Merging partially labelled trees: hardness and a declarative programming solution

International audienceIntraspecific studies often make use of haplotype networks instead of gene genealogies to represent the evolution of a set of genes. Cassens et al. proposed one such network reconstruction method, based on the global maximum parsimony principle, which was later recast by the first author of the present work as the problem of finding a minimum common supergraph of a set of t partially labelled trees. Although algorithms were proposed for solving the problem on two graphs, the complexity of the general problem remains unknown. In this paper, we show that the corresponding decision problem is NP-complete for t = 3. We then propose a declarative programming approach to solving the problem to optimality in practice, as well as a heuristic approach, both based on the IDP system, and assess the performance of both methods on randomly generated data

Lazy Model Expansion: Interleaving Grounding with Search

Finding satisfying assignments for the variables involved in a set of constraints can be cast as a (bounded) model generation problem: search for (bounded) models of a theory in some logic. The state-of-the-art approach for bounded model generation for rich knowledge representation languages, like ASP, FO(.) and Zinc, is ground-and-solve: reduce the theory to a ground or propositional one and apply a search algorithm to the resulting theory. An important bottleneck is the blowup of the size of the theory caused by the reduction phase. Lazily grounding the theory during search is a way to overcome this bottleneck. We present a theoretical framework and an implementation in the context of the FO(.) knowledge representation language. Instead of grounding all parts of a theory, justifications are derived for some parts of it. Given a partial assignment for the grounded part of the theory and valid justifications for the formulas of the non-grounded part, the justifications provide a recipe to construct a complete assignment that satisfies the non-grounded part. When a justification for a particular formula becomes invalid during search, a new one is derived; if that fails, the formula is split in a part to be grounded and a part that can be justified. The theoretical framework captures existing approaches for tackling the grounding bottleneck such as lazy clause generation and grounding-on-the-fly, and presents a generalization of the 2-watched literal scheme. We present an algorithm for lazy model expansion and integrate it in a model generator for FO(ID), a language extending first-order logic with inductive definitions. The algorithm is implemented as part of the state-of-the-art FO(ID) Knowledge-Base System IDP. Experimental results illustrate the power and generality of the approach

Grounding FO and FO(ID) with bounds

Grounding is the task of reducing a first-order theory and finite domain to an equivalent propositional theory. It is used as preprocessing phase in many logic based reasoning systems. Such systems provide a rich first-order input language to a user, and can rely on efficient propositional solvers to perform the actual reasoning. Besides a first-order theory and finite domain, the input for grounders contains in many applications also additional data. By exploiting this data, the size of the grounder's output can often be reduced significantly. A common practice to improve the efficiency of a grounder in this context is by manually adding semantically redundant information to the input theory, indicating where and when the grounder should exploit the data. In this paper we present a method to compute and add such redundant information automatically. Our method therefore simplifies the task of writing input theories that can be grounded efficiently by current systems. We first present our method for classical first-order logic (FO) theories. Then we extend it to FO(ID), the extension of FO with inductive definitions, which allows for more concise and comprehensive input theories. We discuss implementation issues and experimentally validate the practical applicability of our method.status: publishe