Differentiable Inductive Logic Programming for Structured Examples

Abstract

The differentiable implementation of logic yields a seamless combination of symbolic reasoning and deep neural networks. Recent research, which has developed a differentiable framework to learn logic programs from examples, can even acquire reasonable solutions from noisy datasets. However, this framework severely limits expressions for solutions, e.g., no function symbols are allowed, and the shapes of clauses are fixed. As a result, the framework cannot deal with structured examples. Therefore we propose a new framework to learn logic programs from noisy and structured examples, including the following contributions. First, we propose an adaptive clause search method by looking through structured space, which is defined by the generality of the clauses, to yield an efficient search space for differentiable solvers. Second, we propose for ground atoms an enumeration algorithm, which determines a necessary and sufficient set of ground atoms to perform differentiable inference functions. Finally, we propose a new method to compose logic programs softly, enabling the system to deal with complex programs consisting of several clauses. Our experiments show that our new framework can learn logic programs from noisy and structured examples, such as sequences or trees. Our framework can be scaled to deal with complex programs that consist of several clauses with function symbols.

PyReason: Software for Open World Temporal Logic

“…Historically logical rules had mostly been written by domain experts, until early work like Apriori [27] and FOIL [28] to learn association rules from data followed by the emergence of rule mining techniques like causal rule mining [29] and annotated probabilistic temporal logic [24,30,31]. More recently, there has been research on Differentiable Inductive Logic Programming (𝜕ILP) -an inductive rule learning method to learn logical rules from examples [3,16,32]. In the below list 𝑈 𝑛𝑎𝑆𝑒𝑡 and 𝐵𝑖𝑛𝑆𝑒𝑡 are arbitrarily sets of unary and binary predicates relevant to the rules while 𝑝𝑟𝑒𝑑 is always a non-static predicate.…”

“…𝒯 (not necessarily complete) with ordering ⊑. To support contemporary applications in neuro symbolic reasoning [2,3,7,16,17] as well as social network analysis [9,8] we implemented this as a lower semilattice structure. Therefore, we have a single element ⊥ and multiple top elements ⊤ 0 , .…”

“…By implementing the fixpoint operator directly (as opposed to a black box heuristic) the software enables full explainability of the result. As such is the case, this framework captures not only classical logic, but a wide variety of other logic frameworks including fuzzy logic [10,11,12], weighted real valued logic used in logical neural networks [2], van Emden's logic [13], Fitting's bilattice logic [14], various logic frameworks for reasoning over graphs or social networks [9,8,15] (as well as the various network diffusion models captured by those frameworks), and perhaps most importantly, logic frameworks where syntactic structure can be learned using differentiable inductive logic programming [3,16] as well as other neuro symbolic frameworks [17,7]. The key advantages of our approach include the following:…”

“…Further, our framework goes beyond support for fuzzy operators by enabling arbitrary functions that can be used over real values or intervals of reals. This is a key advantage to reasoning about constructs learned with neuro symbolic approaches such as [2,3,16,17,7]. 3.…”

Extensions to Generalized Annotated Logic and an Equivalent Neural Architecture

“…Differentiable inductive logic programming (ILP) [4], [5] is designed to learn logical rules using gradient descent. This approach suffers from scalability issues as it involves generating a large number of "rule templates" and assigning them weights using gradient descent.…”