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501 | Reversible cellular automata are often used to simulate such physical phenomena as gas and fluid dynamics, since they obey the laws of thermodynamics. Such cellular automata have rules specially constructed to be reversible. Such systems have been studied by Tommaso Toffoli, Norman Margolus and others. Several techniqu... |
502 | A special class of cellular automata are totalistic cellular automata. The state of each cell in a totalistic cellular automaton is represented by a number , and the value of a cell at time t depends only on the sum of the values of the cells in its neighborhood at time t − 1. If the state of the cell at time t depend... |
503 | There are many possible generalizations of the cellular automaton concept. |
504 | One way is by using something other than a rectangular grid. For example, if a plane is tiled with regular hexagons, those hexagons could be used as cells. In many cases the resulting cellular automata are equivalent to those with rectangular grids with specially designed neighborhoods and rules. Another variation wou... |
505 | Also, rules can be probabilistic rather than deterministic. Such cellular automata are called probabilistic cellular automata. A probabilistic rule gives, for each pattern at time t, the probabilities that the central cell will transition to each possible state at time t + 1. Sometimes a simpler rule is used; for examp... |
506 | The neighborhood or rules could change over time or space. For example, initially the new state of a cell could be determined by the horizontally adjacent cells, but for the next generation the vertical cells would be used. |
507 | In cellular automata, the new state of a cell is not affected by the new state of other cells. This could be changed so that, for instance, a 2 by 2 block of cells can be determined by itself and the cells adjacent to itself. |
508 | There are continuous automata. These are like totalistic cellular automata, but instead of the rule and states being discrete , continuous functions are used, and the states become continuous . The state of a location is a finite number of real numbers. Certain cellular automata can yield diffusion in liquid patterns i... |
509 | Continuous spatial automata have a continuum of locations. The state of a location is a finite number of real numbers. Time is also continuous, and the state evolves according to differential equations. One important example is reaction–diffusion textures, differential equations proposed by Alan Turing to explain how c... |
510 | There are known examples of continuous spatial automata, which exhibit propagating phenomena analogous to gliders in the Game of Life. |
511 | Graph rewriting automata are extensions of cellular automata based on graph rewriting systems. |
512 | The simplest nontrivial cellular automaton would be one-dimensional, with two possible states per cell, and a cell's neighbors defined as the adjacent cells on either side of it. A cell and its two neighbors form a neighborhood of 3 cells, so there are 23 = 8 possible patterns for a neighborhood. A rule consists of dec... |
513 | These 256 cellular automata are generally referred to by their Wolfram code, a standard naming convention invented by Wolfram that gives each rule a number from 0 to 255. A number of papers have analyzed and compared these 256 cellular automata. The rule 30, rule 90, rule 110, and rule 184 cellular automata are particu... |
514 | Rule 30 exhibits class 3 behavior, meaning even simple input patterns such as that shown lead to chaotic, seemingly random histories. |
515 | Rule 110, like the Game of Life, exhibits what Wolfram calls class 4 behavior, which is neither completely random nor completely repetitive. Localized structures appear and interact in various complicated-looking ways. In the course of the development of A New Kind of Science, as a research assistant to Wolfram in 1994... |
516 | An elementary cellular automaton rule is specified by 8 bits, and all elementary cellular automaton rules can be considered to sit on the vertices of the 8-dimensional unit hypercube. This unit hypercube is the cellular automaton rule space. For next-nearest-neighbor cellular automata, a rule is specified by 25 = 32 bi... |
517 | Cellular automaton rule space allows us to ask the question concerning whether rules with similar dynamical behavior are "close" to each other. Graphically drawing a high dimensional hypercube on the 2-dimensional plane remains a difficult task, and one crude locator of a rule in the hypercube is the number of bit-1 in... |
518 | For larger cellular automaton rule space, it is shown that class 4 rules are located between the class 1 and class 3 rules. This observation is the foundation for the phrase edge of chaos, and is reminiscent of the phase transition in thermodynamics. |
519 | Several biological processes occur—or can be simulated—by cellular automata. |
520 | Some examples of biological phenomena modeled by cellular automata with a simple state space are: |
521 | Additionally, biological phenomena which require explicit modeling of the agents' velocities may be modeled by cellular automata with a more complex state space and rules, such as biological lattice-gas cellular automata. These include phenomena of great medical importance, such as: |
522 | The Belousov–Zhabotinsky reaction is a spatio-temporal chemical oscillator that can be simulated by means of a cellular automaton. In the 1950s A. M. Zhabotinsky discovered that when a thin, homogenous layer of a mixture of malonic acid, acidified bromate, and a ceric salt were mixed together and left undisturbed, fas... |
523 | Probabilistic cellular automata are used in statistical and condensed matter physics to study phenomena like fluid dynamics and phase transitions. The Ising model is a prototypical example, in which each cell can be in either of two states called "up" and "down", making an idealized representation of a magnet. By adjus... |
524 | Cellular automaton processors are physical implementations of CA concepts, which can process information computationally. Processing elements are arranged in a regular grid of identical cells. The grid is usually a square tiling, or tessellation, of two or three dimensions; other tilings are possible, but not yet used.... |
525 | Rule 30 was originally suggested as a possible block cipher for use in cryptography. Two-dimensional cellular automata can be used for constructing a pseudorandom number generator. |
526 | Cellular automata have been proposed for public-key cryptography. The one-way function is the evolution of a finite CA whose inverse is believed to be hard to find. Given the rule, anyone can easily calculate future states, but it appears to be very difficult to calculate previous states. Cellular automata have also b... |
527 | Other problems that can be solved with cellular automata include: |
528 | Cellular automata have been used in generative music and evolutionary music composition and procedural terrain generation in video games. |
529 | For a random starting pattern, these maze-generating cellular automata will evolve into complex mazes with well-defined walls outlining corridors. Mazecetric, which has the rule B3/S1234 has a tendency to generate longer and straighter corridors compared with Maze, with the rule B3/S12345. Since these cellular automato... |
530 | Specific cellular automata rules include: |
531 | In computability theory, the Church–Turing 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 B... |
532 | 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... |
533 | 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 ... |
534 | Since its inception, variations on the original thesis have arisen, including statements about what can physically be realized by a computer in our universe and what can be efficiently computed ). These variations are not due to Church or Turing, but arise from later work in complexity theory and digital physics. The ... |
535 | J. B. Rosser addresses the notion of "effective computability" as follows: "Clearly the existence of CC and RC 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 answ... |
536 | 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 O... |
537 | 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 calculabili... |
538 | 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... |
539 | 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 . But he did not think that the two ideas could be satisfactorily identified "except heuristically". |
540 | 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 to show that the two calculi are equivalent. Church subsequently modified his methods to include... |
541 | Many years later in a letter to Davis , Gödel said that "he was, at the time of these 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 "algori... |
542 | A hypothesis leading to a natural law?: In late 1936 Alan Turing's paper 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... |
543 | 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. |
544 | 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. |
545 | 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 . And in a proof-sketch adde... |
546 | In a few years 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 and Turing had individually proposed that their "formal systems" should be definitions of "effective calculability"; neither framed their st... |
547 | Rosser formally identified the three notions-as-definitions: |
548 | Kleene proposes Thesis I: This left the overt expression of a "thesis" to Kleene. In 1943 Kleene proposed his "Thesis I": |
549 | Since a precise mathematical definition of the term effectively calculable has been wanting, we can take this thesis ... as a definition of it ... |
550 | 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-K... |
551 | 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. |
552 | An attempt to understand the notion of "effective computability" better led Robin Gandy in 1980 to analyze machine computation . Gandy's curiosity about, and analysis of, cellular automata , parallelism, and crystalline automata, led him to propose four "principles ... which it is argued, any machine must satisfy". H... |
553 | 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 behavi... |
554 | The matter remains in active discussion within the academic community. |
555 | 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 R... |
556 | Other formalisms have been proposed for describing effective calculability/computability. Kleene adds to the list the functions "reckonable in the system S1" of Kurt Gödel 1936, and Emil Post's "canonical systems". In the 1950s Hao Wang and Martin Davis greatly simplified the one-tape Turing-machine model . Marvin ... |
557 | 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... |
558 | Proofs in computability theory often invoke the Church–Turing thesis in an informal way to establish the computability of functions while avoiding the 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 a... |
559 | Dirk van Dalen gives the following example for the sake of illustrating this informal use of the Church–Turing thesis: |
560 | Proof: Let A be infinite RE. We list the elements of A effectively, n0, n1, n2, n3, ... |
561 | 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. |
562 | 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 thes... |
563 | 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 c... |
564 | 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.": 101 |
565 | 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 universal Turing machine only suffers a logarithmic slowdown factor in simulating any Turing machine. |
566 | 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 complexity-theoretic Church–Turing thesis or the extended Church–Turing thesis, which is not due to Church or Turing, but r... |
567 | 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, sinc... |
568 | 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 t... |
569 | 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... |
570 | 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. |
571 | The universe is not equivalent to a Turing machine , 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. |
572 | 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 equ... |
573 | There are many other technical possibilities which fall outside or between these three categories, but these serve to illustrate the range of the concept. |
574 | Philosophical aspects of the thesis, regarding both physical and biological computers, are also discussed in Odifreddi's 1989 textbook on recursion theory.: 101–123 |
575 | 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 bus... |
576 | Several computational models allow for the computation of 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, ... |
577 | An analog signal is any continuous-time signal representing some other quantity, i.e., analogous to another quantity. For example, in an analog audio signal, the instantaneous signal voltage varies continuously with the pressure of the sound waves. |
578 | In contrast, a digital signal represents the original time-varying quantity as a sampled sequence of quantized values. Digital sampling imposes some bandwidth and dynamic range constraints on the representation and adds quantization error. |
579 | The term analog signal usually refers to electrical signals; however, mechanical, pneumatic, hydraulic, and other systems may also convey or be considered analog signals. |
580 | An analog signal uses some property of the medium to convey the signal's information. For example, an aneroid barometer uses rotary position as the signal to convey pressure information. In an electrical signal, the voltage, current, or frequency of the signal may be varied to represent the information. |
581 | Any information may be conveyed by an analog signal; such a signal may be a measured response to changes in a physical variable, such as sound, light, temperature, position, or pressure. The physical variable is converted to an analog signal by a transducer. For example, sound striking the diaphragm of a microphone ind... |
582 | An analog signal is subject to electronic noise and distortion introduced by communication channels, recording and signal processing operations, which can progressively degrade the signal-to-noise ratio . As the signal is transmitted, copied, or processed, the unavoidable noise introduced in the signal path will accumu... |
583 | Converting an analog signal to digital form introduces a low-level quantization noise into the signal due to finite resolution of digital systems. Once in digital form, the signal can be transmitted, stored, and processed without introducing additional noise or distortion using error detection and correction. |
584 | Noise accumulation in analog systems can be minimized by electromagnetic shielding, balanced lines, low-noise amplifiers and high-quality electrical components. |
585 | Abstract machines are typically categorized into two types based on the quantity of operations they can execute simultaneously at any given moment: deterministic abstract machines and non-deterministic abstract machines. A deterministic abstract machine is a system in which a particular beginning state or condition alw... |
586 | Turing machines, for example, are some of the most fundamental abstract machines in computer science. These machines conduct operations on a tape of any length. Their instructions provide for both modifying the symbols and changing the symbol that the machine’s pointer is currently at. For example, a rudimentary Turin... |
587 | Any implementation of an abstract machine in the case of physical implementation uses some kind of physical device to execute the instructions of a programming language. An abstract machine, however, can also be implemented in software or firmware at levels between the abstract machine and underlying physical device. |
588 | An abstract machine is, intuitively, just an abstraction of the idea of a physical computer. For actual execution, algorithms must be properly formalised using the constructs offered by a programming language. This implies that the algorithms to be executed must be expressed using programming language instructions. The... |
589 | In the late 1950s, the Association for Computing Machinery and other allied organisations developed many proposals for Universal Computer Oriented Language , such as Conway's machine. The UNCOL concept is good, but it has not been widely used due to the poor performance of the generated code. In many areas of computin... |
590 | Abstract machines for object-oriented programming languages are often stack-based and have special access instructions for object fields and methods. In these machines, memory management is often implicit performed by a garbage collector . Smalltalk-80 , Self , and Java are examples of this implementation. |
591 | A string processing language is a computer language that focuses on processing strings rather than numbers. There have been string processing languages in the form of command shells, programming tools, macro processors, and scripting languages for decades. Using a suitable abstract machine has two benefits: increased e... |
592 | The early abstract machines for functional languages, including the SECD machine and Cardelli's Functional Abstract Machine , defined strict evaluation, also known as eager or call-by-value evaluation, in which function arguments are evaluated before the call and precisely once. Recently, the majority of research has ... |
593 | Predicate calculus is the foundation of logic programming languages. The most well-known logic programming language is Prolog. The rules in Prolog are written in a uniform format known as universally quantified 'Horn clauses', which means to begin the calculation that attempts to discover a proof of the objective. The... |
594 | A generic abstract machine is made up of a memory and an interpreter. The memory is used to store data and programs, while the interpreter is the component that executes the instructions included in programs. |
595 | The interpreter must carry out the operations that are unique to the language it is interpreting. However, given the variety of languages, it is conceivable to identify categories of operations and an "execution mechanism" shared by all interpreters. The interpreter's operations and accompanying data structures are div... |
596 | Operations for processing primitive data: |
597 | Operations and data structures for controlling the sequence of execution of operations; |
598 | Operations and data structures for controlling data transfers; |
599 | Operations and data structures for memory management. |
600 | An abstract machine must contain operations for manipulating primitive data types such as strings and integers. For example, integers are nearly universally considered a basic data type for both physical abstract machines and the abstract machines used by many programming languages. The machine carries out the arithmet... |
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