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Renormalisation and computation II: time cut-off and the Halting Problem

  • YURI I. MANIN (a1)

This is the second instalment in the project initiated in Manin (2012). In the first Part, we argued that both the philosophy and technique of perturbative renormalisation in quantum field theory could be meaningfully transplanted to the theory of computation, and sketched several contexts supporting this view.

In this second part, we address some of the issues raised in Manin (2012) and develop them further in three contexts: a categorification of the algorithmic computations; time cut-off and anytime algorithms; and, finally, a Hopf algebra renormalisation of the Halting Problem.

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J. Baez and M. Stay (2010) Physics, topology, logic and computation: a Rosetta stone. In: B. Coecke (ed.) New Structures for Physics. Springer-Verlag Lecture Notes in Physics 813 95172. (Available at arXiv:0903.0340.)

C. Calude and M. Stay (2008) Most programs stop quickly or never halt. Advances in Applied Mathematics 40 295308.

C. Calude and M. Stay (2006) Natural halting probabilities, partial randomness, and zeta functions. Information and Computation 204 17181739.

P. Gács and A. Levin (1982) Causal nets or what is a deterministic computation? International Journal of Theoretical Physics 21 (12) 961971.

M. Li and P. Vitányi (1993) An Introduction to Kolmogorov Complexity and its Applications, Springer.

Y. Manin (2010) A Course in Mathematical Logic (the second, expanded Edition), Springer-Verlag.

A. Nabutovsky and S. Weinberger (2003) The fractal nature of Riem/Diff I. Geometriae Dedicata 101 145250.

C. P. Schnorr (1974) Optimal enumerations and optimal Gödel numberings. Mathematical Systems Theory 8 (2) 182191.

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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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