Hypercomputation - Hypercomputation Hypercomputation is the theory of methods for the computation of non-recursive functions. The classes of functions which they can compute is studied in the field known as recursion theory. A similar recent term is super-Turing computation, which has been used in the neural network literature to describe machines with various expanded abilities, including the ability to compute directly on real numbers, the ability to carry out infinitely many computations simultaneously, or the ability to carry out computations with exponentially lower complexity than standard Turing machines. Hypercomputation was first introduced by Alan Turing in his 1939 paper Systems of logic based on ordinals, which investigated mathematical systems in which an oracle was available to compute a single arbitrary (non-recursive) function from naturalss to naturals. Other posited.
Church-Turing thesis - formalisms for describing effective computability have been proposed such as register machines, Emil Post's systems, combinatory definability and Markov algorithms (Markov 1960). All these systems have been shown to compute essentially the same functions as Turing machines; systems like this are called Turing-complete. Because all these different attempts of formalizing the concept of algorithm have yielded equivalent results, it is now generally assumed that the Church-Turing thesis is correct. However, the thesis does not have the status of a theorem and cannot be proven; it is conceivable but unlikely that it could be disproven by exhibiting a method which is universally accepted as being an effective algorithm but which cannot be performed on a Turing machine. In fact, the Church-Turing thesis has been so successful, that it is now almost moot..
Computability theory - 2 counters Formal grammar Post system Lambda calculus Partial recursive functions Almost any modern programming language (when given unlimited memory), including: A language with 1 instruction, 1 parameter (see OISC and URISC here) A language with 8 instructions, no parameters (see BrainFuck) Wang tiles Recurrent neural network (finite-precision inputs/outputs/weights, infinite-precision signals initialized to zero) Cellular automaton, including: Conway's Game of Life Cellular automaton with just 1 dimension, 2 states, 3 cells per neighborhood (e.g. rule 110) Non-deterministic Turing machine Probabilistic Turing Machine Quantum computer The last three examples use a slightly different definition of accepting a language. They are said to accept a string if any computation accepts (for non-deterministic), or most computations accept (for probabilistic and quantum). Given these definitions, those machines have the same power as a Turing machine.
Super-Turing computation - with rational numbers. No physical examples of Super-Turing computers are currently known. Classes of computers that might have Super-Turing capabilities in some physical models include: Pulse computers Analog computers Quantum computers See also: hypercomputation.
Recursion theory - machines or processes required to solve them. Much of the field is concerned with different kinds of logical hypercomputation. See also recursive function, arithmetical hierarchy, analytic hierarchy, Church-Turing thesis, recursive set, recursively enumerable set.
Real computation - complement of the Mandelbrot set is only partially decidable". For other such powerful machines, see super-Turing computation and hypercomputation. These hypothetical computing machines can be viewed as idealised analog computers which operate on real numbers and are differential, whereas digital computers are limited to computable numbers and are algebraic. This means that idealised analog computers have a larger information dimension rate (see Information Theory), or potential computing domain, than do digital computers. This in theory enables analog computers to solve problems that are inextricable on digital computers. Computer theorists often refer to these idealised analog computers as real computers (so called because they operate on the set of real numbers), in order to avoid confusion with real-world analog computers. Could a real "real computer" ever be built? Real-world analog computers are.
List of mathematical topics (G-I) - Hankel matrix -- Hardy, G. H -- Hardy space -- Hari, Mata -- Harmonic -- Harmonic analysis -- Harmonic function -- Harmonic mean -- Harmonic oscillator -- Harmonic series -- Harmonic series (music) -- Harsanyi, John -- Hash function -- Hash table -- Hasse diagram -- Hausdorff, Felix -- Hausdorff dimension -- Hausdorff maximality theorem -- Hausdorff space -- Hazard ratio -- Heap -- Heat equation -- Heaviside, Oliver -- Heaviside step function -- Thomas Heath -- Hebrew numerals -- Heisenberg, Werner -- Heisenberg picture -- Helix -- Hellman, Martin -- Heng, Zhang -- Henkin -- Henkin, Leon -- Henstock-Kurzweil Integral -- Henstock-Kurzweil-Stieltjes Integral -- Heptadecagon -- Herbrand, Jacques -- Herbrand theory -- Herbrand universe -- Hereditarily finite set -- Charles Hermite -- Hermite polynomials -- Heron of Alexandria -- Heron's.
List of mathematical logic topics - logic Post correspondence problem Kleene's recursion theorem Recursively enumerable set Recursively enumerable language Decidable language Undecidable language Rice's theorem Effective results in number theory Diophantine set Matiyasevich's theorem Arithmetical hierarchy Subrecursion theory Presburger arithmetic Computational complexity theory Polynomial time Exponential time Complexity classes P and NP NP-complete Hypercomputation Oracle machine Alonzo Church Emil Post Alan Turing Jacques Herbrand Haskell Curry Stephen Cole Kleene Proof theory Metamathematics Sequent Sequent calculus Substructural logics Relevant logic Gerhard Gentzen Mathematical constructivism Nonconstructive proof Intuitionistic logic Type theory Introduction to topos theory Constructivist analysis Finitism Ultraintuitionism Luitzen Egbertus Jan Brouwer.
List of computability and complexity topics - EXPSPACE, EXPTIME, EXPTIME-complete BPP BQP Class NC RP UP (complexity) ZPP Named problems Clique problem Hamiltonian cycle problem Integer factorization Knapsack problem Satisfiability problem 3-satisfiability Subset sum problem Traveling salesman problem Extensions Probabilistic algorithm Non-deterministic Turing machine Probabilistic Turing Machine Hypercomputation.
