57,544 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
Lower Bounds on the Bounded Coefficient Complexity of Bilinear Maps
We prove lower bounds of order for both the problem to multiply
polynomials of degree , and to divide polynomials with remainder, in the
model of bounded coefficient arithmetic circuits over the complex numbers.
These lower bounds are optimal up to order of magnitude. The proof uses a
recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix
multiplication. It reduces the linear problem to multiply a random circulant
matrix with a vector to the bilinear problem of cyclic convolution. We treat
the arising linear problem by extending J. Morgenstern's bound [J. ACM 20, pp.
305-306, 1973] in a unitarily invariant way. This establishes a new lower bound
on the bounded coefficient complexity of linear forms in terms of the singular
values of the corresponding matrix. In addition, we extend these lower bounds
for linear and bilinear maps to a model of circuits that allows a restricted
number of unbounded scalar multiplications.Comment: 19 page
Real-time and Probabilistic Temporal Logics: An Overview
Over the last two decades, there has been an extensive study on logical
formalisms for specifying and verifying real-time systems. Temporal logics have
been an important research subject within this direction. Although numerous
logics have been introduced for the formal specification of real-time and
complex systems, an up to date comprehensive analysis of these logics does not
exist in the literature. In this paper we analyse real-time and probabilistic
temporal logics which have been widely used in this field. We extrapolate the
notions of decidability, axiomatizability, expressiveness, model checking, etc.
for each logic analysed. We also provide a comparison of features of the
temporal logics discussed
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