Conjugacy relations of prefix codes

Yong He, Zhenhe Cui, Zihan Yuan

Abstract: It is shown that, if X and Y are prefix codes and Z is a non-empty language satisfying the condition XZ=ZY<math><mi is="true">X</mi><mi is="true">Z</mi><mo is="true">=</mo><mi is="true">Z</mi><mi is="true">Y</mi></math>, then Z is the union of a non-empty family {Pn}i∈I<math><msub is="true"><mrow is="true"><mo stretchy="false" is="true">{</mo><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub><mo stretchy="false" is="true">}</mo></mrow><mrow is="true"><mi is="true">i</mi><mo is="true">∈</mo><mi is="true">I</mi></mrow></msub></math> of pairwise disjoint prefix sets such that XPi=PiY<math><mi is="true">X</mi><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">i</mi></mrow></msub><mo is="true">=</mo><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mi is="true">i</mi></mrow></msub><mi is="true">Y</mi></math> for all i∈I<math><mi is="true">i</mi><mo is="true">∈</mo><mi is="true">I</mi></math>. Consequently, the conjugacy relations of prefix codes are explored and, under the restriction that both of X and Y are prefix codes, the solutions of the conjugacy equation XZ=ZY<math><mi is="true">X</mi><mi is="true">Z</mi><mo is="true">=</mo><mi is="true">Z</mi><mi is="true">Y</mi></math> for languages are determined. Also, the decidability of the conjugacy problem for finite prefix codes is confirmed.