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Church–Turing thesis

In computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) is a thesis about the nature of computable functions. It states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine. The thesis is named after American mathematician Alonzo Church and the British mathematician Alan Turing. Before the precise definition of computable function, mathematicians often used the informal term effectively calculable to describe functions that are computable by paper-and-pencil methods. In the 1930s, several independent attempts were made to formalize the notion of computability:

 * In 1933, Kurt Gödel, with Jacques Herbrand, formalized the definition of the class of general recursive functions: the smallest class of functions (with arbitrarily many arguments) that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections.
 * In 1936, Alonzo Church created a method for defining functions called the λ-calculus. Within λ-calculus, he defined an encoding of the natural numbers called the Church numerals. A function on the natural numbers is called λ-computable if the corresponding function on the Church numerals can be represented by a term of the λ-calculus.
 * Also in 1936, before learning of Church's work, Alan Turing created a theoretical model for machines, now called Turing machines, that could carry out calculations from inputs by manipulating symbols on a tape. Given a suitable encoding of the natural numbers as sequences of symbols, a function on the natural numbers is called Turing computable if some Turing machine computes the corresponding function on encoded natural numbers.

Church, Kleene, and Turing proved that these three formally defined classes of computable functions coincide: a function is λ-computable if and only if it is Turing computable, and if and only if it is general recursive. This has led mathematicians and computer scientists to believe that the concept of computability is accurately characterized by these three equivalent processes. Other formal attempts to characterize computability have subsequently strengthened this belief (see below).

On the other hand, the Church–Turing thesis states that the above three formally-defined classes of computable functions coincide with the informal notion of an effectively calculable function. Although the thesis has near-universal acceptance, it cannot be formally proven, as the concept of effective calculability is only informally defined.

Since its inception, variations on the original thesis have arisen, including statements about what can physically be realized by a computer in our universe (physical Church-Turing thesis) and what can be efficiently computed (Church–Turing thesis (complexity theory)). These variations are not due to Church or Turing, but arise from later work in complexity theory and digital physics. The thesis also has implications for the philosophy of mind (see below).

Statement in Church's and Turing's words
addresses the notion of "effective computability" as follows: "Clearly the existence of CC and RC (Church's and Rosser's proofs) presupposes a precise definition of 'effective'. 'Effective method' is here used in the rather special sense of a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps". Thus the adverb-adjective "effective" is used in a sense of "1a: producing a decided, decisive, or desired effect", and "capable of producing a result".

In the following, the words "effectively calculable" will mean "produced by any intuitively 'effective' means whatsoever" and "effectively computable" will mean "produced by a Turing-machine or equivalent mechanical device". Turing's "definitions" given in a footnote in his 1938 Ph.D. thesis Systems of Logic Based on Ordinals, supervised by Church, are virtually the same: "$†$ We shall use the expression 'computable function' to mean a function calculable by a machine, and let 'effectively calculable' refer to the intuitive idea without particular identification with any one of these definitions."

The thesis can be stated as: Every effectively calculable function is a computable function. Church also stated that "No computational procedure will be considered as an algorithm unless it can be represented as a Turing Machine". Turing stated it this way: "It was stated ... that 'a function is effectively calculable if its values can be found by some purely mechanical process'. We may take this literally, understanding that by a purely mechanical process one which could be carried out by a machine. The development ... leads to ... an identification of computability$†$ with effective calculability. [$†$ is the footnote quoted above.]"

History
One of the important problems for logicians in the 1930s was the Entscheidungsproblem of David Hilbert and Wilhelm Ackermann, which asked whether there was a mechanical procedure for separating mathematical truths from mathematical falsehoods. This quest required that the notion of "algorithm" or "effective calculability" be pinned down, at least well enough for the quest to begin. But from the very outset Alonzo Church's attempts began with a debate that continues to this day. the notion of "effective calculability" to be (i) an "axiom or axioms" in an axiomatic system, (ii) merely a definition that "identified" two or more propositions, (iii) an empirical hypothesis to be verified by observation of natural events, or (iv) just a proposal for the sake of argument (i.e. a "thesis")?

Circa 1930–1952
In the course of studying the problem, Church and his student Stephen Kleene introduced the notion of λ-definable functions, and they were able to prove that several large classes of functions frequently encountered in number theory were λ-definable. The debate began when Church proposed to Gödel that one should define the "effectively computable" functions as the λ-definable functions. Gödel, however, was not convinced and called the proposal "thoroughly unsatisfactory". Rather, in correspondence with Church (c. 1934–1935), Gödel proposed axiomatizing the notion of "effective calculability"; indeed, in a 1935 letter to Kleene, Church reported that:

"His [Gödel's] only idea at the time was that it might be possible, in terms of effective calculability as an undefined notion, to state a set of axioms which would embody the generally accepted properties of this notion, and to do something on that basis."

But Gödel offered no further guidance. Eventually, he would suggest his recursion, modified by Herbrand's suggestion, that Gödel had detailed in his 1934 lectures in Princeton NJ (Kleene and Rosser transcribed the notes). But he did not think that the two ideas could be satisfactorily identified "except heuristically".

Next, it was necessary to identify and prove the equivalence of two notions of effective calculability. Equipped with the λ-calculus and "general" recursion, Kleene with help of Church and J. Barkley Rosser produced proofs (1933, 1935) to show that the two calculi are equivalent. Church subsequently modified his methods to include use of Herbrand–Gödel recursion and then proved (1936) that the Entscheidungsproblem is unsolvable: there is no algorithm that can determine whether a well formed formula has a beta normal form.

Many years later in a letter to Davis (c. 1965), Gödel said that "he was, at the time of these [1934] lectures, not at all convinced that his concept of recursion comprised all possible recursions". By 1963–1964 Gödel would disavow Herbrand–Gödel recursion and the λ-calculus in favor of the Turing machine as the definition of "algorithm" or "mechanical procedure" or "formal system".

A hypothesis leading to a natural law?: In late 1936 Alan Turing's paper (also proving that the Entscheidungsproblem is unsolvable) was delivered orally, but had not yet appeared in print. On the other hand, Emil Post's 1936 paper had appeared and was certified independent of Turing's work. Post strongly disagreed with Church's "identification" of effective computability with the λ-calculus and recursion, stating:

"Actually the work already done by Church and others carries this identification considerably beyond the working hypothesis stage. But to mask this identification under a definition… blinds us to the need of its continual verification."

Rather, he regarded the notion of "effective calculability" as merely a "working hypothesis" that might lead by inductive reasoning to a "natural law" rather than by "a definition or an axiom". This idea was "sharply" criticized by Church.

Thus Post in his 1936 paper was also discounting Kurt Gödel's suggestion to Church in 1934–1935 that the thesis might be expressed as an axiom or set of axioms.

Turing adds another definition, Rosser equates all three: Within just a short time, Turing's 1936–1937 paper "On Computable Numbers, with an Application to the Entscheidungsproblem" appeared. In it he stated another notion of "effective computability" with the introduction of his a-machines (now known as the Turing machine abstract computational model). And in a proof-sketch added as an "Appendix" to his 1936–1937 paper, Turing showed that the classes of functions defined by λ-calculus and Turing machines coincided. Church was quick to recognise how compelling Turing's analysis was. In his review of Turing's paper he made clear that Turing's notion made "the identification with effectiveness in the ordinary (not explicitly defined) sense evident immediately".

In a few years (1939) Turing would propose, like Church and Kleene before him, that his formal definition of mechanical computing agent was the correct one. Thus, by 1939, both Church (1934) and Turing (1939) had individually proposed that their "formal systems" should be definitions of "effective calculability"; neither framed their statements as theses.

Rosser (1939) formally identified the three notions-as-definitions:

"All three definitions are equivalent, so it does not matter which one is used."

Kleene proposes Thesis I: This left the overt expression of a "thesis" to Kleene. In 1943 Kleene proposed his "Thesis I":

"This heuristic fact [general recursive functions are effectively calculable] ... led Church to state the following thesis. The same thesis is implicit in Turing's description of computing machines. "Thesis I. Every effectively calculable function (effectively decidable predicate) is general recursive [Kleene's italics]"

Since a precise mathematical definition of the term effectively calculable (effectively decidable) has been wanting, we can take this thesis ... as a definition of it ..."

"...the thesis has the character of an hypothesis—a point emphasized by Post and by Church. If we consider the thesis and its converse as definition, then the hypothesis is an hypothesis about the application of the mathematical theory developed from the definition. For the acceptance of the hypothesis, there are, as we have suggested, quite compelling grounds."

The Church–Turing Thesis: Stephen Kleene, in Introduction To Metamathematics, finally goes on to formally name "Church's Thesis" and "Turing's Thesis", using his theory of recursive realizability. Kleene having switched from presenting his work in the terminology of Church-Kleene lambda definability, to that of Gödel-Kleene recursiveness (partial recursive functions). In this transition, Kleene modified Gödel's general recursive functions to allow for proofs of the unsolvability of problems in the Intuitionism of E. J. Brouwer. In his graduate textbook on logic, "Church's thesis" is introduced and basic mathematical results are demonstrated to be unrealizable. Next, Kleene proceeds to present "Turing's thesis", where results are shown to be uncomputable, using his simplified derivation of a Turing machine based on the work of Emil Post. Both theses are proven equivalent by use of "Theorem XXX".

"Thesis I. Every effectively calculable function (effectively decidable predicate) is general recursive."

"Theorem XXX: The following classes of partial functions are coextensive, i.e. have the same members: (a) the partial recursive functions, (b) the computable functions ..."

"Turing's thesis: Turing's thesis that every function which would naturally be regarded as computable is computable under his definition, i.e. by one of his machines, is equivalent to Church's thesis by Theorem XXX."

Kleene, finally, uses for the first time the term the "Church-Turing thesis" in a section in which he helps to give clarifications to concepts in Alan Turing's paper "The Word Problem in Semi-Groups with Cancellation", as demanded in a critique from William Boone.

Later developments
An attempt to understand the notion of "effective computability" better led Robin Gandy (Turing's student and friend) in 1980 to analyze machine computation (as opposed to human-computation acted out by a Turing machine). Gandy's curiosity about, and analysis of, cellular automata (including Conway's game of life), parallelism, and crystalline automata, led him to propose four "principles (or constraints) ... which it is argued, any machine must satisfy". His most-important fourth, "the principle of causality" is based on the "finite velocity of propagation of effects and signals; contemporary physics rejects the possibility of instantaneous action at a distance". From these principles and some additional constraints—(1a) a lower bound on the linear dimensions of any of the parts, (1b) an upper bound on speed of propagation (the velocity of light), (2) discrete progress of the machine, and (3) deterministic behavior—he produces a theorem that "What can be calculated by a device satisfying principles I–IV is computable."

In the late 1990s Wilfried Sieg analyzed Turing's and Gandy's notions of "effective calculability" with the intent of "sharpening the informal notion, formulating its general features axiomatically, and investigating the axiomatic framework". In his 1997 and 2002 work Sieg presents a series of constraints on the behavior of a computor—"a human computing agent who proceeds mechanically". These constraints reduce to:
 * "(B.1) (Boundedness) There is a fixed bound on the number of symbolic configurations a computor can immediately recognize.
 * "(B.2) (Boundedness) There is a fixed bound on the number of internal states a computor can be in.
 * "(L.1) (Locality) A computor can change only elements of an observed symbolic configuration.
 * "(L.2) (Locality) A computor can shift attention from one symbolic configuration to another one, but the new observed configurations must be within a bounded distance of the immediately previously observed configuration.
 * "(D) (Determinacy) The immediately recognizable (sub-)configuration determines uniquely the next computation step (and id [instantaneous description])"; stated another way: "A computor's internal state together with the observed configuration fixes uniquely the next computation step and the next internal state."

The matter remains in active discussion within the academic community.

The thesis as a definition
The thesis can be viewed as nothing but an ordinary mathematical definition. Comments by Gödel on the subject suggest this view, e.g. "the correct definition of mechanical computability was established beyond any doubt by Turing". The case for viewing the thesis as nothing more than a definition is made explicitly by Robert I. Soare, where it is also argued that Turing's definition of computability is no less likely to be correct than the epsilon-delta definition of a continuous function.

Success of the thesis
Other formalisms (besides recursion, the λ-calculus, and the Turing machine) have been proposed for describing effective calculability/computability. Kleene (1952) adds to the list the functions "reckonable in the system S1" of Kurt Gödel 1936, and Emil Post's (1943, 1946) "canonical [also called normal] systems". In the 1950s Hao Wang and Martin Davis greatly simplified the one-tape Turing-machine model (see Post–Turing machine). Marvin Minsky expanded the model to two or more tapes and greatly simplified the tapes into "up-down counters", which Melzak and Lambek further evolved into what is now known as the counter machine model. In the late 1960s and early 1970s researchers expanded the counter machine model into the register machine, a close cousin to the modern notion of the computer. Other models include combinatory logic and Markov algorithms. Gurevich adds the pointer machine model of Kolmogorov and Uspensky (1953, 1958): "... they just wanted to ... convince themselves that there is no way to extend the notion of computable function."

All these contributions involve proofs that the models are computationally equivalent to the Turing machine; such models are said to be Turing complete. Because all these different attempts at formalizing the concept of "effective calculability/computability" have yielded equivalent results, it is now generally assumed that the Church–Turing thesis is correct. In fact, Gödel (1936) proposed something stronger than this; he observed that there was something "absolute" about the concept of "reckonable in S1":

"It may also be shown that a function which is computable ['reckonable'] in one of the systems Si, or even in a system of transfinite type, is already computable [reckonable] in S1. Thus the concept 'computable' ['reckonable'] is in a certain definite sense 'absolute', while practically all other familiar metamathematical concepts (e.g. provable, definable, etc.) depend quite essentially on the system to which they are defined ..."

Informal usage in proofs
Proofs in computability theory often invoke the Church–Turing thesis in an informal way to establish the computability of functions while avoiding the (often very long) details which would be involved in a rigorous, formal proof. To establish that a function is computable by Turing machine, it is usually considered sufficient to give an informal English description of how the function can be effectively computed, and then conclude "by the Church–Turing thesis" that the function is Turing computable (equivalently, partial recursive).

Dirk van Dalen gives the following example for the sake of illustrating this informal use of the Church–Turing thesis:

"Example: Each infinite RE set contains an infinite recursive set.

Proof: Let A be infinite RE. We list the elements of A effectively, n0, n1, n2, n3, ...

From this list we extract an increasing sublist: put m0 = n0, after finitely many steps we find an nk such that nk > m0, put m1 = nk. We repeat this procedure to find m2 > m1, etc. this yields an effective listing of the subset B={m0, m1, m2,...} of A, with the property mi < mi+1.

Claim. B is decidable. For, in order to test k in B we must check if k = mi for some i. Since the sequence of mi's is increasing we have to produce at most k+1 elements of the list and compare them with k. If none of them is equal to k, then k not in B. Since this test is effective, B is decidable and, by Church's thesis, recursive."

In order to make the above example completely rigorous, one would have to carefully construct a Turing machine, or λ-function, or carefully invoke recursion axioms, or at best, cleverly invoke various theorems of computability theory. But because the computability theorist believes that Turing computability correctly captures what can be computed effectively, and because an effective procedure is spelled out in English for deciding the set B, the computability theorist accepts this as proof that the set is indeed recursive.

Variations
The success of the Church–Turing thesis prompted variations of the thesis to be proposed. For example, the physical Church–Turing thesis states: "All physically computable functions are Turing-computable."

The Church–Turing thesis says nothing about the efficiency with which one model of computation can simulate another. It has been proved for instance that a (multi-tape) universal Turing machine only suffers a logarithmic slowdown factor in simulating any Turing machine.

A variation of the Church–Turing thesis addresses whether an arbitrary but "reasonable" model of computation can be efficiently simulated. This is called the feasibility thesis, also known as the (classical) complexity-theoretic Church–Turing thesis or the extended Church–Turing thesis, which is not due to Church or Turing, but rather was realized gradually in the development of complexity theory. It states: "A probabilistic Turing machine can efficiently simulate any realistic model of computation." The word 'efficiently' here means up to polynomial-time reductions. This thesis was originally called computational complexity-theoretic Church–Turing thesis by Ethan Bernstein and Umesh Vazirani (1997). The complexity-theoretic Church–Turing thesis, then, posits that all 'reasonable' models of computation yield the same class of problems that can be computed in polynomial time. Assuming the conjecture that probabilistic polynomial time (BPP) equals deterministic polynomial time (P), the word 'probabilistic' is optional in the complexity-theoretic Church–Turing thesis. A similar thesis, called the invariance thesis, was introduced by Cees F. Slot and Peter van Emde Boas. It states: Reasonable' machines can simulate each other within a polynomially bounded overhead in time and a constant-factor overhead in space." The thesis originally appeared in a paper at STOC'84, which was the first paper to show that polynomial-time overhead and constant-space overhead could be simultaneously achieved for a simulation of a Random Access Machine on a Turing machine.

If BQP is shown to be a strict superset of BPP, it would invalidate the complexity-theoretic Church–Turing thesis. In other words, there would be efficient quantum algorithms that perform tasks that do not have efficient probabilistic algorithms. This would not however invalidate the original Church–Turing thesis, since a quantum computer can always be simulated by a Turing machine, but it would invalidate the classical complexity-theoretic Church–Turing thesis for efficiency reasons. Consequently, the quantum complexity-theoretic Church–Turing thesis states: "A quantum Turing machine can efficiently simulate any realistic model of computation."

Eugene Eberbach and Peter Wegner claim that the Church–Turing thesis is sometimes interpreted too broadly, stating "Though [...] Turing machines express the behavior of algorithms, the broader assertion that algorithms precisely capture what can be computed is invalid". They claim that forms of computation not captured by the thesis are relevant today, terms which they call super-Turing computation.

Philosophical implications
Philosophers have interpreted the Church–Turing thesis as having implications for the philosophy of mind. B. Jack Copeland states that it is an open empirical question whether there are actual deterministic physical processes that, in the long run, elude simulation by a Turing machine; furthermore, he states that it is an open empirical question whether any such processes are involved in the working of the human brain. There are also some important open questions which cover the relationship between the Church–Turing thesis and physics, and the possibility of hypercomputation. When applied to physics, the thesis has several possible meanings:

 * 1) The universe is equivalent to a Turing machine; thus, computing non-recursive functions is physically impossible. This has been termed the strong Church–Turing thesis, or Church–Turing–Deutsch principle, and is a foundation of digital physics.
 * 2) The universe is not equivalent to a Turing machine (i.e., the laws of physics are not Turing-computable), but incomputable physical events are not "harnessable" for the construction of a hypercomputer. For example, a universe in which physics involves random real numbers, as opposed to computable reals, would fall into this category.
 * 3) The universe is a hypercomputer, and it is possible to build physical devices to harness this property and calculate non-recursive functions. For example, it is an open question whether all quantum mechanical events are Turing-computable, although it is known that rigorous models such as quantum Turing machines are equivalent to deterministic Turing machines. (They are not necessarily efficiently equivalent; see above.) John Lucas and Roger Penrose have suggested that the human mind might be the result of some kind of quantum-mechanically enhanced, "non-algorithmic" computation.

There are many other technical possibilities which fall outside or between these three categories, but these serve to illustrate the range of the concept.

Philosophical aspects of the thesis, regarding both physical and biological computers, are also discussed in Odifreddi's 1989 textbook on recursion theory.

Non-computable functions
One can formally define functions that are not computable. A well-known example of such a function is the Busy Beaver function. This function takes an input n and returns the largest number of symbols that a Turing machine with n states can print before halting, when run with no input. Finding an upper bound on the busy beaver function is equivalent to solving the halting problem, a problem known to be unsolvable by Turing machines. Since the busy beaver function cannot be computed by Turing machines, the Church–Turing thesis states that this function cannot be effectively computed by any method.

Several computational models allow for the computation of (Church-Turing) non-computable functions. These are known as hypercomputers. Mark Burgin argues that super-recursive algorithms such as inductive Turing machines disprove the Church–Turing thesis. His argument relies on a definition of algorithm broader than the ordinary one, so that non-computable functions obtained from some inductive Turing machines are called computable. This interpretation of the Church–Turing thesis differs from the interpretation commonly accepted in computability theory, discussed above. The argument that super-recursive algorithms are indeed algorithms in the sense of the Church–Turing thesis has not found broad acceptance within the computability research community.

title: Church–Turing thesis


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Chomsky (surname)

Chomsky (Chomski, Хомскі, Хомский, Хомський, חומסקי, "from (Vyoska) Chomsk/ Khomsk (nearby Brest, now Belarus)") is a surname of Slavic origin. Notable people with the surname include:

 * Alejandro Chomski (born 1968), Argentine film director and screenwriter
 * Aviva Chomsky (born 1957), American historian
 * Carol (Schatz) Chomsky (1930–2008), American linguist and wife of Noam Chomsky
 * Judith Chomsky (born 1942), American human rights lawyer and co-founder of the Juvenile Law Center
 * Marvin J. Chomsky (1929–2022), American television and film director
 * Noam Chomsky (born 1928), American linguist and political activist, professor emeritus at MIT
 * Stanisław Chomski (born 1957), Polish speedway rider and coach
 * William Chomsky (1896–1977), American scholar of Hebrew
 * Pavel Osipovich Khomsky (1925–2016), Soviet and Russian theater director

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Chomsky (surname)


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Computer multitasking

In computing, multitasking is the concurrent execution of multiple tasks (also known as processes) over a certain period of time. New tasks can interrupt already started ones before they finish, instead of waiting for them to end. As a result, a computer executes segments of multiple tasks in an interleaved manner, while the tasks share common processing resources such as central processing units (CPUs) and main memory. Multitasking automatically interrupts the running program, saving its state (partial results, memory contents and computer register contents) and loading the saved state of another program and transferring control to it. This "context switch" may be initiated at fixed time intervals (pre-emptive multitasking), or the running program may be coded to signal to the supervisory software when it can be interrupted (cooperative multitasking).

Multitasking does not require parallel execution of multiple tasks at exactly the same time; instead, it allows more than one task to advance over a given period of time. Even on multiprocessor computers, multitasking allows many more tasks to be run than there are CPUs.

Multitasking is a common feature of computer operating systems since at least the 1960s. It allows more efficient use of the computer hardware; when a program is waiting for some external event such as a user input or an input/output transfer with a peripheral to complete, the central processor can still be used with another program. In a time-sharing system, multiple human operators use the same processor as if it was dedicated to their use, while behind the scenes the computer is serving many users by multitasking their individual programs. In multiprogramming systems, a task runs until it must wait for an external event or until the operating system's scheduler forcibly swaps the running task out of the CPU. Real-time systems such as those designed to control industrial robots, require timely processing; a single processor might be shared between calculations of machine movement, communications, and user interface.

Often multitasking operating systems include measures to change the priority of individual tasks, so that important jobs receive more processor time than those considered less significant. Depending on the operating system, a task might be as large as an entire application program, or might be made up of smaller threads that carry out portions of the overall program.

A processor intended for use with multitasking operating systems may include special hardware to securely support multiple tasks, such as memory protection, and protection rings that ensure the supervisory software cannot be damaged or subverted by user-mode program errors.

The term "multitasking" has become an international term, as the same word is used in many other languages such as German, Italian, Dutch, Romanian, Czech, Danish and Norwegian.

Multiprogramming
In the early days of computing, CPU time was expensive, and peripherals were very slow. When the computer ran a program that needed access to a peripheral, the central processing unit (CPU) would have to stop executing program instructions while the peripheral processed the data. This was usually very inefficient. Multiprogramming is a computing technique that enables multiple programs to be concurrently loaded and executed into a computer's memory, allowing the CPU to switch between them swiftly. This optimizes CPU utilization by keeping it engaged with the execution of tasks, particularly useful when one program is waiting for I/O operations to complete.

The Bull Gamma 60, initially designed in 1957 and first released in 1960, was the first computer designed with multiprogramming in mind. Its architecture featured a central memory and a Program Distributor feeding up to twenty-five autonomous processing units with code and data, and allowing concurrent operation of multiple clusters.

Another such computer was the LEO III, first released in 1961. During batch processing, several different programs were loaded in the computer memory, and the first one began to run. When the first program reached an instruction waiting for a peripheral, the context of this program was stored away, and the second program in memory was given a chance to run. The process continued until all programs finished running.

The use of multiprogramming was enhanced by the arrival of virtual memory and virtual machine technology, which enabled individual programs to make use of memory and operating system resources as if other concurrently running programs were, for all practical purposes, nonexistent.

Multiprogramming gives no guarantee that a program will run in a timely manner. Indeed, the first program may very well run for hours without needing access to a peripheral. As there were no users waiting at an interactive terminal, this was no problem: users handed in a deck of punched cards to an operator, and came back a few hours later for printed results. Multiprogramming greatly reduced wait times when multiple batches were being processed.

Cooperative multitasking
Early multitasking systems used applications that voluntarily ceded time to one another. This approach, which was eventually supported by many computer operating systems, is known today as cooperative multitasking. Although it is now rarely used in larger systems except for specific applications such as CICS or the JES2 subsystem, cooperative multitasking was once the only scheduling scheme employed by Microsoft Windows and classic Mac OS to enable multiple applications to run simultaneously. Cooperative multitasking is still used today on RISC OS systems.

As a cooperatively multitasked system relies on each process regularly giving up time to other processes on the system, one poorly designed program can consume all of the CPU time for itself, either by performing extensive calculations or by busy waiting; both would cause the whole system to hang. In a server environment, this is a hazard that makes the entire environment unacceptably fragile.

Preemptive multitasking
Preemptive multitasking allows the computer system to more reliably guarantee to each process a regular "slice" of operating time. It also allows the system to deal rapidly with important external events like incoming data, which might require the immediate attention of one or another process. Operating systems were developed to take advantage of these hardware capabilities and run multiple processes preemptively. Preemptive multitasking was implemented in the PDP-6 Monitor and Multics in 1964, in OS/360 MFT in 1967, and in Unix in 1969, and was available in some operating systems for computers as small as DEC's PDP-8; it is a core feature of all Unix-like operating systems, such as Linux, Solaris and BSD with its derivatives, as well as modern versions of Windows.

At any specific time, processes can be grouped into two categories: those that are waiting for input or output (called "I/O bound"), and those that are fully utilizing the CPU ("CPU bound"). In primitive systems, the software would often "poll", or "busywait" while waiting for requested input (such as disk, keyboard or network input). During this time, the system was not performing useful work. With the advent of interrupts and preemptive multitasking, I/O bound processes could be "blocked", or put on hold, pending the arrival of the necessary data, allowing other processes to utilize the CPU. As the arrival of the requested data would generate an interrupt, blocked processes could be guaranteed a timely return to execution.

Possibly the earliest preemptive multitasking OS available to home users was Microware's OS-9, available for computers based on the Motorola 6809 such as the TRS-80 Color Computer 2, with the operating system supplied by Tandy as an upgrade for disk-equipped systems. Sinclair QDOS on the Sinclair QL followed in 1984, but it was not a big success. Commodore's Amiga was released the following year, offering a combination of multitasking and multimedia capabilities. Microsoft made preemptive multitasking a core feature of their flagship operating system in the early 1990s when developing Windows NT 3.1 and then Windows 95. In 1988 Apple offered A/UX as a UNIX System V-based alternative to the Classic Mac OS. In 2001 Apple switched to the NeXTSTEP-influenced Mac OS X.

A similar model is used in Windows 9x and the Windows NT family, where native 32-bit applications are multitasked preemptively. 64-bit editions of Windows, both for the x86-64 and Itanium architectures, no longer support legacy 16-bit applications, and thus provide preemptive multitasking for all supported applications.

Real time
Another reason for multitasking was in the design of real-time computing systems, where there are a number of possibly unrelated external activities needed to be controlled by a single processor system. In such systems a hierarchical interrupt system is coupled with process prioritization to ensure that key activities were given a greater share of available process time.

Multithreading
As multitasking greatly improved the throughput of computers, programmers started to implement applications as sets of cooperating processes (e. g., one process gathering input data, one process processing input data, one process writing out results on disk). This, however, required some tools to allow processes to efficiently exchange data.

Threads were born from the idea that the most efficient way for cooperating processes to exchange data would be to share their entire memory space. Thus, threads are effectively processes that run in the same memory context and share other resources with their parent processes, such as open files. Threads are described as lightweight processes because switching between threads does not involve changing the memory context.

While threads are scheduled preemptively, some operating systems provide a variant to threads, named fibers, that are scheduled cooperatively. On operating systems that do not provide fibers, an application may implement its own fibers using repeated calls to worker functions. Fibers are even more lightweight than threads, and somewhat easier to program with, although they tend to lose some or all of the benefits of threads on machines with multiple processors.

Some systems directly support multithreading in hardware.

Memory protection
Essential to any multitasking system is to safely and effectively share access to system resources. Access to memory must be strictly managed to ensure that no process can inadvertently or deliberately read or write to memory locations outside the process's address space. This is done for the purpose of general system stability and data integrity, as well as data security.

In general, memory access management is a responsibility of the operating system kernel, in combination with hardware mechanisms that provide supporting functionalities, such as a memory management unit (MMU). If a process attempts to access a memory location outside its memory space, the MMU denies the request and signals the kernel to take appropriate actions; this usually results in forcibly terminating the offending process. Depending on the software and kernel design and the specific error in question, the user may receive an access violation error message such as "segmentation fault".

In a well designed and correctly implemented multitasking system, a given process can never directly access memory that belongs to another process. An exception to this rule is in the case of shared memory; for example, in the System V inter-process communication mechanism the kernel allocates memory to be mutually shared by multiple processes. Such features are often used by database management software such as PostgreSQL.

Inadequate memory protection mechanisms, either due to flaws in their design or poor implementations, allow for security vulnerabilities that may be potentially exploited by malicious software.

Memory swapping
Use of a swap file or swap partition is a way for the operating system to provide more memory than is physically available by keeping portions of the primary memory in secondary storage. While multitasking and memory swapping are two completely unrelated techniques, they are very often used together, as swapping memory allows more tasks to be loaded at the same time. Typically, a multitasking system allows another process to run when the running process hits a point where it has to wait for some portion of memory to be reloaded from secondary storage.

Programming
Processes that are entirely independent are not much trouble to program in a multitasking environment. Most of the complexity in multitasking systems comes from the need to share computer resources between tasks and to synchronize the operation of co-operating tasks.

Various concurrent computing techniques are used to avoid potential problems caused by multiple tasks attempting to access the same resource.

Bigger systems were sometimes built with a central processor(s) and some number of I/O processors, a kind of asymmetric multiprocessing.

Over the years, multitasking systems have been refined. Modern operating systems generally include detailed mechanisms for prioritizing processes, while symmetric multiprocessing has introduced new complexities and capabilities.

title: Computer multitasking