6,111 research outputs found
Cayley's hyperdeterminant: a combinatorial approach via representation theory
Cayley's hyperdeterminant is a homogeneous polynomial of degree 4 in the 8
entries of a 2 x 2 x 2 array. It is the simplest (nonconstant) polynomial which
is invariant under changes of basis in three directions. We use elementary
facts about representations of the 3-dimensional simple Lie algebra sl_2(C) to
reduce the problem of finding the invariant polynomials for a 2 x 2 x 2 array
to a combinatorial problem on the enumeration of 2 x 2 x 2 arrays with
non-negative integer entries. We then apply results from linear algebra to
obtain a new proof that Cayley's hyperdeterminant generates all the invariants.
In the last section we show how this approach can be applied to general
multidimensional arrays.Comment: 20 page
Having Fun in Learning Formal Specifications
There are many benefits in providing formal specifications for our software.
However, teaching students to do this is not always easy as courses on formal
methods are often experienced as dry by students. This paper presents a game
called FormalZ that teachers can use to introduce some variation in their
class. Students can have some fun in playing the game and, while doing so, also
learn the basics of writing formal specifications in the form of pre- and
post-conditions. Unlike existing software engineering themed education games
such as Pex and Code Defenders, FormalZ takes the deep gamification approach
where playing gets a more central role in order to generate more engagement.
This short paper presents our work in progress: the first implementation of
FormalZ along with the result of a preliminary users' evaluation. This
implementation is functionally complete and tested, but the polishing of its
user interface is still future work
Automatic enumeration of regular objects
We describe a framework for systematic enumeration of families combinatorial
structures which possess a certain regularity. More precisely, we describe how
to obtain the differential equations satisfied by their generating series.
These differential equations are then used to determine the initial counting
sequence and for asymptotic analysis. The key tool is the scalar product for
symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer
Sequence
A simple abstraction of arrays and maps by program translation
We present an approach for the static analysis of programs handling arrays,
with a Galois connection between the semantics of the array program and
semantics of purely scalar operations. The simplest way to implement it is by
automatic, syntactic transformation of the array program into a scalar program
followed analysis of the scalar program with any static analysis technique
(abstract interpretation, acceleration, predicate abstraction,.. .). The
scalars invariants thus obtained are translated back onto the original program
as universally quantified array invariants. We illustrate our approach on a
variety of examples, leading to the " Dutch flag " algorithm
Automated Verification of Practical Garbage Collectors
Garbage collectors are notoriously hard to verify, due to their low-level
interaction with the underlying system and the general difficulty in reasoning
about reachability in graphs. Several papers have presented verified
collectors, but either the proofs were hand-written or the collectors were too
simplistic to use on practical applications. In this work, we present two
mechanically verified garbage collectors, both practical enough to use for
real-world C# benchmarks. The collectors and their associated allocators
consist of x86 assembly language instructions and macro instructions, annotated
with preconditions, postconditions, invariants, and assertions. We used the
Boogie verification generator and the Z3 automated theorem prover to verify
this assembly language code mechanically. We provide measurements comparing the
performance of the verified collector with that of the standard Bartok
collectors on off-the-shelf C# benchmarks, demonstrating their competitiveness
Shaped extensions of singular spectrum analysis
Extensions of singular spectrum analysis (SSA) for processing of
non-rectangular images and time series with gaps are considered. A circular
version is suggested, which allows application of the method to the data given
on a circle or on a cylinder, e.g. cylindrical projection of a 3D ellipsoid.
The constructed Shaped SSA method with planar or circular topology is able to
produce low-rank approximations for images of complex shapes. Together with
Shaped SSA, a shaped version of the subspace-based ESPRIT method for frequency
estimation is developed. Examples of 2D circular SSA and 2D Shaped ESPRIT are
presented
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