Abstract This article explores computable reducibility for presentations of computable linear orders. In particular, this paper investigates the degree structures $\mathbf{P}(\mathcal{L})$ consisting of the $c$-degrees of computable copies of a fixed linear order $\mathcal{L}$, where $\mathcal{L}$ is one of the following: $\omega \cdot k$, $\omega ^{2}$, $\omega +\omega ^{*}$, $\zeta $ or $\omega \cdot \eta $. The main objective is to identify structural differences between these posets, focusing on the existence of least degrees, minimal degrees, universal degrees, minimal covers and the possible number of such covers. Among the notable differences, the structure $\mathbf{P}(\zeta )$ contains degrees with a minimal cover that is not a strong minimal cover, a phenomenon that does not occur in $\mathbf{P}(\omega )$.
Abstract We study the robustness of inference in quantitative bipolar argumentation. Viewing argumentation as a dynamic process and inference in terms of the relative (final) strengths of two arguments, we introduce notions of robust inference and robust inconsistency. Robust inference requires that the arguments’ final strength-based ordering is preserved, under potential changes to a given argumentation graph. When an inference change occurs, the notion of robust inconsistency assesses whether the change of inference persists under subsequent changes, thereby capturing whether the change is ‘here to stay’ or merely the result of ephemeral fluctuations. To analyse situations where robust inference is violated, we introduce pockets of robustness—sets of changes that leave the inference robust when applied to the original graph. Based on these pockets, we define robustness violation explanations. These explanations identify the minimal changes made outside some particular pocket that lead to a change of inference. We further show how our explanations give rise to a gradual notion of robustness. We also discuss and evaluate aspects of the implementation of the notions introduced above and finally demonstrate that our approach to robustness of inference can be straightforwardly applied to labelling-based abstract argumentation.
Abstract We study dimensionality reduction as the construction of a representation accompanied by checkable reconstruction and irredundancy evidence. A representation is logically sparse when no retained predicate is definable from the others in a declared language and semantic model. This notion is distinguished from predictive usefulness, statistical independence and minimum cardinality. The framework uses intuitionistic logic and Heyting semantics to specify admissible definitions, while a sequential deletion algorithm records explicit reconstruction formulas. We prove termination, compositional preservation of the target predicate bank and language-relative irredundancy. Complete finite-algebra search additionally yields an inclusion-minimal generating subset. An executable positive-fragment implementation checks threshold formulas by canonical antichains; a separate finite Kripke implementation tests implication and non-Boolean information states. We develop proof-theoretic adequacy, categorical preservation, certified refinement, neural realization and directional robustness, with explicit conditions for optimization, stability, PAC–Bayes and distribution-shift guarantees. Experiments on three classification benchmarks use twenty paired data splits, three levels of domain-aware descriptors, independent predictive diagnostics and dimension-matched baselines. They quantify exact decoder preservation and the empirical cost of fitting directly on the reduced coordinates. AUC bounds are stated for fixed scores and finite score classes, with their statistical scope made explicit. The resulting method connects representation design with the intuitionistic neural-symbolic programme through auditable proof obligations.
Abstract We study an inverse problem for Dung abstract argumentation. A finite argument set $A$, $|A|=n$, is fixed, an attack relation is unknown, and each observation labels a candidate subset $S\subseteq A$ according to whether it is an extension under a chosen semantics. We formulate prediction from such extension-membership examples in the realizable probably approximately correct model and separate information-theoretic generalization from the computational task of finding a compatible relation. For every extension-based semantics, the Vapnik–Chervonenkis (VC) dimension of the induced hypothesis class satisfies $\operatorname{VC}(\mathcal{H}_\sigma )\le n(n-1)$ under our default irreflexive convention ($n^{2}$ if self-attacks are allowed). For the conflict-free semantics, we prove the exact value $\choose n2$ in the irreflexive case and $\max \{n,\binom n2\}$ when loops are allowed. We also characterize conflict-free observational equivalence and obtain the exact number of distinct loop-allowed classifiers. For the admissible, stable, complete and preferred semantics, an explicit directed gadget yields $\operatorname{VC}(\mathcal{H}_\sigma )\ge \lfloor n/3\rfloor ^{2}$, so their dimensions are $\varTheta (n^{2})$. Bounded in-degree gives sparsity-sensitive upper and lower bounds; below the dense threshold they differ by at most a logarithmic factor. Computationally, we give a direct polynomial-time characterization and witness construction for the conflict-free consistency problem, and self-contained polynomial-size Boolean satisfiability encodings for the admissible and stable consistency problems. These encodings are a zero-loss specialization of earlier argumentation-framework synthesis work rather than a replacement for it. Finally, we formalize noisy observations through agnostic risk and show how one relaxation variable per example turns the encodings into weighted partial maximum satisfiability instances that minimize example-level empirical $0$–$1$ loss. Worked examples, diagrams, convention tables and full proofs make explicit what can be predicted, what cannot be identified and where the statistical and computational claims differ.
Abstract Martínez-Rivillas and de Queiroz gave extensional Kan semantics for the untyped $\lambda$-calculus and later constructed the concrete $K_\infty$ homotopy $\lambda$-model. The two main mathematical results of the present paper are these. First, we show that a smaller front-seed coherence package—whiskering-left/whiskering-right together with an inner-right-front pentagon contraction—already suffices to recover the associator comparison, semantic pentagon and bridge theorems used in the later semantic arguments. Second, we prove explicit global reify, reflect and application formulas for $K_\infty$, with exact coordinatewise identities at every finite stage. We also record two structural clarifications: the recursive all-dimensional continuation of the explicit low-dimensional tower is obtained by a finite packaging phase followed by a uniform equality-generated recursion; and, on a deliberately fixed forward witness language for the classical $\beta /\eta$ separation span, the canonical identity-type higher tower on $K_\infty$ forces all higher non-connection once the two witness classes land at distinct points. The theorem packages are represented in Lean 4, and a citable archive of the development is available at Zenodo 10.5281/zenodo.21744942.
