Bitcoin’s mathematical definition (1)

We want to formulate the data behind Bitcoin more scientifically since it got also attention from the science area. We start with some simple and basic defintions:

  • \(T\): Set of all transactions
  • \(t_{x} \in T\): Transaction \(t\) with id \(X\)
  • \(A\): Set of all addresses
  • \((S_{t_{x}}^{in}, <)\): Bounded lattice-ordered set of all inputs of \(t_{x}\)
  • \((S_{t_{x}}^{out}, <)\): Bounded lattice-ordered set of all outputs of \(t_{x}\)
  • \((r_{i}, a_{i}, v_{i}) \in S_{t_{x}}^{in}: r_{i} \in \mathbb{N}_0, a_{i} \in A, v_{i} \in \mathbb{N}_0^{+}\): \(i\)th element of the inputs where \(r_{i}\) defines the order starting at 0
  • \((r_{j}, a_{j}, v_{j}) \in S_{t_{x}}^{out}: r_{j} \in \mathbb{N}_0, a_{j} \in A, v_{j} \in \mathbb{N}_0^{+}\): \(j\)th element of the outputs where \(r_{j}\) defines the order starting at 0 with the condition: \((r_{i}, a_{i}, v_{i}) \in S_{t_{x}}^{in} . \exists! (r_{j}, a_{j}, v_{j}) \in S_{t_{y}}^{out} \land x \neq y \land a_{i} = a_{j} \land v_{i} = v_{j}\)