108 research outputs found
on the efficiency of convex polyhedra
Abstract The domain of convex polyhedra plays a special role in the collection of numerical domains considered for program analysis and verification. As far as precision is concerned, it would be the most natural choice in many contexts but, due to its worst case exponential complexity, it is sometimes considered an unaffordable option. This has led to a systematic quest for simpler domains that are capable of reasonable precision using less computational resources. There are anyway cases where the use of the domain of convex polyhedra turns out to be feasible, also due to recent progress in their implementation. After reviewing a few known approaches to decrease the amount of resources needed when computing on this domain, we will introduce a couple of novel techniques that can be used to further improve its efficiency, without incurring precision losses
The Parma Polyhedra Library: Toward a Complete Set of Numerical Abstractions for the Analysis and Verification of Hardware and Software Systems
Since its inception as a student project in 2001, initially just for the
handling (as the name implies) of convex polyhedra, the Parma Polyhedra Library
has been continuously improved and extended by joining scrupulous research on
the theoretical foundations of (possibly non-convex) numerical abstractions to
a total adherence to the best available practices in software development. Even
though it is still not fully mature and functionally complete, the Parma
Polyhedra Library already offers a combination of functionality, reliability,
usability and performance that is not matched by similar, freely available
libraries. In this paper, we present the main features of the current version
of the library, emphasizing those that distinguish it from other similar
libraries and those that are important for applications in the field of
analysis and verification of hardware and software systems.Comment: 38 pages, 2 figures, 3 listings, 3 table
An Improved Tight Closure Algorithm for Integer Octagonal Constraints
Integer octagonal constraints (a.k.a. ``Unit Two Variables Per Inequality''
or ``UTVPI integer constraints'') constitute an interesting class of
constraints for the representation and solution of integer problems in the
fields of constraint programming and formal analysis and verification of
software and hardware systems, since they couple algorithms having polynomial
complexity with a relatively good expressive power. The main algorithms
required for the manipulation of such constraints are the satisfiability check
and the computation of the inferential closure of a set of constraints. The
latter is called `tight' closure to mark the difference with the (incomplete)
closure algorithm that does not exploit the integrality of the variables. In
this paper we present and fully justify an O(n^3) algorithm to compute the
tight closure of a set of UTVPI integer constraints.Comment: 15 pages, 2 figure
PURRS: Towards Computer Algebra Support for Fully Automatic Worst-Case Complexity Analysis
Fully automatic worst-case complexity analysis has a number of applications
in computer-assisted program manipulation. A classical and powerful approach to
complexity analysis consists in formally deriving, from the program syntax, a
set of constraints expressing bounds on the resources required by the program,
which are then solved, possibly applying safe approximations. In several
interesting cases, these constraints take the form of recurrence relations.
While techniques for solving recurrences are known and implemented in several
computer algebra systems, these do not completely fulfill the needs of fully
automatic complexity analysis: they only deal with a somewhat restricted class
of recurrence relations, or sometimes require user intervention, or they are
restricted to the computation of exact solutions that are often so complex to
be unmanageable, and thus useless in practice. In this paper we briefly
describe PURRS, a system and software library aimed at providing all the
computer algebra services needed by applications performing or exploiting the
results of worst-case complexity analyses. The capabilities of the system are
illustrated by means of examples derived from the analysis of programs written
in a domain-specific functional programming language for real-time embedded
systems.Comment: 6 page
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