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Logic Minimization
Logic optimization is a process of finding an equivalent representation of the specified logic circuit under one or more specified constraints. This process is a part of a logic synthesis applied in digital electronics and integrated circuit design. Generally, the circuit is constrained to a minimum chip area meeting a predefined response delay. The goal of logic optimization of a given circuit is to obtain the smallest logic circuit that evaluates to the same values as the original one. Usually, the smaller circuit with the same function is cheaper, takes less space, consumes less power, has shorter latency, and minimizes risks of unexpected cross-talk, hazard of delayed signal processing, and other issues present at the nano-scale level of metallic structures on an integrated circuit. In terms of Boolean algebra, the optimization of a complex Boolean expression is a process of finding a simpler one, which would upon evaluation ultimately produce the same results as the or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic Circuit
A logic gate is a device that performs a Boolean function, a logical operation performed on one or more binary inputs that produces a single binary output. Depending on the context, the term may refer to an ideal logic gate, one that has, for instance, zero rise time and unlimited fan-out, or it may refer to a non-ideal physical device (see ideal and real op-amps for comparison). The primary way of building logic gates uses diodes or transistors acting as electronic switches. Today, most logic gates are made from MOSFETs (metal–oxide–semiconductor field-effect transistors). ''From Integrated circuit'' They can also be constructed using vacuum tubes, electromagnetic relays with relay logic, fluidic logic, pneumatic logic, optics, molecules, acoustics, or even mechanical or thermal elements. Logic gates can be cascaded in the same way that Boolean functions can be composed, allowing the construction of a physical model of all of Boolean logic, and therefore, all of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic Friday
The ESPRESSO logic minimizer is a computer program using heuristic and specific algorithms for efficiently reducing the complexity of digital logic gate circuits. ESPRESSO-I was originally developed at IBM by Robert K. Brayton et al. in 1982. and improved as ESPRESSO-II in 1984. Richard L. Rudell later published the variant ESPRESSO-MV in 1986 and ESPRESSO-EXACT in 1987. Espresso has inspired many derivatives. Introduction Electronic devices are composed of numerous blocks of digital circuits, the combination of which performs some required task. The efficient implementation of logic functions in the form of logic gate circuits (such that no more logic gates are used than are necessary) is necessary to minimize production costs, and/or maximize a device's performance. Designing digital logic circuits All digital systems are composed of two elementary functions: memory elements for storing information, and combinational circuits that transform that information. State machines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heuristic
A heuristic or heuristic technique (''problem solving'', '' mental shortcut'', ''rule of thumb'') is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless "good enough" as an approximation or attribute substitution. Where finding an optimal solution is impossible or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution. Heuristics can be mental shortcuts that ease the cognitive load of making a decision. Context Gigerenzer & Gaissmaier (2011) state that sub-sets of ''strategy'' include heuristics, regression analysis, and Bayesian inference. Heuristics are strategies based on rules to generate optimal decisions, like the anchoring effect and utility maximization problem. These strategies depend on using readily accessible, though loosely applicable, information to control problem solving in human beings, machines and abstract i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SAT Solver
In computer science and formal methods, a SAT solver is a computer program which aims to solve the Boolean satisfiability problem (SAT). On input a formula over Boolean data type, Boolean variables, such as "(''x'' or ''y'') and (''x'' or not ''y'')", a SAT solver outputs whether the formula is satisfiability, satisfiable, meaning that there are possible values of ''x'' and ''y'' which make the formula true, or unsatisfiable, meaning that there are no such values of ''x'' and ''y''. In this case, the formula is satisfiable when ''x'' is true, so the solver should return "satisfiable". Since the introduction of algorithms for SAT in the 1960s, modern SAT solvers have grown into complex software, software artifacts involving a large number of heuristics and program optimizations to work efficiently. By a result known as the Cook–Levin theorem, Boolean satisfiability is an NP-completeness, NP-complete problem in general. As a result, only algorithms with exponential worst-case comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Satisfiability
In mathematical logic, a formula is ''satisfiable'' if it is true under some assignment of values to its variables. For example, the formula x+3=y is satisfiable because it is true when x=3 and y=6, while the formula x+1=x is not satisfiable over the integers. The dual concept to satisfiability is validity; a formula is ''valid'' if every assignment of values to its variables makes the formula true. For example, x+3=3+x is valid over the integers, but x+3=y is not. Formally, satisfiability is studied with respect to a fixed logic defining the syntax of allowed symbols, such as first-order logic, second-order logic or propositional logic. Rather than being syntactic, however, satisfiability is a semantic property because it relates to the ''meaning'' of the symbols, for example, the meaning of + in a formula such as x+1=x. Formally, we define an interpretation (or model) to be an assignment of values to the variables and an assignment of meaning to all other non-logical symbol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petrick's Method
In Boolean algebra, Petrick's method (also known as ''Petrick function'' or ''branch-and-bound'' method) is a technique described by Stanley R. Petrick (1931–2006) in 1956 for determining all minimum sum-of-products solutions from a prime implicant chart. Petrick's method is very tedious for large charts, but it is easy to implement on a computer. The method was improved by Insley B. Pyne and Edward Joseph McCluskey in 1962. Algorithm # Reduce the prime implicant chart by eliminating the essential prime implicant rows and the corresponding columns. # Label the rows of the reduced prime implicant chart P_1, P_2, P_3, P_4, etc. # Form a logical function P which is true when all the columns are covered. ''P'' consists of a product of sums where each sum term has the form (P_ + P_ + \cdots + P_), where each P_ represents a row covering column i. # Apply De Morgan's Laws to expand P into a sum of products and minimize by applying the absorption law In algebra, the absorption l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Complexity
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Hierarchy
In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines. It is a resource-bounded counterpart to the arithmetical hierarchy and analytical hierarchy from mathematical logic. The union of the classes in the hierarchy is denoted PH. Classes within the hierarchy have complete problems (with respect to polynomial-time reductions) that ask if quantified Boolean formulae hold, for formulae with restrictions on the quantifier order. It is known that equality between classes on the same level or consecutive levels in the hierarchy would imply a "collapse" of the hierarchy to that level. Definitions There are multiple equivalent definitions of the classes of the polynomial hierarchy. Oracle definition For the oracle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maurice Karnaugh
Maurice Karnaugh (; October 4, 1924 – November 8, 2022) was an American physicist, mathematician, computer scientist, and inventor known for the Karnaugh map used in Boolean algebra. Early life and education Karnaugh earned a B.A in physics from the City College of New York in 1948 and a PhD. in physics from Yale in 1952. He later studied mathematics and physics at City College of New York (1944 to 1948) and transferred to Yale University to complete his B.Sc. (1949), M.Sc. (1950) and Ph.D. in physics with a thesis on ''The Theory of Magnetic Resonance and Lambda-Type Doubling in Nitric-Oxide'' (1952). Career Karnaugh worked at Bell Labs and IBM from 1952 to 1993. From 1980 until 1999 Karnaugh taught computer science at Polytechnic University of New York. At Bell Labs, (1952 to 1966), developing the Karnaugh map (1954) as well as patents for PCM encoding and magnetic logic circuits and coding. He later worked at IBM's Federal Systems Division in Gaithersburg (1966 to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karnaugh Map
A Karnaugh map (KM or K-map) is a diagram that can be used to simplify a Boolean algebra expression. Maurice Karnaugh introduced the technique in 1953 as a refinement of Edward W. Veitch's 1952 Veitch chart, which itself was a rediscovery of Allan Marquand's 1881 ''logical diagram'' or Marquand diagram. They are also known as Marquand–Veitch diagrams, Karnaugh–Veitch (KV) maps, and (rarely) Svoboda charts. An early advance in the history of formal logic methodology, Karnaugh maps remain relevant in the digital age, especially in the fields of logical circuit design and digital engineering. Definition A Karnaugh map reduces the need for extensive calculations by taking advantage of humans' pattern-recognition capability. It also permits the rapid identification and elimination of potential race conditions. The required Boolean results are transferred from a truth table onto a two-dimensional grid where, in Karnaugh maps, the cells are ordered in Gray code, and each cell ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Venn
John Venn, Fellow of the Royal Society, FRS, Fellow of the Society of Antiquaries of London, FSA (4 August 1834 – 4 April 1923) was an English mathematician, logician and philosopher noted for introducing Venn diagrams, which are used in logic, set theory, probability, statistics, and computer science. In 1866, Venn published ''The Logic of Chance'', a groundbreaking book which espoused the frequency theory of probability, arguing that probability should be determined by how often something is forecast to occur as opposed to "educated" assumptions. Venn then further developed George Boole's theories in the 1881 work ''Symbolic Logic'', where he highlighted what would become known as Venn diagrams. Early life John Venn was born on 4 August 1834 in Kingston upon Hull, Yorkshire, to Martha Sykes and Rev. Henry Venn (Church Missionary Society), Henry Venn, who was the rector of the parish of Drypool. His mother died when he was three years old. Venn was descended from a long ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Venn Diagram
A Venn diagram is a widely used diagram style that shows the logical relation between set (mathematics), sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. A Venn diagram uses simple closed curves on a plane to represent sets. The curves are often circles or ellipses. Similar ideas had been proposed before Venn such as by Christian Weise in 1712 (''Nucleus Logicoe Wiesianoe'') and Leonhard Euler in 1768 (''Letters to a German Princess''). The idea was popularised by Venn in ''Symbolic Logic'', Chapter V "Diagrammatic Representation", published in 1881. Details A Venn diagram, also called a ''set diagram'' or ''logic diagram'', shows ''all'' possible logical relations between a finite collection of different sets. These diagrams depict element (mathematics), elements as points in the plane, and sets as r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
