We consider a weakly supervised learning scenario where the supervision signal is generated by a transition function $\sigma$ of labels associated with multiple input instances. We formulate this problem as \emph{multi-instance Partial Label Learning (multi-instance PLL)}, which is an extension to the standard PLL problem. Our problem is met in different fields, including latent structural learning and neuro-symbolic integration. Despite the existence of many learning techniques, limited theoretical analysis has been dedicated to this problem. In this paper, we provide the first theoretical study of multi-instance PLL with possibly an unknown transition $\sigma$. Our main contributions are as follows. Firstly, we propose a necessary and sufficient condition for the learnability of the problem. This condition non-trivially generalizes and relaxes the existing small ambiguity degree in the PLL literature, since we allow the transition to be deterministic. Secondly, we derive Rademacher-style error bounds based on a top-$k$ surrogate loss that is widely used in the neuro-symbolic literature. Furthermore, we conclude with empirical experiments for learning under unknown transitions. The empirical results align with our theoretical findings; however, they also expose the issue of scalability in the weak supervision literature.

Wikidata is a knowledge graph increasingly adopted by many communities for diverse applications. Wikidata statements are annotated with qualifier-value pairs that are used to depict information, such as the validity context of the statement, its causality, provenances, etc. Handling the qualifiers in reasoning is a challenging problem. When defining inference rules (in particular, rules on ontological properties (x subclass of y, z instance of x, etc.)), one must consider the qualifiers, as most of them participate in the semantics of the statements. This poses a complex problem because a) there is a massive number of qualifiers, and b) the qualifiers of the inferred statement are often a combination of the qualifiers in the rule condition. In this work, we propose to address this problem by a) defining a categorization of the qualifiers b) formalizing the Wikidata model with a many-sorted logical language; the sorts of this language are the qualifier categories. We couple this logic with an algebraic specification that provides a means for effectively handling qualifiers in inference rules. Using Wikidata ontological properties, we show how to use the MSL and specification to reason on qualifiers. Finally, we discuss the methodology for practically implementing the work and present a prototype implementation. The work can be naturally extended, thanks to the extensibility of the many-sorted algebraic specification, to cover more qualifiers in the specification, such as uncertain time, recurring events, geographic locations, and others.

We motivate and formally define a new task for fine-tuning rule-like generalization in large language models. It is conjectured that the shortcomings of current LLMs are due to a lack of ability to generalize. It has been argued that, instead, humans are better at generalization because they have a tendency at extracting rules from complex data. We try to recreate this tendency to rule-based generalization. When exposed to tests of analytic intelligence, for example, the visual RAVEN IQ test, human problem-solvers identify the relevant objects in the picture and their relevant attributes and reason based on rules applied to these objects and attributes. Based on the induced rules, they are able to provide a solution to the test. We propose a task that translates this IQ task into language. In this paper, we provide the formal specification for the task and the generative process of its datasets.

Planning as theorem proving in situation calculus was abandoned 50 years ago as an impossible project. But we have developed a Theorem Proving Lifted Heuristic (TPLH) planner that searches for a plan in a tree of situations using the A* search algorithm. It is controlled by a delete relaxation-based domain independent heuristic. We compare TPLH with Fast Downward (FD) and Best First Width Search (BFWS) planners over several standard benchmarks. Since our implementation of the heuristic function is not optimized, TPLH is slower than FD and BFWS. But it computes shorter plans, and it explores fewer states. We discuss previous research on planning within KR\&R and identify related directions. Thus, we show that deductive lifted heuristic planning in situation calculus is actually doable.

Inference and prediction of routes have become of interest over the past decade owing to a dramatic increase in package delivery and ride-sharing services. Given the underlying combinatorial structure and the incorporation of probabilities, route prediction involves techniques from both formal methods and machine learning. One promising approach for predicting routes uses decision diagrams that are augmented with probability values. However, the effectiveness of this approach depends on the size of the compiled decision diagrams. The scalability of the approach is limited owing to its empirical runtime and space complexity. In this work, our contributions are two-fold: first, we introduce a relaxed encoding that uses a linear number of variables with respect to the number of vertices in a road network graph to significantly reduce the size of resultant decision diagrams. Secondly, instead of a stepwise sampling procedure, we propose a single pass sampling-based route prediction. In our evaluations arising from a real-world road network, we demonstrate that the resulting system achieves around twice the quality of suggested routes while being an order of magnitude faster compared to state-of-the-art.

This paper proposes a new AND-OR graph search framework for synthesis of Linear Temporal Logic on finite traces (\LTLf), that overcomes some limitations of previous approaches. Within such framework, we devise a procedure inspired by the Davis-Putnam-Logemann-Loveland (DPLL) algorithm to generate the next available agent-environment moves in a truly depth-first fashion, possibly avoiding exhaustive enumeration or costly compilations. We also propose a novel equivalence check for search nodes based on syntactic equivalence of state formulas. Since the resulting procedure is not guaranteed to terminate, we identify a stopping condition to abort execution and restart the search with state-equivalence checking based on Binary Decision Diagrams (BDD), which we show to be correct. The experimental results show that in many cases the proposed techniques outperform other state-of-the-art approaches. Our implementation Nike competed in the LTLf Realizability Track in the 2023 edition of SYNTCOMP, and won the competition.

The success of Computer Vision (CV) relies heavily on manually annotated data. However, it is prohibitively expensive to annotate images in key domains such as healthcare, where data labeling requires significant domain expertise and cannot be easily delegated to crowd workers. To address this challenge, we propose a neuro-symbolic approach called Rapid, which infers image labeling rules from a small amount of labeled data provided by domain experts and automatically labels unannotated data using the rules. Specifically, Rapid combines pre-trained CV models and inductive logic learning to infer the logic-based labeling rules. Rapid achieves a labeling accuracy of 83.33% to 88.33% on four image labeling tasks with only 12 to 39 labeled samples. In particular, Rapid significantly outperforms finetuned CV models in two highly specialized tasks. These results demonstrate the effectiveness of Rapid in learning from small data and its capability to generalize among different tasks. Code and our dataset are publicly available at https://github.com/Neural-Symbolic-Image-Labeling/

This paper addresses the problem of data-driven model discrimination for unknown switched systems with unknown linear temporal logic (LTL) specifications, representing tasks, that govern their mode sequences, where only sampled data of the unknown dynamics and tasks are available. To tackle this problem, we propose data-driven methods to over-approximate the unknown dynamics and to infer the unknown specifications such that both set-membership models of the unknown dynamics and LTL formulas are guaranteed to include the ground truth model and specification/task. Moreover, we present an optimization-based algorithm for analyzing the distinguishability of a set of learned/inferred model-task pairs as well as a model discrimination algorithm for ruling out model-task pairs from this set that are inconsistent with new observations at run time. Further, we present an approach for reducing the size of inferred specifications to increase the computational efficiency of the model discrimination algorithms.

Evaluating counterfactual statements is a fundamental problem for many approaches to causal reasoning [40]. Such reasoning can for instance be used to explain erroneous system behavior with a counterfactual statement such as'If the input i at the first position of the observed computation π had not been enabled then the system would not have reached an error e.' which can be formalized using the counterfactual operator and the temporal operator F: π ( i) ( Fe). Since the foundational work by Lewis[38] on the formal semantics of counterfactual conditionals, many applications for counterfactuals [28, 5, 34, 46, 3, 15] and some theoretical results on the decidability of the original theory [37] and related notions [20, 2] have been discovered. Still, certain domains have proven elusive for a long time, for instance, theories involving higher-order reasoning and an infinite number of variables. In this paper, we consider a domain that combines both of these aspects: temporal reasoning over infinite sequences. In particular, we consider counterfactual conditionals that relate two properties expressed in temporal logics, such as the temporal property F e from the introductory example. Temporal logics are used ubiquitously as high-level specifications for verification [21, 4] and synthesis [22, 41], and recently have also found use in specifying reinforcement learning tasks [32, 39]. Our work lifts the language of counterfactual reasoning to similar high-level expressions.

Probabilistic programming combines general computer programming, statistical inference, and formal semantics to help systems make decisions when facing uncertainty. Probabilistic programs are ubiquitous, including having a significant impact on machine intelligence. While many probabilistic algorithms have been used in practice in different domains, their automated verification based on formal semantics is still a relatively new research area. In the last two decades, it has attracted much interest. Many challenges, however, remain. The work presented in this paper, probabilistic relations, takes a step towards our vision to tackle these challenges. Our work is based on Hehner's predicative probabilistic programming, but there are several obstacles to the broader adoption of his work. Our contributions here include (1) the formalisation of its syntax and semantics by introducing an Iverson bracket notation to separate relations from arithmetic; (2) the formalisation of relations using Unifying Theories of Programming (UTP) and probabilities outside the brackets using summation over the topological space of the real numbers; (3) the constructive semantics for probabilistic loops using Kleene's fixed-point theorem; (4) the enrichment of its semantics from distributions to subdistributions and superdistributions to deal with the constructive semantics; (5) the unique fixed-point theorem to simplify the reasoning about probabilistic loops; and (6) the mechanisation of our theory in Isabelle/UTP, an implementation of UTP in Isabelle/HOL, for automated reasoning using theorem proving. We demonstrate our work with six examples, including problems in robot localisation, classification in machine learning, and the termination of probabilistic loops.