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.

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