construct illustrative examples

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.

This work addresses the need for a pedagogically effective introduction to real-number computation, targeting instructors, students, and early-stage doctoral researchers. Recognizing that existing surveys often prioritize technical depth over teachability, the authors develop a classroom-oriented exposition grounded in modern models of real computation—such as binary input representations and restricted constants—and integrate insights from ER-completeness theory. Drawing on foundational contributions by Matoušek, Schaefer, and others, they carefully select core results that are both rigorously provable and suitable for instructional settings. The resulting material emphasizes intuitive understanding, practical relevance, and recurring proof techniques, offering a coherent and readily implementable framework for incorporating real-number computation into standard algorithms curricula.

This work addresses the challenge of automatically constructing tight worst-case instances for approximation algorithms operating over continuous input spaces—such as makespan minimization in scheduling and bin packing. We propose a joint optimization framework combining decision-tree compilation with leaf-wise linear programming: first, statically compile the approximation algorithm (e.g., LPT) into an interpretable decision tree; then, formulate and solve a linear program over the input subspace associated with each leaf node to precisely identify inputs that maximize the approximation ratio. Our approach overcomes the scalability limitations of brute-force enumeration while ensuring efficiency, interpretability, and extensibility. Experiments successfully generate the smallest-known tight hard instance for the LPT algorithm in makespan minimization, empirically validating the method’s effectiveness. Moreover, it establishes a novel paradigm for algorithmic diagnosis and refinement through structured, constraint-driven instance generation.

In data-driven algorithm tuning, utility functions often exhibit piecewise discontinuities and Pfaffian structure, undermining theoretical guarantees. Method: This paper proposes the Pfaffian GJ learning framework—a novel generalization of the classical GJ framework from rational functions to the broader class of Pfaffian functions. It integrates Pfaffian theory, geometric complexity analysis, and piecewise modeling techniques to establish rigorous theoretical guarantees for distributed and online learning. Contribution/Results: Unlike prior approaches requiring rationality assumptions on utility functions, our framework is the first to provide provably sound learning guarantees for non-rational—particularly Pfaffian—utility functions. It significantly extends the theoretical applicability boundary and delivers a formally verifiable foundation for automated configuration of diverse parameterized algorithms, including SAT solvers and scheduling policies. The framework ensures statistical consistency and convergence under mild regularity conditions on Pfaffian structures, thereby bridging a critical gap between practical algorithm tuning and foundational learning theory.

This work addresses the problem of explicitly constructing distributions that are nearly unsamplable by restricted computational models, such as low-depth circuits or small-space sources. To this end, we introduce a novel robust extractor that remains effective even when a small number of samples fail to satisfy the standard min-entropy assumption. Leveraging this tool, we unify and generalize existing hardness results, yielding the first explicit distribution that is statistically far from any source samplable by low-degree 𝔽₂ polynomials—specifically, at statistical distance 1−o(1) from the outputs of various restricted sampling models. Our approach also provides a new route toward establishing sampling hardness for AC⁰[⊕] circuits.

This work investigates the relationship between ordered structures—such as thresholds—and tree-like configurations, specifically 2-trees, within real-valued function classes exhibiting large sequential fat-shattering dimension. By integrating techniques from sequential fat-shattering dimension theory, stability analysis of functions, and order properties from combinatorial model theory, the paper introduces a more flexible framework for threshold extraction. This approach substantially improves upon existing bounds, resolving an open problem concerning the upper bound on the dual sequential fat-shattering dimension with at most a double-exponential dependence. In doing so, it corrects a previously flawed proof in the literature and strengthens related results by Anderson–Benedikt and Daskalakis–Golowich.

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 long-standing challenge of explicitly constructing near-optimal lossless rank extractors, weak subspace designs, and strong $s$-blocking sets over small finite fields whose size depends only on the rank or codimension. By integrating tools from function field theory, polynomial identity testing, and Fourier analysis based on $\varepsilon$-biased sets, the authors achieve the first explicit near-optimal constructions of these objects over non-prime fields with $q \geq \mathrm{poly}(s)$. Notably, the resulting strong $s$-blocking set has size $O(s(k - s)q^s)$, improving upon the previous exponential bound $2^{O(s^2 \log s)} q^s k$ and matching the non-explicit optimal asymptotics. The paper also presents the first explicit near-optimal constructions for both lossless rank extractors and weak subspace designs in this setting.

This work addresses the statistical unlearnability of symbolic regression caused by combinatorial explosion in the hypothesis space by analyzing composite function trees built from finitely many smooth operators within the PAC learning framework. For the first time, it establishes a theoretical foundation for their learnability, proving a generalization error bound of 𝒪(Lᵈ/√n) via Rademacher complexity analysis, a Maurer-type vector contraction inequality, and finite union bounding techniques, where L denotes the operator Lipschitz constant and d the tree depth. The result demonstrates that model complexity is governed by depth and Lipschitz constants rather than the exponential growth of symbolic structures. Empirical validation further confirms that the generalization gap correlates strongly with (𝐿̂ᵈ)/√n, challenging conventional understanding of the statistical complexity inherent in symbolic regression.