Abstraction, Problem Decomposition, and Functions

Start Date: 12/15/2019

Course Type: Common Course

Course Link: https://www.coursera.org/learn/abstraction-problem-decomposition-functions

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Abstraction and Problem Decomposition

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Abstraction Carl Jung's definition of abstraction broadened its scope beyond the thinking process to include exactly four mutually exclusive, different complementary psychological functions: sensation, intuition, feeling, and thinking. Together they form a structural totality of the differentiating abstraction process. Abstraction operates in one of these functions when it excludes the simultaneous influence of the other functions and other irrelevancies, such as emotion. Abstraction requires selective use of this structural split of abilities in the psyche. The opposite of abstraction is concretism. "Abstraction" is one of Jung's 57 definitions in Chapter XI of "Psychological Types".
Abstraction Abstraction in philosophy is the process (or, to some, the alleged process) in concept formation of recognizing some set of common features in individuals, and on that basis forming a concept of that feature. The notion of abstraction is important to understanding some philosophical controversies surrounding empiricism and the problem of universals. It has also recently become popular in formal logic under predicate abstraction. Another philosophical tool for discussion of abstraction is thought space.
Functional decomposition Functional Decomposition is a design method intending to produce a non-implementation, architectural description of a computer program. Rather than conjecturing Objects and adding methods to them (OOP), with each Object intending to capture some service of the program, the software architect first establishes a series of functions and types that accomplishes the main processing problem of the computer program, decomposes each to reveal common functions and types, and finally derives Modules from this activity.
Abstraction layer A good abstraction will generalize that which can be made abstract; while allowing specificity where the abstraction breaks down and its successful application requires customization to each unique requirement or problem.
Decomposition method (constraint satisfaction) If solving problems having a decomposition width bounded by a constant is in P, the decomposition leads to a tractable structural restriction. As explained above, tractability requires that two conditions are met. First, if the problem has width bounded by a constant then a decomposition of bounded width can be found in polynomial time. Second, the problem obtained by converting the original problem according to the decomposition is not superpolynomially larger than the original problem, if the decomposition has fixed width.
QR decomposition In linear algebra, a QR decomposition (also called a QR factorization) of a matrix is a decomposition of a matrix "A" into a product "A" = "QR" of an orthogonal matrix "Q" and an upper triangular matrix "R". QR decomposition is often used to solve the linear least squares problem, and is the basis for a particular eigenvalue algorithm, the QR algorithm.
Decomposition method (constraint satisfaction) In constraint satisfaction, a decomposition method translates a constraint satisfaction problem into another constraint satisfaction problem that is binary and acyclic. Decomposition methods work by grouping variables into sets, and solving a subproblem for each set. These translations are done because solving binary acyclic problems is a tractable problem.
Abstraction inversion In computer programming, abstraction inversion is an anti-pattern arising when users of a construct need functions implemented within it but not exposed by its interface. The result is that the users re-implement the required functions in terms of the interface, which in its turn uses the internal implementation of the same functions. This may result in implementing lower-level features in terms of higher-level ones, thus the term 'abstraction inversion'.
Decomposition Five general stages are used to describe the process of decomposition in vertebrate animals: fresh, bloat, active and advanced decay, and dry/remains. The general stages of decomposition are coupled with two stages of chemical decomposition: autolysis and putrefaction. These two stages contribute to the chemical process of decomposition, which breaks down the main components of the body.
Decomposition method (constraint satisfaction) The width of a problem is the width of its minimal-width decomposition. While decompositions of fixed width can be used to efficiently solve a problem, a bound on the width of instances does necessarily produce a tractable structural restriction. Indeed, a fixed width problem has a decomposition of fixed width, but finding it may not be polynomial. In order for a problem of fixed width being efficiently solved by decomposition, one of its decompositions of low width has to be found efficiently. For this reason, decomposition methods and their associated width are defined in such a way not only solving the problem given a fixed-width decomposition of it is polynomial-time, but also finding a fixed width decomposition of a fixed-width problem is polynomial-time.
Decomposition method (constraint satisfaction) By definition, a decomposition method produces a binary acyclic problem; such problems can be solved in time polynomial in its size. As a result, the original problem can be solved by first translating it and then solving the resulting problem; however, this algorithm is polynomial-time only if the decomposition does not increase size superpolynomially. The "width" of a decomposition method is a measure of the size of problem it produced. Originally, the width was defined as the maximal cardinality of the sets of original variables; one method, the hypertree decomposition, uses a different measure. Either way, the width of a decomposition is defined so that decompositions of size bounded by a constant do not produce excessively large problems. Instances having a decomposition of fixed width can be translated by decomposition into instances of size bounded by a polynomial in the size of the original instance.
Leaky abstraction The term "leaky abstraction" was popularized in 2002 by Joel Spolsky. An earlier paper by Kiczales describes some of the issues with imperfect abstractions and presents a potential solution to the problem by allowing for the customization of the abstraction itself.
Constraint satisfaction dual problem Decomposition methods generalize join-tree clustering by grouping variables in such a way the resulting problem has a join tree. Decomposition methods directly associate a tree with problems; the nodes of this tree are associated variables and/or constraints of the original problem. By merging constraints based on this tree, one can produce a problem that has a join tree, and this join tree can be easily derived from the decomposition tree. Alternatively, one can build a binary acyclic problem directly from the decomposition tree.
Abstraction An abstraction can be seen as a compression process, mapping multiple different pieces of constituent data to a single piece of abstract data; based on similarities in the constituent data, for example, many different physical cats map to the abstraction "CAT". This conceptual scheme emphasizes the inherent equality of both constituent and abstract data, thus avoiding problems arising from the distinction between "abstract" and "concrete". In this sense the process of abstraction entails the identification of similarities between objects, and the process of associating these objects with an abstraction (which is itself an object).
Benders decomposition The strategy behind Benders decomposition can be summarized as "divide-and-conquer". That is, in Benders decomposition, the variables of the original problem are divided into two subsets so that a first-stage master problem is solved over the first set of variables, and the values for the second set of variables are determined in a second-stage subproblem for a given first-stage solution. If the subproblem determines that the fixed first-stage decisions are in fact infeasible, then so-called "Benders cuts" are generated and added to the master problem, which is then re-solved until no cuts can be generated. Since Benders decomposition adds new "constraints" as it progresses towards a solution, the approach is called ""row" generation". In contrast, Dantzig–Wolfe decomposition uses ""column" generation".
Functional decomposition For a multivariate function formula_1, functional decomposition generally refers to a process of identifying a set of functions formula_2 such that
Functional decomposition Functional decomposition in systems engineering refers to the process of defining a system in functional terms, then defining lower-level functions and sequencing relationships from these higher level systems functions. The basic idea is to try to divide a system in such a way that each block of a block diagram can be described without an "and" or "or" in the description.
Computational thinking The characteristics that define computational thinking are decomposition, pattern recognition / data representation, generalization/abstraction, and algorithms. By decomposing a problem, identifying the variables involved using data representation, and creating algorithms, a generic solution results. The generic solution is a generalization or abstraction that can be used to solve a multitude of variations of the initial problem.
Abstraction model checking A problem with abstraction model checking is that although the abstraction simulates the real, when the abstraction does not satisfy a property, it does not mean that this property actually fails in the real model. Counter examples are checked against the real state space because we obtain "spurious" counter examples. So a part of the abstraction refinement loop is:
Wahba's problem A number of solutions to the problem have appeared in literature, notably Davenport's q-method, QUEST and singular value decomposition-based methods.