Uncomputability and Physical Law

This paper investigates the role that uncomputability plays in the laws of physics. While uncomputability might seem like an abstract mathematical concept, many questions in physics involve uncomputability. In particular, the form of the energy spectrum of quantum systems capable of universal computation is uncomputable: to answer whether a Hamiltonian system is gapped or gapless in a particular sector requires one to solve the Halting problem. Finally, the problem of determining the most concise expression of physical laws requires one to determine the algorithmic complexity of those laws, and so, the answer to this problem is uncomputable.

[1]  S Lloyd,et al.  A Potentially Realizable Quantum Computer , 1993, Science.

[2]  Claude E. Shannon,et al.  A symbolic analysis of relay and switching circuits , 1938, Transactions of the American Institute of Electrical Engineers.

[3]  Seth Lloyd,et al.  Universal Quantum Simulators , 1996, Science.

[5]  M. W. Johnson,et al.  Quantum annealing with manufactured spins , 2011, Nature.

[6]  Ray J. Solomonoff,et al.  A Formal Theory of Inductive Inference. Part II , 1964, Inf. Control..

[7]  Gregory J. Chaitin,et al.  On the Length of Programs for Computing Finite Binary Sequences , 1966, JACM.

[8]  R. Feynman.  Quantum mechanical computers , 1986 .

[9]  D. G. Cory,et al.  FIRST DIRECT MEASUREMENT OF THE SPIN DIFFUSION RATE IN A HOMOGENOUS SOLID , 1998 .

[10]  C. Ross.  Found , 1869, The Dental register.

[11]  A. Ekert,et al.  Universality in quantum computation , 1995, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[12]  Lloyd,et al.  Almost any quantum logic gate is universal. , 1995, Physical review letters.

[13]  Quantum-mechanical computers and uncomputability. , 1993, Physical review letters.

[14]  S. Smale,et al.  On a theory of computation and complexity over the real numbers; np-completeness , 1989 .

[15]  B. M. Fulk.  MATH , 1992 .

[16]  F. Barahona.  On the computational complexity of Ising spin glass models , 1982 .