Jan Obdrzalek | Searcharxiv

arXiv2016

First order limits of sparse graphs: Plane trees and path-width

Nesetril and Ossona de Mendez introduced the notion of first order convergence as an attempt to unify the notions of convergence for sparse and dense graphs. It is known that there exist first order convergent sequences of graphs with no limit modeling (an analytic representation of the limit). On the positive side, every first order convergent sequence of trees or graphs with no long path (graphs with bounded tree-depth) has a limit modeling. We strengthen these results by showing that every first order convergent sequence of plane trees (trees with embeddings in the plane) and every first order convergent sequence of graphs with bounded path-width has a limit modeling.

math.CO↗

arXiv2015

FO Model Checking of Interval Graphs

We study the computational complexity of the FO model checking problem on interval graphs, i.e., intersection graphs of intervals on the real line. The main positive result is that FO model checking and successor-invariant FO model checking can be solved in time O(n log n) for n-vertex interval graphs with representations containing only intervals with lengths from a prescribed finite set. We complement this result by showing that the same is not true if the lengths are restricted to any set that is dense in an open subset, e.g., in the set $(1, 1 + \varepsilon)$.

cs.DM↗

arXiv2011

Efficient Loop Navigation for Symbolic Execution

Symbolic execution is a successful and very popular technique used in software verification and testing. A key limitation of symbolic execution is in dealing with code containing loops. The problem is that even a single loop can generate a huge number of different symbolic execution paths, corresponding to different number of loop iterations and taking various paths through the loop. We introduce a technique which, given a start location above some loops and a target location anywhere below these loops, returns a feasible path between these two locations, if such a path exists. The technique infers a collection of constraint systems from the program and uses them to steer the symbolic execution towards the target. On reaching a loop it iteratively solves the appropriate constraint system to find out which path through this loop to take, or, alternatively, whether to continue below the loop. To construct the constraint systems we express the values of variables modified in a loop as functions of the number of times a given path through the loop was executed. We have built a prototype implementation of our technique and compared it to state-of-the-art symbolic execution tools on simple programs with loops. The results show significant improvements in the running time. We found instances where our algorithm finished in seconds, whereas the other tools did not finish within an hour. Our approach also shows very good results in the case when the target location is not reachable by any feasible path.

cs.PL↗