Word Problem For Groups - Partial Solution of The Word Problem

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:
f_P(\langle u,v \rangle) = \left\{\begin{matrix} 0 &\mbox{if}\ \langle u,v \rangle \in S \\ \mbox{undefined/does not halt}\ &\mbox{if}\ \langle u,v \rangle \notin S \end{matrix}\right.

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:

g(\langle u,v \rangle) = \left\{\begin{matrix} 0 &\mbox{if}\ \langle u,v \rangle \notin S \\ \mbox{undefined/does not halt}\ &\mbox{if}\ \langle u,v \rangle \in S \end{matrix}\right.

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:

h(x) = \left\{\begin{matrix} 0 &\mbox{if}\ x\neq1\ \mbox{in}\ G \\ \mbox{undefined/does not halt}\ &\mbox{if}\ x=1\ \mbox{in}\ G \end{matrix}\right.

Read more about this topic:  Word Problem For Groups

Famous quotes containing the words partial, solution, word and/or problem:

    “We were soon in the smooth water of the Quakish Lake,... and we had our first, but a partial view of Ktaadn, its summit veiled in clouds, like a dark isthmus in that quarter, connecting the heavens with the earth.”
    —Henry David Thoreau (1817–1862)

    “Coming out, all the way out, is offered more and more as the political solution to our oppression. The argument goes that, if people could see just how many of us there are, some in very important places, the negative stereotype would vanish overnight. ...It is far more realistic to suppose that, if the tenth of the population that is gay became visible tomorrow, the panic of the majority of people would inspire repressive legislation of a sort that would shock even the pessimists among us.”
    —Jane Rule (b. 1931)

    “A poet who makes use of a worse word instead of a better, because the former fits the rhyme or the measure, though it weakens the sense, is like a jeweller, who cuts a diamond into a brilliant, and diminishes the weight to make it shine more.”
    —Horace Walpole (1717–1797)

    “One thing in any case is certain: man is neither the oldest nor the most constant problem that has been posed for human knowledge.”
    —Michel Foucault (1926–1984)