Paper deep dive

Do World Models Learn Global Understanding?

Alexander Detkov, Matt Thomson

Year: 2026Venue: arXiv preprintArea: cs.LGType: PreprintEmbeddings: 52

Intelligence

Status: not_run | Model: - | Prompt: - | Confidence: 0%

Entities (0)

No extracted entities yet.

Relation Signals (0)

No relation signals yet.

Cypher Suggestions (0)

No Cypher suggestions yet.

Abstract

Abstract:AI systems often feel brittle and fragmented. A large language model (LLM) may correctly explain a concept but fail to apply it, or follow safety instructions in one context but not another. This behavior suggests a general failure to lift local information to a global understanding. To gain fundamental insight, we frame "understanding" as learning constraints and propagating their consequences. We construct learning tasks on monoid worlds, sets of states connected by action transitions, where observed training transitions and an unseen constraint jointly determine held-out transitions. Measuring generalization tests whether models can learn global constraints from local transitions and propagate their consequences. We consider inverse, commutativity, composition, and periodicity constraints relevant to spatial and semantic structure. Across attention, recurrent, and state-space architectures, next-state training fits the data but fails to propagate non-trivial constraints. Compositional training, which uses identical paths but hides intermediate states from the input, achieves 96% accuracy on inverse, commutativity, and composition constraints across architectures, yields corresponding improvements in geometric generalization of world models trained on embodied environments and relational generalization in Wikidata-finetuned LLMs. How far do models propagate constraints when inferring an unseen fact may depend on first inferring others? We define proof depth d of a held-out transition, measuring the minimum number of inference rounds to infer the transition, and find that model generalization decreases sharply with proof depth. Increasing compositional path length T improves generalization. These results provide a formal way to investigate global understanding in language and world models and demonstrate that compositional training promotes information propagation and integration.

PDF

Open source PDF →Open local PDF →

Trouble viewing inline? Open PDF directly →

Full Text

49,165 characters extracted from source content.

Expand or collapse full text

Preprint. Under review. DO WORLD MODELS LEARN GLOBAL UNDERSTANDING? Alexander Detkov & Matt Thomson California Institute of Technology Pasadena, CA 91125, USA adetkov,matt@caltech.edu ABSTRACT AI systems often feel frustratingly brittle and fragmented. A large language model (LLM) may correctly explain a concept but fail to apply it, or follow safety instruc- tions in one context but not another. This behavior suggests a general failure to lift local information to a global understanding. To gain fundamental insight, we frame “understanding” as learning constraints and propagating their consequences. We construct learning tasks on monoid worlds, sets of states connected by action transitions(state, action, next state), where observed training transitions and an underlying unseen constraint jointly determine held-out transitions. Measuring generalization tests whether models can learn global constraints from local tran- sitions and propagate their consequences. We consider inverse, commutativity, composition, and periodicity constraints relevant to spatial and semantic struc- ture. Across attention, recurrent, and state-space architectures, next-state training on paths of observed transitions fits the data but fails to propagate non-trivial constraints. Compositional training, which uses identical paths but hides intermedi- ate states from the input, achieves96%accuracy on inverse, commutativity, and composition constraints across architectures, yields corresponding improvements in geometric generalization of world models trained on embodied environments (+42%on propagating commutative constraints) and relational generalization in Wikidata-finetuned LLMs (+73%on composition propagation). How far do models propagate constraints when inferring an unseen fact may depend on first inferring others? We define proof depthdof a held-out transition, measuring the minimum number of inference rounds to infer the transition, and find that model generalization decreases sharply with increasing proof depth. A compositionally trained transformer on lengthT = 2paths generalizes at95%for transitions derivable directly from observations (d = 1) but only at49%for those requiring an additional inference round (d = 2). Increasing compositional path lengthT improves generalization, andT = 4compositional models achieve95%accuracy ond = 2transitions. These results provide a formal way to investigate global understanding in language and world models and demonstrate that compositional training promotes information propagation and integration. 1INTRODUCTION What does it mean for a system to understand? Do language models understand semantics when training on “Tom’s parent is Mary” fails to teach “Mary’s child is Tom” (Berglund et al., 2024)? Do world models understand space if a left rotation followed by a right rotation produces a new scene (Kwon et al., 2026)? In both cases, a constraint (parent and child are converses; left and right rotations are inverses) lets a few seen facts determine many unseen ones. This suggests framing “understanding” as learning constraints and propagating their consequences globally. To study constraint learning and propagation systematically, we construct monoid worlds where constraints are controllable and propagation consequences can be mathematically determined. A monoid world is a set of states with actions defining transitions between states. In the context of space, the set could be points on a line, and the actions,left/right, could correspond to opposite translations. These spatial actions satisfy an inverse constraint: moving left and then right returns you to the original state. Analogously, semantic relations obey constraints like composition: the parent of 1 arXiv:2609.34058v1 [cs.LG] 28 Sep 2026 Preprint. Under review. a parent is a grandparent. We capture these constraints as equalities over action sequences, enabling us to determine from data, manipulate, and directly specify a large class of constraints governing each world. Since our world is precisely defined, we can directly evaluate whether models propagate constraints by constructing held-out sets of transitions which are uniquely determined (semantically entailed) by joint knowledge of the constraint and observed transitions. For example, observing(1, left, 2) entails the unseen transition(2, right, 1)when the constraintleft = right −1 is learned. Generalization determines whether models learn constraints from data and propagate their consequences. Not all consequences of constraints are created equal, and some require multiple inference steps. Consider learning a new fact which sets in motion a chain of factual updates. We frame this formally by defining a miniature proof system and measure the inference complexity of an unseen transition by its proof depthd: the minimum number of rounds in our system to derive the transition. We construct worlds with unseen transitions with varied proof depths and explore how generalization and training dynamics change with the required inference depth. Next-state training on paths of monoid transitions fails to propagate non-trivial constraints across transformer, recurrent, and state-space models. Compositional training, learning action compositions by hiding intermediate state inputs, improves generalization across worlds at an average accuracy of 96%on inverse, commutative, and composition constraints. We apply our monoid framing to study constraint propagation in learning visual environments with world models and relational structure with pretrained large language models (LLMs). We find traditional next-state-style objectives poorly propagate geometric and semantic constraints, while compositional training improves generalization by+42%on commutative constraints and+73%on composition constraints. Finally, we find that generalization decreases sharply with proof depthd, implying that models struggle to iteratively apply constraints. Increasing compositional path length T , however, improves deeper propagation. 2A MONOID FRAMEWORK FOR GLOBAL CONSTRAINTS We construct worlds governed by specified constraints and hold out transitions which are jointly implied by observed transitions and associated constraints. We train models on paths of observed transitions and evaluate them on held-out transitions, measuring a model’s ability to learn global constraints from local data and propagate their consequences. 2.1MONOID WORLD AND GLOBAL CONSTRAINTS A monoid world is a finite set of statesS := [N] := 0,...,N − 1with actionsA := a 0 ,...,a n−1 , each a functiona :S →Son the state space. Consider the world shown in Figure 1a, which has a discrete90 ◦ rotation interpretation with statesS := [4]and actionsL : x7→ x− 1 mod 4andR : x7→ x + 1 mod 4. Actions can be composed to form action sequences, and with the identity action1, form a monoid. Local observations are transitions(s,a,a(s)), which correspond to the change of state s∈S under the action a∈A. In our rotation world, (0,R, 1) is a transition. The global constraints of our monoid world are equality relations between action sequences: two distinct sequences of actions induce the same map on the entire state space. In other words, at every initial state, the paths of transitions for the two action sequences converge to the same terminal state. The global nature comes from the requirement that the constraint on transitions applies uniformly over the entire state space. In our rotation world, we have two essential constraints: inverseLR = RL = 1 and periodicity L 4 = 1, from which all other constraints are derived. When building a monoid world with a specified constraint, we choose a state-space sizeNand randomly sample action transitionsa∈Asuch that the constraint is satisfied. For example, when constructing inverse worldsLR = RL = 1as in Figure 1a, we chooseL : S → Sas a random bijection and deriveRas its inverse. Monoid theory warns us that finite worlds may inadvertently introduce unwanted constraints, so we audit all our worlds by enumeratingu = vglobal constraints for|u| + |v| ≤ 8and reject impure worlds. We explore inverse, commutativity, composition, periodicity, and copy constraints shown as a presentation,⟨Actions| Constraints⟩, in Figure 2. 2 Preprint. Under review. States S= © 0;1;2;3 ª Seen (train) Unseen (test) Global constraint = id S Sample path ( ,,,, ) Unseen transition derivable with constraint. = id S Trained on random paths of seen transitions, tested on unseen. Next-state (T=2) Model Compositional (T=2) Model Evaluation Model True Cross-entropy Prediction Accuracy 123456 Composition length (T) 0.0 0.2 0.4 0.6 0.8 1.0 Unseen transition accuracy Perfect constraint propagation Random guess Next-state 1) Compositional 1001k10k100k Training steps 0.0 0.5 1.0 Transition accuracy Grokking 13£ later Seen Unseen T= 2 2) Figure 1: Toy instantiation of an inverse monoid world with training and evaluation setup. a) A four-state world with actionsLandRsatisfying inverse constraintLR = RL = 1, seen and unseen transitions, and sample training path. b) Constraint propagation uses seen transitions and the constraint to infer unseen transitions. c) Next-state training shows intermediate states, while compositional training does not. Evaluation is generalization over unseen transitions. d) 1) Next-state training fails to propagate inverse constraint, while compositional training succeeds (≥ 93%). 2) Training curves for T = 2 show sharp grokking-like generalization. Plots averaged over 5 seeds. 2.2EVALUATION OF CONSTRAINT PROPAGATION BY ENTAILED HOLD-OUTS We hold out transitions which are jointly inferable from observed training transitions and the world’s global constraint. Formally, inferability is captured by semantic entailment. A transitiont = (s,a,s ′ ) is semantically entailed by constraintCand seen transitionsTif and only if all possible assignments of transitions that satisfyCand agree onTalso agree ont. Consider the cyclic inverse world in Figure 1a. The unseen transitiont = (1,R, 2)is semantically entailed since any world that contains the seen transition (2,L, 1)∈T and the inverse constraint must agree on t. In practice, we perform greedy sampling with semantic entailment checks until we reach a specified hold-out size. 2.3NEXT-STATE AND COMPOSITIONAL TRAINING We train models on paths of (T) transitions,(s 0 ,a 0 ,s 1 ,...,a T−1 ,s T ), wheres t+1 = a t (s t ). At each step we sample uniformly from training transitions originating from the current state. Every transition in the training path belongs to the training set of transitions (no held-out transition is ever shown). Models receive only paths of observed local transitions, never the global constraint directly. Under next-state training, we show intermediate states as inputs and for steps1 ≤ t ≤ Ttrain to predict s t given (s 0 ,a 0 ,s 1 ,...,s t−1 ,a t−1 ) as input (Figure 1c). Under compositional training, we hide intermediate states from the inputs and for1≤ t≤ Ttrain to predicts t given(s 0 ,a 0 ,a 1 ,...,a t−1 )as input, forcing the model to compose actions internally with T controlling the composition length. At T = 1, next-state and compositional training coincide. 3 Preprint. Under review. 2.4MEASURING INFERENCE COMPLEXITY OF A TRANSITION WITH PROOF DEPTH Certain held-out transitions are intuitively harder to infer than others. In the commutative world of Figure 4, both held-out transitions are semantically entailed, but only one can be immediately derived from observed transitions; the other requires first inferring an intermediate transition. We quantify this dependency by proof depth, defined relative to a fixed set of inference rules based on the world’s algebraic constraints. Consider the rules of the commutative world depicted in Figure 4b. Knowing transitions(1,F, 0), (0,R, 2), and(1,R, 3)derives(3,F, 2)since actionsFandRcommute, so the paths1 F −→ 0 R −→ 2and1 R −→ 3 F −→ 2must end at the same state. Constraint cascading then happens in rounds, each round using the derived transitions of previous rounds, then inference rules infer transitions (Figure 4b). The minimal number of rounds to infer a transition is its proof depth. 3CONSTRAINT PROPAGATION IN CONTROLLED MONOID WORLDS We construct monoid worlds withN = 1024states satisfying specified constraints and train models on95%of state-action transitions(s,a,a(s))such that the remaining5%of held-out transitions, which we evaluate model generalization on, are entailed by seen transitions and the world’s constraint. Unless stated otherwise, models have four layers of width 64 and train for150k steps atT = 2with reported results averaged over five seeds. 3.1NEXT-STATE TRAINING FAILS TO PROPAGATE CONSTRAINTS Under a next-state objective, where intermediate states are shown, transformers on inverse worlds fail to generalize with accuracy at chance (Figure 1d.1). Scaling model, world, and compute does not save next-state training, and models continue to fail at propagating inverse constraints (Appendix A). Across attention, recurrent, and state-space architectures, next-state trainingT = 2fails, indistin- guishable from chance, at propagating inverse, commutative, composition, and periodicity constraints in their associated worlds (Figure 2). Only in the trivial copy world where actions are identical (A = B) does next-state training generalize. We find a systematic “copy” failure mode of general constraint propagation by next-state trained models, where models behave as if every world were the copy world. Averaged across architectures and worlds,64%of incorrectly inferred unseen transitions are explained by copy propagation, assuming the unseen(s,A,A(s))transition matches seen(s,B,B(s)), although seen transitions contradict theA = Bconstraint at almost every state (Appendix B). These results suggest the next-state objective poorly propagates constraints from data. 3.2COMPOSITIONAL TRAINING IMPROVES CONSTRAINT PROPAGATION Under a compositional objective, where intermediate states are hidden, transformers successfully propagate inverse constraints, generalizing unseen transitions with≥ 93%accuracy (Figure 1d.1). We find a sharp jump in generalization of98%when varying the composition length fromT = 1, a reduction to next-state training, to T = 2, when training first includes two-action compositions. Across architectures, compositional training at T = 2 enables models to propagate inverse, commu- tativity, and composition constraints with an average accuracy of96%and copy at100%(Figure 2). Extending composition length toT = 4yields nearly identical generalization results of96% averaged over the same worlds (Appendix D). Periodicity constraints, however, are not propagated by next-state or compositional training atT = 2 or 4. Across periodicity worldsA k = 1fork = 3, 4, 5, compositionally trained transformers at1≤ T ≤ 8all fail to generalize, suggesting that the failure is not due to training paths too short to contain the constraint. Compositionally trained models also have reduced “copy” failure modes, hallucinating a copy constraint for36%of failures (Appendix B). Compositional training fits observed transitions an order of magnitude earlier than it recovers held-out ones. On the inverse world, transformers learn100%of seen transitions at1.2k steps but achieve50% generalization at16k, approximately13times as many training steps, evidence of a grokking-like delay (Power et al., 2022). Finally, generalization accuracy, fixing world size, increases monotonically with “constraint evidence”, the number of seen transitions satisfying the constraint (Appendix C). 4 Preprint. Under review. Transformer GRU LSTM SSM Transformer GRU LSTM SSM Transformer GRU LSTM SSM Transformer GRU LSTM SSM Transformer GRU LSTM SSM 0.0 0.2 0.4 0.6 0.8 1.0 Unseen transition accuracy Inverse L;RjLR= RL= 1 ® Commutativity F;RjFR= RF ® Composition A;B;CjAB= C ® Periodicity A;BjA 4 = 1 ® Copy A;BjA= B ® Compositional training improves constraint learning and propagation Perfect constraint propagation Random guess Only compositional learnsBoth failBoth learn Next-state Compositional (Both T=2) Figure 2: Across monoid worlds and architectures, compositional training generalizes better than next-state training with equal optimizer steps. Compositional objective propagates inverse, commutative, and composition constraints with average accuracy of96%, and copy at100%. Next- state training only succeeds on copy and fails on other tested families. Neither propagates periodicity. Points averaged over 5 seeds and bars show standard deviations. 4CONSTRAINT PROPAGATION IN VISION AND LANGUAGE We explore learning and propagating constraints towards geometric generalization in world models trained on visual environments and relational generalization in LLMs trained on knowledge-based tasks. We identify domain-relevant constraints, which often hold only approximately, and build entailed hold-outs which test whether models propagate the constraints. 4.1GEOMETRIC CONSTRAINTS IN VISUAL WORLD MODELS An agent moves (L)eft/(R)ight/(F)orward/(B)ackward through an AI2-THOR virtual environment, generating a dataset of observation-action paths of lengthT = 2(Figure 3a.1) (Kolve et al., 2022). Euclidean geometry imposes constraints on motion requiring that the final destination be independent of the order of translations: stepping forward and then right or right and then forward takes you to the same endpoint. Actions are also invertible since you can undo a previous action by stepping in the opposite direction. In environments with object collisions, these constraints do not hold for all states. We can compute these approximate constraints by identifying paths which start from the same initial observation and recording whether certain action sequences converge to the same endpoint observation. If two action sequences frequently (≥ 50%) map between the same initial and terminal observations, we declare them an approximate constraint. We find the Euclidean constraints: inverse LR = 1 and commutativity FR = RF , which hold roughly 80% of the time. We hold out transitions(obs, action, next obs)that test propagation of inverse and commutative constraints. These hold-outs are taken at states away from object collisions where constraints are valid. Transformer world models are trained to predict the next observation given previous observations and actions. Accuracy is measured by choosing the observation in the dataset with the highest predicted likelihood. Since visual similarity may enable held-out recovery without learning geometric constraints, we compare to a baseline of shifting observation pixels. Under next-observation training, where intermediate observations are shown, world models poorly generalize inverse and commutative constraints, with an accuracy of only50%and51%, respectively, compared to the pixel-shift baseline of41%and44%(Figure 3a.2, 3a.3). Trained compositionally, hiding intermediate observations, world models achieve83%accuracy on inverse and93%on commutative constraints, averaging a +38% improvement over next-observation trained models. To examine recovery without visual shortcuts, we train transformers to predict the next state directly, instead of its observation. Generalization for next-state trained models on inverse and commuta- tive constraints collapses to random guessing, while compositionally trained models achieve94% and98%, respectively (Figure 3a.3). These results suggest next-observation/state training poorly 5 Preprint. Under review. 1. World States Room views Actions Movement Constraints (≈) Inverse Commutativity States Entities Actions (multi-valued) Relations Constraints Inverse Converse Composition 2. Training and evaluation 3. Results Next-observation training (T=2) Compositional training (T=2) Evaluation: inverse ? * Trained with next-token prediction Next-state phrasing (T=2)* Compositional phrasing (T=2)* Evaluation: composition Dana’s parent is Ines, whose parent is Vera. Dana’s parent’s parent is Vera. Dana’s grandparent is ? Inverse Commutative Training StateObservationStateObservation Baseline (pixel-shift) – 41% – 44% Next-observation 0±0%50±2%0±0%51±2% Compositional94±1%83±3%98±1%93±2% Model Training Inverse ConverseComposition Gemma 270M Next-state 0%0%0% Compositional55%1%68% Qwen 600M Next-state 0%0%0% Compositional11%0%79% Gemma 1B Next-state 3%1%0% Compositional61%0%71% Gemma 4B Next-state 3%0%0% Compositional53%3%72% AdaDana BramInes OttoVera grandparent child parent Figure 3: Compositional objectives improve constraint propagation over next-state-style training in vision and language a) 1) AI2-THOR virtual 3D environment, where an embodied agent translates between observed states. InverseLR = RL = 1and commutativeFR = RFconstraints hold approximately due to collisions. 2) Training and evaluation follow Figure 1c with image observations replacing abstract states. 3) Next-observation training marginally outperforms pixel-shifting baseline (+8%) while compositional training improves by (+46%) (observation). Training to predict state, minimizing visual cues, collapses next-state generalization to chance, while compositional training generalizes with96%(state). Results averaged over three seeds. b) 1) Wikidata knowledge graph world, where entities are states and relations are multi-valued actions. 2) LLMs are finetuned with next-token prediction, the objective changing the sentence phrasing. 3) Compositional, not next-state, phrasing propagates inverse constraint at average accuracy of 45% and composition at 73%. propagates inverse and commutative constraints in visual domains while compositional training successfully propagates geometric constraints even without visual cues. 4.2RELATIONAL CONSTRAINTS IN LARGE LANGUAGE MODELS We examine when LLMs, trained on facts, propagate relational constraints without premises supplied in context. We study inverse constraints between successor and predecessor, the converse relation between parent and child, and the composition of parent and grandparent. Unlike inverse relations, converse relations branch, and the child of a parent may be a sibling, not necessarily a strict identity. For each constraint, we extract a Wikidata knowledge graph of512renamed individuals and associated relations (Figure 3b.1). We hold out transitions(person A, relation, person B)jointly entailed by training transitions and constraint. For example, the observed relationship(Dana, parent, Ines)entails the held-out relationship (Ines, child, Dana) under the converse constraint. 6 Preprint. Under review. We finetune pretrained base LLMs (270M to 4B parameters) with next-token prediction on natural- language sentences generated from (T=2) training paths. We compare two phrasings the same underlying path. Next-state phrasing allows models to condition on intermediate entities, e.g., “Dana’s parent is Ines, whose parent is Vera”, while compositional phrasing omits intermediate entities, “Dana’s parent’s parent is Vera” (Figure 3b.2). Unlike previous compositional objectives, next-token prediction over compositional phrasing never uses intermediate entities as targets, so we include (T=1) paths in both conditions. We evaluate on a held-out relation such as(Ines, child, Dana) by prompting “Ines’s child is” and choosing the highest-likelihood name from the graph. For relations with multiple correct answers (converse and grandparent), we report filtered accuracy where all other correct answers (e.g., Dana’s siblings) are excluded from the likelihood comparison. Training with next-state phrasing fails to propagate inverse and composition constraints with an average generalization of1%, marginally better than random guessing (Figure 3b.3). Composi- tional phrasing improves inverse propagation with an average accuracy of45%and composition generalization at 73%. Converse is not propagated by next-state or compositional methods. Composition learning over paths presents a shortcut: while “Dana’s grandparent is Vera” is never shown during training, models see theT = 2path “Dana’s parent is Ines, whose parent is Vera” under next-state phrasing and “Dana’s parent’s parent is Vera” under compositional phrasing. Can propagation still occur when such shortcut paths are removed? We train a transformer onT = 2 paths in the monoid composition worldN = 1024with and without shortcuts (Appendix F). Next-state trained models fail in both settings, while compositionally trained models still recover without shortcuts (50%versus92%with shortcuts). These results suggest that while shortcuts help compositionally trained models recover, they are not required for composition propagation. To understand the effect of branching on converse constraint propagation, we construct anN = 1024 branching world that interpolates between lineβ = 0, where parent and child are true inverses, and binary treeβ = 1, where each parent has two children and only a converse relationship remains (Appendix E). For comparison, parents in our Wikidata world have an average of2.5children. A transformer trained compositionally withT = 2achieves99%onβ = 0, but adding only a10% chance a parent might have another child (β = 0.1) drops accuracy to 63% and≤ 3% for β ≥ 0.75. Compositional phrasing improves inverse and composition constraint propagation but is limited in converse generalization on branching worlds. 5HOW FAR DO CONSTRAINTS PROPAGATE? We measure an unseen transition’s derivation complexity by proof depth and construct commutative monoid worlds of sizeN = 1024with10%of unseen transitions spread nearly uniformly across depthsd = 1 to 8(Figure 4a). Previous results concerned onlyd = 1generalization. We now explore the effect of proof depth on constraint propagation by compositionally trained transformers. We train with composition lengthsT = 1, 2, 3, 4(T = 1is next-state training) for1.5M steps over five seeds. 5.1CONSTRAINT PROPAGATION DECREASES SHARPLY WITH PROOF DEPTH We find generalization of compositionally trained models decreases sharply with proof depth, while next-state trained models do not generalize at any proof depth. A transformer trained with composition lengthT = 2recoversd = 1transitions, whose derivation depends directly on seen transitions, at 95%, whiled = 2recovery, transitions whose derivation also depends on unseend = 1transitions, is only at49%. The steep decreasing trend in generalization continues to deeper transitions withd = 3 at15%,d = 4at5%, andd = 5onward indistinguishable from random chance. Similar step-to-step accuracy ratios over depths1 ≤ d ≤ 4suggest approximate geometric decay with rate0.4± 0.1. Even with compositional training, models struggle to propagate constraints deeply. 5.2PROPAGATION DYNAMICS SUGGEST INTERNAL CONSTRAINT CASCADES Models trained compositionally propagate constraints to infer transitions iteratively by proof depth: firstd = 1transitions are learned, thend = 2,d = 3, and so on (Figure 4d). Transformers trained with composition lengthT = 4reach half their final accuracy ford = 1at26k training steps,d = 2 at50k steps,d = 3at102k, andd = 4at236k, so each depth takes about twice as many steps as the 7 Preprint. Under review. d= 1d= 2 Constraint = Seen (train) Unseen (test) Each step derives unseen transition with constraint and proof depth d counts the steps. d = 1 = d = 2 = 12345678 Proof depth (d) 0.0 0.2 0.4 0.6 0.8 1.0 Unseen transition accuracy Perfect propagation T = 4 T = 3 T = 2 Next-state (T = 1) 10k100k1M Training steps 0.0 0.2 0.4 0.6 0.8 1.0 Unseen transition accuracy d = 1 d = 2 d = 3 d = 4 d = 5 d = 6 d = 7 d = 8 T= 4 Figure 4: Section of a commutative monoid world with unseen transitions of proof depthsd = 1 andd = 2. Proof depth measures propagation complexity and predicts generalization and training dynamics. a) Section of a larger commutative world containing two unseen transitions. b) A constraint cascade iteratively applies a simple derivation rule (here, commutative equations) to infer unseen transitions. Proof depth is the minimal number of iterations. Thed = 1transition is a premise for derivingd = 2c) Generalization decreases geometrically with proof depth, but increasing composition length improves propagation to higher depths. d) Compositional training propagates constraints sequentially in proof-depth order, suggesting an internal constraint cascade. one before (2.1×, geometric mean). Monotonic ordering holds for all1≤ d≤ 8, but spacing beyond d = 4 shrinks, and d = 5 to d = 8 all reach half their final accuracy between 364k and 376k steps. Timing suggests that models may be propagating constraints in a qualitatively similar way to constraint cascades, where local rules are applied repeatedly to derive held-out transitions in order of proof depth d. Unlike general constraint propagation, a cascade predicts that transitions are inferred only after all their derivation premises, the other unseen transitions required for the derivation, are inferred (Figure 4b). We examine150checkpoints for each of5seeds trained over composition lengthsT = 2, 3, 4on the commutative world to find the percentage of inferred transitions missing their derivation premises. To remove timing confounds (on averaged = n + 1is learned afterd = n), we compare to a null which shuffles the identity ofd = ntransitions, preserving the timing but removing any dependency structure. We find that only3%of correct inferences made by the transformer were made without having correctly inferred their derivation premises, compared to11%for the null model, a difference of3.7×. Throughout training, the model’s correct held-out predictions are almost always derivable from seen transitions and other predictions, satisfying the constraint cascade condition. 5.3LONGER COMPOSITION LENGTH ENABLES DEEPER PROPAGATION Training on longer composition lengthsTwith the same number of training steps improves gener- alization at higher proof depths (Figure 4c). IncreasingTfrom2to3maintainsd = 1recovery at ≥ 95%but significantly improvesd = 2by+43%and similarly improves recovery of3 ≤ d ≤ 8 with an average increase of+19%. Raising composition lengthTfrom3to4maintainsd = 1and d = 2recovery while again accumulating improvement for3≤ d≤ 8overT = 3by+14%. Decay curves also change qualitatively, from the geometric-like decay ofT = 2, which is indistinguishable from random guessing byd = 5, to plateau-like behavior atT = 3andT = 4, where generalization 8 Preprint. Under review. appears to flatten out at roughly10%and20%, respectively. These results suggest training on longer composition lengths may be key to deep propagation of constraints. 6DISCUSSION Identical data, learned under a different objective, can turn a model that fails to generalize into one that learns and propagates global constraints. Across attention, recurrent, and state-space architectures, compositionally trained models propagate non-trivial constraints to recover unseen data while next- state trained models do not. Compositional training also shows superior propagation in vision and language domains. Both objectives use identical path data, the only difference is that compositional training hides intermediate states. Global understanding therefore depends on how training requires local observations to be combined. Providing intermediate states encourages models to solve the task locally, while compositional training encourages integration across actions. Training objective, perhaps more than architecture or task domain, determines whether a model propagates certain constraints. Across monoid worlds, constraints that are (not) propagated are con- sistently (not) propagated across the disparate architectures tested. Similarly, constraints propagated compositionally in monoid worlds are also propagated by compositional approaches in visual and language domains. This pattern suggests that controlled monoid worlds capture general learning biases that extend to richer tasks and may provide simplified environments for understanding and designing better-propagating algorithms. Even when models propagate constraints, propagation is shallow, with generalization decreasing sharply with proof depth. Higher proof depth transitions are also recovered later and more slowly. Together with training dynamics, this suggests an internal process resembling a slow, inaccurate constraint cascade. Increasing composition length improves generalization to deeper consequences, suggesting that propagation reach depends on the organization of local information it learns from. Propagating constraint consequences deeply is essential for consistent information integration. 7RELATED WORK LLMs often learn facts without propagating their consequences. The reversal curse shows that training on one direction of a relation can fail to recover its reverse (Berglund et al., 2024). Similar failures exist in implicit reasoning, where models poorly compose relations (Balesni et al., 2025). Changes to training can address individual failures such as reversal (Ma et al., 2026) and composition (Feng et al., 2025) have been proposed, but remain specialized. We frame these failures and improvements in the context of learning and propagating constraints. Across controlled monoid worlds, visual environments, and knowledge-based language tasks, we study how objective determines which constraints models propagate. We further measure how far consequences are propagated using proof depth and explore associated training dynamics. Accurate local predictions do not imply coherent global understanding (Vafa et al., 2024). Previous work improves consistency through reverse-prediction training (Kwon et al., 2026) or architectural modifications that recover group structures in synthetic environments (Keurti et al., 2023). Our constraint propagation framework extends beyond group structure to include semantic constraints such as the converse relation between parent and child. Training without intermediate states has improved generalization in knowledge-graph embedding models (Guu et al., 2015) and discouraged local prediction shortcuts in planning (Bachmann & Nagarajan, 2024). We study how training objective affects constraint propagation across architectures, visual environments, and language tasks. Compositional training improves propagation for many geometric and semantic constraints but fails on others. Even when immediate consequences are recovered, generalization decreases sharply with proof depth. Composition length increases propa- gation reach to higher proof depths, suggesting that how much local information models combine during training shapes how far they propagate what they learn. 9 Preprint. Under review. 8CONCLUSION We frame global understanding as learning constraints from local observations and propagating their consequences deeply. Our monoid worlds provide controlled environments where generalization is entailed by observed data and unobserved global constraints. Across architectures, next-state training fails to propagate non-trivial constraints, while compositional training substantially improves propagation. We apply our constraint framing to visual and language tasks and find corresponding improvements under compositional-style objectives as well as regularities in which constraints propa- gate and under what conditions across domains. Deep propagation remains limited, though longer composition lengths help. These results suggest that global understanding requires training methods that encourage models to integrate local observations and deeply propagate their consequences. 10 Preprint. Under review. AI USE STATEMENT In this work, we used generative AI tools to implement methods, including code that generates the vision and language worlds, extracts the Wikidata graphs, trains and evaluates the models, and organize our code base. Additionally, we used generative AI tools for improving readability, suggest improvements, and proofread drafts. We have reviewed all AI-assisted work. We reviewed and tested all AI-written code and reviewed every AI-suggested edit to the text. We take responsibility for the final content of this work, including text, claims or artifacts produced with the aid of generative AI. REFERENCES Gregor Bachmann and Vaishnavh Nagarajan. The pitfalls of next-token prediction. In Ruslan Salakhutdinov, Zico Kolter, Katherine Heller, Adrian Weller, Nuria Oliver, Jonathan Scarlett, and Felix Berkenkamp (eds.), Proceedings of the 41st International Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research, p. 2296–2318. PMLR, 21–27 Jul 2024. URL https://proceedings.mlr.press/v235/bachmann24a.html. Mikita Balesni, Tomek Korbak, and Owain Evans. Lessons from studying two-hop latent reasoning, 2025. URL https://arxiv.org/abs/2411.16353. Lukas Berglund, Meg Tong, Maximilian Kaufmann, Mikita Balesni, Asa Stickland, Tomek Korbak, and Owain Evans.The reversal curse:LLMs trained on “A is B” fail to learn “B is A”.In International Conference on Learning Representations, 2024. URLhttps://proceedings.iclr.c/paper_files/paper/2024/hash/ 5178b2f2d7c44a390c0777dc77b3f0c-Abstract-Conference.html. Jiahai Feng, Stuart Russell, and Jacob Steinhardt. Extractive structures learned in pretraining enable generalization on finetuned facts. In Aarti Singh, Maryam Fazel, Daniel Hsu, Simon Lacoste-Julien, Felix Berkenkamp, Tegan Maharaj, Kiri Wagstaff, and Jerry Zhu (eds.), Proceedings of the 42nd International Conference on Machine Learning, volume 267 of Proceedings of Machine Learning Research, p. 16853–16883. PMLR, 13–19 Jul 2025. URLhttps://proceedings.mlr. press/v267/feng25m.html. Kelvin Guu, John Miller, and Percy Liang. Traversing knowledge graphs in vector space. In Llu ́ ıs M ` arquez, Chris Callison-Burch, and Jian Su (eds.), Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, p. 318–327, Lisbon, Portugal, September 2015. Association for Computational Linguistics. doi: 10.18653/v1/D15-1038. URLhttps: //aclanthology.org/D15-1038/. Hamza Keurti, Hsiao-Ru Pan, Michel Besserve, Benjamin F Grewe, and Bernhard Sch ̈ olkopf. Ho- momorphism AutoEncoder – learning group structured representations from observed transitions. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett (eds.), Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, p. 16190–16215. PMLR, 23–29 Jul 2023. URL https://proceedings.mlr.press/v202/keurti23a.html. Eric Kolve, Roozbeh Mottaghi, Winson Han, Eli VanderBilt, Luca Weihs, Alvaro Herrasti, Matt Deitke, Kiana Ehsani, Daniel Gordon, Yuke Zhu, Aniruddha Kembhavi, Abhinav Gupta, and Ali Farhadi. AI2-THOR: An interactive 3D environment for visual AI, 2022. URLhttps: //arxiv.org/abs/1712.05474. Soonwoo Kwon, Jin-Young Kim, Hyojun Go, and Kyungjune Baek. Toward stable world models: Measuring and addressing world instability in generative environments. Pattern Recognition, 177: 113351, 2026. doi: 10.1016/j.patcog.2026.113351. URLhttps://doi.org/10.1016/j. patcog.2026.113351. Xutao Ma, Yixiao Huang, Hanlin Zhu, and Somayeh Sojoudi. Breaking the reversal curse in autoregressive language models via identity bridge. In Forty-third International Conference on Machine Learning, 2026. URL https://openreview.net/forum?id=kKUrbW8xJ3. 11 Preprint. Under review. Alethea Power, Yuri Burda, Harri Edwards, Igor Babuschkin, and Vedant Misra. Grokking: Gener- alization beyond overfitting on small algorithmic datasets. CoRR, abs/2201.02177, 2022. URL https://arxiv.org/abs/2201.02177. Keyon Vafa, Justin Y. Chen, Ashesh Rambachan, Jon Kleinberg, and Sendhil Mullainathan. Evaluating the world model implicit in a generative model.In Advances in Neural In- formation Processing Systems, volume 37, p. 26941–26975, 2024.doi:10.52202/ 079017-0846. URLhttps://papers.nips.c/paper_files/paper/2024/hash/ 2f6a6317bada76b26a4f61b70a7db59-Abstract-Conference.html. 12 Preprint. Under review. ASCALING MODEL, WORLD, AND COMPUTE UNDER NEXT-STATE TRAINING DOES NOT ENABLE CONSTRAINT PROPAGATION We explore whether scaling a transformer under next-state training can enable constraint propagation on the inverse world by individually scaling model width from64to320(0.33M to5.6M parameters), world size from1024to8192states, and training time from15k to300k steps. We find that every next-state trained model fits the observed transitions but fails to recover held-out transitions (all at random floor). The compositionally trained control does generalize, achieving97%on held-out inverse transitions. Table 1: The next-state objective does not propagate constraints even after scaling model, world, and compute. Transformers trained on inverse world atT = 2. Every row is next-state training except the last, which trains the same world, training paths, and model but compositionally. Accuracies are average± standard deviation over five seeds. WidthParameters (M)States (N )StepsSeen (%)Unseen (%) Model size640.331024150k100.00.4± 0.5 1281.051024150k100.00.4± 0.5 2563.671024150k100.00.2± 0.4 3205.571024150k100.00.4± 0.5 World size3205.571024150k100.00.4± 0.5 3207.544096150k100.00.1± 0.1 32010.168192150k100.00.1± 0.1 Training time32010.16819215k100.00.1± 0.1 32010.16819275k100.00.1± 0.1 32010.168192150k100.00.1± 0.1 32010.168192300k100.00.1± 0.1 Compositional control32010.168192150k100.097.0± 0.5 BMODELS INCORRECTLY PROPAGATE COPY CONSTRAINTS IN CONFLICT WITH SEEN TRANSITIONS Transformer GRU LSTM SSM Transformer GRU LSTM SSM Transformer GRU LSTM SSM Transformer GRU LSTM SSM 0.0 0.2 0.4 0.6 0.8 1.0 Unseen errors that are another action's target Inverse L;RjLR= RL= 1 ® Commutativity F;RjFR= RF ® Composition A;B;CjAB= C ® Periodicity A;BjA 4 = 1 ® Models propagate a copy constraint the data contradicts Next-state Compositional (Both T=2) Figure 5: Models systematically fail by recovering unseen transitions with another action’s target. When models fail to recover an unseen transition(s,A,A(s)), they instead often predict B(s) for another action B. Points averaged over 5 seeds and bars show standard deviations. When models trained with either the next-state or the compositional objective fail to propagate a constraint, they fail in a systematic way. For an incorrectly answered unseen transition(s,A,A(s)), we count a copy when the model predictsB(s), the target of another actionBfrom the same state. 13 Preprint. Under review. Across inverse, commutative, composition, and periodicity worlds and all four architectures,64% of next-state errors are copies, compared with36%of compositional errors. Copy rate depends on architecture, with the highest rates (all next-state trained) being LSTMs at90%, GRUs at77%, and transformers at 58%. CCOMPOSITIONALLY TRAINED MODELS PROPAGATE CONSTRAINTS IN PROPORTION TO OBSERVED EVIDENCE We vary constraint evidence, the number of seen transitions satisfying the constraint, while fixing the world size. We train a transformer atT = 2over the composition worldN = 1024, varying the amount of held-outC-transitions from10%to97%(evidence levels of 922 and 31, respectively). Next-state trained models do not generalize, while compositionally trained models generalize mono- tonically with evidence:2%,4%,11%,24%,62%, and93%at evidence 31, 62, 128, 256, 512, and 922. 3162128256512922 Constraint evidence (seen instances of AB=C) 0.0 0.2 0.4 0.6 0.8 1.0 Unseen transition accuracy Perfect propagation Compositional Next-state Compositional propagation grows with constraint evidence Figure 6: Compositionally trained transformers recover unseen transitions in proportion to constraint evidence. Transformers trained on the composition worldN = 1024withT = 2paths. Points averaged over 5 seeds and bars show standard deviations. DCOMPOSITION LENGTH HAS MINIMAL EFFECT ON RECOVERY OF IMMEDIATELY DERIVABLE (PROOF DEPTHd = 1) TRANSITIONS We explore whether increasing composition length improves constraint propagation to unseen transi- tions that are immediately derivable from seen ones (d = 1). We train every compositional model of Figure 2 atT = 4in addition toT = 2, keeping the worlds, held-out transitions, and150k training steps fixed. Compositional path length is the only difference. All unseen transitions have proof depth d = 1. Compositionally trained models atT = 4propagate inverse, commutative, and composition constraints with an average accuracy of96%, the same as atT = 2, and copy at100%(Figure 7). Neither T = 2 nor T = 4 propagates periodicity. ECONVERSE CONSTRAINT PROPAGATION DECREASES SHARPLY WITH BRANCHING We explore how branching affects converse constraint propagation by building synthetic family trees that interpolate between purely inverseβ = 0and purely converse constraintsβ = 1. The tree is generated breadth-first withN = 1024, where each parent has one guaranteed child plus another with probabilityβ(expected number of children is1 + β). We hold out52parent and52child relationships (5%) which are jointly entailed by observed transitions and the converse constraint. We train a transformer on T = 2 paths for 150k steps with the compositional or the next-state objective 14 Preprint. Under review. Transformer GRU LSTM SSM Transformer GRU LSTM SSM Transformer GRU LSTM SSM Transformer GRU LSTM SSM Transformer GRU LSTM SSM 0.0 0.2 0.4 0.6 0.8 1.0 Unseen transition accuracy Inverse L;RjLR= RL= 1 ® Commutativity F;RjFR= RF ® Composition A;B;CjAB= C ® Periodicity A;BjA 4 = 1 ® Copy A;BjA= B ® Composition length does not change which constraints propagate Perfect constraint propagation Random guess T=2 T=4 (Both compositional) Figure 7: Composition length has minimal effect on propagating immediately derivable transi- tions. Compositional training atT = 4propagates inverse, commutative, and composition constraints with an average accuracy of96%, the same as atT = 2, and copy at100%. Increasing composition length does not rescue periodicity. Points averaged over 5 seeds and bars show standard deviations. on branching worlds of variousβ. Compositional training on the line world (pure inverseβ = 0) generalizes at99%accuracy, but generalization drops to63%atβ = 0.1,35%atβ = 0.2,13%at β = 0.5, and≤ 3% for β ≥ 0.75 (Figure 8). Next-state training fails for all β. 00.250.50.751 Branching probability ̄ (inverse at 0, converse at 1) 0.0 0.2 0.4 0.6 0.8 1.0 Unseen transition accuracy Perfect propagation Compositional Next-state Converse propagation decreases sharply with branching Figure 8: Converse constraint propagation decreases sharply with branching. Compositional training propagates the inverse constraint (β = 0) at99%, but only a10%chance of a second child (β = 0.1) drops generalization to63%, and forβ ≥ 0.75accuracy drops to≤ 3%. Next-state training fails at every β. Points averaged over 3 seeds and bars show standard deviations. FCOMPOSITION PROPAGATION OCCURS WITHOUT PATH SHORTCUTS We explore whether compositionAB = Cpropagates when shortcut pathss A −→ A(s) B −→ C(s)for a held-out transition(s,C,C(s))are removed from training. We consider the composition monoid worldN = 1024and hold out10%ofC-transitions. We train a transformer onT = 2paths for 150k steps under the next-state or compositional objective with and without shortcut paths in the training dataset. Next-state trained models fail in both settings. Compositionally trained models generalize at 92% accuracy with shortcut paths and 50% without (Table 2). 15 Preprint. Under review. Table 2: Compositional training propagates composition without shortcut paths. Transformers trained on the composition worldN = 1024atT = 2with and without the shortcut paths A −→ A(s) B −→ C(s)of each held-out transition. Accuracies are average±standard deviation over five seeds. ObjectiveShortcut pathsSeen (%)Unseen (%) Next-statewith1000± 0 without1000± 0 Compositionalwith10092± 6 without10050± 10 GPERIODICITY IS NOT PROPAGATED ACROSS COMPOSITION LENGTHS Composition length is varied1≤ T ≤ 8andA k = 1periodicity lengthk = 3, 4, 5overN = 1020 periodicity worlds to see if compositional training can propagate the constraint. A training path contains a full wrap-around loop whenT ≥ k. All unseen transitions have proof depthd = 1. We train transformers with the compositional objective for 150k steps. Every model fits observed transitions but fails to propagate the periodic constraint (all at random floor) even withT ≥ k(Table 3). Table 3: Periodicity is not propagated across composition lengths. All transformers trained compositionally onN = 1020periodicity worlds withA k = 1fork = 3, 4, 5and composition lengths1 ≤ T ≤ 8fail to propagate the periodicity constraint. Accuracies are averaged over five seeds, with standard deviations of at most 0.5%. Unseen (%) at path length T Constraint12345678 A 3 = 10.4± 0.5 0.4± 0.5 0.2± 0.4 0.0± 0.0 0.0± 0.0 0.0± 0.0 0.0± 0.0 0.0± 0.0 A 4 = 10.6± 0.5 0.2± 0.4 0.2± 0.4 0.2± 0.4 0.0± 0.0 0.2± 0.4 0.0± 0.0 0.2± 0.4 A 5 = 10.6± 0.5 0.2± 0.4 0.0± 0.0 0.0± 0.0 0.0± 0.0 0.0± 0.0 0.0± 0.0 0.0± 0.0 16

Instant research discovery