Efficient recursion termination for function-free horn logic

We present an efficient scheme to terminate infinite recursion in function-free Horn logic. In [BW84], Brough and Walker show that a preorder linear resolution with a goal termination strategy is incomplete, i.e. it must miss some answers. Their theory is true if left-recursion is allowed. The crucial assumption underlying Brough and Walker's theory is that the order of literals in a clause should not be altered. This assumption, however, is not necessary in programs that do not contain any extra-logical features such as the 'cut' symbol of Prolog. This is because the order of literals does not affect the correctness of such programs, only their efficiency. In this paper, we show that left-recursion can always be eliminated. The idea is to transform loops of the input set into safe loops, that are left-recursion free. Consequently, the goal termination strategy guarantees to always terminate properly with all possible answers; thus, it is complete in the domain of safe loops. We further show that all rules in a safe loop can be transformed into rules that begin with a base literal. This permits the implementation of a simple scheme to carry out the goal termination strategy more efficiently. The basic idea of this scheme is to distribute the history containing all executed goals over assertions, rather than maintaining it as a centralized data structure. This reduces the amount of work performed during execution

Hypertableau Reasoning for Description Logics

We present a novel reasoning calculus for the description logic SHOIQ^+---a knowledge representation formalism with applications in areas such as the Semantic Web. Unnecessary nondeterminism and the construction of large models are two primary sources of inefficiency in the tableau-based reasoning calculi used in state-of-the-art reasoners. In order to reduce nondeterminism, we base our calculus on hypertableau and hyperresolution calculi, which we extend with a blocking condition to ensure termination. In order to reduce the size of the constructed models, we introduce anywhere pairwise blocking. We also present an improved nominal introduction rule that ensures termination in the presence of nominals, inverse roles, and number restrictions---a combination of DL constructs that has proven notoriously difficult to handle. Our implementation shows significant performance improvements over state-of-the-art reasoners on several well-known ontologies

The Connectivity of Boolean Satisfiability: No-Constants and Quantified Variants

For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. Motivated by research on heuristics and the satisfiability threshold, Gopalan et al. in 2006 studied connectivity properties of the solution graph and related complexity issues for constraint satisfaction problems in Schaefer's framework. They found dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connectivity question that we recently were able to prove. While Gopalan et al. considered CNF(S)-formulas with constants, we here look at two important variants: CNF(S)-formulas without constants, and partially quantified formulas. For the diameter and the st-connectivity question, we prove dichotomies analogous to those of Gopalan et al. in these settings. While we cannot give a complete classification for the connectivity problem yet, we identify fragments where it is in P, where it is coNP-complete, and where it is PSPACE-complete, in analogy to Gopalan et al.'s trichotomy.Comment: superseded by chapter 3 of arXiv:1510.0670

Implementing Groundness Analysis with Definite Boolean Functions

The domain of definite Boolean functions, Def, can be used to express the groundness of, and trace grounding dependencies between, program variables in (constraint) logic programs. In this paper, previously unexploited computational properties of Def are utilised to develop an efficient and succinct groundness analyser that can be coded in Prolog. In particular, entailment checking is used to prevent unnecessary least upper bound calculations. It is also demonstrated that join can be defined in terms of other operations, thereby eliminating code and removing the need for preprocessing formulae to a normal form. This saves space and time. Furthermore, the join can be adapted to straightforwardly implement the downward closure operator that arises in set sharing analyses. Experimental results indicate that the new Def implementation gives favourable results in comparison with BDD-based groundness analyses