Actual causality is increasingly well understood. Recent formal approaches, proposed by Halpern and Pearl, have made this concept mature enough to be amenable to automated reasoning. Actual causality is especially vital for building accountable, explainable systems. Among other reasons, causality reasoning is computationally hard due to the requirements of counterfactuality and the minimality of causes. Previous approaches presented either inefficient or restricted, and domain-specific, solutions to the problem of automating causality reasoning. In this paper, we present a novel approach to formulate different notions of causal reasoning, over binary acyclic models, as optimization problems, based on quantifiable notions within counterfactual computations. We contribute and compare two compact, non-trivial, and sound integer linear programming (ILP) and Maximum Satisfiability (MaxSAT) encodings to check causality. Given a candidate cause, both approaches identify what a minimal cause is. Also, we present an ILP encoding to infer causality without requiring a candidate cause. We show that both notions are efficiently automated. Using models with more than $8000$ variables, checking is computed in a matter of seconds, with MaxSAT outperforming ILP in many cases. In contrast, inference is computed in a matter of minutes.

In learning-assisted theorem proving, one of the most critical challenges is to generalize to theorems unlike those seen at training time. In this paper, we introduce INT, an INequality Theorem proving benchmark, specifically designed to test agents' generalization ability. INT is based on a procedure for generating theorems and proofs; this procedure's knobs allow us to measure 6 different types of generalization, each reflecting a distinct challenge characteristic of automated theorem proving. In addition, unlike prior benchmarks for learning-assisted theorem proving, INT provides a lightweight and user-friendly theorem proving environment with fast simulations, conducive to performing learning-based and search-based research. We introduce learning-based baselines and evaluate them across 6 dimensions of generalization with the benchmark. We then evaluate the same agents augmented with Monte Carlo Tree Search (MCTS) at test time, and show that MCTS can help to prove new theorems.

We study the separation of positive and negative data examples in terms of description logic (DL) concepts and formulas of decidable FO fragments, in the presence of an ontology. In contrast to previous work, we add a signature that specifies a subset of the symbols from the data and ontology that can be used for separation. We consider weak and strong versions of the resulting problem that differ in how the negative examples are treated. Our main results are that (a projective form of) the weak version is decidable in $\mathcal{ALCI}$ while it is undecidable in the guarded fragment GF, the guarded negation fragment GNF, and the DL $\mathcal{ALCFIO}$, and that strong separability is decidable in $\mathcal{ALCI}$, GF, and GNF. We also provide (mostly tight) complexity bounds.

The difference between object-language and metalanguage is crucial for logical analysis, but has yet not been examined for the field of computer science. In this paper the difference is examined with regard to inferential relations. It is argued that inferential relations in a metalanguage (like a calculus for propositional logic) cannot represent conceptual relations of natural language. Inferential relations govern our concept use and understanding. Several approaches in the field of Natural Language Understanding (NLU) and Natural Language Inference (NLI) take this insight in account, but do not consider, how an inference can be assessed as a good inference. I present a logical analysis that can assesss the normative dimension of inferences, which is a crucial part of logical understanding and goes beyond formal understanding of metalanguages.

We present an approach towards the deep, pluralistic logical analysis of argumentative discourse that benefits from the application of state-of-the-art automated reasoning technology for classical higher-order logic. Thanks to its expressivity this logic can adopt the status of a uniform \textit{lingua franca} allowing the encoding of both formalized arguments (their deep logical structure) and dialectical interactions (their attack and support relations). We illustrate this by analyzing an excerpt from an argumentative debate on climate engineering. Another, novel contribution concerns the definition of abstract, language-theoretical foundations for the characterization and assessment of shallow semantical embeddings (SSEs) of non-classical logics in classical higher-order logic, which constitute a pillar stone of our approach. The novel perspective we draw enables more concise and more elegant characterizations of semantical embeddings of logics and logic combinations, which is demonstrated with several examples.

Many systems have been developed in recent years to mine logical rules from large-scale Knowledge Graphs (KGs), on the grounds that representing regularities as rules enables both the interpretable inference of new facts, and the explanation of known facts. Among these systems, the walk-based methods that generate the instantiated rules containing constants by abstracting sampled paths in KGs demonstrate strong predictive performance and expressivity. However, due to the large volume of possible rules, these systems do not scale well where computational resources are often wasted on generating and evaluating unpromising rules. In this work, we address such scalability issues by proposing new methods for pruning unpromising rules using rule hierarchies. The approach consists of two phases. Firstly, since rule hierarchies are not readily available in walk-based methods, we have built a Rule Hierarchy Framework (RHF), which leverages a collection of subsumption frameworks to build a proper rule hierarchy from a set of learned rules. And secondly, we adapt RHF to an existing rule learner where we design and implement two methods for Hierarchical Pruning (HPMs), which utilize the generated hierarchies to remove irrelevant and redundant rules. Through experiments over four public benchmark datasets, we show that the application of HPMs is effective in removing unpromising rules, which leads to significant reductions in the runtime as well as in the number of learned rules, without compromising the predictive performance.

A version of the situation calculus in which situations are represented as first-order terms is presented. Fluents can be computed from the term structure, and actions on the situations correspond to rewrite rules on the terms. Actions that only depend on or influence a subset of the fluents can be described as rewrite rules that operate on subterms of the terms in some cases. If actions are bidirectional then efficient completion methods can be used to solve planning problems. This representation for situations and actions is most similar to the fluent calculus of Thielscher \cite{Thielscher98}, except that this representation is more flexible and more use is made of the subterm structure. Some examples are given, and a few general methods for constructing such sets of rewrite rules are presented. This paper was submitted to FSCD 2020 on December 23, 2019.

Description logics (DLs) are standard knowledge representation languages for modelling ontologies, i.e. knowledge about concepts and the relations between them. Unfortunately, DL ontologies are difficult to learn from data and time-consuming to encode manually. As a result, ontologies for broad domains are almost inevitably incomplete. In recent years, several data-driven approaches have been proposed for automatically extending such ontologies. One family of methods rely on characterizations of concepts that are derived from text descriptions. While such characterizations do not capture ontological knowledge directly, they encode information about the similarity between different concepts, which can be exploited for filling in the gaps in existing ontologies. To this end, several inductive inference mechanisms have already been proposed, but these have been defined and used in a heuristic fashion. In this paper, we instead propose an inductive inference mechanism which is based on a clear model-theoretic semantics, and can thus be tightly integrated with standard deductive reasoning. We particularly focus on interpolation, a powerful commonsense reasoning mechanism which is closely related to cognitive models of category-based induction. Apart from the formalization of the underlying semantics, as our main technical contribution we provide computational complexity bounds for reasoning in EL with this interpolation mechanism.

Enabling machines to legal balancing is a non-trivial task challenged by a multitude of factors some of which are addressed and explored in this work. We propose a holistic approach to formal modeling at different abstraction layers supported by a pluralistic framework in which the encoding of an ethico-legal value and upper ontology is developed in combination with the exploration of a formalization logic, with legal domain knowledge and with exemplary use cases until a reflective equilibrium is reached. Our work is enabled by a meta-logical approach to universal logical reasoning and it applies the recently introduced \logikey\ methodology for designing normative theories for ethical and legal reasoning. The particular focus in this paper is on the formalization and encoding of a value ontology suitable e.g. for explaining and resolving legal conflicts in property law (wild animal cases).

We propose a novel framework seamlessly providing key properties of both neural nets (learning) and symbolic logic (knowledge and reasoning). Every neuron has a meaning as a component of a formula in a weighted real-valued logic, yielding a highly intepretable disentangled representation. Inference is omnidirectional rather than focused on predefined target variables, and corresponds to logical reasoning, including classical first-order logic theorem proving as a special case. The model is end-to-end differentiable, and learning minimizes a novel loss function capturing logical contradiction, yielding resilience to inconsistent knowledge. It also enables the open-world assumption by maintaining bounds on truth values which can have probabilistic semantics, yielding resilience to incomplete knowledge.