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:

    “Both the man of science and the man of art live always at the edge of mystery, surrounded by it. Both, as a measure of their creation, have always had to do with the harmonization of what is new with what is familiar, with the balance between novelty and synthesis, with the struggle to make partial order in total chaos.... This cannot be an easy life.”
    —J. Robert Oppenheimer (1904–1967)

    “What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.”
    —Boris Pasternak (1890–1960)

    “Even the most honest writer lets slip a word too many when he wants to round off a period.”
    —Friedrich Nietzsche (1844–1900)

    “The writer operates at a peculiar crossroads where time and place and eternity somehow meet. His problem is to find that location.”
    —Flannery O’Connor (1925–1964)