130 research outputs found
How hard is it to verify flat affine counter systems with the finite monoid property ?
We study several decision problems for counter systems with guards defined by
convex polyhedra and updates defined by affine transformations. In general, the
reachability problem is undecidable for such systems. Decidability can be
achieved by imposing two restrictions: (i) the control structure of the counter
system is flat, meaning that nested loops are forbidden, and (ii) the set of
matrix powers is finite, for any affine update matrix in the system. We provide
tight complexity bounds for several decision problems of such systems, by
proving that reachability and model checking for Past Linear Temporal Logic are
complete for the second level of the polynomial hierarchy , while
model checking for First Order Logic is PSPACE-complete
Affine Extensions of Integer Vector Addition Systems with States
We study the reachability problem for affine -VASS, which are
integer vector addition systems with states in which transitions perform affine
transformations on the counters. This problem is easily seen to be undecidable
in general, and we therefore restrict ourselves to affine -VASS
with the finite-monoid property (afmp--VASS). The latter have the
property that the monoid generated by the matrices appearing in their affine
transformations is finite. The class of afmp--VASS encompasses
classical operations of counter machines such as resets, permutations,
transfers and copies. We show that reachability in an afmp--VASS
reduces to reachability in a -VASS whose control-states grow
linearly in the size of the matrix monoid. Our construction shows that
reachability relations of afmp--VASS are semilinear, and in
particular enables us to show that reachability in -VASS with
transfers and -VASS with copies is PSPACE-complete. We then focus
on the reachability problem for affine -VASS with monogenic
monoids: (possibly infinite) matrix monoids generated by a single matrix. We
show that, in a particular case, the reachability problem is decidable for this
class, disproving a conjecture about affine -VASS with infinite
matrix monoids we raised in a preliminary version of this paper. We complement
this result by presenting an affine -VASS with monogenic matrix
monoid and undecidable reachability relation
Forward Analysis and Model Checking for Trace Bounded WSTS
We investigate a subclass of well-structured transition systems (WSTS), the
bounded---in the sense of Ginsburg and Spanier (Trans. AMS 1964)---complete
deterministic ones, which we claim provide an adequate basis for the study of
forward analyses as developed by Finkel and Goubault-Larrecq (Logic. Meth.
Comput. Sci. 2012). Indeed, we prove that, unlike other conditions considered
previously for the termination of forward analysis, boundedness is decidable.
Boundedness turns out to be a valuable restriction for WSTS verification, as we
show that it further allows to decide all -regular properties on the
set of infinite traces of the system
Instanton Correction, Wall Crossing And Mirror Symmetry Of Hitchin's Moduli Spaces
We study two instanton correction problems of Hitchin's moduli spaces along
with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli
space can be put into an instanton-corrected form according to physicists
Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set
of special coordinates which can be constructed as Fock-Goncharov coordinates
associated with foliations of quadratic differentials on a Riemann surface. A
wall crossing formula of Kontsevich and Soibelman arises both as a crucial
consistency condition and an effective computational tool. On the other hand
Gross and Siebert have succeeded in determining instanton corrections of
complex structures of Calabi-Yau varieties in the context of mirror symmetry
from a singular affine structure with additional data. We will show that the
two instanton correction problems are equivalent in an appropriate sense via
the identification of the wall crossing formulas in the metric problem with
consistency conditions in the complex structure problem. This result provides
examples of Calabi-Yau varieties where the instanton correction (in the sense
of mirror symmetry) of metrics and complex structures can be determined.Comment: 160 pages. Revised version. References and acknowledgement added.
Minor mistakes and typos corrected. Exposition improve
Vérification relationnelle pour des programmes avec des données entières
Les travaux présentés dans cette thèse sont lies aux problèmes de vérification de l'atteignabilité et de la terminaison de programmes qui manipulent des données entières non-bornées. On décrit une nouvelle méthode de vérification basée sur une technique d'accélération de boucle, qui calcule, de manière exacte, la clôture transitive d'une relation arithmétique. D'abord, on introduit un algorithme d'accélération de boucle qui peut calculer, en quelques secondes, des clôtures transitives pour des relations de l'ordre d'une centaine de variables. Ensuite, on présente une méthode d'analyse de l'atteignabilité, qui manipule des relations entre les variables entières d'un programme, et applique l'accélération pour le calcul des relations entrée-sortie des procédures, de façon modulaire. Une approche alternative pour l'analyse de l'atteignabilité, présentée également dans cette thèse, intègre l'accélération avec l'abstraction par prédicats, afin de traiter le problème de divergence de cette dernière. Ces deux méthodes ont été évaluées de manière pratique, sur un nombre important d'exemples, qui étaient, jusqu'a présent, hors de la portée des outils d'analyse existants. Dernièrement, on a étudié le problème de la terminaison pour certaines classes de boucles de programme, et on a montré la décidabilité pour les relations étudiées. Pour ces classes de relations arithmétiques, on présente un algorithme qui s'exécute en temps au plus polynomial, et qui calcule l'ensemble d'états qui peuvent générer une exécution infinie. Ensuite on a intégré cet algorithme dans une méthode d'analyse de la terminaison pour des programmes qui manipulent des données entières.This work presents novel methods for verification of reachability and termination properties of programs that manipulate unbounded integer data. Most of these methods are based on acceleration techniques which compute transitive closures of program loops. We first present an algorithm that accelerates several classes of integer relations and show that the new method performs up to four orders of magnitude better than the previous ones. On the theoretical side, our framework provides a common solution to the acceleration problem by proving that the considered classes of relations are periodic. Subsequently, we introduce a semi-algorithmic reachability analysis technique that tracks relations between variables of integer programs and applies the proposed acceleration algorithm to compute summaries of procedures in a modular way. Next, we present an alternative approach to reachability analysis that integrates predicate abstraction with our acceleration techniques to increase the likelihood of convergence of the algorithm. We evaluate these algorithms and show that they can handle a number of complex integer programs where previous approaches failed. Finally, we study the termination problem for several classes of program loops and show that it is decidable. Moreover, for some of these classes, we design a polynomial time algorithm that computes the exact set of program configurations from which non-terminating runs exist. We further integrate this algorithm into a semi-algorithmic method that analyzes termination of integer programs, and show that the resulting technique can verify termination properties of several non-trivial integer programs.SAVOIE-SCD - Bib.électronique (730659901) / SudocGRENOBLE1/INP-Bib.électronique (384210012) / SudocGRENOBLE2/3-Bib.électronique (384219901) / SudocSudocFranceF
Verification of Flat FIFO Systems
The decidability and complexity of reachability problems and model-checking for flat counter systems have been explored in detail. However, only few results are known for flat FIFO systems, only in some particular cases (a single loop or a single bounded expression). We prove, by establishing reductions between properties, and by reducing SAT to a subset of these properties that many verification problems like reachability, non-termination, unboundedness are NP-complete for flat FIFO systems, generalizing similar existing results for flat counter systems. We construct a trace-flattable counter system that is bisimilar to a given flat FIFO system, which allows to model-check the original flat FIFO system. Our results lay the theoretical foundations and open the way to build a verification tool for (general) FIFO systems based on analysis of flat subsystems
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