Markov algorithm - Definition and Overview

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 have sufficient power to be a general model of computation, and can thus be shown to be equivalent in power to a Turing machine. Since this model is Turing-complete, Markov algorithms can represent any mathematical expression from its simple notation.

Example

The following example shows the basic operation of a Markov algorithm.

Rules

1. "A" -> "apple"
2. "B" -> "bag"
3. "S" -> "shop"
4. "T" -> "the"
5. "the shop" -> "my brother"

Symbol string

"I bought a B of As from T S."

Algorithm

1. Check the Rules in order from top to bottom to see whether any of the strings to the left of the arrow can be found in the Symbol string.
2. If none are found, stop executing the Algorithm.
3. If one or more is found, replace the leftmost matching text in the Symbol string with the text to the right of the arrow in the first corresponding Rule.
4. Return to step 1 and carry on.

Execution of the algorithm

If the algorithm is applied to the above example, the Symbol string will change in the following manner.

1. "I bought a B of apples from T S."
2. "I bought a bag of apples from T S."
3. "I bought a bag of apples from T shop."
4. "I bought a bag of apples from the shop."
5. "I bought a bag of apples from my brother."

The algorithm will then terminate.

References:

• Caracciolo di Forino, A. String processing languages and generalized Markov algorithms. In Symbol manipulation languages and techniques, D. G. Bobrow (Ed.), North-Holland Publ. Co., Amsterdam, The Netherlands, 1968, pp. 191-206.
• Markov, A.A. 1960. The Theory of Algorithms. American Mathematical Society Translations, series 2, 15, 1-14.