18,528 research outputs found
Computer-aided verification in mechanism design
In mechanism design, the gold standard solution concepts are dominant
strategy incentive compatibility and Bayesian incentive compatibility. These
solution concepts relieve the (possibly unsophisticated) bidders from the need
to engage in complicated strategizing. While incentive properties are simple to
state, their proofs are specific to the mechanism and can be quite complex.
This raises two concerns. From a practical perspective, checking a complex
proof can be a tedious process, often requiring experts knowledgeable in
mechanism design. Furthermore, from a modeling perspective, if unsophisticated
agents are unconvinced of incentive properties, they may strategize in
unpredictable ways.
To address both concerns, we explore techniques from computer-aided
verification to construct formal proofs of incentive properties. Because formal
proofs can be automatically checked, agents do not need to manually check the
properties, or even understand the proof. To demonstrate, we present the
verification of a sophisticated mechanism: the generic reduction from Bayesian
incentive compatible mechanism design to algorithm design given by Hartline,
Kleinberg, and Malekian. This mechanism presents new challenges for formal
verification, including essential use of randomness from both the execution of
the mechanism and from the prior type distributions. As an immediate
consequence, our work also formalizes Bayesian incentive compatibility for the
entire family of mechanisms derived via this reduction. Finally, as an
intermediate step in our formalization, we provide the first formal
verification of incentive compatibility for the celebrated
Vickrey-Clarke-Groves mechanism
Formalization and Validation of Safety-Critical Requirements
The validation of requirements is a fundamental step in the development
process of safety-critical systems. In safety critical applications such as
aerospace, avionics and railways, the use of formal methods is of paramount
importance both for requirements and for design validation. Nevertheless, while
for the verification of the design, many formal techniques have been conceived
and applied, the research on formal methods for requirements validation is not
yet mature. The main obstacles are that, on the one hand, the correctness of
requirements is not formally defined; on the other hand that the formalization
and the validation of the requirements usually demands a strong involvement of
domain experts. We report on a methodology and a series of techniques that we
developed for the formalization and validation of high-level requirements for
safety-critical applications. The main ingredients are a very expressive formal
language and automatic satisfiability procedures. The language combines
first-order, temporal, and hybrid logic. The satisfiability procedures are
based on model checking and satisfiability modulo theory. We applied this
technology within an industrial project to the validation of railways
requirements
Mutual Interlacing and Eulerian-like Polynomials for Weyl Groups
We use the method of mutual interlacing to prove two conjectures on the
real-rootedness of Eulerian-like polynomials: Brenti's conjecture on
-Eulerian polynomials for Weyl groups of type , and Dilks, Petersen, and
Stembridge's conjecture on affine Eulerian polynomials for irreducible finite
Weyl groups.
For the former, we obtain a refinement of Brenti's -Eulerian polynomials
of type , and then show that these refined Eulerian polynomials satisfy
certain recurrence relation. By using the Routh--Hurwitz theory and the
recurrence relation, we prove that these polynomials form a mutually
interlacing sequence for any positive , and hence prove Brenti's conjecture.
For , our result reduces to the real-rootedness of the Eulerian
polynomials of type , which were originally conjectured by Brenti and
recently proved by Savage and Visontai.
For the latter, we introduce a family of polynomials based on Savage and
Visontai's refinement of Eulerian polynomials of type . We show that these
new polynomials satisfy the same recurrence relation as Savage and Visontai's
refined Eulerian polynomials. As a result, we get the real-rootedness of the
affine Eulerian polynomials of type . Combining the previous results for
other types, we completely prove Dilks, Petersen, and Stembridge's conjecture,
which states that, for every irreducible finite Weyl group, the affine descent
polynomial has only real zeros.Comment: 28 page
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