Benjamin Przybocki

Guilherme V. Toledo, Benjamin Przybocki, Yoni Zohar

We consider the closure of three classical combination properties, namely, stable infiniteness, gentleness and shininess (or, equivalently for decidable theories, strong politeness), under intersection and combinability. We compute every possible intersection, and then compute the maximal set of theories that can be combined with each resulting intersection. We iterate this process until no new sets are identified. How many properties will we end up with?

David P. Woodruff, Vincent Cohen-Addad, Lalit Jain et al.

Recent advances in large language models (LLMs) have opened new avenues for accelerating scientific research. While models are increasingly capable of assisting with routine tasks, their ability to contribute to novel, expert-level mathematical discovery is less understood. We present a collection of case studies demonstrating how researchers have successfully collaborated with advanced AI models, specifically Google's Gemini-based models (in particular Gemini Deep Think and its advanced variants), to solve open problems, refute conjectures, and generate new proofs across diverse areas in theoretical computer science, as well as other areas such as economics, optimization, and physics. Based on these experiences, we extract common techniques for effective human-AI collaboration in theoretical research, such as iterative refinement, problem decomposition, and cross-disciplinary knowledge transfer. While the majority of our results stem from this interactive, conversational methodology, we also highlight specific instances that push beyond standard chat interfaces. These include deploying the model as a rigorous adversarial reviewer to detect subtle flaws in existing proofs, and embedding it within a "neuro-symbolic" loop that autonomously writes and executes code to verify complex derivations. Together, these examples highlight the potential of AI not just as a tool for automation, but as a versatile, genuine partner in the creative process of scientific discovery.

Benjamin Przybocki, John Mackey, Marijn J. H. Heule et al.

Ramsey-good graphs are graphs that contain neither a clique of size $s$ nor an independent set of size $t$. We study doubly saturated Ramsey-good graphs, defined as Ramsey-good graphs in which the addition or removal of any edge necessarily creates an $s$-clique or a $t$-independent set. We present a method combining SAT solving with bespoke LLM-generated code to discover infinite families of such graphs, answering a question of Grinstead and Roberts from 1982. In addition, we use LLMs to generate and formalize correctness proofs in Lean. This case study highlights the potential of integrating automated reasoning, large language models, and formal verification to accelerate mathematical discovery. We argue that such tool-driven workflows will play an increasingly central role in experimental mathematics.

Benjamin Przybocki, Bernardo Subercaseaux, Marijn J. H. Heule

Bounded Variable Addition (BVA) is a central preprocessing method in modern state-of-the-art SAT solvers. We provide a graph-theoretic characterization of which 2-CNF encodings can be constructed by an idealized BVA algorithm. Based on this insight, we prove new results about the behavior and limitations of BVA and its interaction with other preprocessing techniques. We show that idealized BVA, plus some minor additional preprocessing (e.g., equivalent literal substitution), can reencode any 2-CNF formula with $n$ variables into an equivalent 2-CNF formula with $(\tfrac{\lg(3)}{4}+o(1))\,\tfrac{n^2}{\lg n}$ clauses. Furthermore, we show that without the additional preprocessing the constant factor worsens from $\tfrac{\lg(3)}{4} \approx 0.396$ to $1$, and that no reencoding method can achieve a constant below $0.25$. On the other hand, for the at-most-one constraint on $n$ variables, we prove that idealized BVA cannot reencode this constraint using fewer than $3n-6$ clauses, a bound that we prove is achieved by actual implementations. In particular, this shows that the product encoding for at-most-one, which uses $2n+o(n)$ clauses, cannot be constructed by BVA regardless of the heuristics used. Finally, our graph-theoretic characterization of BVA allows us to leverage recent work in algorithmic graph theory to develop a drastically more efficient implementation of BVA that achieves a comparable clause reduction on random monotone 2-CNF formulas.

Andrew Krapivin, Benjamin Przybocki, Bernardo Subercaseaux

We present several novel encodings for cardinality constraints, which use fewer clauses than previous encodings and, more importantly, introduce new generally applicable techniques for constructing compact encodings. First, we present a CNF encoding for the $\text{AtMostOne}(x_1,\dots,x_n)$ constraint using $2n + 2 \sqrt{2n} + O(\sqrt[3]{n})$ clauses, thus refuting the conjectured optimality of Chen's product encoding. Our construction also yields a smaller monotone circuit for the threshold-2 function, improving on a 50-year-old construction of Adleman and incidentally solving a long-standing open problem in circuit complexity. On the other hand, we show that any encoding for this constraint requires at least $2n + \sqrt{n+1} - 2$ clauses, which is the first nontrivial unconditional lower bound for this constraint and answers a question of Kučera, Savický, and Vorel. We then turn our attention to encodings of $\text{AtMost}_k(x_1,\dots,x_n)$, where we introduce "grid compression", a technique inspired by hash tables, to give encodings using $2n + o(n)$ clauses as long as $k = o(\sqrt[3]{n})$ and $4n + o(n)$ clauses as long as $k = o(n)$. Previously, the smallest known encodings were of size $(k+1)n + o(n)$ for $k \le 5$ and $7n - o(n)$ for $k \ge 6$.