Wikipedia

Markov algorithm

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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.

Refal is a programming language based on Markov algorithm.

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. If the applied rule was a terminating one, stop executing the Algorithm.
  5. Return to step 1 and carry on.

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"
  6. "a never used" -> ."terminating rule"

Symbol string

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

Execution

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

  1. "I bought a bag of As from T S."
  2. "I bought a bag of apples from T S."
  3. "I bought a bag of apples from the S."
  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.

Another Example

These rules give a more interesting example. They rewrite marked binary numbers to their unary counterparts. For example: a1101101101b will be rewritten to a string of 877 consecutive ones.

  1. "0xb" -> "b0"
  2. "1xb" -> "b1"
  3. "0x1" -> "10x"
  4. "0x0" -> "00x"
  5. "1x1" -> "11x"
  6. "1x0" -> "01x"
  7. "a0" -> "a0x"
  8. "a1" -> "a1x"
  9. "ab" -> "u"
  10. "u1" -> "1ju"
  11. "u0" -> "ju"
  12. "j1" -> "11j"
  13. "j" -> ""
  14. "u" -> ""

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.
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