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Markov's principle (also known as the Leningrad principle [1]), named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below. The principle is logically valid classically, but not in intuitionistic constructive mathematics. However, many particular instances of it are ...
In theoretical computer science, a Markov algorithm is a string rewriting system that uses grammar-like rules to operate on strings of symbols. Markov algorithms have been shown to be Turing-complete , which means that they are suitable as a general model of computation and can represent any mathematical expression from its simple notation.
More precisely Markov's theorem can be stated as follows: [2] [3] given two braids represented by elements , ′ in the braid groups ,, their closures are equivalent links if and only if ′ can be obtained from applying to a sequence of the following operations:
In this context, the Markov property indicates that the distribution for this variable depends only on the distribution of a previous state. An example use of a Markov chain is Markov chain Monte Carlo, which uses the Markov property to prove that a particular method for performing a random walk will sample from the joint distribution.
In probability theory and ergodic theory, a Markov operator is an operator on a certain function space that conserves the mass (the so-called Markov property). If the underlying measurable space is topologically sufficiently rich enough, then the Markov operator admits a kernel representation. Markov operators can be linear or non-linear.
The term Markov assumption is used to describe a model where the Markov property is assumed to hold, such as a hidden Markov model. A Markov random field extends this property to two or more dimensions or to random variables defined for an interconnected network of items. [1] An example of a model for such a field is the Ising model.
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The "Markov" in "Markov decision process" refers to the underlying structure of state transitions that still follow the Markov property. The process is called a "decision process" because it involves making decisions that influence these state transitions, extending the concept of a Markov chain into the realm of decision-making under uncertainty.