order theory

This work addresses the automatic inference of closed-form bounds for recursively defined functions—such as operator fixed points or solutions to functional equations—arising in program cost analysis, loop acceleration, and hybrid system verification. We introduce the *B-bound abstract domain*, which approximates numerical functions via conjunctions of predefined bounding functions, enabling synthesis of highly nonlinear invariants. To systematically lift Galois connections from value domains to function spaces, we design an *abstract domain functor*. Our approach integrates constraint-driven abstract construction, higher-order abstract interpretation, operator fixed-point theory, and symbolic-numerical dimensionality reduction. Experiments demonstrate that the framework efficiently handles multivariate, piecewise, and non-discrete functions; significantly improves nonlinear invariant inference; simplifies transition function design; and achieves end-to-end automation across diverse verification and analysis tasks.

This paper addresses the lack of deep semantic characterization at the type level in substructural type systems—such as ordered and linear types. We introduce the first parametric monadic logical relations framework, parameterized by algebraic structures (e.g., monoids, commutative monoids), unifying the modeling of ordering and linearity constraints. This constitutes the first systematic extension of parametricity theory to substructural type systems. We formally prove that, for ordered types, standard functions—such as list concatenation, reversal, folding, and tree traversal—are uniquely implementable; for linear list identity types, only input permutations are admissible. These results establish foundational theorems of substructural parametricity and yield several strong uniqueness characterizations. Our framework provides a novel paradigm for semantic coherence and inhabitant analysis in substructural type systems.

Conventional definitions of false positives (FPs) and false negatives (FNs) fail for model selection in non-Boolean structured domains—such as ranking, clustering, and causal inference—where models lack a natural Boolean logic structure. Method: This paper introduces the first framework that formalizes model classes as partially ordered sets (posets), integrating poset theory, multiple hypothesis testing, and structural risk minimization to define and control generalized false positive error for non-Boolean structures—including permutations and directed acyclic graphs (DAGs). Contribution/Results: (1) It establishes natural, interpretable analogues of FP/FN errors for non-Boolean models; (2) it provides a unified framework for controlling the false positive rate (FPR) under partial orders; and (3) it enables statistically reliable and computationally feasible model selection in high-dimensional, complex structured spaces. By grounding statistical inference in poset-based falsifiability, the framework substantially extends the applicability of classical multiple testing theory beyond Boolean hypotheses.

This paper addresses the limited applicability of traditional valuation algebras by proposing a generalized information algebra framework axiomatized on abstract problem systems—not merely variable subsets. The framework models information as compositional and extractable modules, thereby unifying a broader class of information structures and relationships. Methodologically, it integrates abstract algebra, lattice theory, category-theoretic concepts, and stochastic mapping theory to systematically construct an algebraic system supporting local computation, duality, information ordering, compactness, and continuity, while generalizing the Dempster–Shafer theory. Key contributions include: (i) establishing the first universal axiomatic system for information algebras over arbitrary problem systems; (ii) proving that classical valuation algebras arise as a special case under this framework; and (iii) enabling unified modeling, approximate computation, and semantically complete representation of uncertain information.

This work addresses the algebraic classification and tractability boundary of constraint satisfaction problems (CSPs). Focusing on conservative CSP templates, we develop a unified theoretical framework linking absorptive algebras to local consistency solvability—the first systematic characterization of their deep interplay. Using subpower-finite algebraic modeling, absorptive subalgebra analysis, and polynomial-time reductions, we rigorously establish a complexity dichotomy theorem for conservative CSPs: every template is either solvable in polynomial time via local consistency algorithms or NP-complete. Our result provides a complete tractability criterion for conservative CSPs and strengthens the expressive power of algebraic methods in CSP classification. Moreover, it furnishes a rigorous theoretical foundation and decision framework for consistency-based solving approaches.

This work addresses the challenge of efficiently identifying partial optimality in preorder optimization—specifically, detecting pairs of elements that cannot satisfy a given preorder relation in any optimal solution. We propose novel partial optimality conditions and develop an efficient verification algorithm grounded in combinatorial optimization and graph theory. The resulting method substantially enhances the ability to recognize non-preorderable element pairs, significantly increasing the proportion of such pairs that can be pruned efficiently on both real-world and synthetic datasets. This advancement provides a more powerful tool for preorder inference with direct applications in bioinformatics and social network analysis.

This work addresses the limitations of existing Taylor expansion theory, which struggles to apply to classical web-based models of linear logic such as Köthe spaces and finiteness spaces, particularly when dealing with non-positive coefficients and partial summation structures. The paper introduces a general web-based semantic framework that accommodates partial summation and, for the first time, extends Taylor expansion theory to settings involving non-positive coefficients. This unified approach encompasses coherence spaces, probabilistic coherence spaces, finiteness spaces, and Köthe spaces. By integrating semantic tools from linear logic, differential λ-calculus, sequence space theory, and absolute convergence analysis of formal power series, the authors demonstrate that all major web-based models satisfy a generalized form of Taylor expansion, thereby broadening the mathematical foundations and applicability of differential program semantics.

This work investigates the relationship between ordered structures—such as thresholds—and tree-like configurations, specifically 2-trees, within real-valued function classes exhibiting large sequential fat-shattering dimension. By integrating techniques from sequential fat-shattering dimension theory, stability analysis of functions, and order properties from combinatorial model theory, the paper introduces a more flexible framework for threshold extraction. This approach substantially improves upon existing bounds, resolving an open problem concerning the upper bound on the dual sequential fat-shattering dimension with at most a double-exponential dependence. In doing so, it corrects a previously flawed proof in the literature and strengthens related results by Anderson–Benedikt and Daskalakis–Golowich.

The formalization of finite element methods in Rocq lacks foundational libraries for monomial orders and graded orders. Method: We develop the first comprehensive formal library, rigorously defining binary relations, total orders, monomial orders, and graded structures; we introduce a higher-order operator framework to uniformly construct and verify four canonical graded orders—lex, grevlex, and others—and design a proof reuse mechanism to systematically derive order properties. Contribution/Results: The library comprises over 700 mechanically verified lemmas, covering definitions, core operators, and essential properties—including well-foundedness and compatibility—thereby significantly enhancing both the efficiency and trustworthiness of finite element method formalization within Rocq.

This paper investigates the preservation of decidability under disjoint combinations of first-order theories. Addressing classical combination conditions—stability-infinity, shininess, strong politeness, and tameness—we introduce, for the first time, Galois connections to characterize their intrinsic algebraic structure, thereby constructing induced complete lattice models that systematically unify existing and novel combination frameworks. Our approach precisely determines the maximal sets of theories extendable by each condition, thereby refuting several long-standing open conjectures. Moreover, within this algebraic framework, we derive a family of new combination theorems, providing a unified foundation for constructing broader classes of decidable theory combinations. The work integrates model-theoretic, order-theoretic, and categorical perspectives, enabling a formal reconstruction and boundary characterization of combination properties.