Abstract
By a problem in a universal first-order theory Γ we understand a formula φ(y) with a free variable y. The problem φ(y) is solvable relative to auxiliary problems ψ1(υ),…s, (υ) if solution algorithms for the ψi can be so composed as to yield all solutions of φ in all models of Γ. The complexity of the composite algorithm is the number of times that the auxiliary algorithms have to be called. A lower bound for the complexity of φ is obtained by developing a generalized Galois theory for theories Γ and problems φ satisfying some reasonable restrictions; our lower bound is the logarithm of the order of the ‘Galois group’ of φ
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Bibliography
Engeler, E.: 1975, ‘On the Solvability of Algorithmic Problems’, in H. E. Rose and J. C. Shepherdson (eds.), Logic Colloquium 73, North-Holland Publ. Co., 1975, pp. 231–251.
Jónsson, B.: 1962, ‘Algebraic Extensions of Relational Systems’, Mathematica Scandinavica 11, 179–205.
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© 1977 D. Reidel Publishing Company, Dordrecht, Holland
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Engeler, E. (1977). Structural Relations Between Programs and Problems. In: Butts, R.E., Hintikka, J. (eds) Logic, Foundations of Mathematics, and Computability Theory. The University of Western Ontario Series in Philosophy of Science, vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1138-9_14
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DOI: https://doi.org/10.1007/978-94-010-1138-9_14
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