Related papers: G\"odel Mirror: A Formal System For Contradiction-…
Circular and non-wellfounded proofs have become an increasingly popular tool for metalogical treatments of systems with forms of induction and/or recursion. In this work we investigate the expressivity of a variant CT of G\"odel's system T…
The present dissertation introduces the research project on HOLMS (\textbf{HOL} Light Library for \textbf{M}odal \textbf{S}ystems), a growing modular framework for modal reasoning within the HOL Light proof assistant. To provide an…
Control systems are an integral part of almost every engineering and physical system and thus their accurate analysis is of utmost importance. Traditionally, control systems are analyzed using paper-and-pencil proof and computer simulation…
We introduce the concept of the self-referencing causal cycle (abbreviated RECALL) - a mechanism that enables large language models (LLMs) to bypass the limitations of unidirectional causality, which underlies a phenomenon known as the…
We introduce SELF-DISCOVER, a general framework for LLMs to self-discover the task-intrinsic reasoning structures to tackle complex reasoning problems that are challenging for typical prompting methods. Core to the framework is a…
AI-driven autoformalization of mathematics is advancing rapidly. However, the type checker of a proof assistant guarantees only the logical correctness of proofs; it does not verify whether propositions and definitions faithfully capture…
This paper investigates the ability of transformer-based models to learn structural recursion from examples. Recursion is a universal concept in both natural and formal languages. Structural recursion is central to the programming language…
G\"odel's argument for the First Incompleteness Theorem is, structurally, a proof by contradiction. This article intends to reframe the argument by, first, isolating an additional assumption the argument relies on, and then, second, arguing…
In this paper, we argue that formal systems of first order Arithmetic that admit Goedelian undecidable propositions validly are abnormally non-constructive. We argue that, in such systems, the strong representation of primitive recursive…
In this project, a rather complete proof-theoretical formalization of Lambek Calculus (non-associative with arbitrary extensions) has been ported from Coq proof assistent to HOL4 theorem prover, with some improvements and new theorems.…
Despite the syntactic fluency of Large Language Models (LLMs), ensuring their logical correctness in high-stakes domains remains a fundamental challenge. We present a neurosymbolic framework that combines LLMs with SMT solvers to produce…
A semantic analysis of formal systems is undertaken, wherein the duality of their symbolic definition based on the "State of Doing" and "State of Being" is brought out. We demonstrate that when these states are defined in a way that opposes…
We introduce a framework for learning continuous neural representations of formal specifications by distilling the geometry of their semantics into a latent space. Existing approaches rely either on symbolic kernels -- which preserve…
Large language models can change answers under harmless edits that matter in practice: RAG outputs flip when passages are reordered, fine-tuning erodes invariances learned at pretraining, debate or chain-of-thought prompts take…
In his paper on the incompleteness theorems, G\"odel seemed to say that a direct way of constructing a formula that says of itself that it is unprovable might involve a faulty circularity. In this note, it is proved that 'direct'…
Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate…
Formal mathematical reasoning remains a critical challenge for artificial intelligence, hindered by limitations of existing benchmarks in scope and scale. To address this, we present FormalMATH, a large-scale Lean4 benchmark comprising…
Most discussions of G\"odel's theorems fall into one of two types: either they emphasize perceived philosophical, cultural "meanings" of the theorems, and perhaps sketch some of the ideas of the proofs, usually relating G\"odel's proofs to…
RL-trained Lean theorem provers mode-collapse at inference time: on miniF2F-test with DeepSeek-Prover-V1.5-RL, doubling the i.i.d.\ sampling budget from $k{=}32$ to $k{=}64$ produces zero additional solved theorems (42/244 in both cases). A…
Non-wellfounded proof systems impose a global condition called the global trace condition (GTC) on a derivation tree to ensure soundness. Providing a categorical characterisation of the GTC that guarantees soundness remains challenging due…