57 research outputs found
Verifying Safety Properties With the TLA+ Proof System
TLAPS, the TLA+ proof system, is a platform for the development and
mechanical verification of TLA+ proofs written in a declarative style requiring
little background beyond elementary mathematics. The language supports
hierarchical and non-linear proof construction and verification, and it is
independent of any verification tool or strategy. A Proof Manager uses backend
verifiers such as theorem provers, proof assistants, SMT solvers, and decision
procedures to check TLA+ proofs. This paper documents the first public release
of TLAPS, distributed with a BSD-like license. It handles almost all the
non-temporal part of TLA+ as well as the temporal reasoning needed to prove
standard safety properties, in particular invariance and step simulation, but
not liveness properties
Balanced Schnyder woods for planar triangulations: an experimental study with applications to graph drawing and graph separators
In this work we consider balanced Schnyder woods for planar graphs, which are
Schnyder woods where the number of incoming edges of each color at each vertex
is balanced as much as possible. We provide a simple linear-time heuristic
leading to obtain well balanced Schnyder woods in practice. As test
applications we consider two important algorithmic problems: the computation of
Schnyder drawings and of small cycle separators. While not being able to
provide theoretical guarantees, our experimental results (on a wide collection
of planar graphs) suggest that the use of balanced Schnyder woods leads to an
improvement of the quality of the layout of Schnyder drawings, and provides an
efficient tool for computing short and balanced cycle separators.Comment: Appears in the Proceedings of the 27th International Symposium on
Graph Drawing and Network Visualization (GD 2019
Trace Spaces: an Efficient New Technique for State-Space Reduction
State-space reduction techniques, used primarily in model-checkers, all rely
on the idea that some actions are independent, hence could be taken in any
(respective) order while put in parallel, without changing the semantics. It is
thus not necessary to consider all execution paths in the interleaving
semantics of a concurrent program, but rather some equivalence classes. The
purpose of this paper is to describe a new algorithm to compute such
equivalence classes, and a representative per class, which is based on ideas
originating in algebraic topology. We introduce a geometric semantics of
concurrent languages, where programs are interpreted as directed topological
spaces, and study its properties in order to devise an algorithm for computing
dihomotopy classes of execution paths. In particular, our algorithm is able to
compute a control-flow graph for concurrent programs, possibly containing
loops, which is "as reduced as possible" in the sense that it generates traces
modulo equivalence. A preliminary implementation was achieved, showing
promising results towards efficient methods to analyze concurrent programs,
with very promising results compared to partial-order reduction techniques
Lower bounds on the dilation of plane spanners
(I) We exhibit a set of 23 points in the plane that has dilation at least
, improving the previously best lower bound of for the
worst-case dilation of plane spanners.
(II) For every integer , there exists an -element point set
such that the degree 3 dilation of denoted by in the domain of plane geometric spanners. In the
same domain, we show that for every integer , there exists a an
-element point set such that the degree 4 dilation of denoted by
The
previous best lower bound of holds for any degree.
(III) For every integer , there exists an -element point set
such that the stretch factor of the greedy triangulation of is at least
.Comment: Revised definitions in the introduction; 23 pages, 15 figures; 2
table
Morphing Schnyder drawings of planar triangulations
We consider the problem of morphing between two planar drawings of the same
triangulated graph, maintaining straight-line planarity. A paper in SODA 2013
gave a morph that consists of steps where each step is a linear morph
that moves each of the vertices in a straight line at uniform speed.
However, their method imitates edge contractions so the grid size of the
intermediate drawings is not bounded and the morphs are not good for
visualization purposes. Using Schnyder embeddings, we are able to morph in
linear morphing steps and improve the grid size to
for a significant class of drawings of triangulations, namely the class of
weighted Schnyder drawings. The morphs are visually attractive. Our method
involves implementing the basic "flip" operations of Schnyder woods as linear
morphs.Comment: 23 pages, 8 figure
Small grid embeddings of 3-polytopes
We introduce an algorithm that embeds a given 3-connected planar graph as a
convex 3-polytope with integer coordinates. The size of the coordinates is
bounded by . If the graph contains a triangle we can
bound the integer coordinates by . If the graph contains a
quadrilateral we can bound the integer coordinates by . The
crucial part of the algorithm is to find a convex plane embedding whose edges
can be weighted such that the sum of the weighted edges, seen as vectors,
cancel at every point. It is well known that this can be guaranteed for the
interior vertices by applying a technique of Tutte. We show how to extend
Tutte's ideas to construct a plane embedding where the weighted vector sums
cancel also on the vertices of the boundary face
On the Area Requirements of Planar Greedy Drawings of Triconnected Planar Graphs
In this paper we study the area requirements of planar greedy drawings of
triconnected planar graphs. Cao, Strelzoff, and Sun exhibited a family
of subdivisions of triconnected plane graphs and claimed that every planar
greedy drawing of the graphs in respecting the prescribed plane
embedding requires exponential area. However, we show that every -vertex
graph in actually has a planar greedy drawing respecting the
prescribed plane embedding on an grid. This reopens the
question whether triconnected planar graphs admit planar greedy drawings on a
polynomial-size grid. Further, we provide evidence for a positive answer to the
above question by proving that every -vertex Halin graph admits a planar
greedy drawing on an grid. Both such results are obtained by
actually constructing drawings that are convex and angle-monotone. Finally, we
consider -Schnyder drawings, which are angle-monotone and hence greedy
if , and show that there exist planar triangulations for
which every -Schnyder drawing with a fixed requires
exponential area for any resolution rule
Automatic Verification Of TLA+ Proof Obligations With SMT Solvers
International audienceTLA+ is a formal specification language that is based on ZF set theory and the Temporal Logic of Actions TLA. The TLA+ proof system TLAPS assists users in deductively verifying safety properties of TLA+ specifications. TLAPS is built around a proof manager, which interprets the TLA+ proof language, generates corresponding proof obligations, and passes them to backend verifiers. In this paper we present a new backend for use with SMT solvers that supports elementary set theory, functions, arithmetic, tuples, and records. Type information required by the solvers is provided by a typing discipline for TLA+ proof obligations, which helps us disambiguate the translation of expressions of (untyped) TLA+, while ensuring its soundness. Preliminary results show that the backend can help to significantly increase the degree of automation of certain interactive proofs
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