Skip to content

#057 abaabbbaba=babaa

Original presentation

\( \left\langle a, b \;\middle|\; a b a^{2} b^{3} a b a = b a b a^{2} \right\rangle \)

Modified presentation

\( \left\langle a, b, c, d \;\middle|\; a b a^{2} c a = b a b a^{2},\; c = b^{3} a b,\; d = a b a^{2} c a c a c \right\rangle \)

Complete rewriting system

Using recursive path order (RPOCmp) with \(a < d < c < b\):

\( \begin{aligned} b^{3} a b &\rightarrow c \\ b a b a^{2} &\rightarrow a b a^{2} c a \\ a b a^{2} c a c a c &\rightarrow d \\ b^{3} a c &\rightarrow c b^{2} a b \\ b^{3} a^{2} b a^{2} c a &\rightarrow c a b a^{2} \\ c a^{2} &\rightarrow d a \\ b a b a d &\rightarrow a b a^{2} c d \\ b d &\rightarrow d a c \\ b^{3} a d &\rightarrow c a b a^{2} c a c \\ b^{3} a^{2} b a^{2} c d &\rightarrow c a b a d \\ b^{3} a^{2} b a^{2} d a &\rightarrow c a b a^{3} \\ c a d &\rightarrow d^{2} \\ a b a^{2} d^{3} a &\rightarrow d a^{2} \\ b^{3} a^{2} b a^{2} d^{2} &\rightarrow c a b a^{2} d \\ a b a^{2} d^{4} &\rightarrow d a d \end{aligned} \)