construct canonical forms

Traditional graph representations face significant challenges in graph isomorphism testing and symmetry-aware visualization due to high computational complexity and low efficiency. This work proposes “graph linear notation”—a complete graph invariant derived from canonical form algorithms—and establishes it, for the first time, as an equivalent definition for finite graphs. This representation not only substantially simplifies graph isomorphism comparison and symmetry-aware visualization but also naturally accommodates the extension and application of classical graph-theoretic concepts, such as coloring and paths, within its framework. By unifying these capabilities, the proposed notation offers a highly efficient and coherent new paradigm for structural graph analysis.

UniMath lacks computational support for algebraic structures—particularly W-types—and syntactic term algebras. Method: We develop a general algebraic library formalizing multi-sorted signatures and equational systems; implement, for the first time in UniMath, a computationally tractable subclass of W-types; construct, via explicit categorical methods, the univalent category of single-sorted ground term algebras; and rigorously prove its universality as the initial object in the category of algebras. We further establish an equivalence between ground term algebras and homotopical W-types. Contribution/Results: The library enables computable term algebras without recourse to general inductive definitions. We validate its effectiveness on foundational examples from universal algebra and propositional logic. This work substantially extends UniMath’s capacity to formalize and compute with structured mathematical objects, bridging abstract algebraic semantics and homotopy-theoretic type theory.

This work investigates the computational complexity of the Tensor Orbit-Closure Intersection (TOCI) problem under group actions, which underlies fundamental questions in group-action equivalence testing and separability of invariant polynomials. Method: We introduce the algebraic complexity class TOCI and develop a novel algebraic reduction framework grounded in geometric invariant theory and representation theory. Contributions: We establish that tensor network equivalence is TOCI-complete—the first completeness result for TOCI. We further construct a polynomial-time many-one reduction from Graph Isomorphism (GI) to TOCI, proving GI ⊆ TOCI and thereby providing the first rigorous lower bound for TOCI. Moreover, our framework unifies and explains the complexity of longstanding open problems—including noncommutative Polynomial Identity Testing (PIT) and quantum state classification—by reducing them to TOCI. This work constitutes the first systematic complexity-theoretic treatment of orbit-closure intersection problems.

This paper addresses the characterization of graph homomorphism indistinguishability over structurally restricted graph classes—such as graphs of bounded treewidth, bounded tree depth, and planar graphs—within a unified algebraic framework. Method: It introduces, for the first time, a synthesis of homomorphism tensor embeddings with group representation theory and linear maps between tensor subspaces, integrating homomorphism counting, tensor algebra, and feasibility theory for linear equation systems. Contribution/Results: (1) It provides exact algebraic characterizations of homomorphism indistinguishability over bounded-treewidth, bounded-pathwidth, and bounded-tree-depth graph classes; (2) it establishes its equivalence to the feasibility of specific linear systems (e.g., $Ax = b$); and (3) it unifies the algebraic foundations underlying logical equivalences (e.g., $C^k$, fixed-point logic with counting) and algebraic equivalences (e.g., Weisfeiler–Leman variants, matrix rank conditions), thereby resolving an open problem posed by Dell et al. concerning the characterization of bounded-pathwidth graphs.

This work addresses the problem of modeling invariant functions over symmetric matrices (under conjugation by permutations) and point clouds (under rotations, reflections, and point permutations). Methodologically, inspired by Galois theory, we construct the first lightweight universal approximator framework that yields separating invariant features of dimensionality only $O(n^2)$ for symmetric matrices and further optimizes to $O(n)$ for point clouds—breaking the bottleneck of traditional high-dimensional invariant representations. Our theoretical foundation integrates invariant algebra and generating sets of rational function fields, coupled with DeepSets architecture and orbit-separation analysis under group actions. Experiments on molecular property regression and point cloud distance prediction empirically validate almost-everywhere orbit separation, enabling universal approximation of weighted graph functions. The proposed framework significantly enhances both expressive power and computational efficiency of invariant representations.

This work investigates the algebraic–combinatorial mechanisms underlying graph isomorphism discrimination and introduces the framework of “separating modules,” a polynomial vector space grounded in the representation theory of symmetric groups. By employing complexity measures such as support size, symmetric circuit size, and multiplicity, it establishes equivalences with subgraph counting (support size \(k\) corresponds to order \(O(k)\)) and the Weisfeiler–Leman algorithm (circuit size \(n^{\Theta(k)}\) corresponds to \(\Theta(k)\)-WL). The central contribution provides the first intrinsic characterization of multiplicity separation: two graphs are distinguishable if and only if their automorphism groups have distinct cycle indices. Furthermore, the paper demonstrates that the multiplicity barrier is strictly stronger than the occurrence barrier and connects invariant polynomials to the graph reconstruction conjecture and finite-type invariants.

This work addresses the long-standing challenge that tensor trains lack a unique canonical form over arbitrary fields—including finite fields—hindering irreducibility testing and standardization. The paper presents the first construction of a unique canonical form for tensor trains valid over any field, combining algebraic structural analysis with a polynomial-time reduction algorithm. This framework enables direct extraction of the canonical representation, principal indices, and corresponding values from a full tensor. Beyond establishing a theoretical upper bound on compression size, the approach substantially extends the theoretical foundations and applicability of tensor networks beyond real or complex fields, demonstrating their efficacy as a universal formalism across diverse algebraic settings.

This study addresses the equivalence verification problem between two fundamental representations of finite closure systems—implicational and intersectional canonical bases—specifically, whether an intersectional basis fully captures all closed sets generated by a given set of implications. By integrating techniques from computational complexity theory, formal concept analysis, and functional dependency theory, the work establishes for the first time that this problem is coNP-complete, even when restricted to acyclic implication sets with premises of size at most three. This result precisely characterizes the computational complexity of verifying completeness in closure system representations, rules out the existence of output-polynomial algorithms even in restricted settings such as acyclic convex geometries, and provides new lower bounds for related problems including characteristic model identification.

This work proposes the Graphical Algebraic Geometry (GAG) framework, which for the first time rigorously formalizes polynomials, ideals, and affine varieties from commutative algebra using a diagrammatic language. By integrating tools from category theory, (co)span semantics, and algebraic geometry, GAG establishes a universal and complete compositional reasoning system for polynomial constraint satisfaction problems (#CSP). The core contributions include establishing a formal correspondence between #CSP and graph rewriting, uncovering a deep connection between GAG and the qudit ZH quantum graphical calculus, and proving that constraint rewriting in GAG is #P-hard. Furthermore, it is shown that computing amplitudes in the qudit ZH calculus requires only a constant number of oracle queries to GAG, thereby opening a novel pathway for efficient modeling of quantum computations.

This work proposes a novel and efficient method for computing isolated regular solutions of multivariate polynomial systems with composable structure. By introducing variable substitutions, the original system is reduced to a lower-dimensional system in intermediate variables. The approach integrates a probabilistic symbolic homotopy algorithm, algebraic independence analysis, and the Chevalley–Shephard–Todd theorem to handle cases invariant under finite reflection groups. It is the first systematic exploitation of composability in polynomial systems to substantially reduce symbolic solving complexity. The algorithm achieves arithmetic complexity polynomial in both the input size and the number of solutions. Experimental results on benchmark systems—including those invariant under symmetric groups, hyperoctahedral groups, and exceptional reflection groups—demonstrate its superior computational efficiency.