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The Strength of Abstraction with Predicative Comprehension
Frege's theorem says that second-order Peano arithmetic is interpretable in
Hume's Principle and full impredicative comprehension. Hume's Principle is one
example of an abstraction principle, while another paradigmatic example is
Basic Law V from Frege's Grundgesetze. In this paper we study the strength of
abstraction principles in the presence of predicative restrictions on the
comprehension schema, and in particular we study a predicative Fregean theory
which contains all the abstraction principles whose underlying equivalence
relations can be proven to be equivalence relations in a weak background
second-order logic. We show that this predicative Fregean theory interprets
second-order Peano arithmetic.Comment: Forthcoming in Bulletin of Symbolic Logic. Slight change in title
from previous version, at request of referee
Computational Arithmetic Geometry I: Sentences Nearly in the Polynomial Hierarchy
We consider the average-case complexity of some otherwise undecidable or open
Diophantine problems. More precisely, consider the following: (I) Given a
polynomial f in Z[v,x,y], decide the sentence \exists v \forall x \exists y
f(v,x,y)=0, with all three quantifiers ranging over N (or Z). (II) Given
polynomials f_1,...,f_m in Z[x_1,...,x_n] with m>=n, decide if there is a
rational solution to f_1=...=f_m=0. We show that, for almost all inputs,
problem (I) can be done within coNP. The decidability of problem (I), over N
and Z, was previously unknown. We also show that the Generalized Riemann
Hypothesis (GRH) implies that, for almost all inputs, problem (II) can be done
via within the complexity class PP^{NP^NP}, i.e., within the third level of the
polynomial hierarchy. The decidability of problem (II), even in the case m=n=2,
remains open in general.
Along the way, we prove results relating polynomial system solving over C, Q,
and Z/pZ. We also prove a result on Galois groups associated to sparse
polynomial systems which may be of independent interest. A practical
observation is that the aforementioned Diophantine problems should perhaps be
avoided in the construction of crypto-systems.Comment: Slight revision of final journal version of an extended abstract
which appeared in STOC 1999. This version includes significant corrections
and improvements to various asymptotic bounds. Needs cjour.cls to compil
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