Scientists form hypotheses and experimentally test them. If a hypothesis fails (is refuted), scientists try to explain the failure to eliminate other hypotheses. We introduce similar explanation techniques for inductive logic programming (ILP). We build on the ILP approach learning from failures. Given a hypothesis represented as a logic program, we test it on examples. If a hypothesis fails, we identify clauses and literals responsible for the failure. By explaining failures, we can eliminate other hypotheses that will provably fail. We introduce a technique for failure explanation based on analysing SLD-trees. We experimentally evaluate failure explanation in the Popper ILP system. Our results show that explaining failures can drastically reduce learning times.

Over the last years, there has been increasing research on the scaling behaviour of statistical relational representations with the size of the domain, and on the connections between domain size dependence and lifted inference. In particular, the asymptotic behaviour of statistical relational representations has come under scrutiny, and projectivity was isolated as the strongest form of domain size independence. In this contribution we show that every probabilistic logic program under the distribution semantics is asymptotically equivalent to a probabilistic logic program consisting only of range-restricted clauses over probabilistic facts. To facilitate the application of classical results from finite model theory, we introduce the abstract distribution semantics, defined as an arbitrary logical theory over probabilistic facts to bridge the gap to the distribution semantics underlying probabilistic logic programming. In this representation, range-restricted logic programs correspond to quantifier-free theories, making asymptotic quantifier results avilable for use. We can conclude that every probabilistic logic program inducing a projective family of distributions is in fact captured by this class, and we can infer interesting consequences for the expressivity of probabilistic logic programs as well as for the asymptotic behaviour of probabilistic rules.

Since the seminal work of Hintikka on epistemic logic [28], of Von Wright on the logic of preference [55, 56] and of Cohen & Levesque on the logic of intention [19], many formal logics for reasoning about cognitive attitudes of agents such as knowledge and belief [24], preference [32, 48], desire [23], intention [44, 30] and their combination [38, 54] have been proposed. Generally speaking, these logics are nothing but formal models of rational agency relying on the idea that an agent endowed with cognitive attitudes makes decisions on the basis of what she believes and of what she desires or prefers. The idea of describing rational agents in terms of their epistemic and motivational attitudes is something that these logics share with classical decision theory and game theory. Classical decision theory and game theory provide a quantitative account of individual and strategic decision-making by assuming that agents' beliefs and desires can be respectively modeled by subjective probabilities and utilities. Qualitative approaches to individual and strategic decision-making have been proposed in AI [16, 22] to characterize criteria that a rational agent should adopt for making decisions when she cannot build a probability distribution over the set of possible events and her preference over the set of possible outcomes cannot be expressed by a utility function but only by a qualitative ordering over the outcomes.

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in nondeterministic polynomial time. We also explore specializations like nominal letrec-matching for expressions, for DAGs, and for garbage-free expressions and determine their complexity. Finally, we also provide a nominal unification algorithm for higher-order expressions with recursive let and atom-variables, where we show that it also runs in nondeterministic polynomial time.

Labeled data for imitation learning of theorem proving in large libraries of formalized mathematics is scarce as such libraries require years of concentrated effort by human specialists to be built. This is particularly challenging when applying large Transformer language models to tactic prediction, because the scaling of performance with respect to model size is quickly disrupted in the data-scarce, easily-overfitted regime. We propose PACT ({\bf P}roof {\bf A}rtifact {\bf C}o-{\bf T}raining), a general methodology for extracting abundant self-supervised data from kernel-level proof terms for co-training alongside the usual tactic prediction objective. We apply this methodology to Lean, an interactive proof assistant which hosts some of the most sophisticated formalized mathematics to date. We instrument Lean with a neural theorem prover driven by a Transformer language model and show that PACT improves theorem proving success rate on a held-out suite of test theorems from 32\% to 48\%.

Circuit representations are becoming the lingua franca to express and reason about tractable generative and discriminative models. In this paper, we show how complex inference scenarios for these models that commonly arise in machine learning -- from computing the expectations of decision tree ensembles to information-theoretic divergences of deep mixture models -- can be represented in terms of tractable modular operations over circuits. Specifically, we characterize the tractability of a vocabulary of simple transformations -- sums, products, quotients, powers, logarithms, and exponentials -- in terms of sufficient structural constraints of the circuits they operate on, and present novel hardness results for the cases in which these properties are not satisfied. Building on these operations, we derive a unified framework for reasoning about tractable models that generalizes several results in the literature and opens up novel tractable inference scenarios.

We develop Stratified Shortest Solution Imitation Learning (3SIL) to learn equational theorem proving in a deep reinforcement learning (RL) setting. The self-trained models achieve state-of-the-art performance in proving problems generated by one of the top open conjectures in quasigroup theory, the Abelian Inner Mapping (AIM) Conjecture. To develop the methods, we first use two simpler arithmetic rewriting tasks that share tree-structured proof states and sparse rewards with the AIM problems. On these tasks, 3SIL is shown to significantly outperform several established RL and imitation learning methods. The final system is then evaluated in a standalone and cooperative mode on the AIM problems. The standalone 3SIL-trained system proves in 60 seconds more theorems (70.2%) than the complex, hand-engineered Waldmeister system (65.5%). In the cooperative mode, the final system is combined with the Prover9 system, proving in 2 seconds what standalone Prover9 proves in 60 seconds.

Vampire has been for a long time the strongest first-order automated theorem prover, widely used for hammer-style proof automation in ITPs such as Mizar, Isabelle, HOL and Coq. In this work, we considerably improve the performance of Vampire in hammering over the full Mizar library by enhancing its saturation procedure with efficient neural guidance. In particular, we employ a recursive neural network classifying the generated clauses based only on their derivation history. Compared to previous neural methods based on considering the logical content of the clauses, this leads to large real-time speedup of the neural guidance. The resulting system shows good learning capability and achieves state-of-the-art performance on the Mizar library, while proving many theorems that the related ENIGMA system could not prove in a similar hammering evaluation.

Egalitarian considerations play a central role in many areas of social choice theory. Applications of egalitarian principles range from ensuring everyone gets an equal share of a cake when deciding how to divide it, to guaranteeing balance with respect to gender or ethnicity in committee elections. Yet, the egalitarian approach has received little attention in judgment aggregation -- a powerful framework for aggregating logically interconnected issues. We make the first steps towards filling that gap. We introduce axioms capturing two classical interpretations of egalitarianism in judgment aggregation and situate these within the context of existing axioms in the pertinent framework of belief merging. We then explore the relationship between these axioms and several notions of strategyproofness from social choice theory at large. Finally, a novel egalitarian judgment aggregation rule stems from our analysis; we present complexity results concerning both outcome determination and strategic manipulation for that rule.

The chase is a well-established family of algorithms used to materialize Knowledge Bases (KBs), like Knowledge Graphs (KGs), to tackle important tasks like query answering under dependencies or data cleaning. A general problem of chase algorithms is that they might perform redundant computations. To counter this problem, we introduce the notion of Trigger Graphs (TGs), which guide the execution of the rules avoiding redundant computations. We present the results of an extensive theoretical and empirical study that seeks to answer when and how TGs can be computed and what are the benefits of TGs when applied over real-world KBs. Our results include introducing algorithms that compute (minimal) TGs. We implemented our approach in a new engine, and our experiments show that it can be significantly more efficient than the chase enabling us to materialize KBs with 17B facts in less than 40 min on commodity machines.