apply fixed-point theory

This work addresses the constructive problem of fixed-point iteration for continuous self-maps on complete lattices by introducing a purely algebraic reasoning framework, termed AIC, which eschews traditional index-based analytic methods in favor of equational logic to express fixed-point iterations as algebraic identities. The framework not only provides an algebraic reconstruction of classical results such as the Tarski–Kantorovich principle and its k-induction generalizations but also yields a novel fixed-point theorem based on limsup and liminf constructions. Furthermore, it reveals the incompleteness of finite axiom systems for this setting and demonstrates the necessity of infinitary axioms. All theoretical developments are formalized in Isabelle/HOL and fully verified automatically via Sledgehammer, thereby establishing precise completeness boundaries for the AIC system.

Computing fixed points of non-monotonic operators—such as those arising from negation-as-failure or hypothetical updates—is notoriously challenging, as traditional monotonic methods do not apply and existing approximation techniques are either imprecise or computationally expensive. This work introduces, for the first time, controlled incompleteness into approximate fixed-point computation, integrating abstract interpretation, Approximation Fixpoint Theory (AFT), lattice theory, and partitioning-based optimization to devise a practical algorithm. The proposed method guarantees termination, polynomial-time complexity, and soundness over finite lattices while substantially improving approximation precision. Empirical evaluations demonstrate its effectiveness: deployed as an accelerating preprocessor in Answer Set Programming and applied to speculative program analysis, it significantly reduces rollback frequency, thereby validating both its efficiency and practical utility.

This study investigates the computational expressivity of the intuitionistic proof system μLJ—featuring a least fixed-point operator—and its cyclic extension CμLJ, aiming to precisely characterize the class of first-order total functions they represent. Methodologically, it integrates sequent calculus, realizability semantics, computability modeling, reverse mathematics, and ordinal analysis. The main contributions are: (i) the first rigorous proof that, under the “proofs-as-programs” paradigm, both μLJ and CμLJ exactly capture the first-order total functions provably total in the second-order arithmetic subsystem Π²₁-CA₀; (ii) the establishment of computational equivalence between cyclic and standard fixed-point proof systems; (iii) the development of a novel computability semantics tailored to cyclic proofs; and (iv) a reverse-mathematical foundation for the Knaster–Tarski fixed-point theorem. These results resolve a long-standing open problem on compactness boundaries, providing the first exact upper and lower bounds.

This work addresses the construction of verifiable witnesses for fixed-point games over lattices to determine whether the least fixed point of a function satisfies a given lower bound. By employing a Galois connection to unify the “logical universe” and the “behavioral universe”—two distinct lattice structures—it establishes, for the first time, a bidirectional correspondence between strategies and witnesses in both primal and dual fixed-point games. The proposed framework extends the applicability of fixed-point verification to novel domains such as certifying lower bounds on termination probabilities in probabilistic systems. It has been successfully applied to verify distinguishing formulas for bisimulation, behavioral metrics, and lower bounds on termination probabilities in Markov chains, thereby demonstrating its generality and effectiveness.

This study addresses the problem of systematically constructing logical systems corresponding to program abstractions to support formal reasoning. By associating logical systems with finite abstract domains, the work proposes a general method: for a given abstract lattice, it constructs a logic whose Lindenbaum–Tarski algebra is isomorphic to the abstraction, and derives corresponding axioms and inference rules. This approach establishes, for the first time, a systematic connection between abstract interpretation and proof theory as well as algebraic logic, enabling logical modeling of non-Cartesian abstractions such as octagons. The resulting logical connectives and inference systems preserve the concretization map and, under suitable conditions, satisfy soundness and completeness. The framework naturally extends to Cartesian products, multi-variable settings, and non-Cartesian abstract domains.

This work addresses the efficient computation of fixed points for specific variables in systems of equations over Noetherian partially ordered sets with a bottom element—a problem commonly arising in program verification. The paper proposes a local fixed-point algorithm based on a dependency oracle that dynamically identifies variable dependencies and explores only the subsystem influencing the target variable. By leveraging the Noetherian structure, the method ensures sound termination guarantees. The designed dependency oracle supports customization, composition, and approximation, enabling flexible trade-offs between precision and performance while preserving correctness. Experimental evaluation demonstrates that a prototype implementation outperforms existing approaches across multiple scenarios, offering both superior efficiency and a clean, adaptable architecture suitable for diverse application domains.

This work addresses the challenge of preserving and verifying structural correctness—such as dimensional consistency, stratification, escape properties, and numeric representations—during compilation of ML-family languages. To this end, it introduces an internal scaffolding mechanism based on fixed-point combinators, integrating closed negative types and fractional types to encode program semantic structure into MLIR intermediate representations at the middle end of compilation. Leveraging categorical constructions, the approach enables accompanying verification without requiring developers to explicitly engage with category theory, thereby ensuring structural integrity throughout the entire compilation pipeline. By exploiting MLIR’s dialect system, attribute infrastructure, and static single-assignment form, proof artifacts remain amenable to continuous toolchain validation as code is progressively lowered, achieving, for the first time, end-to-end verifiable preservation of structural properties.

This work investigates the query complexity of finding Tarski fixed points on high-dimensional lattices. We introduce the first algorithmic framework based on secure partial information functions and directly leverage this function to construct an efficient fixed-point search algorithm. On the four-dimensional lattice $[n]^4$, our algorithm achieves a query complexity of $O(\log^2 n)$, matching the known lower bound. For the general $k$-dimensional case, we improve the best-known upper bound to $O(\log^{\lceil (k-1)/3 \rceil + 1} n)$, substantially reducing the number of required queries.