Partial Solution of The Word Problem
The word problem for a recursively presented group can be partially solved in the following sense:
-
- Given a recursive presentation P = ⟨X|R⟩ for a group G, define:
- then there is a partial recursive function fP such that:
- Given a recursive presentation P = ⟨X|R⟩ for a group G, define:
More informally, there is an algorithm that halts if u=v, but does not do so otherwise.
It follows that to solve the word problem for P it is sufficient to construct a recursive function g such that:
However u=v in G if and only if uv−1=1 in G. It follows that to solve the word problem for P it is sufficient to construct a recursive function h such that:
Read more about this topic: Word Problem For Groups
Famous quotes containing the words partial, solution, word and/or problem:
“There is no luck in literary reputation. They who make up the final verdict upon every book are not the partial and noisy readers of the hour when it appears; but a court as of angels, a public not to be bribed, not to be entreated, and not to be overawed, decides upon every mans title to fame. Only those books come down which deserve to last.”
—Ralph Waldo Emerson (18031882)
“I herewith commission you to carry out all preparations with regard to ... a total solution of the Jewish question in those territories of Europe which are under German influence.... I furthermore charge you to submit to me as soon as possible a draft showing the ... measures already taken for the execution of the intended final solution of the Jewish question.”
—Hermann Goering (18931946)
“Denotation by means of sounds and markings is a remarkable abstraction. Three letters designate God for me; several lines a million things. How easy becomes the manipulation of the universe here, how evident the concentration of the intellectual world! Language is the dynamics of the spiritual realm. One word of command moves armies; the word liberty entire nations.”
—Novalis [Friedrich Von Hardenberg] (17721801)
“The problem is that we attempt to solve the simplest questions cleverly, thereby rendering them unusually complex. One should seek the simple solution.”
—Anton Pavlovich Chekhov (18601904)