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Counter Machine
A counter machine is an abstract machine used in a formal logic and theoretical computer science to model computation. It is the most primitive of the four types of register machines. A counter machine comprises a set of one or more unbounded ''registers'', each of which can hold a single non-negative integer, and a list of (usually sequential) arithmetic and control instructions for the machine to follow. The counter machine is typically used in the process of designing parallel algorithms in relation to the mutual exclusion principle. When used in this manner, the counter machine is used to model the discrete time-steps of a computational system in relation to memory accesses. By modeling computations in relation to the memory accesses for each respective computational step, parallel algorithms may be designed in such a matter to avoid interlocking, the simultaneous writing operation by two (or more) threads to the same memory address. Basic features For a given counter machin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Machine
An abstract machine is a computer science theoretical model that allows for a detailed and precise analysis of how a computer system functions. It is analogous to a mathematical function in that it receives inputs and produces outputs based on predefined rules. Abstract machines vary from literal machines in that they are expected to perform correctly and independently of hardware. Abstract machines are “machines” because they allow step-by-step execution of programmes; they are “ abstract” because they ignore many aspects of actual ( hardware) machines. A typical abstract machine consists of a definition in terms of input, output, and the set of allowable operations used to turn the former into the latter. They can be used for purely theoretical reasons as well as models for real-world computer systems. In the theory of computation, abstract machines are often used in thought experiments regarding computability or to analyse the complexity of algorithms. This use of abstr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Program
A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components. A computer program in its human-readable form is called source code. Source code needs another computer program to execute because computers can only execute their native machine instructions. Therefore, source code may be translated to machine instructions using the language's compiler. ( Assembly language programs are translated using an assembler.) The resulting file is called an executable. Alternatively, source code may execute within the language's interpreter. If the executable is requested for execution, then the operating system loads it into memory and starts a process. The central processing unit will soon switch to this process so it can fetch, decode, and then execute each machine instruction. If the source code is requested for execution, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel Numbering
In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was developed by Kurt Gödel for the proof of his incompleteness theorems. () A Gödel numbering can be interpreted as an encoding in which a number is assigned to each symbol of a mathematical notation, after which a sequence of natural numbers can then represent a sequence of symbols. These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic. Since the publishing of Gödel's paper in 1931, the term "Gödel numbering" or "Gödel code" has been used to refer to more general assignments of natural numbers to mathematical objects. Simplified overview Gödel noted that each statement within a system can be represented by a natural number (its ''Gödel number''). The significance of this was th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Turing Machine
In computer science, a universal Turing machine (UTM) is a Turing machine that can simulate an arbitrary Turing machine on arbitrary input. The universal machine essentially achieves this by reading both the description of the machine to be simulated as well as the input to that machine from its own tape. Alan Turing introduced the idea of such a machine in 1936–1937. This principle is considered to be the origin of the idea of a stored-program computer used by John von Neumann in 1946 for the "Electronic Computing Instrument" that now bears von Neumann's name: the von Neumann architecture. Martin Davis, ''The universal computer : the road from Leibniz to Turing'' (2017) In terms of computational complexity, a multi-tape universal Turing machine need only be slower by logarithmic factor compared to the machines it simulates. Introduction Every Turing machine computes a certain fixed partial computable function from the input strings over its alphabet. In that sense it be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Recursive Function
In mathematical logic and computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "computable" in an intuitive sense – as well as in a formal one. If the function is total, it is also called a total recursive function (sometimes shortened to recursive function). In computability theory, it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines (this is one of the theorems that supports the Church–Turing thesis). The μ-recursive functions are closely related to primitive recursive functions, and their inductive definition (below) builds upon that of the primitive recursive functions. However, not every total recursive function is a primitive recursive function—the most famous example is the Ackermann function. Other equivalent classes of functions are the functions of lambda calculus and the functions tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Recursion
In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined before entering the loop). Primitive recursive functions form a strict subset of those general recursive functions that are also total functions. The importance of primitive recursive functions lies in the fact that most computable functions that are studied in number theory (and more generally in mathematics) are primitive recursive. For example, addition and division, the factorial and exponential function, and the function which returns the ''n''th prime are all primitive recursive. In fact, for showing that a computable function is primitive recursive, it suffices to show that its time complexity is bounded above by a primitive recursive function of the input size. It is hence not that easy to devise a computable function that is ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion (computer Science)
In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science. Most computer programming languages support recursion by allowing a function to call itself from within its own code. Some functional programming languages (for instance, Clojure) do not define any looping constructs but rely solely on recursion to repeatedly call code. It is proved in computability theory that these recursive-only languages are Turing complete; this means that they are as powerful (they can be used to solve the same problems) as imperative languages based on control structures such as and . Repeatedly calling a function from within itself may cause the call stack to have a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
μ Operator
In computability theory, the μ-operator, minimization operator, or unbounded search operator searches for the least natural number with a given property. Adding the μ-operator to the five primitive recursive operators makes it possible to define all computable functions. Definition Suppose that R(''y'', ''x''1, ..., ''x''''k'') is a fixed (''k''+1)-ary relation on the natural numbers. The μ-operator "μ''y''", in either the unbounded or bounded form, is a "number theoretic function" defined from the natural numbers to the natural numbers. However, "μ''y''" contains a ''predicate'' over the natural numbers that delivers ''true'' when the predicate is satisfied and ''false'' when it is not. The ''bounded'' μ-operator appears earlier in Kleene (1952) ''Chapter IX Primitive Recursive Functions, §45 Predicates, prime factor representation'' as: :"\mu y_ R(y). \ \ \mbox \ y |
Turing Machine Equivalents
A Turing machine is a hypothetical computing device, first conceived by Alan Turing in 1936. Turing machines manipulate symbols on a potentially infinite strip of tape according to a finite table of rules, and they provide the theoretical underpinnings for the notion of a computer algorithm. While none of the following models have been shown to have more power than the single-tape, one-way infinite, multi-symbol Turing-machine model, their authors defined and used them to investigate questions and solve problems more easily than they could have if they had stayed with Turing's ''a''-machine model. Machines equivalent to the Turing machine model Turing equivalence 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 (i.e. more mathematical funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Recursive Function
In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined before entering the loop). Primitive recursive functions form a strict subset of those general recursive functions that are also total functions. The importance of primitive recursive functions lies in the fact that most computable functions that are studied in number theory (and more generally in mathematics) are primitive recursive. For example, addition and division, the factorial and exponential function, and the function which returns the ''n''th prime are all primitive recursive. In fact, for showing that a computable function is primitive recursive, it suffices to show that its time complexity is bounded above by a primitive recursive function of the input size. It is hence not that easy to devise a computable function that is ''n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
One-instruction Set Computer
A one-instruction set computer (OISC), sometimes called an ultimate reduced instruction set computer (URISC), is an abstract machine that uses only one instructionobviating the need for a machine language opcode. With a judicious choice for the single instruction and given infinite resources, an OISC is capable of being a universal computer in the same manner as traditional computers that have multiple instructions. OISCs have been recommended as aids in teaching computer architecture and have been used as computational models in structural computing research. The first carbon nanotube computer is a 1-bit one-instruction set computer (and has only 178 transistors). Machine architecture In a Turing-complete model, each memory location can store an arbitrary integer, anddepending on the modelthere may be arbitrarily many locations. The instructions themselves reside in memory as a sequence of such integers. There exists a class of universal computers with a single instruction ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Register-machine Model
Although some authors use the name "register machine" synonymously with "counter machine", this article will give details and examples of only of the most primitive speciesthe "counter machine"of the genus "register machine." Within the species "counter machine" there are a number of varieties: the models of Hermes (1954), Kaphengst (1957), Ershov (1958), Péter (1958), Minsky (1961) and Minsky (1967), Melzak (1961), Lambek (1961), Shepherdson and Sturgis (1963), and Schönhage (1980). These models will be described in more detail in the following. The models in more detail 1954: Hermes' model Shepherdson and Sturgis observe that "the proof of this universality f digital computers to Turing machines... seems to have been first written down by Hermes, who showed in --their reference numberhow an idealized computer could be programmed to duplicate the behavior of any Turing machine" (Shepherdson and Sturgis, p. 219). Shepherdson and Sturgis observe that: :"Kaphengst' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
