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Spektralproblem and completeness of logical decision problems

  • Section V: Spektralproblem
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Book cover Logic and Machines: Decision Problems and Complexity (LaM 1983)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 171))

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Abstract

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.

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  1. Lehrstuhl Informatik II Universität, D-4600, Dortmund

    Egon Börger

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  1. Egon Börger
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E. Börger G. Hasenjaeger D. Rödding

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Börger, E. (1984). Spektralproblem and completeness of logical decision problems. In: Börger, E., Hasenjaeger, G., Rödding, D. (eds) Logic and Machines: Decision Problems and Complexity. LaM 1983. Lecture Notes in Computer Science, vol 171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-13331-3_50

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  • DOI: https://doi.org/10.1007/3-540-13331-3_50

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