Finite Open-World Query Answering with Number Restrictions (Extended Version)

Open-world query answering is the problem of deciding, given a set of facts, conjunction of constraints, and query, whether the facts and constraints imply the query. This amounts to reasoning over all instances that include the facts and satisfy the constraints. We study finite open-world query answering (FQA), which assumes that the underlying world is finite and thus only considers the finite completions of the instance. The major known decidable cases of FQA derive from the following: the guarded fragment of first-order logic, which can express referential constraints (data in one place points to data in another) but cannot express number restrictions such as functional dependencies; and the guarded fragment with number restrictions but on a signature of arity only two. In this paper, we give the first decidability results for FQA that combine both referential constraints and number restrictions for arbitrary signatures: we show that, for unary inclusion dependencies and functional dependencies, the finiteness assumption of FQA can be lifted up to taking the finite implication closure of the dependencies. Our result relies on new techniques to construct finite universal models of such constraints, for any bound on the maximal query size.Comment: 59 pages. To appear in LICS 2015. Extended version including proof

Detecting Decidable Classes of Finitely Ground Logic Programs with Function Symbols

In this article, we propose a new technique for checking whether the bottom-up evaluation of logic programs with function symbols terminates. The technique is based on the definition of mappings from arguments to strings of function symbols, representing possible values which could be taken by arguments during the bottom-up evaluation. Starting from mappings, we identify mapping-restricted arguments, a subset of limited arguments, namely arguments that take values from finite domains. Mapping-restricted programs, consisting of rules whose arguments are all mapping restricted, are terminating under the bottom-up computation, as all of its arguments take values from finite domains. We show that mappings can be computed by transforming the original program into a unary logic program: this allows us to establish decidability of checking if a program is mapping restricted. We study the complexity of the presented approach and compare it to other techniques known in the literature. We also introduce an extension of the proposed approach that is able to recognize a wider class of logic programs. The presented technique provides a significant improvement, as it can detect terminating programs not identified by other criteria proposed so far. Furthermore, it can be combined with other techniques to further enlarge the class of programs recognized as terminating under the bottom-up evaluation. </jats:p

Using linear constraints for logic program termination analysis

It is widely acknowledged that function symbols are an important feature in answer set programming, as they make modeling easier, increase the expressive power, and allow us to deal with infinite domains. The main issue with their introduction is that the evaluation of a program might not terminate and checking whether it terminates or not is undecidable. To cope with this problem, several classes of logic programs have been proposed where the use of function symbols is restricted but the program evaluation termination is guaranteed. Despite the significant body of work in this area, current approaches do not include many simple practical programs whose evaluation terminates. In this paper, we present the novel classes of rule-bounded and cycle-bounded programs, which overcome different limitations of current approaches by performing a more global analysis of how terms are propagated from the body to the head of rules. Results on the correctness, the complexity, and the expressivity of the proposed approach are provided.Comment: Under consideration in Theory and Practice of Logic Programming (TPLP