Constraint-Guided Reasoning in Language Models

Large language models (LLMs) are increasingly used as formalizers: given a reasoning problem expressed in natural language, they produce a formal specification that a symbolic solver can then solve, with guarantees relative to that specification. Many specifications are runnable yet semantically wrong, encoding incorrect constraints, domains, objectives, predicates, or goals, and solver-error feedback dries up once execution succeeds. Purely LLM-based diagnosis, lacking a formal ground truth, often fails to spot these errors, and its ungrounded critiques cause regressions in 11.8% of initially correct specifications in our experiments. We propose DRAGO, a repair framework that replaces this ungrounded critique with deterministic, formalism-specific diagnosis, and retains a rewrite only when the diagnosed error weight strictly decreases under a reference-free acceptance predicate. Across five benchmarks spanning logical reasoning (SMT-LIB2), constraint programming (MiniZinc), and planning (PDDL), DRAGO cuts regressions by nearly an order of magnitude, to 1.3%, while improving answer accuracy by 12.9 percentage points over zero-shot formalization and 8.7 percentage points over LLM-diagnostic repair, averaged across benchmarks. We also uncover two rendering defects in Planetarium that cause 50% (3,086 instances) of its floor-tile split to be inconsistent with the reference formalizations, and release a corrected split.

Real-world molecular design remains challenging for large language model (LLM)-based agents. It requires them to interpret design contexts, satisfy multiple constraints, identify infeasible specifications, and reason over multi-step tool outputs. Existing benchmarks do not capture this complexity, focusing instead on explicit and narrow constraints, only feasible problems, and single-path solutions. To address this gap, we propose MolDesignBench, a scenario-grounded benchmark that more closely reflects real-world molecular design for evaluating tool-augmented LLM agents. MolDesignBench comprises 2K generation and optimization instances that combine implicit requirements embedded in design narratives with explicit property and functional-group constraints, including infeasible cases, and require the effective use of 17 specialized chemistry tools. Experiments across diverse frontier LLMs reveal low success rates--with the best achieving only $\sim43$%--and frequent failures in implicit-constraint reasoning, infeasibility detection, and tool reasoning. The corresponding fine-grained failure-mode analysis identifies implicit constraint interpretation and infeasibility detection as the primary bottlenecks, establishing MolDesignBench as a rigorous testbed to guide future research on chemical agents. The benchmark, tool interface, and evaluation code are publicly available.

While Large Language Models have improved rapidly, many fundamental questions remain about how to evaluate the knowledge and reasoning abilities they acquire, and how such evaluations relate to the learning signals used in pre-training. In this paper, we propose ModelLog, a declarative probabilistic framework for pre-training evaluation that makes the semantic structure of model behavior explicit and provides new formal tools for relating evaluation to learning. ModelLog specifies evaluation targets as symbolic constraints over token-level predictions and measures how strongly a model's distribution satisfies those constraints. We explore the framework through a new suite of tasks targeting negation, mutual exclusivity, and consistency, finding systematic failures that are difficult to characterize through token likelihood or answer accuracy alone. We further show that these evaluation scores can also be interpreted as losses, whose gradients reflect logical strength, informativeness, and variable-level sensitivity. This links evaluation and learning through a shared semantics, suggesting evaluation methods that diagnose model behavior while also helping to clarify the semantic structure of learning.

Many robotic tasks are temporally extended and demand precise specifications of subgoals, constraints, and their temporal ordering. Yet human operators typically communicate such tasks in natural language, which is inherently ambiguous, underspecified, and context dependent. Translating human instructions into formal task specifications, such as Linear Temporal Logic (LTL), is therefore essential for verifiable and safe robotic execution. Existing LLM-based translators attempt to bridge this gap through open-ended reasoning or post-hoc constraint enforcement, but the former may violate domain constraints, whereas the latter can disrupt the reasoning needed for novel instructions. This paper proposes Self-Constrained Reasoning (SCR), a framework that mitigates this trade-off by internalizing structural knowledge into the model's decision-making process rather than imposing it as an external filter. By combining a structural constraint representation with a hierarchical decision-making formulation, SCR guides reasoning within a formally grounded space while preserving adaptability to unseen instructions. Experiments show that SCR improves both domain-constraint satisfaction and generalization, providing an effective and interpretable approach for translating human intent into verifiable specifications for robotic execution.

Reasoning in large language models (LLMs) is often grounded in human text, human demonstrations, and human-generated rationales. For equilibrium reasoning in complex games, however, relying on human data can be suboptimal. In fact, human play is often guided by intuition and heuristics and can deviate substantially from game equilibrium. This discrepancy is amplified in games with mixed-strategy equilibria, where human data is heavily biased toward pure strategies. Consequently, conditioning LLMs on this data yields weak game strategies. To grant LLMs the reasoning capacity in games, in this work, we study how to elicit equilibrium play using solver output. We propose Mixed-Strategy Decision Tree (MDT), which articulates the silent optimality of the equilibrium into sparse strategic rules that both humans and LLMs could understand. Using solver output rather than human annotation allows us to extend the input to arbitrarily new states and continuations. We instantiate this study on No-Limit Texas Hold'em by querying a solver oracle for over 250 million mixed-strategy decisions; MDT together with other techniques \textbf{reduces the $\ell_1$ distance to the equilibrium by $52.6\%$} across $8$ different LLM configurations. A Route-only ablation tests the incremental contribution of the shadow-based contrast, while complete River-endgame and Liar's Dice experiments evaluate strategic fidelity and portability beyond the original NLH communication setting.

When does a committed intermediate stage in an LLM reasoning pipeline earn its cost? Constrained Path Reasoning (CPR) pairs a source-aware path hypothesis with stage-level accounting. Search generates provisional states; trusted or validated invariants can constrain hard, while other proposals remain soft and revisable. CPR predicts that task-compatible commitments can factor transitions, concentrate candidate mass, induce regularity, and expose feedback when their gains exceed propagated error and execution cost. The formalism covers discrete commitments and continuous flows and measures effective branching, endpoint concentration, and cost per usable output. Across 1,180 generated QCQPs and 40 engineered degenerate polynomial instances (2,140 endpoints), residual triage recovers 63.0% of repair-all's additional feasible yield with 17.7% of its attempts. Fixed-LLM accounting (270 unique calls shared across nested arms) finds usable yield of 41.1% direct, 90.0% after formalization and deterministic execution, 20.0% after one-shot convexification, and 21.1% for the full path. In 120 paired-condition calls, a two-action rollback rule reaches 90% usable yield versus 36.7% for the feedback-conditioned selector. Two endpoint probes separate source from validation: a 72-output cross-trajectory transplant reduces entropy and acceptable mass; a 24-output same-call self-proposal pilot gives unchanged two-repeat collision entropy, 25.0% versus 8.3% usable yield, and 1/8 deterministically confirmed endpoint checks. Model-generated states supply hypotheses; trusted execution earns constraint strength.

Larger language models consistently outperform smaller ones on reasoning benchmarks, yet the reasoning differences underlying this gap remain underexplored. Across benchmarks in mathematics, physics, chemistry, and programming, we observe stable performance gaps: averaged over datasets, Qwen3-32B outperforms Qwen3-8B by 6.43%, while GPT-OSS-120B exceeds GPT-OSS-20B by 7.38%. To study the reasoning differences behind these gains, we develop AdvCluster, an automated framework that identifies questions where the larger model shows a stable advantage, extracts fine-grained advantage descriptions from paired reasoning traces produced by larger and smaller models, and organizes them through semantic clustering with quantitative evaluation and selection guided by a reviewer model. Our analysis yields a systematic taxonomy of larger model reasoning advantages, spanning both common advantages that recur across domains and specialized advantages associated with particular domains. Across these patterns, a recurring theme is Constraint-Guided Reasoning: larger models are better at identifying explicit and implicit constraints, organizing them into structured reasoning, and using them to rule out infeasible paths and verify intermediate steps.

This paper addresses two limitations of large language models (LLMs) in solving complex problems: (1) their reasoning processes exhibit Bayesian-like stochastic generation, where each token is sampled from a context-dependent probability distribution, leading to inherently random decision trajectories rather than deterministic planning; (2) the reasoning and decision-making mechanisms are statically decoupled, meaning dynamically retrieved domain knowledge fails to dynamically adjust the underlying reasoning strategy. These dual deficiencies result in initial decisions lacking strategic anchoring and reasoning chains often failing to converge on correct solutions, as stochastic generation lacks mechanisms for trajectory correction or knowledge-guided optimization during sequential reasoning. To resolve these issues, we propose a problem-solving method integrated into the LLM's generation process to guide reasoning. This method, compatible with numerous LLMs and featuring reusable solutions, is grounded in a novel Heuristic-Classification-of-Thoughts prompting schema (HCoT). HCoT synergizes the LLM's reasoning ability with a structured problem space via a heuristic classification model that controls the reasoning process and provides reusable abstract solutions. Evaluated on two complex inductive reasoning tasks with ill-defined search spaces, HCoT outperforms existing approaches (e.g., Tree-of-Thoughts and Chain-of-Thoughts prompting) in performance. On the well-structured 24 Game task, HCoT demonstrates significantly higher token efficiency compared to the state-of-the-art Tree-of-Thoughts-Breadth-First-Search. In terms of both accuracy and token usage, HCoT achieves a Pareto frontier balance, offering a strong trade-off between performance and computational cost.

Asked whether to walk or drive to a car wash 50 m away, most language models say walk, forgetting that the car has to be there. Such failures are usually measured by accuracy on questions where the hidden constraint applies. We show that this is misleading: a model can answer these questions correctly without reasoning about the constraint at all, simply by favoring the cautious option. We instead ask whether a model's decision changes when the constraint is removed. We test this at three levels of control: log-probability sweeps on open-weight models, a 500-prompt stress test, and CORE, a new human-validated benchmark of minimal constraint-present/constraint-absent pairs. The picture is consistent. The constraint nudges decisions rather than governing them. Models still miss presence constraints like the car wash, yet elsewhere they apply constraints that are not there. As a result, standard accuracy flatters all ten models we evaluate and reorders their ranking, and prompting fixes that look effective largely vanish under paired scoring. Claims about hidden-constraint reasoning need paired evidence.

Large Language Models (LLMs) have demonstrated great capabilities across diverse natural language tasks; yet their ability to solve abstraction and optimization problems with constraints remains scarcely explored. In this paper, we investigate whether LLMs can reason and optimize under the physical and operational constraints of Optimal Power Flow (OPF) problem. We introduce a challenging evaluation setup that requires a set of fundamental skills such as reasoning, structured input handling, arithmetic, and constrained optimization. Our evaluation reveals that SoTA LLMs fail in most of the tasks, and that reasoning LLMs still fail in the most complex settings. Our findings highlight critical gaps in LLMs' ability to handle structured reasoning under constraints, and this work provides a rigorous testing environment for developing more capable LLM assistants that can tackle real-world power grid optimization problems.

Solving topological grid puzzles requires reasoning over global spatial invariants such as connectivity, loop closure, and region symmetry and remains challenging for even the most powerful large language models (LLMs). To study these abilities under controlled settings, we introduce TopoBench, a benchmark of six puzzle families across three difficulty levels. We evaluate strong reasoning LLMs on TopoBench and find that even frontier models solve fewer than one quarter of hard instances, with two families nearly unsolved. To investigate whether these failures stem from reasoning limitations or from difficulty extracting and maintaining spatial constraints, we annotate 750 chain of thought traces with an error taxonomy that surfaces four candidate causal failure modes, then test them with targeted interventions simulating each error type. These interventions show that certain error patterns like premature commitment and constraint forgetting have a direct impact on the ability to solve the puzzle, while repeated reasoning is a benign effect of search. Finally we study mitigation strategies including prompt guidance, cell-aligned grid representations and tool-based constraint checking, finding that the bottleneck lies in extracting constraints from spatial representations and not in reasoning over them. Code and data are available at github.com/mayug/topobench-benchmark.

Large language models are increasingly applied to operational decision-making where the underlying structure is constrained optimization. Existing benchmarks evaluate whether LLMs can formulate optimization problems as solver code, but leave open a complementary question. Can LLMs directly produce correct solutions to fully specified constrained optimization problems without access to a solver? We introduce ConstraintBench, a benchmark for evaluating LLMs on direct constrained optimization across 10 operations research domains, with all ground-truth solutions verified by the Gurobi solver. Each task presents a natural-language scenario with entities, constraints, and an optimization objective; the model must return a structured solution that a deterministic verifier checks against every constraint and the solver-proven optimum. We evaluate six frontier models on 200 tasks and find that feasibility, not optimality, is the primary bottleneck. The best model achieves only 65.0% feasibility, yet feasible solutions average 89 to 96% of the Gurobi-optimal objective. No model exceeds 30.5% on joint feasibility and optimality within 0.1% of the solver reference. Per-domain analysis shows large variation in difficulty, with average feasibility spanning from 85.0% in the facility location domain to 0.8% in the crew assignment domain. Further, systematic failure modes include duration constraint misunderstanding, entity hallucination, and a feasibility-optimality decoupling in facility location and vehicle routing where models achieve high feasibility but 0% optimality. ConstraintBench and all evaluation infrastructure will be publicly released.

Large language models consistently fail the "car wash problem," a viral reasoning benchmark requiring implicit physical constraint inference. We present a variable isolation study (n=20 per condition, 6 conditions, 120 total trials) examining which prompt architecture layers in a production system enable correct reasoning. Using Claude 3.5 Sonnet with controlled hyperparameters (temperature 0.7, top_p 1.0), we find that the STAR (Situation-Task-Action-Result) reasoning framework alone raises accuracy from 0% to 85% (p=0.001, Fisher's exact test, odds ratio 13.22). Adding user profile context via vector database retrieval provides a further 10 percentage point gain, while RAG context contributes an additional 5 percentage points, achieving 100% accuracy in the full-stack condition. These results suggest that structured reasoning scaffolds -- specifically, forced goal articulation before inference -- matter substantially more than context injection for implicit constraint reasoning tasks.

Natural language understanding requires interleaving textual and logical reasoning, yet large language models often fail to perform such reasoning reliably. Existing neurosymbolic systems combine LLMs with solvers but remain limited to fully formalizable tasks such as math or program synthesis, leaving natural documents with only partial logical structure unaddressed. We introduce Logitext, a neurosymbolic language that represents documents as natural language text constraints (NLTCs), making partial logical structure explicit. We develop an algorithm that integrates LLM-based constraint evaluation with satisfiability modulo theory (SMT) solving, enabling joint textual-logical reasoning. Experiments on a new content moderation benchmark, together with LegalBench and Super-Natural Instructions, show that Logitext improves both accuracy and coverage. This work is the first that treats LLM-based reasoning as an SMT theory, extending neurosymbolic methods beyond fully formalizable domains.

Reasoning-specialized models like OpenAI's 5.1 and DeepSeek-V3.2 allocate substantial inference compute to extended chain-of-thought (CoT) traces, yet reasoning tokens incur significant costs. How do different reasoning modalities of code, natural language, hybrid, or none do perform under token constraints? We introduce a framework that constrains models to reason exclusively through code, comments, both, or neither, then systematically ablates token budgets to 10%, 30%, 50%, and 70% of optimal. We evaluate four frontier models (GPT-5.1, Gemini 3 Flash, DeepSeek-V3.2, Grok 4.1) across mathematical benchmarks (AIME, GSM8K, HMMT). Our findings reveal: (1) truncated reasoning can hurt as DeepSeek-V3.2 achieves 53% with no reasoning but only 17% with truncated CoT at 50% budget; (2) code degrades gracefully as Gemini's comments collapse to 0% while code maintains 43-47%; (3) hybrid reasoning underperforms single modalities; (4) robustness is model-dependent as Grok maintains 80-90% at 30% budget where OpenAI and DeepSeek collapse to 7-27%. These results suggest incomplete reasoning chains actively mislead models, with implications for deploying reasoning-specialized systems under resource constraints.