/sci/ - Science & Math

Let [math]\mathbf{0}, \mathbf{1}[/math] be the constant 0 and constant 1 sequences. Define [math]\tau: \{0,1\}^\mathbb{N} \to \{0,1\}^\mathbb{N}[/math] by [math]\tau (\overbrace{1 \dots 1}^{n-1 \textrm{ times}}, 0, x_{n+1}, x_{n+2} \dots) = (\overbrace{0 \dots 0}^{n-1 \textrm{ times}}, 1, x_{n+1}, x_{n+2} \dots)[/math] and [math]\tau(\mathbf{1}) = \mathbf{0}[/math]. Define also [math]\phi: \{0,1\}^\mathbb{N} \to \mathbb{Z}[/math] by [math]\phi(x) = \min \{ n \mid x_n = 0 \} - 2[/math]; we leave [math]\phi(\mathbf{1})[/math] undefined.

Is there an explicit formula for [math]\phi \circ \tau[/math] and [math]\phi \circ \tau^{-1}[/math]?