solve functional equations

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.

Optimization and feasibility checking for algebraic functions involving radicals, rational expressions, and similar non-polynomial terms pose significant challenges due to the “curse of dimensionality” inherent in conventional layered auxiliary-variable lifting. Method: This paper introduces a compact polynomial reconstruction framework that replaces each algebraic function with a single new variable, coupled with an integrated algorithmic pipeline grounded in real algebraic geometry: implicitization, branch-isolating inequality generation, and Positivstellensatz/SOS-based feasibility verification. Contribution/Results: By avoiding redundant variable proliferation, the method substantially improves modeling efficiency and scalability of polynomial programming formulations. On enhanced classical benchmarks, it achieves up to 50× speedup over naive reconstruction. The approach provides both theoretical guarantees and a practical toolkit for efficiently translating algebraic programs into polynomial optimization problems.

Traditional accounts of “function” narrowly construe it as either a mapping or a structure-preserving property, failing to capture its generative role in syntactic construction and semantic interpretation. Method: The paper proposes a generative reconceptualization—defining function as a dual-structured generative principle capable of both syntactic item formation and semantic interpretation. To formalize this, it introduces Construction-Defining Functions (CDFs), establishing the first axiomatic framework that jointly ensures generativity and compositionality. The framework is modeled categorically using initial algebras and endofunctors, integrating type theory, model theory, and formal semantics. Contribution/Results: This work achieves the first cross-paradigmatic unification of structural generation mechanisms across logic, linguistic semantics, and computability theory. It provides a foundational redefinition of “function,” advancing deep integration between logical and linguistic theories while offering a rigorous, mathematically grounded basis for functional abstraction in formal systems.

This work investigates the effective decomposition of higher-arity relations over finite domains into binary relations, with applications in constraint satisfaction problems, clone theory, and relational databases. By interpreting relations as partially defined graphs of multi-valued functions and leveraging functional completeness and function decomposition techniques from many-valued logic, the authors first reduce arbitrary high-arity relations to ternary ones. Subsequently, they achieve full binarization through a transformation that replaces disjunctions with existential quantifiers. This approach yields an elementary and computationally efficient constructive proof, unifying and rederiving Pierce’s reduction theorem for finite domains. Moreover, it demonstrates that the graph of any Sheffer function suffices to generate all relations, thereby establishing a computable theoretical foundation for the aforementioned fields.

Existing machine learning frameworks suffer from insufficient formalization of objective functions and lack a unified, cross-domain behavioral design paradigm. Method: We propose an equation-constrained compositional function modeling approach for learners, constructing task graphs and compositional semantic graphs to enable model-agnostic behavioral specification and optimization. We introduce a novel task-oriented pattern language framework and the “manipulator” task paradigm, supporting end-to-end, architecture-agnostic, and adversarial-training-free minimal editing of data attributes. Contribution/Results: Theoretically, our work integrates formal methods and theoretical computer science principles. Empirically, we demonstrate precise, controllable, and interpretable behavioral editing on small-scale models under stable training—without stochastic sampling or data intervention—yielding significant improvements in deployment efficiency and formal verifiability.

This work addresses the construction of most general solutions for parameterized constraint formulas of the form ∃x₁…∃xₙ φ(x₁,…,xₙ,y₁,…,yₘ) within theories 𝒯 that admit elimination of specific existential quantifiers, where φ is a quantifier-free conjunction of literals and the yᵢ are parameters. By introducing conditional function symbols that capture “if-then-else” constructs, the authors generalize existing results on the existence of most general unifiers in discriminator clusters. Integrating parameterized constraint solving with algebraic semantic characterizations, they establish a unified framework for constructing most general solutions. This approach substantially broadens the scope of applicability compared to prior methods, and its effectiveness and generality are demonstrated through illustrative examples.

This work addresses the challenges posed by higher-order functions in mathematical optimization modeling, which often lead to unnatural LaTeX output and inefficient constraint verification. To overcome these issues, the authors propose an egglog-based optimization approach that performs desugaring reconstruction on the λ-calculus intermediate representation of JijModeling 2, thereby recovering comprehension-like syntactic structures. By integrating Henkin-style constants with Datalog-inspired rules, the method enables declarative, multi-step constraint checking. A custom cost model and equality saturation techniques are introduced to enhance performance significantly: the generated LaTeX aligns more closely with conventional mathematical notation, and complex constraint validation—previously requiring minutes or failing to terminate—now completes within seconds.

This work addresses the challenge of efficiently synthesizing a strategy that realizes the largest satisfiable subset of specifications in LTLf synthesis when multiple specifications cannot be simultaneously satisfied. The authors propose a fully symbolic algorithm that, through a single fixed-point computation, directly associates realizable sets of objectives with states in the product game, thereby avoiding the enumeration of exponentially many subsets. By introducing Boolean objective variables and exploiting objective monotonicity, the method compactly encodes all possible objective combinations, significantly improving computational efficiency. Experimental results demonstrate that the approach achieves up to two orders of magnitude speedup compared to baseline enumeration-based methods.

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 study addresses the challenge in axiomatic design of accurately translating customer needs and constraints into a minimal and independent set of primary functional requirements (FRs). Focusing on the problem definition phase, it systematically elucidates the nature, invariance, and formulation principles of primary FRs. Building upon Nam P. Suh’s theoretical framework and integrating insights from complexity theory and requirements engineering, the work establishes—for the first time—the objectivity and uniqueness of primary FRs, clarifies common misconceptions, and critically examines the applicability boundaries of large language models in this context. The research provides designers with a clear, actionable methodology for constructing primary FRs, thereby significantly enhancing the rigor of problem definition and the likelihood of successful design outcomes.