Supporting Abstraction when Model Checking ASM

Model checking as a method for automatic tool support for verification highly stimulates industry's interests. It is limited, however, with respect to the size of the systems' state space. In earlier work, we developed an interface between the ASM Workbench and the SMV model checker that allows model checking of finite ASM models. In this work, we add a means for abstraction in case the model to be checked is infinite and therefore not feasible for the model checking approach. We facilitate the ASM specification language (ASM-SL) with a notion for abstract types and introduce an interface between ASM-SL and Multiway Decision Graphs (MDGs). MDGs are capable of representing transition systems with abstract types and functions and provide the functionality necessary for symbolic model checking. Our interface maps abstract ASM models into MDGs in a semantic preserving way. It provides a very simple means for generating abstract models that are infinite but can be checked by a model checker based on MDGs

LCF-style Platform based on Multiway Decision Graphs

AbstractThe combination of state exploration approach (mainly model checking) and deductive reasoning approach (theorem proving) promises to overcome the limitation and to enhance the capabilities of each. In this paper, we are interested in defining a platform for Multiway Decision Graphs (MDGs) in LCF-style theorem prover. We define a platform to represent the MDG operations: conjunction, disjunction, relational product and prune-by-subsumption as a set of inference rules. Based on this platform, the reachability analysis is implemented as a conversion that uses the MDG theory within the HOL theorem prover. Finally, we present some experimental results to show the performance of the MDG operations of our platform

Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs

We propose polynomial-time algorithms that sparsify planar and bounded-genus graphs while preserving optimal or near-optimal solutions to Steiner problems. Our main contribution is a polynomial-time algorithm that, given an unweighted graph $G$ embedded on a surface of genus $g$ and a designated face $f$ bounded by a simple cycle of length $k$, uncovers a set $F \subseteq E(G)$ of size polynomial in $g$ and $k$ that contains an optimal Steiner tree for any set of terminals that is a subset of the vertices of $f$. We apply this general theorem to prove that: * given an unweighted graph $G$ embedded on a surface of genus $g$ and a terminal set $S \subseteq V(G)$, one can in polynomial time find a set $F \subseteq E(G)$ that contains an optimal Steiner tree $T$ for $S$ and that has size polynomial in $g$ and $|E(T)|$; * an analogous result holds for an optimal Steiner forest for a set $S$ of terminal pairs; * given an unweighted planar graph $G$ and a terminal set $S \subseteq V(G)$, one can in polynomial time find a set $F \subseteq E(G)$ that contains an optimal (edge) multiway cut $C$ separating $S$ and that has size polynomial in $|C|$. In the language of parameterized complexity, these results imply the first polynomial kernels for Steiner Tree and Steiner Forest on planar and bounded-genus graphs (parameterized by the size of the tree and forest, respectively) and for (Edge) Multiway Cut on planar graphs (parameterized by the size of the cutset). Additionally, we obtain a weighted variant of our main contribution

Integrating MDG variable ordering in a VHDL-MDG design verification system

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal

Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte