13 research outputs found
End-to-End Differentiable Proving
We introduce neural networks for end-to-end differentiable proving of queries
to knowledge bases by operating on dense vector representations of symbols.
These neural networks are constructed recursively by taking inspiration from
the backward chaining algorithm as used in Prolog. Specifically, we replace
symbolic unification with a differentiable computation on vector
representations of symbols using a radial basis function kernel, thereby
combining symbolic reasoning with learning subsymbolic vector representations.
By using gradient descent, the resulting neural network can be trained to infer
facts from a given incomplete knowledge base. It learns to (i) place
representations of similar symbols in close proximity in a vector space, (ii)
make use of such similarities to prove queries, (iii) induce logical rules, and
(iv) use provided and induced logical rules for multi-hop reasoning. We
demonstrate that this architecture outperforms ComplEx, a state-of-the-art
neural link prediction model, on three out of four benchmark knowledge bases
while at the same time inducing interpretable function-free first-order logic
rules.Comment: NIPS 2017 camera-ready, NIPS 201
Generalizing completeness results for loop checks in logic programming
AbstractLoop checking is a mechanism for pruning infinite SLD-derivations. In (Bol, Apt and Klop, 1991) simple loop checks were introduced and their soundness, completeness and relative strength was studied. Since no sound and complete simple loop check exists even in the absence of function symbols, subclasses of programs were determined for which the (sound) loop checks introduced by Bol are complete.In this paper, the Generalization Theorem is proved. This theorem presents a method to extend (under certain conditions) a class of programs for which a given loop check is complete to a larger class, for which the loop check is still complete. Then this theorem is applied to the results of Bol, giving rise to stronger completeness theorems.It appears that unnecessary complications in the proof of the theorem can be avoided by introducing a normal form for SLD-derivations, allowing only certain most general unifiers. This normal form might have other applications than those in the area of loop checking
On Redundancy Elimination Tolerant Scheduling Rules
In (Ferrucci, Pacini and Sessa, 1995) an extended form of resolution, called
Reduced SLD resolution (RSLD), is introduced. In essence, an RSLD derivation is
an SLD derivation such that redundancy elimination from resolvents is performed
after each rewriting step. It is intuitive that redundancy elimination may have
positive effects on derivation process. However, undesiderable effects are also
possible. In particular, as shown in this paper, program termination as well as
completeness of loop checking mechanisms via a given selection rule may be
lost. The study of such effects has led us to an analysis of selection rule
basic concepts, so that we have found convenient to move the attention from
rules of atom selection to rules of atom scheduling. A priority mechanism for
atom scheduling is built, where a priority is assigned to each atom in a
resolvent, and primary importance is given to the event of arrival of new atoms
from the body of the applied clause at rewriting time. This new computational
model proves able to address the study of redundancy elimination effects,
giving at the same time interesting insights into general properties of
selection rules. As a matter of fact, a class of scheduling rules, namely the
specialisation independent ones, is defined in the paper by using not trivial
semantic arguments. As a quite surprising result, specialisation independent
scheduling rules turn out to coincide with a class of rules which have an
immediate structural characterisation (named stack-queue rules). Then we prove
that such scheduling rules are tolerant to redundancy elimination, in the sense
that neither program termination nor completeness of equality loop check is
lost passing from SLD to RSLD.Comment: 53 pages, to appear on TPL
An analysis of loop checking mechanisms for logic programs
AbstractWe systematically study loop checking mechanisms for logic programs by considering their soundness, completeness, relative strength and related concepts. We introduce a natural concept of a simple loop check and prove that no sound and complete simple loop check exists, even for programs without function symbols. Then we introduce a number of sound simple loop checks and identify natural classes of Prolog programs without function symbols for which they are complete. In these classes a limited form of recursion is allowed. As a by-product we obtain an implementation of the closed world assumption of Reiter (1978) and a query evaluation algorithm for these classes of logic programs