73 research outputs found
Counterexample-Guided Polynomial Loop Invariant Generation by Lagrange Interpolation
We apply multivariate Lagrange interpolation to synthesize polynomial
quantitative loop invariants for probabilistic programs. We reduce the
computation of an quantitative loop invariant to solving constraints over
program variables and unknown coefficients. Lagrange interpolation allows us to
find constraints with less unknown coefficients. Counterexample-guided
refinement furthermore generates linear constraints that pinpoint the desired
quantitative invariants. We evaluate our technique by several case studies with
polynomial quantitative loop invariants in the experiments
Applying machine learning to the problem of choosing a heuristic to select the variable ordering for cylindrical algebraic decomposition
Cylindrical algebraic decomposition(CAD) is a key tool in computational
algebraic geometry, particularly for quantifier elimination over real-closed
fields. When using CAD, there is often a choice for the ordering placed on the
variables. This can be important, with some problems infeasible with one
variable ordering but easy with another. Machine learning is the process of
fitting a computer model to a complex function based on properties learned from
measured data. In this paper we use machine learning (specifically a support
vector machine) to select between heuristics for choosing a variable ordering,
outperforming each of the separate heuristics.Comment: 16 page
Certified Computer Algebra on top of an Interactive Theorem Prover
Contains fulltext :
35027.pdf (publisher's version ) (Open Access
Green's functions for parabolic systems of second order in time-varying domains
We construct Green's functions for divergence form, second order parabolic
systems in non-smooth time-varying domains whose boundaries are locally
represented as graph of functions that are Lipschitz continuous in the spatial
variables and 1/2-H\"older continuous in the time variable, under the
assumption that weak solutions of the system satisfy an interior H\"older
continuity estimate. We also derive global pointwise estimates for Green's
function in such time-varying domains under the assumption that weak solutions
of the system vanishing on a portion of the boundary satisfy a certain local
boundedness estimate and a local H\"older continuity estimate. In particular,
our results apply to complex perturbations of a single real equation.Comment: 25 pages, 0 figur
Solving Geometric Problems with Real Quantifier Elimination
Many problems arising in real geometry can be formulated as first-order formulas. Thus quantifier elimination can be used to solve these problems. In this note, we discuss the applicability of implemented quantifier elimination algorithms for solving geometrical problems. In particular, we demonstrate how the tools of redlog can be applied to solve a real implicitization problem, namely the Enneper surface
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