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6,901 | For parallel computing, using a programming model instead of a language is common. The reason is that details of the parallel hardware leak into the abstractions used to program the hardware. This causes the programmer to have to map patterns in the algorithm onto patterns in the execution model . As a consequence, ... |
6,902 | Some programming language researchers criticise the notion of paradigms as a classification of programming languages, e.g. Harper, and Krishnamurthi. They argue that many programming languages cannot be strictly classified into one paradigm, but rather include features from several paradigms. See Comparison of multi-p... |
6,903 | Different approaches to programming have developed over time. Classification of each approach was either described at the time the approach was first developed, but often not until some time later, retrospectively. An early approach consciously identified as such is structured programming, advocated since the mid 1960s... |
6,904 | Machine code is the lowest-level of computer programming as it is machine instructions that define behavior at the lowest level of abstract possible for a computer. As it is the most prescriptive way to code it is classified as imperative. |
6,905 | It is sometimes called the first-generation programming language. |
6,906 | Assembly language introduced mnemonics for machine instructions and memory addresses. Assembly is classified as imperative and is sometimes called the second-generation programming language. |
6,907 | In the 1960s, assembly languages were developed to support library COPY and quite sophisticated conditional macro generation and preprocessing abilities, CALL to subroutine, external variables and common sections , enabling significant code re-use and isolation from hardware specifics via the use of logical operators s... |
6,908 | Procedural languages, also called the third-generation programming languages are the first described as high-level languages. They support vocabulary related to the problem being solved. For example, |
6,909 | These languages are classified as procedural paradigm. They directly control the step by step process that a computer program follows. The efficacy and efficiency of such a program is therefore highly dependent on the programmer's skill. |
6,910 | In attempt to improve on procedural languages, object-oriented programming languages were created, such as Simula, Smalltalk, C++, Eiffel, Python, PHP, Java, and C#. In these languages, data and methods to manipulate the data are in the same code unit called an object. This encapsulation ensures that the only way that... |
6,911 | There is controversy raised by Alexander Stepanov, Richard Stallman and other programmers, concerning the efficacy of the OOP paradigm versus the procedural paradigm. The need for every object to have associative methods leads some skeptics to associate OOP with software bloat; an attempt to resolve this dilemma came t... |
6,912 | Although most OOP languages are third-generation, it is possible to create an object-oriented assembler language. High Level Assembly is an example of this that fully supports advanced data types and object-oriented assembly language programming – despite its early origins. Thus, differing programming paradigms can b... |
6,913 | A declarative programming program describes what the problem is, not how to solve it. The program is structured as a set of properties to find in the expected result, not as a procedure to follow. Given a database or a set of rules, the computer tries to find a solution matching all the desired properties. An archetype... |
6,914 | Functional programming is a subset of declarative programming. Programs written using this paradigm use functions, blocks of code intended to behave like mathematical functions. Functional languages discourage changes in the value of variables through assignment, making a great deal of use of recursion instead. |
6,915 | The logic programming paradigm views computation as automated reasoning over a body of knowledge. Facts about the problem domain are expressed as logic formulas, and programs are executed by applying inference rules over them until an answer to the problem is found, or the set of formulas is proved inconsistent. |
6,916 | Symbolic programming is a paradigm that describes programs able to manipulate formulas and program components as data. Programs can thus effectively modify themselves, and appear to "learn", making them suited for applications such as artificial intelligence, expert systems, natural-language processing and computer ga... |
6,917 | Differentiable programming structures programs so that they can be differentiated throughout, usually via automatic differentiation. |
6,918 | Literate programming, as a form of imperative programming, structures programs as a human-centered web, as in a hypertext essay: documentation is integral to the program, and the program is structured following the logic of prose exposition, rather than compiler convenience. |
6,919 | Symbolic techniques such as reflection, which allow the program to refer to itself, might also be considered as a programming paradigm. However, this is compatible with the major paradigms and thus is not a real paradigm in its own right. |
6,920 | Classification of the principal programming paradigms |
6,921 | How programming paradigms evolve and get adopted? |
6,922 | The structured chart subject to these constraints, particularly the loop constraint implying a single exit , may however use additional variables in the form of bits in order to keep track of information that the original program represents by the program location. The construction was based on Böhm's programming lang... |
6,923 | The theorem forms the basis of structured programming, a programming paradigm which eschews goto commands and exclusively uses subroutines, sequences, selection and iteration. |
6,924 | The theorem is typically credited: 381 to a 1966 paper by Corrado Böhm and Giuseppe Jacopini. David Harel wrote in 1980 that the Böhm–Jacopini paper enjoyed "universal popularity",: 381 particularly with proponents of structured programming. Harel also noted that "due to its rather technical style is apparently more... |
6,925 | Harel also writes that the more generic name was proposed by H.D. Mills as "The Structure Theorem" in the early 1970s.: 381 |
6,926 | This version of the theorem replaces all the original program's control flow with a single global while loop that simulates a program counter going over all possible labels in the original non-structured program. Harel traced the origin of this folk theorem to two papers marking the beginning of computing. One is the ... |
6,927 | Donald Knuth criticized this form of the proof, which results in pseudocode like the one below, by pointing out that the structure of the original program is completely lost in this transformation.: 274 Similarly, Bruce Ian Mills wrote about this approach that "The spirit of block structure is a style, not a language.... |
6,928 | The proof in Böhm and Jacopini's paper proceeds by induction on the structure of the flow chart.: 381 Because it employed pattern matching in graphs, the proof of Böhm and Jacopini's was not really practical as a program transformation algorithm, and thus opened the door for additional research in this direction. |
6,929 | The Reversible Structured Program Theorem is an important concept in the field of reversible computing. It posits that any computation achievable by a reversible program can also be accomplished through a reversible program using only a structured combination of control flow constructs such as sequences, selections, an... |
6,930 | For the Structured Program Theorem, both local and global methods of proof are known. However, for its reversible version, while a global method of proof is recognized, a local approach similar to that undertaken by Böhm and Jacopini is not yet known. This distinction is an example that underscores the challenges and n... |
6,931 | The Böhm–Jacopini proof did not settle the question of whether to adopt structured programming for software development, partly because the construction was more likely to obscure a program than to improve it. On the contrary, it signalled the beginning of the debate. Edsger Dijkstra's famous letter, "Go To Statement C... |
6,932 | Some academics took a purist approach to the Böhm–Jacopini result and argued that even instructions like break and return from the middle of loops are bad practice as they are not needed in the Böhm–Jacopini proof, and thus they advocated that all loops should have a single exit point. This purist approach is embodied ... |
6,933 | Edward Yourdon notes that in the 1970s there was even philosophical opposition to transforming unstructured programs into structured ones by automated means, based on the argument that one needed to think in structured programming fashion from the get go. The pragmatic counterpoint was that such transformations benefit... |
6,934 | The direct application of the Böhm–Jacopini theorem may result in additional local variables being introduced in the structured chart, and may also result in some code duplication. The latter issue is called the loop and a half problem in this context. Pascal is affected by both of these problems and according to empir... |
6,935 | In 1973, S. Rao Kosaraju proved that it's possible to avoid adding additional variables in structured programming, as long as arbitrary-depth, multi-level breaks from loops are allowed. Furthermore, Kosaraju proved that a strict hierarchy of programs exists, nowadays called the Kosaraju hierarchy, in that for every int... |
6,936 | A simpler result from Kosaraju's paper is that a program is reducible to a structured program if and only if it does not contain a loop with two distinct exits. Reducibility was defined by Kosaraju, loosely speaking, as computing the same function and using the same "primitive actions" and predicates as the original p... |
6,937 | McCabe actually found that these four graphs are not independent when appearing as subgraphs, meaning that a necessary and sufficient condition for a program to be non-structured is for its CFG to have as subgraph one of any subset of three of these four graphs. He also found that if a non-structured program contains o... |
6,938 | McCabe's characterization of the forbidden graphs for structured programming can be considered incomplete, at least if the Dijkstra's D structures are considered the building blocks.: 274–275 |
6,939 | Up to 1990 there were quite a few proposed methods for eliminating gotos from existing programs, while preserving most of their structure. The various approaches to this problem also proposed several notions of equivalence, which are stricter than simply Turing equivalence, in order to avoid output like the folk theore... |
6,940 | In the 1980s IBM researcher Harlan Mills oversaw the development of the COBOL Structuring Facility, which applied a structuring algorithm to COBOL code. Mills's transformation involved the following steps for each procedure. |
6,941 | This construction can be improved by converting some cases of the selection statement into subprocedures. |
6,942 | Material not yet covered above: |
6,943 | The machine operates on an infinite memory tape divided into discrete cells, each of which can hold a single symbol drawn from a finite set of symbols called the alphabet of the machine. It has a "head" that, at any point in the machine's operation, is positioned over one of these cells, and a "state" selected from a f... |
6,944 | The Turing machine was invented in 1936 by Alan Turing, who called it an "a-machine" . It was Turing's doctoral advisor, Alonzo Church, who later coined the term "Turing machine" in a review. With this model, Turing was able to answer two questions in the negative: |
6,945 | Thus by providing a mathematical description of a very simple device capable of arbitrary computations, he was able to prove properties of computation in general—and in particular, the uncomputability of the Entscheidungsproblem . |
6,946 | Turing machines proved the existence of fundamental limitations on the power of mechanical computation. While they can express arbitrary computations, their minimalist design makes them too slow for computation in practice: real-world computers are based on different designs that, unlike Turing machines, use random-acc... |
6,947 | Turing completeness is the ability for a computational model or a system of instructions to simulate a Turing machine. A programming language that is Turing complete is theoretically capable of expressing all tasks accomplishable by computers; nearly all programming languages are Turing complete if the limitations of f... |
6,948 | A Turing machine is an idealised model of a central processing unit that controls all data manipulation done by a computer, with the canonical machine using sequential memory to store data. Typically, the sequential memory is represented as a tape of infinite length on which the machine can perform read and write oper... |
6,949 | In the context of formal language theory, a Turing machine is capable of enumerating some arbitrary subset of valid strings of an alphabet. A set of strings which can be enumerated in this manner is called a recursively enumerable language. The Turing machine can equivalently be defined as a model that recognises vali... |
6,950 | Given a Turing machine M and an arbitrary string s, it is generally not possible to decide whether M will eventually produce s. This is due to the fact that the halting problem is unsolvable, which has major implications for the theoretical limits of computing. |
6,951 | The Turing machine is capable of processing an unrestricted grammar, which further implies that it is capable of robustly evaluating first-order logic in an infinite number of ways. This is famously demonstrated through lambda calculus. |
6,952 | A Turing machine that is able to simulate any other Turing machine is called a universal Turing machine . Another mathematical formalism, lambda calculus, with a similar "universal" nature was introduced by Alonzo Church. Church's work intertwined with Turing's to form the basis for the Church–Turing thesis. This thesi... |
6,953 | In his 1948 essay, "Intelligent Machinery", Turing wrote that his machine consisted of: |
6,954 | ...an unlimited memory capacity obtained in the form of an infinite tape marked out into squares, on each of which a symbol could be printed. At any moment there is one symbol in the machine; it is called the scanned symbol. The machine can alter the scanned symbol, and its behavior is in part determined by that symbol... |
6,955 | The Turing machine mathematically models a machine that mechanically operates on a tape. On this tape are symbols, which the machine can read and write, one at a time, using a tape head. Operation is fully determined by a finite set of elementary instructions such as "in state 42, if the symbol seen is 0, write a 1; i... |
6,956 | More explicitly, a Turing machine consists of: |
6,957 | In the 4-tuple models, erasing or writing a symbol and moving the head left or right are specified as separate instructions. The table tells the machine to erase or write a symbol or move the head left or right, and then assume the same or a new state as prescribed, but not both actions and in the same instructi... |
6,958 | Every part of the machine and its actions is finite, discrete and distinguishable; it is the unlimited amount of tape and runtime that gives it an unbounded amount of storage space. |
6,959 | The 7-tuple for the 3-state busy beaver looks like this : |
6,960 | In the words of van Emde Boas , p. 6: "The set-theoretical object provides only partial information on how the machine will behave and what its computations will look like." |
6,961 | For instance, |
6,962 | The most common convention represents each "Turing instruction" in a "Turing table" by one of nine 5-tuples, per the convention of Turing/Davis in The Undecidable, p. 126–127 and Davis p. 152): |
6,963 | Other authors p. 119, Hopcroft and Ullman p. 158, Stone p. 9) adopt a different convention, with new state qm listed immediately after the scanned symbol Sj: |
6,964 | For the remainder of this article "definition 1" will be used. |
6,965 | In the following table, Turing's original model allowed only the first three lines that he called N1, N2, N3 . He allowed for erasure of the "scanned square" by naming a 0th symbol S0 = "erase" or "blank", etc. However, he did not allow for non-printing, so every instruction-line includes "print symbol Sk" or "erase" ,... |
6,966 | Any Turing table can be constructed from the above nine 5-tuples. For technical reasons, the three non-printing or "N" instructions can usually be dispensed with. For examples see Turing machine examples. |
6,967 | Less frequently the use of 4-tuples are encountered: these represent a further atomization of the Turing instructions , Boolos & Jeffrey , Davis-Sigal-Weyuker ); also see more at Post–Turing machine. |
6,968 | The word "state" used in context of Turing machines can be a source of confusion, as it can mean two things. Most commentators after Turing have used "state" to mean the name/designator of the current instruction to be performed—i.e. the contents of the state register. But Turing made a strong distinction between a re... |
6,969 | Thus the state of progress of the computation at any stage is completely determined by the note of instructions and the symbols on the tape. That is, the state of the system may be described by a single expression consisting of the symbols on the tape followed by Δ and then by the note of instructions. This expressio... |
6,970 | Earlier in his paper Turing carried this even further: he gives an example where he placed a symbol of the current "m-configuration"—the instruction's label—beneath the scanned square, together with all the symbols on the tape ; this he calls "the complete configuration" . To print the "complete configuration" on one l... |
6,971 | A variant of this is seen in Kleene where Kleene shows how to write the Gödel number of a machine's "situation": he places the "m-configuration" symbol q4 over the scanned square in roughly the center of the 6 non-blank squares on the tape and puts it to the right of the scanned square. But Kleene refers to "q4" itse... |
6,972 | Example: total state of 3-state 2-symbol busy beaver after 3 "moves" : |
6,973 | This means: after three moves the tape has ... 000110000 ... on it, the head is scanning the right-most 1, and the state is A. Blanks can be part of the total state as shown here: B01; the tape has a single 1 on it, but the head is scanning the 0 to its left and the state is B. |
6,974 | "State" in the context of Turing machines should be clarified as to which is being described: the current instruction, or the list of symbols on the tape together with the current instruction, or the list of symbols on the tape together with the current instruction placed to the left of the scanned symbol or to the r... |
6,975 | Turing's biographer Andrew Hodges has noted and discussed this confusion. |
6,976 | To the right: the above table as expressed as a "state transition" diagram. |
6,977 | Usually large tables are better left as tables . They are more readily simulated by computer in tabular form . However, certain concepts—e.g. machines with "reset" states and machines with repeating patterns —can be more readily seen when viewed as a drawing. |
6,978 | Whether a drawing represents an improvement on its table must be decided by the reader for the particular context. |
6,979 | The reader should again be cautioned that such diagrams represent a snapshot of their table frozen in time, not the course of a computation through time and space. While every time the busy beaver machine "runs" it will always follow the same state-trajectory, this is not true for the "copy" machine that can be provid... |
6,980 | The diagram "progress of the computation" shows the three-state busy beaver's "state" progress through its computation from start to finish. On the far right is the Turing "complete configuration" at each step. If the machine were to be stopped and cleared to blank both the "state register" and entire tape, these "co... |
6,981 | Many machines that might be thought to have more computational capability than a simple universal Turing machine can be shown to have no more power ). They might compute faster, perhaps, or use less memory, or their instruction set might be smaller, but they cannot compute more powerfully . |
6,982 | A Turing machine is equivalent to a single-stack pushdown automaton that has been made more flexible and concise by relaxing the last-in-first-out requirement of its stack. In addition, a Turing machine is also equivalent to a two-stack PDA with standard LIFO semantics, by using one stack to model the tape left of t... |
6,983 | At the other extreme, some very simple models turn out to be Turing-equivalent, i.e. to have the same computational power as the Turing machine model. |
6,984 | Common equivalent models are the multi-tape Turing machine, multi-track Turing machine, machines with input and output, and the non-deterministic Turing machine as opposed to the deterministic Turing machine for which the action table has at most one entry for each combination of symbol and state. |
6,985 | Read-only, right-moving Turing machines are equivalent to DFAs . |
6,986 | For practical and didactical intentions the equivalent register machine can be used as a usual assembly programming language. |
6,987 | A relevant question is whether or not the computation model represented by concrete programming languages is Turing equivalent. While the computation of a real computer is based on finite states and thus not capable to simulate a Turing machine, programming languages themselves do not necessarily have this limitation. ... |
6,988 | Early in his paper Turing makes a distinction between an "automatic machine"—its "motion ... completely determined by the configuration" and a "choice machine": |
6,989 | ...whose motion is only partially determined by the configuration ... When such a machine reaches one of these ambiguous configurations, it cannot go on until some arbitrary choice has been made by an external operator. This would be the case if we were using machines to deal with axiomatic systems. |
6,990 | Turing does not elaborate further except in a footnote in which he describes how to use an a-machine to "find all the provable formulae of the calculus" rather than use a choice machine. He "suppose that the choices are always between two possibilities 0 and 1. Each proof will then be determined by a sequence of choi... |
6,991 | This is indeed the technique by which a deterministic Turing machine can be used to mimic the action of a nondeterministic Turing machine; Turing solved the matter in a footnote and appears to dismiss it from further consideration. |
6,992 | An oracle machine or o-machine is a Turing a-machine that pauses its computation at state "o" while, to complete its calculation, it "awaits the decision" of "the oracle"—an entity unspecified by Turing "apart from saying that it cannot be a machine" , The Undecidable, p. 166–168). |
6,993 | As Turing wrote in The Undecidable, p. 128 : |
6,994 | It is possible to invent a single machine which can be used to compute any computable sequence. If this machine U is supplied with the tape on the beginning of which is written the string of quintuples separated by semicolons of some computing machine M, then U will compute the same sequence as M. |
6,995 | This finding is now taken for granted, but at the time it was considered astonishing. The model of computation that Turing called his "universal machine"—"U" for short—is considered by some ) to have been the fundamental theoretical breakthrough that led to the notion of the stored-program computer. |
6,996 | Turing's paper ... contains, in essence, the invention of the modern computer and some of the programming techniques that accompanied it. |
6,997 | In terms of computational complexity, a multi-tape universal Turing machine need only be slower by logarithmic factor compared to the machines it simulates. This result was obtained in 1966 by F. C. Hennie and R. E. Stearns. |
6,998 | It is often believed that Turing machines, unlike simpler automata, are as powerful as real machines, and are able to execute any operation that a real program can. What is neglected in this statement is that, because a real machine can only have a finite number of configurations, it is nothing but a finite-state machi... |
6,999 | There are a number of ways to explain why Turing machines are useful models of real computers: |
7,000 | A limitation of Turing machines is that they do not model the strengths of a particular arrangement well. For instance, modern stored-program computers are actually instances of a more specific form of abstract machine known as the random-access stored-program machine or RASP machine model. Like the universal Turing ma... |
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