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We consider a class of formula equations in first-order logic, Horn formula equations, which are defined by a syntactic restriction on the occurrences of predicate variables. Horn formula equations play an important role in many…

We introduce Open Horn Type Theory (OHTT), an extension of dependent type theory with two primitive judgment forms: coherence and gap, subject to a mutual exclusion law. Unlike classical or intuitionistic negation, gap is not defined via…

We study topology, particularly compactness, as an extension of Shulman's work on constructive mathematics via affine logic, while allowing propositional impredicativity. We introduce a notion of compactness in affine logic and prove the…

Infinitesimal contraction analysis, wherein global asymptotic convergence results are obtained from local dynamical properties, has proven to be a powerful tool for applications in biological, mechanical, and transportation systems. Thus…

The tensor train decomposition decomposes a tensor into a "train" of 3-way tensors that are interconnected through the summation of auxiliary indices. The decomposition is stable, has a well-defined notion of rank and enables the user to…

Based on rectangle theory of formal concept and set covering theory, the concept reduction preserving binary relations is investigated in this paper. It is known that there are three types of formal concepts: core concepts, relative…

Distilling from a knowledge base only the part that is relevant to a subset of alphabet, which is recognized as forgetting, has attracted extensive interests in AI community. In standard propositional logic, a general algorithm of…

Horn clauses and first-order resolution are commonly used to implement type classes in Haskell. Several corecursive extensions to type class resolution have recently been proposed, with the goal of allowing (co)recursive dictionary…

The expressiveness of propositional non-clausal (NC) formulas is exponentially richer than that of clausal formulas. Yet, clausal efficiency outperforms non-clausal one. Indeed, a major weakness of the latter is that, while Horn clausal…

Do reasoning models have "Aha!" moments? Prior work suggests that models like DeepSeek-R1-Zero undergo sudden mid-trace realizations that lead to accurate outputs, implying an intrinsic capacity for self-correction. Yet, it remains unclear…

A fundamental issue in machine learning is the robustness of the model with respect to changes in the input. In natural language processing, models typically contain a first embedding layer, transforming a sequence of tokens into vector…

The restricted strong convexity is an effective tool for deriving globally linear convergence rates of descent methods in convex minimization. Recently, the global error bound and quadratic growth properties appeared as new competitors. In…

Considerable attention has been given to the problem of non-monotonic reasoning in a belief function framework. Earlier work (M. Ginsberg) proposed solutions introducing meta-rules which recognized conditional independencies in a…

Large crowdsourced datasets are widely used for training and evaluating neural models on natural language inference (NLI). Despite these efforts, neural models have a hard time capturing logical inferences, including those licensed by…

This work investigates preserving and reversing unimodality and convexity properties for sequences under transformations defined by sign-regular kernels. It is shown that these transformations only preserve these properties if the kernels…

Explicit reasoning models are trained to produce intermediate reasoning traces before final answers, but downstream fine-tuning is often performed on ordinary instruction-response data that contains no such traces. We show that this…

Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this paper, we will survey the use of structural proof theory as an alternative foundation. Researchers have been using…

Large Reasoning Models (LRMs) perform strongly in complex reasoning tasks via Chain-of-Thought (CoT) prompting, but often suffer from verbose outputs, increasing computational overhead. Existing fine-tuning-based compression methods either…

Constrained Horn Clauses (CHCs) are an intermediate program representation that can be generated by several verification tools, and that can be processed and solved by a number of Horn solvers. One of the main challenges when using CHCs in…

Recent studies on frequent itemset mining algorithms resulted in significant performance improvements. However, if the minimal support threshold is set too low, or the data is highly correlated, the number of frequent itemsets itself can be…