computational complexity theory. Although some of the general problems are #P-hard problems, the special cases solved are not themselves #P-hard, and thus do not prove FP = #P. Holographic algorithms have some similarities with quantum computation, but are completely classical. == Holant problems == Holographic algorithms exist in the context of Holant problems, which generalize counting constraint satisfaction problems (#CSP). A #CSP instance is a hypergraph G=(V,E) called the constraint graph. Each hyperedge represents a variable and each vertex v {\displaystyle v} is assigned a constraint f v . {\displaystyle f_{v}.} A vertex is connected to an hyperedge if the constraint on the vertex involves the variable on the hyperedge. The counting problem is to compute ∑ σ : E → { 0 , 1 } ∏ v ∈ V f v ( σ | E ( v ) ) , ( 1 ) {\displaystyle \sum _{\sigma :E\to \{0,1\}}\prod _{v\in V}f_{v}(\sigma |_{E(v)}),~~~~~~~~~~(1)} which is a sum over all variable assignments, the product of every constraint, where the inputs to the constraint f v {\displaystyle f_{v}} are the variables on the incident hyperedges of v {\displaystyle v} . A Holant problem is like a #CSP except the input must be a graph, not a hypergraph. Restricting the class of input graphs in this way is indeed a generalization. Given a #CSP instance, replace each hyperedge e of size s with a vertex v of degree s with edges incident to the vertices contained in e. The constraint on v is the equality function of arity s. This identifies all of the variables on the edges incident to v, which is the same effect as the single variable on the hyperedge e. In the context of Holant problems, the expression in (1) is called the Holant after a related exponential sum introduced by Valiant. == Holographic reduction == A standard technique in complexity theory is a many-one reduction, where an instance of one problem is reduced to an instance of another (hopefully simpler) problem. However, holographic reductions between two computational problems preserve the sum of solutions without necessarily preserving correspondences between solutions. For instance, the total number of solutions in both sets can be preserved, even though individual problems do not have matching solutions. The sum can also be weighted, rather than simply counting the number of solutions, using linear basis vectors. === General example === It is convenient to consider holographic reductions on bipartite graphs. A general graph can always be transformed it into a bipartite graph while preserving the Holant value. This is done by replacing each edge in the graph by a path of length 2, which is also known as the 2-stretch of the graph. To keep the same Holant value, each new vertex is assigned the binary equality constraint. Consider a bipartite graph G=(U,V,E) where the constraint assigned to every vertex u ∈ U {\displaystyle u\in U} is f u {\displaystyle f_{u}} and the constraint assigned to every vertex v ∈ V {\displaystyle v\in V} is f v {\displaystyle f_{v}} . Denote this counting problem by Holant ( G , f u , f v ) . {\displaystyle {\text{Holant}}(G,f_{u},f_{v}).} If the vertices in U are viewed as one large vertex of degree |E|, then the constraint of this vertex is the tensor product of f u {\displaystyle f_{u}} with itself |U| times, which is denoted by f u ⊗ | U | . {\displaystyle f_{u}^{\otimes |U|}.} Likewise, if the vertices in V are viewed as one large vertex of degree |E|, then the constraint of this vertex is f v ⊗ | V | . {\displaystyle f_{v}^{\otimes |V|}.} Let the constraint f u {\displaystyle f_{u}} be represented by its weighted truth table as a row vector and the constraint f v {\displaystyle f_{v}} be represented by its weighted truth table as a column vector. Then the Holant of this constraint graph is simply f u ⊗ | U | f v ⊗ | V | . {\displaystyle f_{u}^{\otimes |U|}f_{v}^{\otimes |V|}.} Now for any complex 2-by-2 invertible matrix T (the columns of which are the linear basis vectors mentioned above), there is a holographic reduction between Holant ( G , f u , f v ) {\displaystyle {\text{Holant}}(G,f_{u},f_{v})} and Holant ( G , f u T ⊗ ( deg u ) , ( T − 1 ) ⊗ ( deg v ) f v ) . {\displaystyle {\text{Holant}}(G,f_{u}T^{\otimes (\deg u)},(T^{-1})^{\otimes (\deg v)}f_{v}).} To see this, insert the identity matrix T ⊗ | E | ( T − 1 ) ⊗ | E | {\displaystyle T^{\otimes |E|}(T^{-1})^{\otimes |E|}} in between f u ⊗ | U | f v ⊗ | V | {\displaystyle f_{u}^{\otimes |U|}f_{v}^{\otimes |V|}} to get f u ⊗ | U | f v ⊗ | V | {\displaystyle f_{u}^{\otimes |U|}f_{v}^{\otimes |V|}} = f u ⊗ | U | T ⊗ | E | ( T − 1 ) ⊗ | E | f v ⊗ | V | {\displaystyle =f_{u}^{\otimes |U|}T^{\otimes |E|}(T^{-1})^{\otimes |E|}f_{v}^{\otimes |V|}} = ( f u T ⊗ ( deg u ) ) ⊗ | U | ( f v ( T − 1 ) ⊗ ( deg v ) ) ⊗ | V | . {\displaystyle =\left(f_{u}T^{\otimes (\deg u)}\right)^{\otimes |U|}\left(f_{v}(T^{-1})^{\otimes (\deg v)}\right)^{\otimes |V|}.} Thus, Holant ( G , f u , f v ) {\displaystyle {\text{Holant}}(G,f_{u},f_{v})} and Holant ( G , f u T ⊗ ( deg u ) , ( T − 1 ) ⊗ ( deg v ) f v ) {\displaystyle {\text{Holant}}(G,f_{u}T^{\otimes (\deg u)},(T^{-1})^{\otimes (\deg v)}f_{v})} have exactly the same Holant value for every constraint graph. They essentially define the same counting problem. === Specific examples === ==== Vertex covers and independent sets ==== Let G be a graph. There is a 1-to-1 correspondence between the vertex covers of G and the independent sets of G. For any set S of vertices of G, S is a vertex cover in G if and only if the complement of S is an independent set in G. Thus, the number of vertex covers in G is exactly the same as the number of independent sets in G. The equivalence of these two counting problems can also be proved using a holographic reduction. For simplicity, let G be a 3-regular graph. The 2-stretch of G gives a bipartite graph H=(U,V,E), where U corresponds to the edges in G and V corresponds to the vertices in G. The Holant problem that naturally corresponds to counting the number of vertex covers in G is Holant ( H , OR 2 , EQUAL 3 ) . {\displaystyle {\text{Holant}}(H,{\text{OR}}_{2},{\text{EQUAL}}_{3}).} The truth table of OR2 as a row vector is (0,1,1,1). The truth table of EQUAL3 as a column vector is ( 1 , 0 , 0 , 0 , 0 , 0 , 0 , 1 ) T = [ 1 0 ] ⊗ 3 + [ 0 1 ] ⊗ 3 {\displaystyle (1,0,0,0,0,0,0,1)^{T}={\begin{bmatrix}1\\0\end{bmatrix}}^{\otimes 3}+{\begin{bmatrix}0\\1\end{bmatrix}}^{\otimes 3}} . Then under a holographic transformation by [ 0 1 1 0 ] , {\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}},} OR 2 ⊗ | U | EQUAL 3 ⊗ | V | {\displaystyle {\text{OR}}_{2}^{\otimes |U|}{\text{EQUAL}}_{3}^{\otimes |V|}} = ( 0 , 1 , 1 , 1 ) ⊗ | U | ( [ 1 0 ] ⊗ 3 + [ 0 1 ] ⊗ 3 ) ⊗ | V | {\displaystyle =(0,1,1,1)^{\otimes |U|}\left({\begin{bmatrix}1\\0\end{bmatrix}}^{\otimes 3}+{\begin{bmatrix}0\\1\end{bmatrix}}^{\otimes 3}\right)^{\otimes |V|}} = ( 0 , 1 , 1 , 1 ) ⊗ | U | [ 0 1 1 0 ] ⊗ | E | [ 0 1 1 0 ] ⊗ | E | ( [ 1 0 ] ⊗ 3 + [ 0 1 ] ⊗ 3 ) ⊗ | V | {\displaystyle =(0,1,1,1)^{\otimes |U|}{\begin{bmatrix}0&1\\1&0\end{bmatrix}}^{\otimes |E|}{\begin{bmatrix}0&1\\1&0\end{bmatrix}}^{\otimes |E|}\left({\begin{bmatrix}1\\0\end{bmatrix}}^{\otimes 3}+{\begin{bmatrix}0\\1\end{bmatrix}}^{\otimes 3}\right)^{\otimes |V|}} = ( ( 0 , 1 , 1 , 1 ) [ 0 1 1 0 ] ⊗ 2 ) ⊗ | U | ( ( [ 0 1 1 0 ] [ 1 0 ] ) ⊗ 3 + ( [ 0 1 1 0 ] [ 0 1 ] ) ⊗ 3 ) ⊗ | V | {\displaystyle =\left((0,1,1,1){\begin{bmatrix}0&1\\1&0\end{bmatrix}}^{\otimes 2}\right)^{\otimes |U|}\left(\left({\begin{bmatrix}0&1\\1&0\end{bmatrix}}{\begin{bmatrix}1\\0\end{bmatrix}}\right)^{\otimes 3}+\left({\begin{bmatrix}0&1\\1&0\end{bmatrix}}{\begin{bmatrix}0\\1\end{bmatrix}}\right)^{\otimes 3}\right)^{\otimes |V|}} = ( 1 , 1 , 1 , 0 ) ⊗ | U | ( [ 0 1 ] ⊗ 3 + [ 1 0 ] ⊗ 3 ) ⊗ | V | {\displaystyle =(1,1,1,0)^{\otimes |U|}\left({\begin{bmatrix}0\\1\end{bmatrix}}^{\otimes 3}+{\begin{bmatrix}1\\0\end{bmatrix}}^{\otimes 3}\right)^{\otimes |V|}} = NAND 2 ⊗ | U | EQUAL 3 ⊗ | V | , {\displaystyle ={\text{NAND}}_{2}^{\otim