17,780 research outputs found
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
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
Efficient computation of exact solutions for quantitative model checking
Quantitative model checkers for Markov Decision Processes typically use
finite-precision arithmetic. If all the coefficients in the process are
rational numbers, then the model checking results are rational, and so they can
be computed exactly. However, exact techniques are generally too expensive or
limited in scalability. In this paper we propose a method for obtaining exact
results starting from an approximated solution in finite-precision arithmetic.
The input of the method is a description of a scheduler, which can be obtained
by a model checker using finite precision. Given a scheduler, we show how to
obtain a corresponding basis in a linear-programming problem, in such a way
that the basis is optimal whenever the scheduler attains the worst-case
probability. This correspondence is already known for discounted MDPs, we show
how to apply it in the undiscounted case provided that some preprocessing is
done. Using the correspondence, the linear-programming problem can be solved in
exact arithmetic starting from the basis obtained. As a consequence, the method
finds the worst-case probability even if the scheduler provided by the model
checker was not optimal. In our experiments, the calculation of exact solutions
from a candidate scheduler is significantly faster than the calculation using
the simplex method under exact arithmetic starting from a default basis.Comment: In Proceedings QAPL 2012, arXiv:1207.055
Definable Ellipsoid Method, Sums-of-Squares Proofs, and the Isomorphism Problem
The ellipsoid method is an algorithm that solves the (weak) feasibility and
linear optimization problems for convex sets by making oracle calls to their
(weak) separation problem. We observe that the previously known method for
showing that this reduction can be done in fixed-point logic with counting
(FPC) for linear and semidefinite programs applies to any family of explicitly
bounded convex sets. We use this observation to show that the exact feasibility
problem for semidefinite programs is expressible in the infinitary version of
FPC. As a corollary we get that, for the isomorphism problem, the
Lasserre/Sums-of-Squares semidefinite programming hierarchy of relaxations
collapses to the Sherali-Adams linear programming hierarchy, up to a small loss
in the degree
Definable ellipsoid method, sums-of-squares proofs, and the isomorphism problem
The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex sets. We use this observation to show that the exact feasibility problem for semidefinite programs is expressible in the infinitary version of FPC. As a corollary we get that, for the graph isomorphism problem, the Lasserre/Sums-of-Squares semidefinite programming hierarchy of relaxations collapses to the Sherali-Adams linear programming hierarchy, up to a small loss in the degree. © 2018 ACM.Peer ReviewedPostprint (author's final draft
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