15 research outputs found
Abstract Fixpoint Computations with Numerical Acceleration Methods
Static analysis by abstract interpretation aims at automatically proving
properties of computer programs. To do this, an over-approximation of program
semantics, defined as the least fixpoint of a system of semantic equations,
must be computed. To enforce the convergence of this computation, widening
operator is used but it may lead to coarse results. We propose a new method to
accelerate the computation of this fixpoint by using standard techniques of
numerical analysis. Our goal is to automatically and dynamically adapt the
widening operator in order to maintain precision
Improving Strategies via SMT Solving
We consider the problem of computing numerical invariants of programs by
abstract interpretation. Our method eschews two traditional sources of
imprecision: (i) the use of widening operators for enforcing convergence within
a finite number of iterations (ii) the use of merge operations (often, convex
hulls) at the merge points of the control flow graph. It instead computes the
least inductive invariant expressible in the domain at a restricted set of
program points, and analyzes the rest of the code en bloc. We emphasize that we
compute this inductive invariant precisely. For that we extend the strategy
improvement algorithm of [Gawlitza and Seidl, 2007]. If we applied their method
directly, we would have to solve an exponentially sized system of abstract
semantic equations, resulting in memory exhaustion. Instead, we keep the system
implicit and discover strategy improvements using SAT modulo real linear
arithmetic (SMT). For evaluating strategies we use linear programming. Our
algorithm has low polynomial space complexity and performs for contrived
examples in the worst case exponentially many strategy improvement steps; this
is unsurprising, since we show that the associated abstract reachability
problem is Pi-p-2-complete
Enforcing Termination of Interprocedural Analysis
Interprocedural analysis by means of partial tabulation of summary functions
may not terminate when the same procedure is analyzed for infinitely many
abstract calling contexts or when the abstract domain has infinite strictly
ascending chains. As a remedy, we present a novel local solver for general
abstract equation systems, be they monotonic or not, and prove that this solver
fails to terminate only when infinitely many variables are encountered. We
clarify in which sense the computed results are sound. Moreover, we show that
interprocedural analysis performed by this novel local solver, is guaranteed to
terminate for all non-recursive programs --- irrespective of whether the
complete lattice is infinite or has infinite strictly ascending or descending
chains
Computing Invariants with Transformers: Experimental Scalability and Accuracy
International audienceUsing abstract interpretation, invariants are usually obtained by solving iteratively a system of equations linking preconditions according to program statements. However, it is also possible to abstract first the statements as transformers, and then propagate the preconditions using the transformers. The second approach is modular because procedures and loops can be abstracted once and for all, avoiding an iterative resolution over the call graph and all the control flow graphs. However, the transformer approach based on polyhedral abstract domains encurs two penalties: some invariant accuracy may be lost when computing transformers, and the execution time may increase exponentially because the dimension of a transformer is twice the dimension of a precondition. The purposes of this article are 1) to measure the benefits of the modular approach and its drawbacks in terms of execution time and accuracy using significant examples and a newly developed benchmark for loop invariant analysis, ALICe, 2) to present a new technique designed to reduce the accuracy loss when computing transformers, 3) to evaluate experimentally the accuracy gains this new technique and other previously discussed ones provide with ALICe test cases and 4) to compare the executions times and accuracies of different tools, ASPIC, ISL, PAGAI and PIPS. Our results suggest that the transformer-based approach used in PIPS, once improved with transformer lists, is as accurate as the other tools when dealing with the ALICe benchmark. Its modularity nevertheless leads to shorter execution times when dealing with nested loops and procedure calls found in real applications
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
Transfer Function Synthesis without Quantifier Elimination
Traditionally, transfer functions have been designed manually for each
operation in a program, instruction by instruction. In such a setting, a
transfer function describes the semantics of a single instruction, detailing
how a given abstract input state is mapped to an abstract output state. The net
effect of a sequence of instructions, a basic block, can then be calculated by
composing the transfer functions of the constituent instructions. However,
precision can be improved by applying a single transfer function that captures
the semantics of the block as a whole. Since blocks are program-dependent, this
approach necessitates automation. There has thus been growing interest in
computing transfer functions automatically, most notably using techniques based
on quantifier elimination. Although conceptually elegant, quantifier
elimination inevitably induces a computational bottleneck, which limits the
applicability of these methods to small blocks. This paper contributes a method
for calculating transfer functions that finesses quantifier elimination
altogether, and can thus be seen as a response to this problem. The
practicality of the method is demonstrated by generating transfer functions for
input and output states that are described by linear template constraints,
which include intervals and octagons.Comment: 37 pages, extended version of ESOP 2011 pape
Enhancing the Compilation of Synchronous Dataflow Programs with a Combined Numerical-Boolean Abstraction
RR version = http://hal.inria.fr/hal-00780521/enInternational audienceIn this paper, we propose an enhancement of the compilation of synchronous programs with a combined numerical-Boolean abstraction. While our approach applies to synchronous dataflow languages in general, here, we consider the SIGNAL language for illustration. In the new abstraction, every signal in a program is associated with a pair of the form ( clock, value ), where clock is a Boolean function and value is a Boolean or numeric function. Given the performance level reached by recent progress in Satisfiability Modulo Theory (SMT), we use an SMT solver to reason on this abstraction. Through sample examples, we show how our solution is used to determine absence of reaction captured by empty clocks; mutual exclusion captured by two or more clocks whose associated signals never occur at the same time; or hierarchical control of component activations via clock inclusion. We also show that the analysis improves the quality of the code generated automatically by a compiler, e.g., a code with smaller footprint, or a code executed more efficiently thanks to optimizations enabled by the new abstraction. The implementation of the whole approach includes a translator of synchronous programs towards the standard input format of SMT solvers, and an ad hoc SMT solver that integrates advanced functionalities to cope with the issues of interest in this wor