3,714 research outputs found
Aligator.jl - A Julia Package for Loop Invariant Generation
We describe the Aligator.jl software package for automatically generating all
polynomial invariants of the rich class of extended P-solvable loops with
nested conditionals. Aligator.jl is written in the programming language Julia
and is open-source. Aligator.jl transforms program loops into a system of
algebraic recurrences and implements techniques from symbolic computation to
solve recurrences, derive closed form solutions of loop variables and infer the
ideal of polynomial invariants by variable elimination based on Gr\"obner basis
computation
Splitting Proofs for Interpolation
We study interpolant extraction from local first-order refutations. We
present a new theoretical perspective on interpolation based on clearly
separating the condition on logical strength of the formula from the
requirement on the com- mon signature. This allows us to highlight the space of
all interpolants that can be extracted from a refutation as a space of simple
choices on how to split the refuta- tion into two parts. We use this new
insight to develop an algorithm for extracting interpolants which are linear in
the size of the input refutation and can be further optimized using metrics
such as number of non-logical symbols or quantifiers. We implemented the new
algorithm in first-order theorem prover VAMPIRE and evaluated it on a large
number of examples coming from the first-order proving community. Our
experiments give practical evidence that our work improves the state-of-the-art
in first-order interpolation.Comment: 26th Conference on Automated Deduction, 201
Reflected Spectrally Negative Stable Processes and their Governing Equations
This paper explicitly computes the transition densities of a spectrally
negative stable process with index greater than one, reflected at its infimum.
First we derive the forward equation using the theory of sun-dual semigroups.
The resulting forward equation is a boundary value problem on the positive
half-line that involves a negative Riemann-Liouville fractional derivative in
space, and a fractional reflecting boundary condition at the origin. Then we
apply numerical methods to explicitly compute the transition density of this
space-inhomogeneous Markov process, for any starting point, to any desired
degree of accuracy. Finally, we discuss an application to fractional Cauchy
problems, which involve a positive Caputo fractional derivative in time
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