Language models (LMs) can generate code but cannot guarantee its correctness—often producing outputs that violate type safety, program invariants, or other semantic properties. Constrained decoding offers a solution by restricting generation to only produce programs that satisfy user-defined properties. However, existing methods are either limited to syntactic constraints or rely on brittle, ad hoc encodings of semantic properties over token sequences rather than program structure. We present ChopChop , the first programmable framework for constraining the output of LMs with respect to semantic properties. ChopChop introduces a principled way to construct constrained decoders based on analyzing the space of programs a prefix represents. It formulates this analysis as a realizability problem which is solved via coinduction, connecting token-level generation with structural reasoning over programs. We demonstrate ChopChop ’s generality by using it to enforce (1) equivalence to a reference program and (2) type safety. Across a range of models and tasks, ChopChop improves success rates while maintaining practical decoding latency.

Large Language Models (LLMs) are often asked to generate structured outputs that obey precise syntactic rules, such as code snippets or formatted data. Grammar-constrained decoding (GCD) can guarantee that LLM outputs matches such rules by masking out tokens that will provably lead to outputs that do not belong to a specified context-free grammar (CFG). To guarantee soundness, GCD algorithms have to compute how a given LLM subword tokenizer can ``align'' with the tokens used by a given context-free grammar and compute token masks based on this information. Doing so efficiently is challenging and existing GCD algorithms require tens of minutes to preprocess common grammars. We present a new GCD algorithm together with an implementation that offers 17.71x faster offline preprocessing than existing approaches while preserving state-of-the-art efficiency in online mask computation.

Formally verifying system properties is one of the most effective ways of improving system quality, but its high manual effort requirements often render it prohibitively expensive. Tools that automate formal verification by learning from proof corpora to synthesize proofs have just begun to show their promise. These tools are effective because of the richness of the data the proof corpora contain. This richness comes from the stylistic conventions followed by communities of proof developers, together with the powerful logical systems beneath proof assistants. However, this richness remains underexploited, with most work thus far focusing on architecture rather than on how to make the most of the proof data. This article systematically explores how to most effectively exploit one aspect of that proof data: identifiers. We develop the Passport approach, a method for enriching the predictive Coq model used by an existing proof-synthesis tool with three new encoding mechanisms for identifiers: category vocabulary indexing, subword sequence modeling, and path elaboration. We evaluate our approach’s enrichment effect on three existing base tools: ASTactic, Tac, and Tok. In head-to-head comparisons, Passport automatically proves 29% more theorems than the best-performing of these base tools. Combining the three tools enhanced by the Passport approach automatically proves 38% more theorems than combining the three base tools. Finally, together, these base tools and their enhanced versions prove 45% more theorems than the combined base tools. Overall, our findings suggest that modeling identifiers can play a significant role in improving proof synthesis, leading to higher-quality software.

$\newcommand{\floor}[1]{\left\lfloor {#1} \right\rfloor} \renewcommand{\Re}{\mathbb{R}}$ Tverberg's theorem states that a set of $n$ points in $\Re^d$ can be partitioned into $\floor{n/(d+1)}$ sets with a common intersection. A point in this intersection (aka Tverberg point) is a centerpoint of the input point set, and the Tverberg partition provides a compact proof of this, which is algorithmically useful. Unfortunately, computing a Tverberg point exactly requires $n^{O(d^2)}$ time. We provide several new approximation algorithms for this problem, which improve either the running time or quality of approximation, or both. In particular, we provide the first strongly polynomial (in both $n$ and $d$) approximation algorithm for finding a Tverberg point.

Correlation Clustering is a powerful graph partitioning model that aims to cluster items based on the notion of similarity between items. An instance of the Correlation Clustering problem consists of a graph G (not necessarily complete) whose edges are labeled by a binary classifier as similar and dissimilar. Classically, we are tasked with producing a clustering that minimizes the number of disagreements: an edge is in disagreement if it is a similar edge and is present across clusters or if it is a dissimilar edge and is present within a cluster. Define the disagreements vector to be an n dimensional vector indexed by the vertices, where the v-th index is the number of disagreements at vertex v. Recently, Puleo and Milenkovic (ICML '16) initiated the study of the Correlation Clustering framework in which the objectives were more general functions of the disagreements vector. In this paper, we study algorithms for minimizing \ellq norms (q >= 1) of the disagreements vector for both arbitrary and complete graphs. We present the first known algorithm for minimizing the \ellq norm of the disagreements vector on arbitrary graphs and also provide an improved algorithm for minimizing the \ell_q norm (q >= 1) of the disagreements vector on complete graphs. We also study an alternate cluster-wise local objective introduced by Ahmadi, Khuller and Saha (IPCO '19), which aims to minimize the maximum number of disagreements associated with a cluster. We present an improved (2 + \eps) approximation algorithm for this objective.