p-adic forcing method

This study investigates the computational complexity of $p$-adic optimization problems beyond the binary ($p=2$) case. By generalizing Baker’s forcing method to arbitrary primes $p$, the authors establish a reduction framework linking integer optimization to $p$-adic optimization. Leveraging this framework, they provide the first unified proof that several prominent $p$-adic models—including $p$-adic linear regression, 2-adic dynamic neural networks, and various van der Put–based $p$-adic neural network architectures—are all NP-hard. This work bridges $p$-adic analysis, computational complexity theory, and neural network modeling, significantly extending the applicability of forcing techniques and offering a foundational complexity-theoretic characterization for $p$-adic machine learning.

This paper investigates the computational complexity of deciding satisfiability of systems of linear equations over the $p$-adic number field $mathbb{Q}_p$, subject to $p$-adic valuation constraints—such as $v_p(x) = c$ or inequalities. Employing algebraic reduction leveraging $p$-adic structure, Hensel’s lemma, divide-and-conquer techniques, and integer linear programming, we fully resolve the open cases $p = 2, 3$ left by Guépin et al.: we prove NP-completeness for $p = 3$, and devise an exact $O(n^3)$ polynomial-time algorithm for $p = 2$. We further generalize the results to joint valuation constraints across multiple primes and to systems involving linear valuation inequalities, all decidable in polynomial time. The key innovation lies in establishing a precise correspondence between valuation constraints and the underlying linear algebraic structure, thereby overcoming longstanding complexity barriers specific to small primes.

This paper investigates the root cause of undecidability for global questions in arithmetic—such as P vs. NP—not as a consequence of computational limitations, but as an intrinsic feature of logical structure: impredicativity arising from quantifier-like constructions that induce reflexive hierarchies and hyperfinite reflection, thereby triggering Feferman-style undecidability. Methodologically, the analysis integrates Gödel arithmetization, reflection principles, Feferman’s provability theory, and model-theoretic semantics. It establishes, for the first time, that the inherent difficulty of complexity-theoretic statements stems from the impredicative logical architecture of arithmetized assertions, rather than from features of particular computational models. The results demonstrate that undecidability of uniform complexity statements is model-independent, offering a novel structural explanation—grounded in impredicativity—for longstanding metamathematical problems. (136 words)

We introduce $p$-adic manifold learning, propose an algorithm to solve it, and propose benchmark tasks from impartial games.

This work investigates the tractability and computational hardness of Promise Constraint Satisfaction Problems (PCSPs) over Boolean domains. By introducing Fourier-analytic techniques into the PCSP framework—specifically leveraging influence measures of Boolean functions in conjunction with random 2-to-1 minors and sharp threshold theory—the study uncovers two universal mechanisms that govern whether a given problem is efficiently solvable or computationally intractable: the preservation of coordinate influences and the existence of sharp thresholds. This approach extends the prevailing paradigm for ordered PCSPs and, for the first time, establishes a clear dichotomy of tractability within broader classes of Boolean functions, including unate functions and polynomial threshold functions, thereby yielding new complexity-theoretic characterizations.

This work presents the first assumption-free formalization of Cauchy real numbers in Cubical Agda, circumventing reliance on the axiom of choice, setoid bookkeeping, or explicit universe-level management—issues that commonly hinder constructive real number constructions in intuitionistic mathematics. Building upon the higher inductive-inductive types introduced in Homotopy Type Theory, the construction leverages Cubical Agda’s native support for higher inductive types to yield a fully type-checked, non-vacuous, and postulate-free development of the reals. This approach not only resolves longstanding challenges related to redundancy and universe complexity but also establishes a robust foundation for machine-verified constructive analysis.

This work addresses the limitations of MCSat in solving complex SMT problems involving nonlinear integer and real arithmetic by reformulating it as a theory-agnostic proof system. By formally capturing key implementation mechanisms from the Yices2 solver, the authors derive a unified and general framework of MCSat inference rules, which they instantiate across multiple theories—including propositional logic, nonlinear real arithmetic, and uninterpreted functions. This approach not only integrates core design choices of modern SMT solvers but also establishes the first unified MCSat calculus supporting multiple theories, substantially enhancing its expressiveness and applicability. The effectiveness of the proposed framework is demonstrated through representative examples.

This work investigates when feasible arithmetic theories—such as those containing \( S^1_2 \)—can efficiently prove the bounded consistency of their extensions by true sentences, i.e., whether they can polynomially simulate such extensions. By integrating techniques from bounded arithmetic, interpretability theory, and the Busy Beaver function, we identify relative consistency strength as the decisive criterion governing the possibility of simulation. We further demonstrate that Busy Beaver assertions act as intrinsic barriers to simulation: if a base theory cannot polynomially simulate a given extension, then for all sufficiently large \( k \), it also fails to simulate the extension augmented with the assertion \( \mathrm{BB}(k) \). These findings offer a novel perspective on the interplay between proof complexity and simulability among formal theories.