604 research outputs found
Colouring set families without monochromatic k-chains
A coloured version of classic extremal problems dates back to Erd\H{o}s and
Rothschild, who in 1974 asked which -vertex graph has the maximum number of
2-edge-colourings without monochromatic triangles. They conjectured that the
answer is simply given by the largest triangle-free graph. Since then, this new
class of coloured extremal problems has been extensively studied by various
researchers. In this paper we pursue the Erd\H{o}s--Rothschild versions of
Sperner's Theorem, the classic result in extremal set theory on the size of the
largest antichain in the Boolean lattice, and Erd\H{o}s' extension to
-chain-free families.
Given a family of subsets of , we define an
-colouring of to be an -colouring of the sets without
any monochromatic -chains . We
prove that for sufficiently large in terms of , the largest
-chain-free families also maximise the number of -colourings. We also
show that the middle level, , maximises the
number of -colourings, and give asymptotic results on the maximum
possible number of -colourings whenever is divisible by three.Comment: 30 pages, final versio
On Verifying Complex Properties using Symbolic Shape Analysis
One of the main challenges in the verification of software systems is the
analysis of unbounded data structures with dynamic memory allocation, such as
linked data structures and arrays. We describe Bohne, a new analysis for
verifying data structures. Bohne verifies data structure operations and shows
that 1) the operations preserve data structure invariants and 2) the operations
satisfy their specifications expressed in terms of changes to the set of
objects stored in the data structure. During the analysis, Bohne infers loop
invariants in the form of disjunctions of universally quantified Boolean
combinations of formulas. To synthesize loop invariants of this form, Bohne
uses a combination of decision procedures for Monadic Second-Order Logic over
trees, SMT-LIB decision procedures (currently CVC Lite), and an automated
reasoner within the Isabelle interactive theorem prover. This architecture
shows that synthesized loop invariants can serve as a useful communication
mechanism between different decision procedures. Using Bohne, we have verified
operations on data structures such as linked lists with iterators and back
pointers, trees with and without parent pointers, two-level skip lists, array
data structures, and sorted lists. We have deployed Bohne in the Hob and Jahob
data structure analysis systems, enabling us to combine Bohne with analyses of
data structure clients and apply it in the context of larger programs. This
report describes the Bohne algorithm as well as techniques that Bohne uses to
reduce the ammount of annotations and the running time of the analysis
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