7,251 research outputs found
Polynomial Size Analysis of First-Order Shapely Functions
We present a size-aware type system for first-order shapely function
definitions. Here, a function definition is called shapely when the size of the
result is determined exactly by a polynomial in the sizes of the arguments.
Examples of shapely function definitions may be implementations of matrix
multiplication and the Cartesian product of two lists. The type system is
proved to be sound w.r.t. the operational semantics of the language. The type
checking problem is shown to be undecidable in general. We define a natural
syntactic restriction such that the type checking becomes decidable, even
though size polynomials are not necessarily linear or monotonic. Furthermore,
we have shown that the type-inference problem is at least semi-decidable (under
this restriction). We have implemented a procedure that combines run-time
testing and type-checking to automatically obtain size dependencies. It
terminates on total typable function definitions.Comment: 35 pages, 1 figur
Hyperelliptic Theta-Functions and Spectral Methods: KdV and KP solutions
This is the second in a series of papers on the numerical treatment of
hyperelliptic theta-functions with spectral methods. A code for the numerical
evaluation of solutions to the Ernst equation on hyperelliptic surfaces of
genus 2 is extended to arbitrary genus and general position of the branch
points. The use of spectral approximations allows for an efficient calculation
of all characteristic quantities of the Riemann surface with high precision
even in almost degenerate situations as in the solitonic limit where the branch
points coincide pairwise. As an example we consider hyperelliptic solutions to
the Kadomtsev-Petviashvili and the Korteweg-de Vries equation. Tests of the
numerics using identities for periods on the Riemann surface and the
differential equations are performed. It is shown that an accuracy of the order
of machine precision can be achieved.Comment: 16 pages, 8 figure
Validating plans with continuous effects
A critical element in the use of PDDL2.1, the modelling language developed for the International Planning Competition series, has been the common understanding of the semantics of the language. The fact that this has been implemented in plan validation software was vital to the progress of the competition. However, the validation of plans using actions with continuous effects presents new challenges (that precede the challenges presented by planning with those effects). In this paper we review the need for continuous effects, their semantics and the problems that arise in validation of plans that include them. We report our progress in implementing the semantics in an extended version of the plan validation software
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
On the Termination Problem for Probabilistic Higher-Order Recursive Programs
In the last two decades, there has been much progress on model checking of
both probabilistic systems and higher-order programs. In spite of the emergence
of higher-order probabilistic programming languages, not much has been done to
combine those two approaches. In this paper, we initiate a study on the
probabilistic higher-order model checking problem, by giving some first
theoretical and experimental results. As a first step towards our goal, we
introduce PHORS, a probabilistic extension of higher-order recursion schemes
(HORS), as a model of probabilistic higher-order programs. The model of PHORS
may alternatively be viewed as a higher-order extension of recursive Markov
chains. We then investigate the probabilistic termination problem -- or,
equivalently, the probabilistic reachability problem. We prove that almost sure
termination of order-2 PHORS is undecidable. We also provide a fixpoint
characterization of the termination probability of PHORS, and develop a sound
(but possibly incomplete) procedure for approximately computing the termination
probability. We have implemented the procedure for order-2 PHORSs, and
confirmed that the procedure works well through preliminary experiments that
are reported at the end of the article
Linear Optimization with Cones of Moments and Nonnegative Polynomials
Let A be a finite subset of N^n and R[x]_A be the space of real polynomials
whose monomial powers are from A. Let K be a compact basic semialgebraic set of
R^n such that R[x]_A contains a polynomial that is positive on K. Denote by
P_A(K) the cone of polynomials in R[x]_A that are nonnegative on K. The dual
cone of P_A(K) is R_A(K), the set of all A-truncated moment sequences in R^A
that admit representing measures supported in K. Our main results are: i) We
study the properties of P_A(K) and R_A(K) (like interiors, closeness, duality,
memberships), and construct a convergent hierarchy of semidefinite relaxations
for each of them. ii) We propose a semidefinite algorithm for solving linear
optimization problems with the cones P_A(K) and R_A(K), and prove its
asymptotic and finite convergence; a stopping criterion is also given. iii) We
show how to check whether P_A(K) and R_A(K) intersect affine subspaces; if they
do, we show to get get a point in the intersections; if they do not, we prove
certificates for the non-intersecting
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