The TPTP World is the well established infrastructure that supports research, development, and deployment of Automated Theorem Proving (ATP) systems. The TPTP World supports a range of classical logics, and since release v9.0.0 has supported non-classical logics. This paper provides a self-contained comprehensive overview of the TPTP World infrastructure for ATP in non-classical logics: the non-classical language extension, problems and solutions, and tool support. A detailed description of use of the infrastructure for quantified normal multi-modal logic is given.

Large vision language models exhibit notable limitations on Geometry Problem Solving (GPS) because of their unreliable diagram interpretation and pure natural-language reasoning. A recent line of work mitigates this by using symbolic solvers: the model directly generates a formal program that a geometry solver can execute. However, this direct program generation lacks intermediate reasoning, making the decision process opaque and prone to errors. In this work, we explore a new approach that integrates Chain-of-Thought (CoT) with formal language. The model interleaves natural language reasoning with incremental emission of solver-executable code, producing a hybrid reasoning trace in which critical derivations are expressed in formal language. To teach this behavior at scale, we combine (1) supervised fine-tuning on an 11K newly developed synthetic dataset with interleaved natural language reasoning and automatic formalization, and (2) solver-in-the-loop reinforcement learning that jointly optimizes both the CoT narrative and the resulting program through outcome-based rewards. Built on Qwen2.5-VL-7B, our new model, named GF-Reasoner, achieves up to 15% accuracy improvements on standard GPS benchmarks, surpassing both 7B-scale peers and the much larger model Qwen2.5-VL-72B. By exploiting high-order geometric knowledge and offloading symbolic computation to the solver, the generated reasoning traces are noticeably shorter and cleaner. Furthermore, we present a comprehensive analysis of method design choices (e.g., reasoning paradigms, data synthesis, training epochs, etc.), providing actionable insights for future research.

Answer Set Programming (ASP) is often hindered by the grounding bottleneck: large Herbrand universes generate ground programs so large that solving becomes difficult. Many methods employ ad-hoc heuristics to improve grounding performance, motivating the need for a more formal and generalizable strategy. We introduce the notion of diminution, defined as a selected subset of the Herbrand universe used to generate a reduced ground program before solving. We give a formal definition of diminution, analyze its key properties, and study the complexity of identifying it. We use a specific encoding that enables off-the-shelf ASP solver to evaluate candidate subsets. Our approach integrates seamlessly with existing grounders via domain predicates. In extensive experiments on five benchmarks, applying diminutions selected by our strategy yields significant performance improvements, reducing grounding time by up to 70% on average and decreasing the size of grounding files by up to 85%. These results demonstrate that leveraging diminutions constitutes a robust and general-purpose approach for alleviating the grounding bottleneck in ASP.

Lifted inference rules exploit symmetries for fast reasoning in statistical rela-tional models. Computational complexity of these rules is highly dependent onthe choice of the constraint language they operate on and therefore coming upwith the right kind of representation is critical to the success of lifted inference.In this paper, we propose a new constraint language, called setineq, which allowssubset, equality and inequality constraints, to represent substitutions over the vari-ables in the theory. Our constraint formulation is strictly more expressive thanexisting representations, yet easy to operate on. We reformulate the three mainlifting rules: decomposer, generalized binomial and the recently proposed singleoccurrence for MAP inference, to work with our constraint representation. Exper-iments on benchmark MLNs for exact and sampling based inference demonstratethe effectiveness of our approach over several other existing techniques.

Python has become the dominant language for general-purpose programming, yet it lacks robust tools for formal verification. In contrast, programmers working in languages such as C benefit from mature model checkers, for example CBMC, which enable exhaustive symbolic reasoning and fault localisation. The inherent complexity of Python, coupled with the verbosity and low-level nature of existing transpilers (e.g., Cython), have historically limited the applicability of formal verification to Python programs. In this paper, we propose PyVeritas, a novel framework that leverages Large Language Models (LLMs) for high-level transpilation from Python to C, followed by bounded model checking and MaxSAT-based fault localisation in the generated C code. PyVeritas enables verification and bug localisation for Python code using existing model checking tools for C. Our empirical evaluation on two Python benchmarks demonstrates that LLM-based transpilation can achieve a high degree of accuracy, up to 80--90% for some LLMs, enabling effective development environment that supports assertion-based verification and interpretable fault diagnosis for small yet non-trivial Python programs.

This paper introduces a new family of cognitive modal logics designed to formalize conjectural reasoning: a modal system in which cognitive contexts extend known facts with hypothetical assumptions to explore their consequences. Unlike traditional doxastic and epistemic systems, conjectural logics rely on a principle, called Axiom C ($φ\rightarrow \Boxφ$), that ensures that all established facts are preserved across hypothetical layers. While Axiom C was dismissed in the past due to its association with modal collapse, we show that the collapse only arises under classical and bivalent assumptions, and specifically in the presence of Axiom T. Hence we avoid Axiom T and adopt a paracomplete semantic framework, grounded in Weak Kleene logic or Description Logic, where undefined propositions coexist with modal assertions. This prevents the modal collapse and guarantees a layering to distinguish between factual and conjectural statements. Under this framework we define new modal systems, e.g., KC and KDC, and show that they are complete, decidable, and robust under partial knowledge. Finally, we introduce a dynamic operation, $\mathsf{settle}(φ)$, which formalizes the transition from conjecture to accepted fact, capturing the event of the update of a world's cognitive state through the resolution of uncertainty.

Given a relational specification between inputs and outputs as a logic formula, the problem of functional synthesis is to automatically synthesize a function from inputs to outputs satisfying the relation. Recently, a rich line of work has emerged tackling this problem for specifications in different theories, from Boolean to general first-order logic. In this paper, we launch an investigation of this problem for the theory of Pres-burger Arithmetic, that we call Presburger Functional Synthesis (PFnS). We show that PFnS can be solved in EXPTIME and provide a matching exponential lower bound. This is unlike the case for Boolean functional synthesis (BFnS), where only conditional exponential lower bounds are known. Further, we show that PFnS for one input and one output variable is as hard as BFnS in general. We then identify a special normal form, called PSyNF, for the specification formula that guarantees poly-time and poly-size solvability of PFnS. We prove several properties of PSyNF, including how to check and compile to this form, and conditions under which any other form that guarantees poly-time solvability of PFnS can be compiled in poly-time to PSyNF. Finally, we identify a syntactic normal form that is easier to check but is exponentially less succinct than PSyNF.

We consider the problem of implementing deontic modal logic. We show how (deontic) modal operators can be expressed elegantly using default negation (negation-as-failure) and strong negation present in answer set programming (ASP). We propose using global constraints of ASP to represent obligations and impermissibilities of deontic modal logic. We show that our proposed representation results in the various paradoxes of deontic modal logic being elegantly resolved.

Differentiable inductive logic programming (ILP) techniques have proven effective at finding approximate rule-based solutions to link prediction and node classification problems on knowledge graphs; however, the common assumption of chain-like rule structure can hamper the performance and interpretability of existing approaches. We introduce GLIDR, a differentiable rule learning method that models the inference of logic rules with more expressive syntax than previous methods. GLIDR uses a differentiable message passing inference algorithm that generalizes previous chain-like rule learning methods to allow rules with features like branches and cycles. GLIDR has a simple and expressive rule search space which is parameterized by a limit on the maximum number of free variables that may be included in a rule. Explicit logic rules can be extracted from the weights of a GLIDR model for use with symbolic solvers. We demonstrate that GLIDR can significantly outperform existing rule learning methods on knowledge graph completion tasks and even compete with embedding methods despite the inherent disadvantage of being a structure-only prediction method. We show that rules extracted from GLIDR retain significant predictive performance, and that GLIDR is highly robust to training data noise. Finally, we demonstrate that GLIDR can be chained with deep neural networks and optimized end-to-end for rule learning on arbitrary data modalities.

This dialog paper offers a preview and provides a foretaste of an upcoming work on the axiomatization of basic interactive algorithms. The modern notion of algorithm was elucidated in the 1930s--1950s. It was axiomatized a quarter of a century ago as the notion of ``sequential algorithm'' or ``classical algorithm''; we prefer to call it ``basic algorithm" now. The axiomatization was used to show that for every basic algorithm there is a behaviorally equivalent abstract state machine. It was also used to prove the Church-Turing thesis as it has been understood by the logicians. Starting from the 1960s, the notion of algorithm has expanded -- probabilistic algorithms, quantum algorithms, etc. -- prompting introduction of a much more ambitious version of the Church-Turing thesis commonly known as the ``physical thesis.'' We emphasize the difference between the two versions of the Church-Turing thesis and illustrate how nondeterministic and probabilistic algorithms can be viewed as basic algorithms with appropriate oracles. The same view applies to quantum circuit algorithms and many other classes of algorithms.