Computability — Logical and Recursive Complexity
The basis of this short course is the strong analogy between programs and proofs (of their specifications). The main theme is the classification of computable number-theoretic functions according to the logical complexity of their formal specification or termination proofs. A significant sub-branch of mathematical logic has grown around this theme since the 1950’s and new ideas are presently giving rise to further developments. The methods employed are chiefly those from proof theory, particularly “normalization” as presented in the accompanying lectures of Helmut Schwichtenberg, and “ordinal assignments”. Since program-termination corresponds to well-foundedness of computation trees, it is hardly surprising that transfinite ordinals and their constructive representations play a crucial role, measuring the logical complexity of programs and of the functions which they compute.
KeywordsInduction Hypothesis Turing Machine Recursive Function Accumulation Rule Proof Theory
Unable to display preview. Download preview PDF.
- , G. KreiselOn the interpretation of non-finitist proofs II, Journ. of Symbolic Logic 17, 43–58.Google Scholar
-  A. Grzegorczyk, Some classes of recursive functions, Rozprawy Matem. IV, Warsaw.Google Scholar
-  J.W. Robbin, Subrecursive hierarchies, Ph.D. Princeton.72, 204–236Google Scholar
-  W.W. Tait, Normal derivability in classical logic, in J. Barwise Ed., Springer Lecture Notes in Math. 72, 204–236.Google Scholar
-  R.L. Constable, Extending and refining hierarchies of computable functions, T.R.25, Computer Science Dept., Univ. of Wisconsin.Google Scholar
-  K. Schütte, Proof theory, Springer Verlag, Berlin.Google Scholar
-  J. Paris and L. Harrington, A mathematical incompleteness in Peano arithmetic, ibid 1133–1142.Google Scholar
-  C.S. Copestake and S.S. Wainer, A proof theoretic approach to the termination of computer programs, Report 26.88, Centre for Theoretical Computer Science, Leeds University.Google Scholar
-  M.V. Eairtlough and S.S. Wainer, Ordinal complexity of recursive definitions, Report 28.89, Centre for Theoretical Computer Science, Leeds University.Google Scholar