We study the problem of generating interesting integer sequences with a combinatorial interpretation. For this we introduce a two-step approach. In the first step, we generate first-order logic sentences which define some combinatorial objects, e.g., undirected graphs, permutations, matchings etc. In the second step, we use algorithms for lifted first-order model counting to generate integer sequences that count the objects encoded by the first-order logic formulas generated in the first step. For instance, if the first-order sentence defines permutations then the generated integer sequence is the sequence of factorial numbers n!. We demonstrate that our approach is able to generate interesting new sequences by showing that a non-negligible fraction of the automatically generated sequences can actually be found in the Online Encyclopaedia of Integer Sequences (OEIS) while generating many other similar sequences which are not present in OEIS and which are potentially interesting. A key technical contribution of our work is the method for generation of first-order logic sentences which is able to drastically prune the space of sentences by discarding large fraction of sentences which would lead to redundant integer sequences.

Relational learning for knowledge base completion has been receiving considerable attention. Intuitively, rule-based strategies are clearly appealing, given their transparency and their ability to capture complex relational dependencies. In practice, however, pure rule-based strategies are currently not competitive with state-of-the-art methods, which is a reflection of the fact that (i) learning high-quality rules is challenging, and (ii) classical entailment is too brittle to cope with the noisy nature of the learned rules and the given knowledge base. In this paper, we introduce STRiKE, a new approach for relational learning in knowledge bases which addresses these concerns. Our contribution is three-fold. First, we introduce a new method for learning stratified rule bases from relational data. Second, to use these rules in a noise-tolerant way, we propose a strategy which extends k-entailment, a recently introduced cautious entailment relation, to stratified rule bases. Finally, we introduce an efficient algorithm for reasoning based on k-entailment.

Collective inference is a popular approach for solving tasks as knowledge graph completion within the statistical relational learning field. There are many existing solutions for this task, however, each of them is subjected to some limitation, either by restriction to only some learning settings, lacking interpretability of the model or theoretical test error bounds. We propose an approach based on cautious inference process which uses first-order rules and provides PAC-style bounds.