functional equation solving

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 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.

Traditional coequational methods suffer from limitations in expressiveness and usability. This work proposes “equational path constraints” as an algebraic alternative: by assigning a pair of values to each path in a coalgebra and enforcing their equality, it algebraically characterizes finite-behavior properties, thereby enabling an axiomatic definition of covarieties and the construction of final coalgebras. The approach establishes a connection with coequations in the setting of monads and provides an upper bound on the number of colors required. Combining category theory, coalgebraic techniques, and Adámek–Barr final sequence constructions, the method is successfully validated across multiple case studies—including automata commutativity, differential equations, bi-infinite streams, and modal frame conditions—demonstrating its effectiveness and broad applicability.

This paper characterizes the class BFF₂—type-2 functions computable by second-order polynomial-time oracle Turing machines. To this end, it introduces the first exact algebraic characterization of BFF₂ via higher-order term rewriting: by defining a cost-size interpretation, it establishes that higher-order rewrite systems induced by polynomially bounded interpretations precisely capture all and only the functions in BFF₂. The approach models type-2 function semantics as higher-order term reduction and integrates second-order polynomial bounding analysis to yield a purely rewriting-based characterization of BFF₂. The main contribution is the first tight correspondence between higher-order rewriting and BFF₂, providing a novel algebraic framework for complexity analysis, termination verification, and feasibility certification of higher-order programs. This result lays a theoretical foundation for reasoning about computational resources in higher-order functional computation.

This paper establishes the first systematic algebraic framework for valued promise constraint satisfaction problems (valued PCSPs). To capture structural relationships among such problems, it introduces *valued minions*—a novel algebraic object—and proves that the existence of a minion homomorphism between two valued PCSPs is equivalent to polynomial-time reducibility between them, thereby establishing a general theorem linking algebraic homomorphisms to computational reductions. This framework extends universal algebraic methods to the valued promise setting for the first time, unifying explanations of classical inapproximability thresholds and yielding a strong inapproximability result for linear equations over finite fields: even when instances are *almost satisfiable*, they cannot be approximated beyond the random assignment threshold. Key innovations include: (i) the formal definition of valued minions; (ii) the proof of equivalence between minion homomorphisms and polynomial-time reductions; and (iii) the paradigmatic transfer of algebraic tools from classical CSPs to valued PCSPs.

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 why the Kullback–Leibler (KL) divergence is uniquely suited for inference by formalizing inference as the selection of a minimal element within a preorder of positive measures, where divergences serve merely as numerical representations. Building on the axiom of reconstruction invariance—which requires that inference outcomes remain unchanged under equivalent problem formulations—the authors show that KL divergence emerges uniquely without invoking additional assumptions. This framework unifies maximum entropy, Bayesian updating, and exponential family estimation, extending classical axiomatic characterizations from finite alphabets to general measurable spaces. By integrating category theory, f-divergence theory, preorder structures, and Čencov’s category of statistical models, the paper establishes a rigorous mathematical foundation wherein inference operators arise naturally as covariant functors, applicable uniformly across both discrete and continuous settings.

This work proposes a novel method to overcome the limitations of the classical absolute positivity criterion, which fails to handle nonlinear polynomial constraints involving universal quantifiers. Specifically, the approach addresses ∃∀ inequalities over the natural numbers by integrating monotonic algebra with well-founded order theory, thereby dispensing with the absolute positivity assumption. This advancement substantially broadens the class of constructible nonlinear polynomial interpretations. Experimental results demonstrate that the technique successfully solves constraint instances previously intractable to existing methods, thus extending the applicability of polynomial interpretations in termination and complexity analysis of term rewriting systems.

This work addresses the entanglement of information propagation and contraction mechanisms in functional interpretations by introducing an algebraic structure called the “information nucleus” to formally disentangle these two aspects. The information nucleus precisely captures affine information propagation, while a formula-indexed contraction structure handles repeated assumptions. This approach unifies and extends classical interpretations such as Dialectica and Herbrand, yielding a functional interpretation framework applicable to both affine and contraction-containing fragments of finite-type arithmetic. The proposed framework further supports the systematic incorporation of auxiliary principles like continuity, enabling precise tracking of informational content and providing extracted realizers with refined mechanisms for handling challenges and choices.

Existing quadratic constraint approaches for characterizing neural network activation functions are overly conservative, limiting the precision of reachability and safety analyses. This work proposes a domain-dependent framework for verifiable quadratic inequalities: it generates candidate constraints via local sampling and employs sum-of-squares (SOS) certificates to ensure global validity, yielding tight and sound quadratic representations for scalar nonlinearities. The method transcends the limitations of conventional sector or slope bounds by incorporating neuron-wise dependencies and local bound refinement—particularly for ReLU networks—to reduce conservatism. It is compatible with convex quadratic programming, semialgebraic set descriptions, and integral quadratic constraint (IQC) techniques. Experiments demonstrate that the framework significantly improves analysis accuracy for smooth activations such as tanh and extends effectively to systems involving saturation-type nonlinearities.