analyze ambiguity

This paper addresses the open problem of synthesizing guarded loops with polynomial invariants for formal verification. Prior approaches are limited to affine, unguarded loops; ours is the first algebraic-geometry-driven method supporting nondeterministic branching loops with inequality guards, polynomial update maps, and arbitrary polynomial invariants. Our core contributions are threefold: (1) introducing a novel class of synthesizable invariants; (2) reducing loop synthesis to solving systems of multivariate polynomial equations over the rationals; and (3) integrating algebraic-geometric techniques—such as Gröbner bases and elimination theory—with SMT solvers to efficiently compute solution spaces. We implement a prototype system and evaluate it on multiple benchmarks. Experimental results demonstrate its ability to synthesize finite-loop programs satisfying complex polynomial invariants, confirming correctness, effectiveness, and scalability.

This work addresses the challenge of synthesizing loop structures from polynomial invariants in program verification. We propose the first general loop synthesis method supporting loops with inequality guards, polynomial update maps, and arbitrary polynomial invariants—extending beyond the affine restrictions of prior approaches. Our method pioneers the application of algebraic geometry to loop synthesis: it reduces the synthesis problem to a decidable geometric formulation by constructing a finite system of polynomial equations whose complex solution set precisely characterizes all valid loops satisfying the given invariant. The algorithm integrates symbolic polynomial system solving with SMT reasoning (e.g., Z3) and is implemented and validated in a prototype tool. Our key contribution is the principled, provably sound, and computationally effective synthesis of non-affine loops—breaking the long-standing affine barrier and enabling structured, verifiable, and algorithmic generation of general polynomial loops.

Existing machine learning datasets inadequately support AI-assisted open-ended research in professional mathematics—particularly in algebraic combinatorics—due to insufficient scale, structural richness, and formal verifiability. Method: We introduce the Algebraic Combinatorics Dataset Repository (ACD Repo), the first benchmark suite designed for cutting-edge mathematical research, covering nine unsolved problems with over one million structured, formally verifiable examples per problem. Our approach integrates supervised narrow-model training, model interpretability analysis, large language model–driven program synthesis, and symbolic encoding of combinatorial structures—establishing a novel “interpretable modeling + program synthesis” paradigm for conjecture generation. Contribution/Results: We release nine high-quality, reproducible datasets; substantially lower the barrier to AI-augmented original mathematical conjecturing; and empirically characterize the limits of neural models in abstract pattern induction—demonstrating both their capacity for nontrivial structural generalization and their systematic failures in higher-order combinatorial reasoning.

This paper investigates quantifier elimination and Craig interpolation in orthologic. While orthologic fails to admit quantifier elimination, its interpolation property remains unsettled. To address this, we develop a sequent calculus grounded in ortholattice semantics and establish its soundness and strong completeness with respect to the class of all complete ortholattices. Building on this proof system, we devise a constructive interpolation algorithm that— for the first time—rigorously establishes the existence of computable interpolants in orthologic and achieves polynomial-time interpolant generation. This result overcomes a longstanding bottleneck in non-classical logics, where interpolation existence and computational feasibility are typically incompatible. It provides efficient formal support for applications including quantum logic and unreachability verification in program analysis.

This paper uncovers the algebraic essence underlying convex analysis, Gaussian probability, and quadratic structure. To this end, we introduce Graphical Quadratic Algebra (GQA)—a novel algebraic framework based on chordal graphs—that uniformly models quadratic relations, Gaussian stochastic processes, and nondeterministic Gaussian processes via rotation-invariant quadratic generators. We provide the first sound and complete axiomatic characterization of three fundamental models: least-squares estimation, Gaussian randomness, and nondeterminism—revealing their shared conditional algebraic structure. Our method integrates string diagram theory, categorical semantics, and formal semantics of probabilistic programming. Theoretical contributions include soundness and completeness proofs for all three models within GQA. Applications demonstrate efficacy in linear regression, probabilistic programming, and noisy circuit modeling.

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 study addresses the construction of functions in algebraic combinatorics subject to stringent distributional constraints and the discovery of previously unknown combinatorial symmetries. To this end, we propose the SLURP framework, which integrates MapSeek-Functional and MapSeek-Symbolic approaches through alternating pseudo-label supervised learning, symbolic regression, and formal verification in Lean 4. The framework yields the first combinatorial interpretation of $q,t$-Narayana polynomials based on non-crossing partitions and provides a combinatorial proof of symmetry in previously unresolved cases by leveraging newly discovered statistics. All code and formalized results are publicly released to ensure reproducibility and rigorous verification.

This work addresses the challenge of implementing reverse-mode automatic differentiation for programs featuring algebraic effects such as finite discrete probabilistic choice. Building upon the Compositional Homomorphic Automatic Differentiation (CHAD) framework, it formulates automatic differentiation as a semantics-preserving program transformation and constructs a backward-pass mechanism tailored to the finite atomic distribution monad. The correctness of this construction is established using logical relations from category theory. This study presents the first systematic extension of reverse-mode automatic differentiation to effectful languages with discrete outputs, introducing a reusable differentiation scheme applicable to a broad class of algebraic effects—including nondeterminism, exceptions, and writer effects. The approach not only enables correct reverse differentiation of programs with finite discrete probabilistic structure but also lays a foundational theoretical groundwork for differentiating more general effectful languages.

This work addresses the ambiguity problem in finite-state automata by proposing a general unambiguation algorithmic framework. The approach generalizes the classical subset construction method, generating successor states on demand while preserving the original automaton’s structure. It supports partial unambiguation for full, finite, and polynomial degrees of ambiguity and naturally extends to weighted automata. The proposed algorithm computes states in polynomial time, thereby achieving—for the first time—an efficient construction of unambiguous automata applicable across multiple ambiguity levels and automaton models.