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Propositional proof systems, the consistency of first order theories and the complexity of computations. (English) Zbl 0696.03029
Let S be a finitely axiomatized theory. The authors consider the sequence $$Con_ S(\underline n)$$, where $$Con_ S(\underline n)$$ denotes the statement that there is no proof of contradiction in S whose length is $$\leq n$$. They formulate nine statements, e.g. the following ones:
(1) There exists a finitely axiomatized fragment T of true arithmetic such that for every finitely axiomatized consistent theory S, there exists a polynomial p such that, for every $$n\in {\mathbb{N}}$$, there is a proof of $$Con_ S(\underline n)$$ in T, whose length is $$\leq p(n).$$
(6) There exists a finitely axiomatized fragment T of true arithmetic such that, for every finitely axiomatized consistent theory S, there exists a deterministic Turing machine M and a polynomial p such that for any given n, M constructs a proof of $$Con_ S(\underline n)$$ in T, in time $$\leq p(n).$$
Then it is proved that statements (1)-(5) and (6)-(9) are equivalent, respectively. The problem arises whether (1)-(9) are true or not. This problem is discussed in the last part of the paper.
Reviewer: N.Both

##### MSC:
 03F20 Complexity of proofs 03D10 Turing machines and related notions 03D15 Complexity of computation (including implicit computational complexity) 03B05 Classical propositional logic 03B10 Classical first-order logic 03F30 First-order arithmetic and fragments
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