Spektralproblem and completeness of logical decision problems

  • Egon Börger
Section V: Spektralproblem
Part of the Lecture Notes in Computer Science book series (LNCS, volume 171)


We develop a simple general technique for minimal logical implementation of Turing machine programs. This implementation provides for completeness of logical decision problems which via the implementation correspond to limited or unlimited halting problems.

We exemplify our method by a new, simple proof for the automata-theoretical characterization of spectra of arbitrary order n as nondeterministically in n-fold exponential time Turing recognizable sets. We show that via our implementation technique the underlying completeness phenomenon is literally the same as in the Church/Turing theorem on the ∑1-completeness of Hilbert's Entscheidungsproblem, as in Trachtenbrot's theorem on the ∑1-completeness of the finite satisfiability problem for first order logic, but also as in Cook's theorem on the NP-completeness of the decision problem for propositional logic and in many well known lower complexity bound results for logical decision problems.

The Spektralproblem is in the title and in the center of our discussion because it has been officially formulated 33 years ago by the founder of the Institut für mathematische Logik und Grundlagenforschung of the University of Münster where our symposium takes place today. For this reason we include also a summary of the history of the Spektralproblem from 1951 to today, which at some points touches also the history of that Institut.


Turing Machine Recursive Function Type Formula Characterization Theorem Register Machine 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Egon Börger
    • 1
  1. 1.Lehrstuhl Informatik II UniversitätDortmund

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