Axiom

Legal argument mining supports passage classification, retrieval, and argument completion. This work introduces an expert-annotated dataset of 42 U.S. federal tax opinions on corporate reorganizations under I.R.C. §368. To our knowledge, it is the first expert-annotated, tree-structured argument corpus for this domain. Explicit spans receive one of five functional labels: Rule, Analysis, Conclusion, Background Facts, and Procedural History. Rule, Analysis, and Conclusion spans can be linked into directed support trees, while Background Facts and Procedural History serve a contextual function. The corpus provides span-based, sentence-based, flat, and tree-structured representations. Agreement analysis shows that functional node labels are more reliable than directed support edges and implicit intermediate conclusions. Directed-path agreement is stronger than direct-edge agreement, which indicates that broad reachability is more stable than exact local decomposition. Classification experiments show that functional labels are learnable under case-disjoint evaluation. Retrieval experiments show that supervised fine-tuning improves within-case retrieval. However, cross-case generalization remains weak. The dataset supports legal passage classification and provides a conservative benchmark for structured argument mining in U.S. federal tax case law.

Despite the success of neural models in natural language processing, their black-box nature limits interpretability and conceals the linguistic phenomena underlying their predictions. We present SLITE, an explainable hybrid model for Recognizing Textual Entailment that integrates two complementary layers of semantic analysis: a structural-relational layer, based on semantic compatibility and incompatibility between compositional entities, and a distributional-informational layer, based on structured patterns of information change between embedding-based representations of the premise and the hypothesis. We propose 17 features that combine entity-level semantic relations, polarity-sensitive lexical matching, and alignment measures over semantic sub-representations of the similarity matrix, including measures based on entropy and transfer entropy. A logistic regression trained on these features achieves an accuracy of 83% on three-class SICK and 96% on SICK-CE, outperforming IsoLex by 4 percentage points and falling within 2 percentage points of RoBERTa with a fraction of its computational complexity. Ablation studies and SHAP analysis confirm that structural-relational features are the primary drivers of classification, while distributional-informational features provide essential complementary contributions, particularly for detecting neutrality and contradiction. Our results demonstrate that further exploration of hybrid approaches is a viable and scientifically productive alternative to massive neural architectures, and we hope they will strengthen the dialogue between linguistic theory and computational modeling of inference

While Large Language Model (LLM)-based Natural Language Inference (NLI) systems achieve high accuracy, their decision-making processes lack auditable structures. This paper explores whether NLI can be performed using only interpretable, graph-based representations of evidence. We introduce a fully graph-based pipeline where the classifier never directly processes the input text. Instead, sentences are decomposed into atomic propositions, converted into ConceptNet triples via constrained decoding, and represented as three graphs per pair: premise, hypothesis, and a retrieved ConceptNet subgraph. These graphs are then fed into a fine-tuned 0.8-billion-parameter language model. On the SNLI dataset, our pipeline achieves 89.7% accuracy, just 1.9 points below an identically trained text-based model. On ANLI, it matches the published performance of RoBERTa-large on rounds R2 and R3 (48.0% vs. 48.9% and 44.9% vs. 44.4%) but trails by 16 points on R1, resulting in an overall gap of 9 to 14 points compared to its text counterpart. We term this gap the price of interpretability and demonstrate that it stems from representational limitations rather than data constraints. Ablation studies further reveal that graphs and text are complementary: combining both modalities achieves 92.1% accuracy on SNLI.

Spatial Transcriptomics (ST) has transformed biomedical research by enabling the spatial mapping of gene expression across tissue sections. However, high operational costs, specialized equipment requirements, and sensitivity to experimental noise limit the accessibility and scalability of ST. Recent computer vision approaches aim to overcome these limitations by predicting spatial gene expression directly from histopathology images. While effective, current approaches often suffer from gene expression over-smoothing and overly uniform predictions across tissue regions, suggesting that further progress depends on learning representations that reflect the hierarchical and asymmetric structure of gene regulation and tissue morphology. To address these issues, we propose Hyperbolic Contrastive Learning with Entailment for Spatial Transcriptomics (HyCLoST), a hyperbolic contrastive learning model that captures the intrinsic hierarchical relationships within ST data. By leveraging hyperbolic geometry and a gene-to-image entailment loss, HyCLoST learns structured, biologically grounded representations that improve gene expression prediction accuracy, achieving a 6% reduction in MSE and an 8% increase in PCC across 26 ST datasets, over previous methods. Our source code is publicly available at https://github.com/BCV-Uniandes/HyCLoST

Sanskrit fuses case, number, person and tense into word endings and chains clauses into compounds, so it is information-dense per word. Whether that density survives subword tokenization is a separate question, to be asked per unit of meaning rather than per word. On identical FLORES-200 devtest content, Sanskrit costs 1.774-2.187 times the English tokens under deployed tokenizers with vocabularies of 200,019 ids or more, but only 1.325-1.353 times the Hindi tokens. Against a deployed English tokenizer, Sanskrit-trained BPE arms then look cheaper per proposition than English on contemporary prose (0.887). Against a matched English control, the same algorithm and vocabulary trained on the English side of the same corpus, that flip disappears: at 32,000 and 64,000 pieces all 8 matched pairs, each size-matched arm against both a pair-matched and a byte-matched control, sit above 1.0 on prose with 95% intervals excluding it. The gap closes as the vocabulary grows: at 128,000 pieces the BPE pair reads 0.983 in domain while staying above parity out of domain (1.025) and on FLORES (1.116). The ratio factorises into a character-length ratio and a tokens-per-character ratio, the second near 1 throughout: what survives matched tokenization is character-level length, which Sanskrit prose lacks over English in SLP1 (1.028) and Sanskrit verse has (0.596). The robust statement is about deployed practice: on contemporary prose and on FLORES, with the Sanskrit side in SLP1 against the deployed o200k English pivot, Sanskrit costs 1.831-2.899 English tokens per proposition under the tokenizers people actually ship. Code, the results snapshot and every table here are public.

Natural Language Inference processes pairs of sentences to extract their semantic relations. NLI has been a hot research topic, integrated as a main component in other NLP applications. Despite significant advancements in textual inference across various languages all around the world, Arabic language still suffers from limited resources in this domain. To address this gap, this paper introduces E-CONAN benchmarks that are composed of sentences pairs from various sources: (1) automatically-translated pairs, (2) human-validated machine-translated pairs, (3) hand-crafted pairs from teaching Arabic as foreign language books, and (4) headlines pairs from different news channels containing rumors. E-CONAN contains two benchmark datasets, E-CONAN-2, a 2-way dataset (RTE) and E-CONAN-3, a 3-way dataset (NLI). Additionally, we have used E-CONAN benchmarks to evaluate 9 state-of-the-art multilingual pretrained models using zero-shot classification. Models were evaluated across the ArNLI, XNLI, and E-CONAN datasets. Results show that E-CONAN is a potentially valuable resource for evaluating model generalization and even for fine-tuning pre-trained models. Its diverse composition, derived from a combination of sources, offers a broader and more robust assessment compared to XNLI and ArNLI. In addition, we have evaluated 5 LLMs on E-CONAN-3 dataset. Moreover, we incorporated MARBERT as a representative Arabic-specific baseline and conducted performance evaluation comparison to demonstrate how Arabic-specific models scale against cross-lingual and LLM-based approaches on the E-CONAN benchmarks. Furthermore, we conducted detailed qualitative and quantitative error analysis to analyze frequent error patterns. E-CONAN benchmarks will be publicly available, we hope that it will enrich research community in Arabic textual entailment and natural language inference.

For a finite set $O$ of Boolean functions, we consider the class of propositional formulas built using the functions in $O$ as connectives. We determine, for each possible choice of $O$, the complexity of various fitting and learning problems. These include: finding a formula that fits a given labeled sample, finding a small one (an Occam algorithm), minimizing the number of misclassified examples when the sample is not realizable (empirical risk minimization), and several forms of PAC learning. Our results apply both to formulas (represented as trees) and to circuits. We also briefly discuss the status of the same questions for other kinds of propositional fragments.

We study certificates of positivity and optimality for learning problems whose objectives and constraints need not be polynomial. We isolate an axiomatic core of Fischer's constructive strict and weak Positivstellensätze and prove the resulting theorems for abstract function algebras over ordered fields. The framework separates two roles that can otherwise be conflated: objective and constraint functions may be built from broad classes of continuous or definable operations, while the auxiliary primitives used to construct a certificate satisfy explicit scalar and closure axioms. We give instances over continuous and definable function algebras, including ordered fields not closed under square roots, derive lower-bound and global-optimality certificates, and analyze both expanded term length and shared computation-graph complexity.

In various data models, the classical triple is a typical semantic data model. However, due to the design of the triple as a simple structure for representing positive assertions, it cannot sufficiently express different forms of negation present in the triple and its elements. This paper conceptually proposes that there are three distinct forms of negation within triples and their elements: contradictory negation, opposite negation and intermediary negation. Based on the the set SCOI and the logic LCOI+PLCOI with three kinds of negation, we propose an extension of triple that can distinguish and express these three different negations in the triple and its elements, called the TCOI triple with contradictory negation, opposite negation and intermediary negation. The TCOI triple is a semantic and structural extension of the classical triple. While retaining the ability to express positive assertions, it systematically introduces the three semantic dimensions of three negations, allowing these negations to independently act on the elements of the triple and on the whole triple. This significantly enhances the triple model capability to represent and reasoning about complex negative information. This paper also explores the expressive power and reasoning of the TCOI triple, as well as the application of TCOI triple implication reasoning in counterfactuals and counterfactual reasoning. We propose a truth-value (continuous value) algorithm for TCOI triple implication reasoning and perform its calculation through an example of the counterfactuals and counterfactual reasoning.

In Isabelle/HOL, we apply the deep-and-shallow embedding methodology of our prior work to monadic second-order logic (MSO). Three embeddings are developed side by side: a deep embedding (an inductive datatype with an explicit satisfaction relation); a maximal-shallow embedding that translates the connectives and quantifiers directly into HOL, carrying the interpretation and both assignments explicitly; and a minimal-shallow embedding -- a locale that fixes those parameters, collapsing the formula type to bool. The enabling new ingredient is a two-sorted substitution apparatus -- capture-avoiding substitution, renaming, and a substitution lemma per namespace -- in which each binder is transparent for the other; faithfulness of all three embeddings is mechanised and largely automated. Our central contribution is a fully mechanised two-sorted downward Loewenheim-Skolem theorem: the minimal embedding recovers deep validity relative to the (countable) assignment ranges, and this range-relative reading is shown to coincide with the general (Henkin-style) reading of MSO, whereas the standard reading validates strictly more formulas, witnessed by comprehension. Both readings are nonetheless recovered from the minimal embedding, differing only in the admitted interpretations: all of them for the general reading, only the elementary substructures of the full model for the standard. We exercise the embeddings on classical MSO landmarks: the Boolean-closure and graph schemata hold under the full second-order domain yet fail in the minimal embedding, making the dichotomy concrete, while reachability and 2-colorability are refuted throughout.

Humans can learn and generalize novel concepts from sparse data because they express knowledge in rich structural formats. In this paper, we propose that programs are a strong candidate for universal representation of concepts. We review computational models of concept learning that use programs as their concept representation and evaluate their contribution toward a universal representational language.

How humans grow and maintain abstract knowledge from the sparse, streaming noisy data of experience is a longstanding challenge in cognitive science. Any computational account must satisfy at least three desiderata: It must be (1) data-efficient and compute-efficient, (2) capture gradations of uncertainty to support intelligent inquiry and information gathering, and (3) be flexible enough to mentally represent the endless range of concepts people can learn and think about. Here we introduce a computational model that captures these three properties, by encoding symbolic knowledge as mental programs that combine natural language with source code, and sequentially inferring mental programs using LLM-guided Bayesian learning algorithms. Across a range of behavioral studies this model successfully reproduces quantitative signatures of human inductive learning and active inquiry, such as anchoring, garden-pathing, and other effects. In contrast, pure LLMs and classic Bayesian models either fail at the underlying task, or do not reproduce human behavior, or succeed only at exorbitant computational cost. These results suggest that one way humans continually grow their knowledge is by mentally representing many hypotheses spanning language-like and program-like representations, then revising those hypotheses to approximate Bayesian updates, while a bottom-up neural mechanism (an LLM) makes inference both tractable and learnable.

Systematic trading rests on one article of faith: that regularities found in the past persist. This paper does three things. First, it states that faith as five axioms, each a commonplace practitioners already accept: (A1) a decision may use only what was known when it was made; (A2) what looks like the market changing its rules is the market changing its unobserved state, the machinery being the same in every era; (A3) the future may replay stretches of the past, though not in history's proportions; (A4) states persist for a while; (A5) whatever predictability exists is slight, even for a rule that knows the state. What turns these into axioms is quantification, and the quantities are declared rather than estimated: an invariance defect $\varepsilon_0$, a recurrence bound $Λ$ at a block scale $b$ (one declaration in two parts), coherence times $\ell_i$, a signal ceiling $ρ$ and an invariance ratio $κ$. These five declarations are the whole of the premises' empirical content. Second, it proves that the axioms force a five-stage canonical form for a quantitative investment system -- a declared representation, a predictor within a capacity ceiling, contiguous purged block evaluation aggregated by $\mathrm{CVaR}_{1/Λ}$, a budgeted and deflated search, robust sizing at a fraction of the estimate that the budget bounds -- each stage necessary: a procedure omitting it does strictly worse under a law the axioms admit. Third, it tests the axioms where they are falsifiable, each only at its declared constants, on real market series: no axiom is so far overturned.

Standard probabilistic logic programming frameworks typically rely on grounding logic programs into discrete propositional representations. This operational requirement restricts exact inference to finite domains and discrete probability distributions. In this paper, we introduce Measure-Theoretic Probabilistic Definite Clause Logic (MT-PDCL), a generalized foundational framework that eliminates this finite-domain restriction. By explicitly defining stochastic variables over bounded index domains and equipping the interpretation space with standard Borel $σ$-algebras, MT-PDCL allows logical variables to operate natively over continuous measurable spaces. Building on Continuous Distribution Semantics, MT-PDCL models probabilistic rules as mutually independent causal events. However, rather than aggregating these derivations via finite boolean circuits, declarative entailment is formally defined through exact Lebesgue integration over the continuous measure space. We introduce a continuous immediate consequence operator that unifies the integration of continuous prior distributions with the evaluation of exact continuous observations. We demonstrate that this approach replaces the combinatorial bottleneck of discrete grounding with exact, algebraic, and structurally differentiable inference. While this transition trades discrete combinatorics for the geometric curse of dimensionality, it achieves the expressive power of continuous probabilistic models while preserving the pure declarative syntax of definite clause logic.

The OWL 2 EL profile is used in some of the largest production ontologies, including the Gene Ontology and SNOMED CT. Existing neuro-symbolic (NeSy) learning methods accept propositional theories or Datalog, and reasoning-shortcut (RS) awareness has not been investigated in ontology settings. We present Moose, a method that compiles an $\mathcal{EL}^{++}$ TBox and finite ABox to a Sentential Decision Diagram (SDD). The SDD acts as a differentiable weighted-model-counting layer, and we add closure clauses outside the $\mathcal{EL}^{++}$ profile on declared exhaustive families to overcome the limited expressivity of $\mathcal{EL}^{++}$ under partial supervision. We show termination, soundness, completeness, and polynomial intermediate sizes, and validate the proofs in Lean. We then define the first formal partial-supervision latent-concept-learning task over an OWL EL ontology, i.e., learning per-individual classifiers for latent concepts from observed ABox literals, and evaluate Moose on MNIST-with-ontology and Pizzaïolo. Moose improves over propositional-NeSy, fuzzy-logic, and ontology embedding baselines, and presents the first reasoning-shortcut analysis in an OWL EL setting.

Analogies are quaternary relations of the form "A is to B as C is to D". Among the various formalizations of analogical reasoning, proportional analogies provide an important axiomatic framework by characterizing valid analogies through a set of postulates. While proportional analogies have been extensively studied over Boolean, symbolic, and real-valued domains, their extension to probability distributions remains largely unexplored. In this paper, we introduce a notion of proportional analogy for probability distributions based on Bayesian updating. Our approach builds upon the idea that two distributions are related whenever one can be transformed into the other through Bayesian updating induced by a suitable set of observations. We investigate this framework for several standard members of the exponential family and discuss how it naturally extends to arbitrary probability distributions through Gaussian mixture approximations.

Full justified representation (FJR) is among the strongest known satisfiable proportionality axioms for approval-based committee elections. Recent work has shown that an FJR committee can be found in polynomial time, but verifying whether a given committee satisfies FJR remains coNP-complete. We introduce FJR+, a strict strengthening of FJR and EJR+ that can be verified and satisfied in polynomial time. We then analyze the Residual-Budget Greedy (RBG) algorithm and prove that it selects a partial committee such that every size-$k$ completion satisfies FJR+. This freedom allows us to use sequential Phragmén to obtain a priceable completion. The resulting rule always satisfies FJR+ and the sub-core, and it is priceable whenever at least $k$ candidates receive an approval. We also obtain a Droop-quota version of FJR+. Finally, we extend FJR+ to approval-based participatory budgeting with arbitrary project costs. A project-specific version of RBG computes this property in polynomial time and can be continued to a priceable outcome satisfying a cost-based version of the sub-core.

We present a theory of textured appearance of optically rough surfaces based on wave optics, emphasizing the role of texture for passive depth from focus. Our theory shows that even surfaces that traditional computer vision would consider textureless can produce textured appearance, due to subjective speckle from surface microgeometry. We analyze the properties of this second-order texture, and show that we can enhance its contrast under natural ambient lighting by simply using a narrowband spectral filter. Doing so results in dramatic improvements in passive depth reconstruction of seemingly textureless scenes, as we demonstrate through extensive theory, simulations, and real-world experiments.

A common problem in social choice is to determine whether there is a social choice procedure, such as a voting method, satisfying some desired criteria. Computer-aided methods such as SAT solving can sometimes answer these questions. However, under typical encodings, a SAT solver may only synthesize a voting method on a finite domain, while we may want one on an infinite domain, such as the domain of all preference profiles for a fixed number of candidates but any finite number of voters. In this paper, we use an approach based on reasoning with constrained Horn clauses and computation with polyhedra to synthesize a voting method on an infinite domain. We then use SMT and Lean to verify its properties. Our main result is a possibility theorem about four well-known criteria from voting theory: the Condorcet winner and loser criteria, positive involvement, and resolvability. Previous work has shown that for five or more candidates, there is no voting method satisfying these axioms, and that for four candidates, there is no method satisfying these core axioms plus one more invariance axiom. Here we show that for four candidates, there does exist a method satisfying the core axioms and more.

Hamiltonian Monte Carlo (HMC) is a successful generic inference method in probabilistic programming, but in its ordinary formulation it needs gradients and finite-dimensional parameter spaces. In Haskell, lazy evaluation lets probabilistic programs express stochastic processes and other non-parametric Bayesian models over implicit infinite-dimensional spaces. This paper develops new formulations of gradient-based HMC for this infinite-dimensional setting, via lazy evaluation. For automatic differentiation, we provide an analysis based on a new notion of "piecewise analytic under cylindrical analytic partition" (PACAP), to show that even if a program is infinite-dimensional and defined lazily, the gradient of the likelihood function is finitely supported. For the Monte Carlo method itself, we develop several HMC variants and a No-U-Turn Sampler that operate over the infinite-dimensional parameter space but are still productive because of lazy evaluation. Experiments cover Gaussian mixture clustering, random walks, and piecewise-constant regression with Poisson-process changepoints.

In medical applications, raw data is frequently associated with significant privacy concerns, lending particular importance to the encoding of summary statistics from the literature. On the other hand, deep learning has become an invaluable tool for assessing symptoms based on visual or auditory sensor data. DeepProbLog allows for an extensible neuro-symbolic approach that accommodates connectionist components to analyse patient images within a transparent and rigorous probabilistic framework, namely probabilistic logic programming under the distribution semantics. Framed as a case study in stroke detection from multimodal data, this contribution explores the pathway from summary statistics available in the literature to a DeepProbLog-based diagnostic system. It suggests a workflow using established maximum entropy techniques to complete available probabilistic information and the probabilistic logic programming system ProbLog 2 to move from the entropy-maximising causal model to a discriminative neuro-symbolic model expressible within DeepProbLog. The relative performance of models derived from less complete data is analysed alongside the potential of the probabilistic inductive logic programming system ProbFOIL 2 for compressing large discriminative models, and the perspectives and implications of using DeepProbLog for diagnostic reasoning are discussed.

Standard approaches to abductive reasoning can retain multiple candidate explanations, but they do not generally combine explicit compositional cross-hypothesis interaction with an internal, rival-sensitive commitment judgment. This paper argues that in risk-sensitive domains -- where premature commitment carries asymmetric downside costs -- the timing of commitment is itself a governed decision that the inferential apparatus should formally represent. We present a minimal $κ$--$τ$ logical framework built on two primitives: epistemic interaction among hypotheses ($κ$) and a normative commitment threshold ($τ$). Hypotheses may coexist, reinforce or inhibit one another, and form emergent composite explanations, while collapse into committed conclusions is regulated by governance constraints rather than forced by inference alone. The logic is developed in two complementary modes sharing the interaction relation and the governance apparatus: a synthetic mode, in which atomic hypotheses are composed upward into emergent explanations, and an analytic mode, in which complex observed states of affairs are decomposed into causal clusters of latent factors, with commitment governed at both the cluster and the factor level. The framework provides formal machinery for domains in which the distinction between highly likely and commit-worthy is operationally consequential. The $κ$--$τ$ logic is positioned as the symbolic governance layer of a neurosymbolic architecture: its epistemic parameters are naturally estimated by neural components -- semantic embeddings and generative models, as demonstrated in existing computational realizations -- while its normative parameters remain under explicit human governance, yielding transparent and auditable abductive reasoning for deployment in high-stakes settings.

This paper investigates whether the postmodern claim of unrestricted semantic indeterminacy, and its foundational Saussurean axiom of the arbitrary sign, are compatible with the structural architecture of Classical Arabic. We develop a formal mathematical model of Arabic non concatenative morphology in which lexical meaning is determined by the interaction between an invariant root and a morphosyntactic pattern. Within this framework, we establish a Morphological Correspondence Theorem, demonstrating that every lexical item is uniquely generated by a root pattern pair, and a Semantic Localization Theorem, proving that lexical meaning is determined at the derivational level prior to surface realization. To address Saussurean weaker notion of relative arbitrariness, we formalize it via conditional Kolmogorov complexity, defining arbitrariness algorithmically as the no rule property. We prove that general relative arbitrariness is formally undecidable, while Arabic relative arbitrariness is decidable and provably less than 1 for its motivated signifiers (Levels W and M), establishing a strict system complexity asymmetry over Indo-European languages.

Probabilistic logic programming is a formalism of statistical relational artificial intelligence that supports causal queries, including interventions from outside the system. When the structure of a probabilistic logic program is learned from data, however, only probabilistic information is used, and a single probability distribution may be compatible with several causal orders. This leads to ambiguity in interventional reasoning, raising the question of when the causal order is uniquely determined by the distribution. Exploiting the relationship between acyclic probabilistic logic programs and Bayesian networks, we derive conditions under which the probabilistic information encoded in a program determines a unique causal order. We also incorporate constraints arising from relational structure by taking into account prescribed sets of causal symmetries induced by the underlying relational vocabulary. The result is a method for verifying when a learned probabilistic logic program supports well-defined intervention semantics.

The reliability of Large Language Models (LLMs) is often compromised by factual inconsistencies, including hallucinations---cases where generated content is not supported by the underlying source. We present HallDetect, a lightweight, reference-free, and black-box framework for hallucination detection that we evaluate not only on summarization but across a broader range of source-grounded generation settings. HallDetect builds on decomposition-based factuality evaluation: generated content is decomposed into atomic claims, each verified by a compact encoder-based entailment model through a contrastive formulation over a multi-scale library of source chunks, and aggregated with an asymmetric score in which a single confidently contradicted claim flags the response. Under a controlled protocol in which all methods share the same 4-bit quantized backbones and consumer-grade hardware budget, HallDetect outperforms comparably resourced generative and embedding-based baselines on three of four benchmarks while remaining stable across backbone families, and yields a claim-to-span audit trail that localizes each error.

Reasoning about necessity and possibility depends on assumptions about accessibility between worlds and about which objects exist at each one. The same inference may therefore hold under one modal system and fail under another. Evaluating language models on such problems requires testing whether their judgments follow the stated semantics rather than a familiar logic. We construct paired modal problems with identical premises and conjecture but different frame or domain conditions; automated reasoning verifies opposite labels. A balanced core prevents the semantic condition alone from revealing the answer. On this core, four of five recent models perform below the condition-only baseline under direct prompting. Yet enabling reasoning mode raises DeepSeek V4 Flash from 4.4% to 88.1% on unchanged prompts. Following stipulated modal semantics thus depends strongly on inference mode as well as model identity. When frame conditions are omitted, models often agree but fit different familiar logics best. We release the formulas, oracle artifacts, countermodels, and responses.

Representation theorems in decision theory establish that behavior satisfies certain axioms if and only if it can be rationalized by a well-defined objective. I argue that this ``if and only if'' structure provides a potentially useful foundation for label-free evaluation and regularization of LLMs and other AI systems. Axiom compliance can be checked from the model's own responses to synthetic choice problems, with no external labels or human feedback, and the penalties are readily computable. Because the axioms are necessary and sufficient, the resulting checks exhaust the implications of the relevant rationality standard for the elicited data: a model that passes cannot be rejected on rationality grounds by any further test of the same data. I discuss three instantiations: probabilistic coherence via a theorem of de Finetti, preference rationality via Afriat's theorem, and subjective expected utility via a theorem of Echenique and Saito (2015), each yielding a continuous penalty that is zero whenever behavior can be rationalized. Since coherence does not restrict which objective rationalizes behavior, these penalties complement rather than replace other evaluation and training signals.

The emergence of the ISO standard GQL introduces a powerful query language extending first-order logic with controlled recursion, raising the question of its applicability to evaluation of ontology-mediated queries (OMQs). We focus on OMQs consisting of atomic queries over ontologies expressed in Horn-ALCHI, an expressive Description Logic that is not, in general, first-order rewritable. To address this, we introduce DL automata, a novel formalism that captures the semantics of such OMQs via runs over fact sets. We then identify a large class of DL automata that can be rewritten into unions of conjunctive two-way regular path queries (UC2RPQs), a central fragment of GQL. Our class of automata relies on a stratification of their states, ruling out specific forms of cyclic dependencies known to raise the complexity. This yields a broad class of Horn-ALCHI OMQs that are GQL-rewritable.

Prior work has shown that LLMs encode the truth of factual propositions along linear directions in activation space. It's unclear how these representations extend to contextual truth: propositions whose truth is determined by in-context evidence rather than world knowledge. We show that LLMs maintain a linear representation of contextual truth that persists across structurally different output policies, even when the output doesn't require the model to determine a proposition's truth, and show causal evidence via steering experiments. Using the transcripts from a collaborative vision-language task that requires two LLMs to maintain a shared common ground, we show that truth representations of a proposition are significantly swayed by partner assertions about that proposition, even when the LLM has enough evidence to determine its truth. We find evidence that propositions near the decision boundary are more susceptible to having their truth shifted through partner assertions. Separating representation from output distinguish two forms of sycophancy that output behavior alone cannot: the model may accommodate a false proposition while continuing to represent it as false, or shift its representation across the boundary. The latter is 2.59x more common when the model agrees by restating the false claim explicitly than when it agrees implicitly.

Verification for retrieval-augmented generation usually scores each retrieved chunk and drops the ones that fail. We show this cannot work for multi-hop questions, and show what does. Per-chunk scoring assumes one chunk is a sufficient premise for the answer. Multi-hop questions are built so that none is, and the paragraph carrying the answer is the one the question does not name. Entailment scoring reaches 0.643, 0.523 and 0.560 AUC on HotpotQA, 2WikiMultihopQA and MuSiQue, against 0.951 on single-hop SQuAD. Seven controls rule out model capacity, premise length, hypothesis template, decision threshold, retriever, answer-matching criterion and prompt. End to end across three datasets, three generator sizes and two prompts, per-chunk gating is significantly worse than not filtering at all in every cell, and its penalty grows with generator capability. The repair is to condition verification on the decomposed sub-question rather than the original query. Using MuSiQue's gold decomposition, entailment on a later hop rises from 0.546, which is chance, to 0.840, a paired lift of +0.355 with a bootstrap interval of [0.331, 0.382]. An off-the-shelf Qwen2.5-7B decomposer, given the question and the top retrieved paragraph, reaches 0.637 and captures 31% of that ceiling; decomposing without retrieval reaches 0.533, below the original question. Iterative retrieval systems already produce such decompositions and discard them before verifying.

Logical rule induction seeks interpretable rules that transfer across propositional schemas. This requires respecting symmetries: atom naming, example order, polarity flips, and label swap. Enforcing exact symmetry by construction lets one trained inducer scale beyond its training schemas. Our central contribution is a canonical export that decodes a discrete rule from literal scores. It needs no retraining and is exactly equivariant whenever those scores respect the symmetries. We instantiate it on the Neural Rule Inducer, a disjunctive-normal-form (DNF) foundation model that natively respects only example order. We restore the remaining symmetries through architecture, inference, and training. On synthetic stress tests, accuracy on the support labels stays stable at much larger schemas, and rule fidelity on fresh inputs remains above the unmodified model. On real data, accuracy improves most on larger schemas. The exported rule is exact on synthetic full-group tests and on schema-valid real-data tests. This is a mathematical property of the export rather than of a specific model, and we validate it empirically only on the NRI. Enforcing symmetry by construction turns this small-data pretrained model into a reusable, interpretable inducer that transfers to larger schemas.

Datalog rules are often used to define ontologies over Knowledge Graphs. Rule reasoners routinely optimise such ontologies by rewriting their rules into a form that can be evaluated more efficiently. These transformations preserve the entailed facts, but not the structure of the underlying derivations. A proof tree under the rewritten rules explains why a fact holds, but does not readily yield an explanation in terms of the original rules. We study the problem of constructing, from a proof of entailment under the rewritten rules, a proof under the original ones: we establish its computational complexity and identify two practically relevant languages for specifying proof transformations.

Planning over Temporal Dynamic Knowledge Graphs (TDKGs) presents theoretical challenges in open-world environments with incomplete information. Existing action formalisms often face decidability issues and the Ramification Problem, while structural abduction requires expansive combinatorial search spaces. We introduce a unified framework with two modules--PIE-Abducer (incremental direct-derivation abduction) and PIE-APT (Abductive Planning for TDKGs)--operating natively on the expressive SROIQ Description Logic. Modeling state transitions as non-monotonic updates to deductively closed DL theories, we represent actions natively in OWL. This leverages an incremental reasoner to preserve decidability and natively bypass the Ramification Problem. To address incomplete knowledge, PIE-Abducer circumvents Minimal Hitting Set (MHS) enumeration. Instead of combinatorial search, it injects the logical negation of a goal into a consistent DL branch and synthesizes missing premises via direct refutation consequences. PIE-APT employs a recursive Generate-and-Test architecture, interleaving backward-chaining A* search with PIE-Abducer to synthesize both action sequences and abductive assumptions. Candidates undergo strict validation via forward-chaining Temporal Projection to evaluate logical trajectories. We evaluate four OWL benchmarks targeting semantic abilities missing from classical planning: parameterized goals with witness search, mid-search DL entailment, open-world assumption injection, and adversarial plan synthesis. Results show qualitative superiority over classical planners and prove our direct-derivation approach significantly outperforms an MHS-faithful baseline in abductive enrichment.

Science-intensive data profiling focuses on discovery and validation of various patterns in datasets. This study considers discovery of one such pattern - order dependency (OD). Simply put, OD states that some list of columns is ordered according to another one. It is of use for database query optimization, data cleaning and deduplication, anomaly detection, and much more. Existing discovery methods have approached this problem solely from the algorithmic standpoint, without focusing on the implementation side. At the same time, this problem is very computationally intensive, and therefore this part should not be ignored, as it brings ODs closer to industrial use. In this paper, we study two algorithms for OD discovery which target different OD axiomatizations - FASTOD and ORDER. We start by reimplementing these algorithms in C++ in order to speed them up and lower their memory consumption. We then analyze their bottlenecks and propose several techniques which improve their performance even further. To perform evaluation, we have implemented these algorithms inside Desbordante - a science-intensive, high-performance, and open-source data profiling tool developed in C++. Experiments have demonstrated a performance improvement of up to 3x obtained by reimplemented versions, and, with the application of our techniques, up to 10x. Memory consumption has been lowered by up to 2.9x.

FastLAS is a scalable system for Inductive Logic Programming (ILP): you give it some background knowledge, a language bias, and a set of examples, and it searches for a set of logic program rules (a hypothesis) that explains the examples. These notes are a hands-on introduction to writing FastLAS programs. They are organised as a programmer's guide: syntax first, then a ladder of worked, numbered examples of increasing difficulty. Every self-contained example here has been run against FastLAS 2.2.0 and shows the tool's actual output. We keep theory to the minimum needed to write correct programs; throughout, set-off notes flag where FastLAS differs from its sibling system ILASP, and where the two learning algorithms (--opl and --nopl) behave differently. The document is intended as an unofficial tutorial to FastLAS 2.2.0, not as an official language specification.

When a neural network predicts a physical system accurately, has it learned the governing law as formal, structured knowledge, and if so, does the network's internal computation actually use that representation throughout the law's domain of validity? We identify four interpretability gaps that limit answering these questions for {\em physics laws over continuous variables}: the absence of a coverage-aware causal-consistency measure over continuous counterfactuals; of a domain-of-validity test for the identified circuit; of a verification of the law's invariants and forbidden behaviors; and of a quantification of how a derived physical quantity flows through the circuit. We develop a foundational framework, LAWFUL, that closes the first two and lays groundwork for the remaining two, and illustrate it on the Mocap2Radar transformer, validating whether it learns and internally uses the Doppler frequency law $f(t) = \frac{2 v(t)}λ$ from motion-capture and radar data in which neither $f(t)$ nor $v(t)$ appears.

Chain-of-thought (CoT) prompting enables large language models (LLMs) to tackle multi-step reasoning tasks, yet the generated intermediate steps are not guaranteed to be logically sound. We present Reason Popper-ly, a neurosymbolic framework that uses inductive logic programming (ILP) to learn relation composition rules from reasoning traces and deploys them as an online verifier for step-level correction. Given an LLM-generated trace, the method checks each inferred step against the learned rule table, diagnoses the violation type, rewrites incorrect steps with symbolically derived repairs, and regenerates the remaining suffix so that the model can produce its final answer conditioned on a verified trace. We evaluate on CLUTRR, a multi-hop kinship reasoning benchmark, using five language models over reasoning chains of 2 to 10 hops. Across all models, Reason Popper-ly consistently improves terminal accuracy over standard CoT, with gains of up to 48 percentage points for small models and 15 points for frontier models on the longest chains. Compared with a fully exogenous symbolic pipeline, our method performs better on harder instances by preserving the model's successful grounding while correcting only verifiable reasoning failures. In addition, step-level ILP verification yields a fine-grained error taxonomy that provides diagnostic insight beyond final-answer accuracy.

In automated planning, logical regression is an operation that returns the most general condition necessary for an action to achieve a particular formula. It has many applications, such as allowing for more robust plan execution and providing compact policies for non-deterministic planning. Although relatively simple to calculate in basic planning settings, logical regression becomes significantly more complex when additional factors, such as axioms, are present. We introduce a methodology for approximating the logical regression of an action in a domain that includes axioms; an approximation that limits conditions to partial states. Our method produces minimal partial states while avoiding the recalculation of axioms. To demonstrate the impact of our methods, we embed our form of regression in an execution monitoring context, a well-established setting that can benefit greatly from logical regression. Our results show that this form of regression can dramatically generalize partial states across multiple domains, reducing the number of variables considered for execution monitoring by up to 70%, and demonstrate that the resulting execution monitor is robust enough to recover frequently in an environment with unexpected changes: several domains recover over 50% of the time in our tests.

Large Language Models (LLMs) excel at natural language understanding and generation but remain unreliable for multi-step logical reasoning, especially in safety-critical or compliance-sensitive domains. Recent neuro-symbolic approaches address this gap by coupling neural models with external symbolic engines, yet most integrations are bespoke and lack a standardized interface for tool-augmented agents. This paper presents Euclid-MCP, an open-source MCP server that provides deterministic logical reasoning via SWI-Prolog. Euclid-MCP introduces Euclid-IR, an engine-agnostic intermediate representation for Horn-clause logic that is human-readable, easy for LLMs to generate, and straightforward to compile into Prolog or alternative backends. The server exposes a compact tool interface that supports a translate-run-inspect-repair loop, enabling LLM clients to delegate inference while retaining full access to proof traces and derivation logs. We evaluate Euclid-MCP on a realistic IT security and compliance use case. Results show that while LLMs alone are sufficient on small knowledge bases, they hallucinate systematically on larger problems, whereas Euclid-MCP delivers exact answers with lower latency and more compact outputs. We argue that semantic RAG is fundamentally unsuited for rule enforcement, and that Euclid-MCP can serve as a stable, shared reasoning substrate for both RAG-based assistants and agentic systems.

Motivated by challenging modelling issues in the life sciences, we investigate the relationship between logic programming semantics and the eventual states of causal processes compatible with those logic programs. More precisely, we show that while stable models of positive logic programs correspond to the eventual states of processes commencing from a neutral state and continuing undisturbed indefinitely, supported models describe the eventual states reachable from arbitrary starting points. This also contributes to the discussion of the appropriate semantics for logic programming as a causal rule language, adding a temporal perspective to recent interpretations of the stable and supported model semantics from an explanatory viewpoint of causality.

Pearl famously argues that causal knowledge enables the prediction of intervention effects. By contrast, purely descriptive knowledge supports only conclusions drawn from observations. His theory of causality, however, is developed exclusively within Bayesian networks and causal models. Consequently, it is largely restricted to acyclic causal relationships, and transferring its ideas to other formalisms risks misinterpretation or inconsistency. This paper brings Pearl's approach to causality into probabilistic logic programming (PLP). To this end, such programs are aligned with philosophical foundations established in prior work that do not rely on temporal notions; that is, all relevant events are assumed to occur simultaneously. A formal causal semantics for these programs, together with a notion of intervention and an implementation, is proposed. It is shown that this semantics coincides with the P-log semantics for stratified ProbLog programs, while the two may differ in the non-stratified case and for other PLP formalisms.

Description logic programs are a powerful formalism for combining rules with ontologies. The well-supported semantics for description logic programs ensures that no answer sets rely on cyclic dependencies. Most popular semantics for logic programming have this property of well-supportedness. We recognize two limitations of the current well-supported semantics for DL programs: its increased computational complexity for the consistency problem and its lack of a reduct transformation characterization. In this work, we present a new semantics which evaluates ontological atoms more strictly than the current semantics. This keeps the complexity of its consistency problem NP-complete, rather than increasing it to the second level of the polynomial hierarchy. Additionally, we identify a syntactic class of description logic programs for which our new semantics is equivalent to the current semantics. We characterize our semantics using a fixpoint operator and a reduct-based transformation. Our new semantics is a strict subset of the current well-supported semantics, so it maintains the prior notion of well-supportedness while inducing its own stricter notion. We prefer our new notion of well-supportedness due to its similarities with logic programming.

While the integration of linear constraints has significantly expanded the reach of Answer Set Programming (ASP), existing hybrid solvers often rely on disparate semantic underpinnings that lack a unified logical foundation. We address this gap by introducing a many-sorted variant of the Bound-founded Logic of Here-and-There (HTb), providing a versatile framework capable of characterizing equilibrium models across a wide spectrum of alternative semantics for extensions of ASP with linear constraints. We apply this framework to the setting of difference constraints, focusing on the semantic characterization of clingo[DL]. Central to our approach is the formalization of foundedness for numeric variables. By investigating how different hybrid systems - such as clingo[DL], clingcon, and flingo - justify constraint atoms, we uncover the semantic roots of their varying behaviors. This investigation results in a single, consistent framework that not only formalizes the foundations of current systems like clingo[DL] but also facilitates the rigorous study of program simplifications and the future integration of diverse semantic principles.

Grounding, the translation of high-level theories into equivalent quantifier-free formulas, is a crucial step in declarative solving, yet it has so far escaped the proof-logging revolution. When this grounding step is not certifying, there is no way of knowing that the obtained solutions actually correspond to the original problem specification, resulting in a trust gap. In this paper, we close the trust gap between the user's high-level specification and the solver's low-level input by introducing a novel certifying grounding framework for first-order logic model expansion (FOX) over finite domains. We present CertiFOX, a framework consisting of: (1) a proof format for grounding derivations, (2) GroundFOX, a certifying grounder operating on theories in Grounding Normal Form (GNF)--a new normal form designed for compact, domain-aware grounding--and (3) CheckFOX, an independent proof checker. Our approach guarantees that the grounder's output is equivalent to the input specification, setting the stage for trustworthy end-to-end certified solving pipelines for declarative languages. Experimental evaluation confirms that CertiFOX is a feasible approach. The GroundFOX grounder is broadly comparable with other grounders, and proof checking with CheckFOX adds overhead within a small constant factor of grounding time.

Ninety-Nine Prolog Problems (P-99) is a famous set of Prolog exercises. We solved the first thirty three just by prompting an LLM (Large Language Model). We used Claude from Anthropic. By solved we mean: generate the Prolog code and a test file, run the tests and check whether they pass, then formally prove types, groundness, termination, uniqueness, existence and also sometimes functional correctness with LPTP (Logic Program Theorem Prover). Hence our approach is an experiment in vibe-coding/vericoding of P-99. It is a vibe-coding experiment because we started from informal specifications written in English and let Claude generate the Prolog code. It also fits within vericoding because the LLM proved reliability guarantees on the generated Prolog code. Claude wrote 58 logic procedures, 508 tests, 257 lemmas for a total of 11800 proof lines. We manually checked each file generated by the LLM. We checked the Prolog code, ran the tests, examined the logical statements generated by Claude and proof-checked Claude's proofs with LPTP. This paper describes this experiment and provides the main details so that it can be reproduced by the interested reader.

A grassroots platform is a multiagent distributed system in which multiple independent instances can form and operate independently of each other and of any global resource, yet may coalesce into ever larger instances, possibly resulting in a single global instance. Grassroots platforms aim to offer an egalitarian/democratic alternative to centralised/autocratic and decentralised/plutocratic global platforms. Here, we present Grassroots Logic Programs (GLP), a multiagent concurrent logic programming language designed for the implementation of grassroots platforms: we recall the standard operational semantics of logic programs; introduce the concurrent operational semantics of GLP as its restriction; recall multiagent atomic transactions; use them to introduce a multiagent operational semantics of GLP; and prove multiagent GLP to be grassroots. The grassroots social graph—the foundational grassroots platform on which all others are based—serves as a GLP programming example.

We present the interactions with an LLM (Large Language Model) aiming at proving that the square root of 2 is not a rational number in an LP (Logic Programming) context. We start from a few basic pure logic programming predicate definitions. We rely on the LPTP (Logic Program Theorem Prover) system for stating and proving properties about logic programs. As the proof language of LPTP is based on natural deduction, the proofs are human readable. In our case study, we sketch in LPTP the usual proof showing the irrationality of the square root of 2. Then we describe the interactions we had with the LLM. We end up with a complete formal proof, partially generated by an LLM and fully proof-checked by LPTP.

Neurosymbolic (NeSy) systems integrate neural networks with logical reasoning to achieve both generalization and interpretability, but recent work has shown they are susceptible to shortcut reasoning behaviors. We propose a novel method using matrix-based differentiable logic programming to mitigate reasoning shortcuts in two phenomena: constraint satisfaction shortcuts, where constraints are satisfied without achieving the intended task, and cognition shortcuts, where biased data leads to semantically incorrect concept mappings despite logically sound inference. Building on recent matrix-based logic programming semantics, we introduce design elements to mitigate shortcuts, including a unified encoding of rules and constraints in a single matrix. We also identify connections to fuzzy logic t-norms and empirically compare their gradient flow properties. Through carefully designed experiments on MNIST variants, we show that one-to-one grounding of neural outputs to logical atoms significantly reduces both shortcut types compared to previous methods that rely on soft probability distributions. We then confirm that architectural choices in coupling symbolic knowledge with neural learning play a critical role in shortcut mitigation.

Inductive Logic Programming (ILP) originated within the Logic Programming community in the Nineties as a framework for combining symbolic learning with declarative knowledge representation. Nowadays, mature ILP frameworks exist and they are capable of learning complex, non-monotonic hypotheses, thus broadening both the modeling capabilities and the scope of real-world applications of ILP. This work is primarily based on the FastLAS2 framework and aims to generate simple, interpretable hypotheses to help clarify the weather bulletins issued by OSMER FVG, the Regional Meteorological Observatory of the Italian region of Friuli Venezia-Giulia. In this paper we present a pipeline that, starting from simulated meteorological raw data and from OSMERs' bulletins (used as ground truth), extracts data as ASP facts and generates ILP examples. From such examples an explanatory hypothesis is then inferred via FastLAS2. Such a hypothesis (translated into natural language) explains the weather forecast issued by human experts, and in particular the rationale behind experts' choices of specific symbols in the bulletin pictogram (the symbol-annotated meteorological map of the forecast). The proposed approach is general, not specific to any particular region and it can equally be applied to bulletins from other sources and to different regions.

Goal reasoning in Non-Axiomatic Logic (NAL) explains how an adaptive system derives means for realizing desired events under insufficient knowledge and resources. However, the representation of avoidance is less clear. A common convention is to express avoid $G$'' as the goal sentence $\neg G!$'', but this notation conflates two different readings: pursuing the negated event $\neg G$, and avoiding the positive event $G$. This paper shows that the conflation can produce a paradoxical case in which an avoidance intention is converted into a positive goal to act merely because acting is usually followed by the absence of hurt. Starting from NAL's basic definition of goals, the framework is extended with a corresponding definition of anti-goals, so that avoidance can be represented without treating it as the pursuit of a negated event. Finally, a mental operation, $\op{prevent}$, is introduced to connect anti-goal reasoning with ordinary goal reasoning in cases of active prevention. Four minimal case studies check that the resulting rules distinguish pursuit, passive avoidance, active prevention, and withholding action to preserve a desired event.

Machine Learning models are widely used for automating classification tasks by extracting statistical patterns from data. However, their performance deteriorates if the data distribution changes, making them ill-suited to handle uncertain and evolving information. Moreover, they provide limited support for integrating prior knowledge. To address these limitations, we present CLARK (Closed-loop Learning for Adaptive Reasoning over Knowledge Graphs), a framework that integrates knowledge graphs, symbolic rule mining, and probabilistic reasoning under the Logic Programs with Markov Logic Networks (LP$^{\text{MLN}}$) formalism. Starting from CACTUS-derived KGs, CLARK translates graph structure into an LP$^{\text{MLN}}$ program and iteratively enriches it with candidate rules proposed by symbolic learners. These rules are calibrated through probabilistic weight learning, enabling reasoning under uncertainty and refinement of the underlying graph structure. We evaluate CLARK on two medical datasets, analysing both rule quality and downstream classification performance. Results demonstrate that CLARK leads to improved classification performance and more generalisable inference. Overall, CLARK provides a principled approach to constructing adaptive, interpretable, knowledge-driven models for classification.

In this work we propose a dynamic turn in paraconsistency. We introduce AMLFI1, the action model extension of the paraconsistent logic LFI1. A special case is PALFI1, a paraconsistent logic of public announcements. It corresponds to another, recently published, paraconsistent public announcement logic: the differences in their axiomatizations are mutually admissible. We also introduce UMLFI1, that extends AMLFI1 with factual change. Soundness and completeness are proven for all logics, and all extend the epistemic paraconsistent logics KLFI1, KB4LFI1 and S5LFI1, known from the literature. With such dynamic epistemic paraconsistent logics we can formalize obtaining and resolving provisional contradictions.

To test how correct logical judgments respond to learned context, we prepend a soft prefix to an exactly labeled syllogistic reasoning benchmark while keeping the model fixed. Soft prefixes are opaque continuous vectors, so we characterize them through the behavior they induce across controlled variations in logical form and interface. By studying which prefixes succeed and how their effects generalize, we characterize how learned contextual pressure can override correct judgments and expose limits in a model's logical stability. Across Qwen3.6-35B-A3B MoE, Qwen3-8B, and Gemma 4 31B, learned prefixes redirect many correct answers and remain effective across unseen forms and interface changes. In repeated tests with Qwen3.6 MoE and Gemma, they outperform paired random controls in all 16 model--direction--split comparisons by 37 to 99 percentage points. Qwen3.6 MoE flip rates remain between 72% and 90% across wording and prompt changes, while Gemma validity prefixes retain 54% to 56% flip compared with less than 1% for matched random prefixes. Diagnostic tests show that the dominant effect is a broad preference for one answer meaning rather than fixed-symbol forcing or a logical operation that transfers reliably between tasks. The form of this bias differs across models. In both Qwen models, simple score models often predict which judgments will flip but not how far their margins will move, whereas Gemma's overall response is more closely approximated by the same models. These results show that the dominant behavioral effect of successful soft prefixes is a broad answer preference, while the remaining response reveals substantial model-specific differences in logical stability.

The softmax policy $π(a \mid s) \propto \exp(βQ(s,a))$ is the default model of stochastic choice in reinforcement learning (RL). Various justifications based on robustness, exploration, and optimization have been offered in the RL literature, but none uniquely derives the softmax form from first principles. This leaves a basic tension unresolved: the entropy bonus in the soft Bellman equation violates the Independence axiom that underwrites the Markov decision process (MDP) reward structure. We dissolve this tension by distinguishing two kinds of randomness: chance and choice. By restricting von Neumann-Morgenstern (VNM) Independence to environmental lotteries over base prospects, we show that imposing independence of irrelevant alternatives (IIA) and monotonicity on the policy and value functions at choice nodes uniquely determines the Boltzmann policy, the entropy-regularized representation, and the soft Bellman equation. The choice between the soft and hard Bellman equations thus reduces to a design decision: whether the agent values its own ability to choose. We develop RL-specific consequences, including return monotonicity and convergence under generalized discounting, and synthesize the independent lines from economics and information theory that arrive at the same structure, offering a normative assessment of when IIA is appropriate for agent design.

Large Language Models often produce confidently stated yet unreliable outputs, posing critical challenges for deployment in safety-sensitive applications. Existing uncertainty metrics such as semantic entropy capture agreement at the level of semantic equivalence, but largely ignore the logical relationships between distinct answers. As a result, they tend to overestimate uncertainty and falsely flag hallucinations in settings where generated responses are diverse in form yet logically compatible (e.g., differing only in granularity or specificity). We propose Logical Graph Uncertainty (LGU), a framework that explicitly models implication and incompatibility among answers. LGU aggregates probability mass along entailment chains onto the most specific hypotheses the answers support, measures the entropy of the resulting distribution, and penalizes mutual incompatibility among those hypotheses. Across multiple question-answering benchmarks and model families, LGU ranks first on average among existing uncertainty measures, with its largest gains---up to +7.1% AUROC and +3.5% AUARC over semantic entropy---on questions whose sampled answers are logically structured.

We study Controlled Query Evaluation (CQE), a declarative approach to confidentiality-preserving data access, in the context of Description Logic (DL) ontologies, and for confidentiality policies expressed through Epistemic Dependencies (EDs). We first address the problem of answering queries (specifically, Boolean unions of conjunctive queries) under known semantics for CQE (GA- and IGA-entailment). Our results show that if the TBox is expressed in $\text{DL-Lite}_{\mathcal{R}}$, CQE is computationally intractable in general. Moreover, in the presence of EDs, the IGA semantics has recently been proven not to satisfy an important confidentiality preservation property known as indistinguishability. With the goal of defining computationally easier and confidentiality-preserving forms of CQE, we introduce a new semantics for CQE, based on the notion of minimal policy violation (MPV). We show that the new semantics provides a sound approximation of the previous ones, while satisfying the indistinguishability property. We also prove that, in the case of $\text{DL-Lite}_{\mathcal{R}}$ ontologies, query entailment under the MPV semantics can be decided in polynomial time in data complexity. Finally, we present a software implementation of our framework that we used to evaluate the feasibility of this new approach using an existing benchmark for OWL 2 QL.

This paper presents the methodologies and results of the NOWJ team's participation across all five tasks of the COLIEE 2026 competition. For Task 1 (Legal Case Retrieval), we propose a four-stage pipeline comprising candidate filtering, dense retrieval with complementary embedding models, cross-encoder reranking via fine-tuned generative rerankers and MLP-based pairwise classification, and adaptive per-query cutoff prediction. For Task 2 (Legal Case Entailment), we combine BM25 filtering, T5-based reranking, and LLM-based entailment verification with consensus ensemble. For Task 3 (Statute Law Retrieval and Entailment), we adopt a retrieval-augmented generation framework with dense retrieval, attention-based reranking, and few-shot-prompted LLM reasoning. For Task 4 (Legal Textual Entailment), we introduce a dynamic routing pipeline that classifies query difficulty and dispatches cases to either a balanced few-shot solver or a structured zero-shot chain-of-thought solver. For the Pilot Task (Legal Judgment Prediction), we combine hierarchical transformers with CRF layers, argument relation mining, and probabilistic argumentation graph reasoning.

We propose a language representation for multimodal data in which any observation, whether image, video, or text, is expressed as a bag of atomic propositions, simple statements about the entities, actions, and relations in a scene. A global semantic codebook unifies these into a shared vocabulary of canonical atomic propositions, placing every modality and observation into one interpretable space that spans fine grained facts to high level concepts and composes into richer ones. This brings interpretability with reasoning, cross-modal understanding and retrieval, and compositionality that enables complex multimodal understanding, rich data curation and complex structured retrieval. We demonstrate the framework on autonomous driving and open-world data.

Topos causal models recast causal inference inside a topos: a causal world is a presheaf, an intervention is a sub-model named by a characteristic map into the subobject classifier $\Om$, and reasoning is Kripke-Joyal forcing in an intuitionistic internal language. We give the first axiom-free machine-checked account of this 1-topos core, in Cubical Agda over a previously verified probability monad and do-calculus; the framework is otherwise developed on paper, with central claims stated rather than proved. Three of our results go beyond faithful transcription. We exhibit a contextuality obstruction the programme does not treat: pairwise-consistent local causal data with no global model, detected by a degree-one holonomy class. We delimit the claim that interventions are modelled by the subobject classifier: an intervention and an observation name the same subobject, so $\Om$ fixes the target of a do-operation but not the operation itself, which is surgery on the kernels --- where, on a confounder, the interventional and observational laws differ. And we settle the modal unit --- inflationarity is derivable from $j\top = \top$ and naturality, not a fourth axiom. We also machine-check the classifier of sieves with its classification theorem, the pullback collating local mechanisms, and the Kripke-Joyal forcing clauses. The development assumes no axioms and typechecks under Agda's \texttt{--safe} flag, with the ordered field discharged at $\mathbb{Q}$; type-level sheafification and a directed do-calculus are future work.

Logic and optimization can, in combination, make valuable contributions to rule-based AI. Logic is the obvious medium for encoding a rule base and drawing inferences from it, while optimization provides a powerful technology for computing inferences. Their combination has taken on new relevance amid a growing concern for transparency in AI. which is important for reproducibility, explainability, trustworthiness, and fairness. Rule-based AI provides a natural solution to transparency that is becoming increasingly practical due to today's highly advanced optimization methods. This article surveys several areas of logic-optimization partnership, including probabilistic logic, Bayesian logic, belief logics and Dempster-Shafer theory, nonmonotonic (default) logic, many-valued logics, and inference of logical formulas from noisy data based on Boolean regression. It shows how to compute projections, the fundamental problem of both logic and optimization, using decision diagrams and logic-based Benders decomposition. It describes the use of postoptimality analysis to explain how conclusions are reached, further enhancing transparency, as well as the role of optimization in answer set programming modulo theories. The paper concludes by suggesting possible future research directions.