Related papers: G\"odel Mirror: A Formal System For Contradiction-…
This article describes the *Confluence Framework*, a novel framework for proving and disproving confluence using a divide-and-conquer modular strategy, and its implementation in CONFident. Using this approach, we are able to automatically…
The investigations on higher-order type theories and on the related notion of parametric polymorphism constitute the technical counterpart of the old foundational problem of the circularity (or impredicativity) of second and higher order…
The chase is a sound, complete, but possibly non-terminating algorithm for reasoning with existential rules (aka. tuple-generating dependencies), a highly expressive knowledge representation language. Although the procedure appears simple,…
If we apply an extension of the Deduction meta-Theorem to Goedel's meta-reasoning of "undecidability", we can conclude that Goedel's formal system of Arithmetic is not omega-consistent. If we then take the standard interpretation…
We introduce a~paraconsistent modal logic $\mathbf{K}\mathsf{G}^2$, based on G\"{o}del logic with coimplication (bi-G\"{o}del logic) expanded with a De Morgan negation $\neg$. We use the logic to formalise reasoning with graded, incomplete…
There are two well known systems formalizing total recursion beyond primitive recursion (\textbf{PR}), system \textbf{T} by G\"odel and system \textbf{F} by Girard and Reynolds. system \textbf{T} defines recursion on typed objects and can…
Statement autoformalization acts as a critical bridge between human mathematics and formal mathematics by translating natural language problems into formal language. While prior works have focused on data synthesis and diverse training…
We propose ProofNet++, a neuro-symbolic framework that enhances automated theorem proving by combining large language models (LLMs) with formal proof verification and self-correction mechanisms. Current LLM-based systems suffer from…
Discovering mathematical models that characterize the observed behavior of dynamical systems remains a major challenge, especially for systems in a chaotic regime. The challenge is even greater when the physics underlying such systems is…
G\"odel logic with the projection operator Delta (G_Delta) is an important many-valued as well as intermediate logic. In contrast to classical logic, the validity and the satisfiability problems of G_Delta are not directly dual to each…
We revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal)…
Large language models (LLMs) struggle with formal domains that require rigorous logical deduction and symbolic reasoning, such as mathematical proof generation. We propose a neuro-symbolic approach that combines LLMs' generative strengths…
We look at non-classical negations and their corresponding adjustment connectives from a modal viewpoint, over complete distributive lattices, and apply a very general mechanism in order to offer adequate analytic proof systems to logics…
Most language models currently available are prone to self-contradiction during dialogues. To mitigate this issue, this study explores a novel contradictory dialogue processing task that aims to detect and modify contradictory statements in…
Recent work has shown that integrating large language models (LLMs) with theorem provers (TPs) in neuro-symbolic pipelines helps with entailment verification and proof-guided refinement of explanations for natural language inference (NLI).…
The logarithmic divergence is an extension of the Bregman divergence motivated by optimal transport and a generalized convex duality, and satisfies many remarkable properties. Using the geometry induced by the logarithmic divergence, we…
Matching logic is a formalism for specifying, and reasoning about, mathematical structures, using patterns and pattern matching. Growing in popularity, it has been used to define many logical systems such as separation logic with recursive…
We introduce a paraconsistent expansion of the G\"{o}del logic with a De Morgan negation $\neg$ and modalities $\blacksquare$ and $\blacklozenge$. We equip it with Kripke semantics on frames with two (possibly fuzzy) relations: $R^+$ and…
This paper investigates whether contemporary AI architectures employing deep recursion, meta-learning, and self-referential mechanisms provide evidence of machine consciousness. Integrating philosophical history, cognitive science, and AI…
This study introduces a hybrid neuro-symbolic framework that achieves deterministic detection of statutory inconsistency in complex law. We use the U.S. Internal Revenue Code (IRC) as a case study because its complexity makes it a fertile…