Abstract
The concept of a Turing machine has been well defined. It would be sufficient to show that Bitcoin uses a dual-stack architecture that acts as a dual counter machine. Such systems have already been demonstrated as being Turing complete. We demonstrate that Bitcoin script is a minimal family of which \( \lambda \lambda \) and R are members. Further, using the compositional product rule and the iteration rule, we demonstrate that Bitcoin scripting is Turing complete with the limitations imposed on any real-world computer. The limitation is that there cannot be an infinite tape. Iterations can be simulated using an “unrolled” loop function with allocation to the “Alt” stack. As the product rule says, if A, B are machines, then A.B is also a machine. The iteration rule shows that if A is a machine, then (A) is also a machine. Further, the minimum power of A under which the observed square of the final configuration is blank. The consequence of such rules is that for every partial recursive function of in variables we can show that it can be evaluated by a machine of the proposed family.
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Notes
- 1.
Marvin Minsky, Computation: Finite and Infinite Machines, Prentice–Hall, Inc., N.J., 1967. See Chapter 8, Section 8.2 “Unsolvability of the Halting Problem.”
- 2.
Here \( \Omega \) is equivalent to \( \Omega ^{{\prime }} -\Omega _{R} \).
- 3.
- 4.
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Bitcoin scripting
Bitcoin Wiki Webpage. https://en.Bitcoin.it/wiki/Script. Accessed 13 Mar 2019
Davide De Rosa Webpage. https://davidederosa.com/basic-blockchain-programming/bitcoin-script-language-part-one/. Accessed 13 Mar 2019
Fortran programming
Fortran Wiki Webpage. http://fortranwiki.org/fortran/show/HomePage. Accessed 13 Mar 2019
GNU Webpage. https://gcc.gnu.org/fortran/. Accessed 13 Mar 2019
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Wright, C.S. (2020). A Proof of Turing Completeness in Bitcoin Script. In: Bi, Y., Bhatia, R., Kapoor, S. (eds) Intelligent Systems and Applications. IntelliSys 2019. Advances in Intelligent Systems and Computing, vol 1037. Springer, Cham. https://doi.org/10.1007/978-3-030-29516-5_23
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