The paper investigates the computational complexity of grounded and preferred semantics in computably finitary abstract argumentation frameworks (AFs). It maps key decision problems—credulous acceptance, skeptical acceptance, extension existence, uniqueness, and non‑empty existence—showing that for grounded semantics many are Σ₁⁰-complete or trivial, while for preferred semantics the problems fall into Σ₁⁰, Σ₂⁰, Σ₁ⁱ, and Σ₂ⁱ classes. The results delineate exactly where finitarity reduces complexity to the arithmetical hierarchy and where second‑order quantification pushes problems back into the analytical hierarchy.
arXiv:2606. 14688v1 Announce Type: cross Abstract: AI systems coupled to proof assistants now generate formal mathematics at scale, and the gap between what a checker can verify and what a mathematician would value has become the binding constraint.
arXiv:2608. 15147v1 Announce Type: new Abstract: Machine intelligence has conquered the symbolic world but stalled at the physical one.
arXiv:2601. 18747v2 Announce Type: replace-cross Abstract: Modern AI agents increasingly rely on search infrastructure to execute complex, neuro-symbolic reasoning workflows.
arXiv:2303. 04203v4 Announce Type: replace Abstract: The theory of computation was built to answer Turing's question: what is effectively calculable by an unbounded, immortal, disembodied agent following rules?
The paper introduces a semantic elevation operator that transforms static semantic questions about a program into dynamic questions about whether a property is preserved after the program rewrites itself. It proves that for intensional transformations the elevated property remains undecidable, showing that the class of non‑verifiable properties is closed under this operator and that repeated application climbs the arithmetical hierarchy to Υ02‑completeness. The work also demonstrates that no finite tower of verifiers can provide an unconditional certificate of preservation, and suggests a categorical perspective for future exploration.
The paper introduces a hierarchy of four increasingly strict faithfulness criteria—discrimination, logical admissibility, monotonic logical faithfulness, and probabilistic logical faithfulness—for evaluating knowledge base completion models, particularly when the target is a description logic knowledge base. It demonstrates that ranking accuracy alone does not guarantee logical faithfulness and shows that current embedding models fail to satisfy any of the criteria across the hierarchy. The authors provide a formal grounding for the strongest criterion using relative model counts and evaluate several models on εL ontologies, revealing gaps between performance metrics and logical correctness.
We propose ZX-Calculus (Knowledge Evolution Calculus), a conservative extension of Martin-Lof Dependent Type Theory (MLTT) integrating trace-indexed types, presheaf non-monotone semantics, and constructive AGM belief revision. A Coq mechanisation accompanies the paper (34 complete proofs; zero admits for the two central results).
arXiv:2609.36279v1 Announce Type: cross Abstract: The shallow embedding of higher-order modal logic in classical higher-order logic, used in Benzm\"uller and Scott's Notes on G\"odel's and Scott's va...
arXiv:2608. 30273v1 Announce Type: cross Abstract: The exact Shannon capacity is unknown for every odd cycle beyond the five-cycle $C_5$, making odd cycles a central open problem in zero-error information theory.
arXiv:2609. 11326v2 Announce Type: replace-cross Abstract: We ask whether it can be certified algorithmically that a self-modifying computational system preserves a safety property at its next step (preservation) and along its whole evolution (persistence).
arXiv:2607. 19573v1 Announce Type: cross Abstract: Structural generalization has been measured repeatedly by several benchmarks, yet it has never been formally defined.
arXiv:2606. 28639v2 Announce Type: replace-cross Abstract: We establish the mathematical limits of AGI safety in two forms: verifying a fixed system, and verifying that a certified safety property persists once the system self-modifies.