9,779 research outputs found
Lex-Partitioning: A New Option for BDD Search
For the exploration of large state spaces, symbolic search using binary
decision diagrams (BDDs) can save huge amounts of memory and computation time.
State sets are represented and modified by accessing and manipulating their
characteristic functions. BDD partitioning is used to compute the image as the
disjunction of smaller subimages.
In this paper, we propose a novel BDD partitioning option. The partitioning
is lexicographical in the binary representation of the states contained in the
set that is represented by a BDD and uniform with respect to the number of
states represented. The motivation of controlling the state set sizes in the
partitioning is to eventually bridge the gap between explicit and symbolic
search.
Let n be the size of the binary state vector. We propose an O(n) ranking and
unranking scheme that supports negated edges and operates on top of precomputed
satcount values. For the uniform split of a BDD, we then use unranking to
provide paths along which we partition the BDDs. In a shared BDD representation
the efforts are O(n). The algorithms are fully integrated in the CUDD library
and evaluated in strongly solving general game playing benchmarks.Comment: In Proceedings GRAPHITE 2012, arXiv:1210.611
Higher-order CIS codes
We introduce {\bf complementary information set codes} of higher-order. A
binary linear code of length and dimension is called a complementary
information set code of order (-CIS code for short) if it has
pairwise disjoint information sets. The duals of such codes permit to reduce
the cost of masking cryptographic algorithms against side-channel attacks. As
in the case of codes for error correction, given the length and the dimension
of a -CIS code, we look for the highest possible minimum distance. In this
paper, this new class of codes is investigated. The existence of good long CIS
codes of order is derived by a counting argument. General constructions
based on cyclic and quasi-cyclic codes and on the building up construction are
given. A formula similar to a mass formula is given. A classification of 3-CIS
codes of length is given. Nonlinear codes better than linear codes are
derived by taking binary images of -codes. A general algorithm based on
Edmonds' basis packing algorithm from matroid theory is developed with the
following property: given a binary linear code of rate it either provides
disjoint information sets or proves that the code is not -CIS. Using
this algorithm, all optimal or best known codes where and are shown to be -CIS for all
such and , except for with and with .Comment: 13 pages; 1 figur
Invariant Generation through Strategy Iteration in Succinctly Represented Control Flow Graphs
We consider the problem of computing numerical invariants of programs, for
instance bounds on the values of numerical program variables. More
specifically, we study the problem of performing static analysis by abstract
interpretation using template linear constraint domains. Such invariants can be
obtained by Kleene iterations that are, in order to guarantee termination,
accelerated by widening operators. In many cases, however, applying this form
of extrapolation leads to invariants that are weaker than the strongest
inductive invariant that can be expressed within the abstract domain in use.
Another well-known source of imprecision of traditional abstract interpretation
techniques stems from their use of join operators at merge nodes in the control
flow graph. The mentioned weaknesses may prevent these methods from proving
safety properties. The technique we develop in this article addresses both of
these issues: contrary to Kleene iterations accelerated by widening operators,
it is guaranteed to yield the strongest inductive invariant that can be
expressed within the template linear constraint domain in use. It also eschews
join operators by distinguishing all paths of loop-free code segments. Formally
speaking, our technique computes the least fixpoint within a given template
linear constraint domain of a transition relation that is succinctly expressed
as an existentially quantified linear real arithmetic formula. In contrast to
previously published techniques that rely on quantifier elimination, our
algorithm is proved to have optimal complexity: we prove that the decision
problem associated with our fixpoint problem is in the second level of the
polynomial-time hierarchy.Comment: 35 pages, conference version published at ESOP 2011, this version is
a CoRR version of our submission to Logical Methods in Computer Scienc
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