Statistical relational models provide compact encodings of probabilistic dependencies in relational domains, but result in highly intractable graphical models. The goal of lifted inference is to carry out probabilistic inference without needing to reason about each individual separately, by instead treating exchangeable, undistinguished objects as a whole. In this paper, we study the domain recursion inference rule, which, despite its central role in early theoretical results on domain-lifted inference, has later been believed redundant. We show that this rule is more powerful than expected, and in fact significantly extends the range of models for which lifted inference runs in time polynomial in the number of individuals in the domain. This includes an open problem called S4, the symmetric transitivity model, and a first-order logic encoding of the birthday paradox. We further identify new classes S2FO2 and S2RU of domain-liftable theories, which respectively subsume FO2 and recursively unary theories, the largest classes of domain-liftable theories known so far, and show that using domain recursion can achieve exponential speedup even in theories that cannot fully be lifted with the existing set of inference rules.

In this work, we introduce a machine learning based method to search for adynamic proof within these proof systems.

Polynomial inequalities lie at the heart of many mathematical disciplines. In this paper, we consider the fundamental computational task of automatically searching for proofs of polynomial inequalities. We adopt the framework of semi-algebraic proof systems that manipulate polynomial inequalities via elementary inference rules that infer new inequalities from the premises. These proof systems are known to be very powerful, but searching for proofs remains a major difficulty. In this work, we introduce a machine learning based method to search for a dynamic proof within these proof systems. We propose a deep reinforcement learning framework that learns an embedding of the polynomials and guides the choice of inference rules, taking the inherent symmetries of the problem as an inductive bias. We compare our approach with powerful and widely-studied linear programming hierarchies based on static proof systems, and show that our method reduces the size of the linear program by several orders of magnitude while also improving performance. These results hence pave the way towards augmenting powerful and well-studied semi-algebraic proof systems with machine learning guiding strategies for enhancing the expressivity of such proof systems.

Machine learning for early prediction in medicine has recently shown breakthrough performance, however, the focus on improving prediction accuracy has led to a neglect of faithful explanations that are required to gain the trust of medical practitioners. The goal of this paper is to teach LLMs to follow medical consensus guidelines step-by-step in their reasoning and prediction process. Since consensus guidelines are ubiquitous in medicine, instantiations of verbalized medical inference rules to electronic health records provide data for fine-tuning LLMs to learn consensus rules and possible exceptions thereof for many medical areas. Consensus rules also enable an automatic evaluation of the model's inference process regarding its derivation correctness (evaluating correct and faithful deduction of a conclusion from given premises) and value correctness (comparing predicted values against real-world measurements). We exemplify our work using the complex Sepsis-3 consensus definition. Our experiments show that small fine-tuned models outperform one-shot learning of considerably larger LLMs that are prompted with the explicit definition and models that are trained on medical texts including consensus definitions. Since fine-tuning on verbalized rule instantiations of a specific medical area yields nearly perfect derivation correctness for rules (and exceptions) on unseen patient data in that area, the bottleneck for early prediction is not out-of-distribution generalization, but the orthogonal problem of generalization into the future by forecasting sparsely and irregularly sampled clinical variables. We show that the latter results can be improved by integrating the output representations of a time series forecasting model with the LLM in a multimodal setup.

We consider a setting in which a principal gets to choose which game from some given set is played by a group of agents. The principal would like to choose a game that favors one of the players, the social preferences of the players, or the principal's own preferences. Unfortunately, given the potential multiplicity of equilibria, it is conceptually unclear how to tell which of even any two games is better. Oesterheld et al. (2022) propose that we use assumptions about outcome correspondence -- i.e., about how the outcomes of different games relate -- to allow comparisons in some cases. For example, it seems reasonable to assume that isomorphic games are played isomorphically. From such assumptions we can sometimes deduce that the outcome of one game G' is guaranteed to be better than the outcome of another game G, even if we do not have beliefs about how each of G and G' will be played individually. Following Oesterheld et al., we then call G' a safe improvement on G. In this paper, we study how to derive safe improvement relations. We first show that if we are given a set of games and arbitrary assumptions about outcome correspondence between these games, deriving safe improvement relations is co-NP-complete. We then study the (in)completeness of a natural set of inference rules for outcome correspondence. We show that in general the inference rules are incomplete. However, we also show that under natural, generally applicable assumptions about outcome correspondence the rules are complete.

Statistical relational models provide compact encodings of probabilistic dependencies in relational domains, but result in highly intractable graphical models. The goal of lifted inference is to carry out probabilistic inference without needing to reason about each individual separately, by instead treating exchangeable, undistinguished objects as a whole. In this paper, we study the domain recursion inference rule, which, despite its central role in early theoretical results on domain-lifted inference, has later been believed redundant. We show that this rule is more powerful than expected, and in fact significantly extends the range of models for which lifted inference runs in time polynomial in the number of individuals in the domain. This includes an open problem called S4, the symmetric transitivity model, and a first-order logic encoding of the birthday paradox. We further identify new classes S2FO2 and S2RU of domain-liftable theories, which respectively subsume FO2 and recursively unary theories, the largest classes of domain-liftable theories known so far, and show that using domain recursion can achieve exponential speedup even in theories that cannot fully be lifted with the existing set of inference rules.

Statistical relational models provide compact encodings of probabilistic dependencies in relational domains, but result in highly intractable graphical models. The goal of lifted inference is to carry out probabilistic inference without needing to reason about each individual separately, by instead treating exchangeable, undistinguished objects as a whole. In this paper, we study the domain recursion inference rule, which, despite its central role in early theoretical results on domain-lifted inference, has later been believed redundant.

Large Language Models (LLMs) often exhibit limited logical coherence, mapping premises to conclusions without adherence to explicit inference rules. We propose Proof-Carrying Reasoning with LLMs (PCRLLM), a framework that constrains reasoning to single-step inferences while preserving natural language formulations. Each output explicitly specifies premises, rules, and conclusions, thereby enabling verification against a target logic. This mechanism mitigates trustworthiness concerns by supporting chain-level validation even in black-box settings. Moreover, PCRLLM facilitates systematic multi-LLM collaboration, allowing intermediate steps to be compared and integrated under formal rules. Finally, we introduce a benchmark schema for generating large-scale step-level reasoning data, combining natural language expressiveness with formal rigor.

The paper proposes to treat agent awareness as a form of knowledge, breaking the tradition in the existing literature on awareness. It distinguishes the de re and de dicto forms of such knowledge. The work introduces two modalities capturing these forms and formally specifies their meaning using a version of 2D-semantics. The main technical result is a sound and complete logical system describing the interplay between the two proposed modalities and the standard "knowledge of the fact" modality.