Obtaining human-like performance in NLP is often argued to require compositional generalisation. Whether neural networks exhibit this ability is usually studied by training models on highly compositional synthetic data. However, compositionality in natural language is much more complex than the rigid, arithmetic-like version such data adheres to, and artificial compositionality tests thus do not allow us to determine how neural models deal with more realistic forms of compositionality. In this work, we re-instantiate three compositionality tests from the literature and reformulate them for neural machine translation (NMT). Our results highlight that: i) unfavourably, models trained on more data are more compositional; ii) models are sometimes less compositional than expected, but sometimes more, exemplifying that different levels of compositionality are required, and models are not always able to modulate between them correctly; iii) some of the non-compositional behaviours are mistakes, whereas others reflect the natural variation in data. Apart from an empirical study, our work is a call to action: we should rethink the evaluation of compositionality in neural networks and develop benchmarks using real data to evaluate compositionality on natural language, where composing meaning is not as straightforward as doing the math. 1 1 Introduction Although the successes of deep neural networks in natural language processing (NLP) are astounding and undeniable, they are still regularly criticised for lacking the powerful generalisation capacities that characterise human intelligence. A frequently mentioned concept in such critiques is compositionality : the ability to build up the meaning of a complex expression by combining the meanings of its parts (e.g. Partee, 1984). Compositionality is assumed 1 The data and code are available at https://github. to play an essential role in how humans understand language, but whether neural networks also exhibit this property has since long been a topic of vivid debate (e.g. Fodor and Pylyshyn, 1988; Smolensky, 1990; Marcus, 2003; Nefdt, 2020). Studies about the compositional abilities of neural networks consider almost exclusively models trained on synthetic datasets, in which compositionality can be ensured and isolated (e.g. Lake and Baroni, 2018; Hupkes et al., 2020). 2 In such tests, the interpretation of expressions is computed completely locally : every subpart is evaluated independently  without taking into account any external context  and the meaning of the whole expression is then formed by combining the meanings of its parts in a bottom-up fashion. This protocol matches the type of compositionality observed in arithmetic: the meaning of (3 + 5) is always 8 , independent of the context it occurs in. However, as exemplified by the sub-par performance of symbolic models that allow only strict, local protocols, compositionality in natural domains is far more intricate than this rigid, arithmetic-like variant of compositionality. Natural language seems very compositional, but at the same time, it is riddled with cases that are difficult to interpret with a strictly local interpretation of compositionality. Sometimes, the meaning of an expression does not derive from its parts (e.g. for idioms), but the parts themselves are used compositionally in other contexts. In other cases, the meaning of an expression does depend on its parts in a compositional way, but arriving at this meaning requires a more global approach because the meanings of the parts need to be disambiguated by information from elsewhere. For instance, consider the meaning of homonyms (these dates are perfect for our dish/wedding), potentially idiomatic expressions (the child kicked the bucket off the pavement), 2 Apart from Raunak et al. (2019), work on compositionality and natural' language considers highly structured subsets of language (e.g. Kim and Linzen, 2020; Keysers et al., 2019). or scope ambiguities (every human likes a cat). This paradoxical tension between local and global forms of compositionality inspired many debates on the compositionality of natural language. Likewise, it impacts the evaluation of compositionality in NLP models. On the one hand, local compositionality seems necessary for robust and reliable generalisation. Yet, at the same time, global compositionality is needed to appropriately address the full complexity of language, which makes evaluating compositionality of state-of-the-art models in the wild' a complicated endeavour. In this work, we face this challenge head-on. We concentrate on the domain of neural machine translation (NMT), which is paradigmatically close to the tasks typically considered for compositionality tests, where the target represents the meaning' of the input. 3 Furthermore, MT is an important domain of NLP, for which compositional generalisation is important to produce more robust translations and train adequate models for low-resource languages (see, e.g. Chaabouni et al., 2021). As an added advantage, compositionality is traditionally well studied and motivated for MT (Rosetta, 1994; Janssen and Partee, 1997; Janssen, 1998). We reformulate three theoretically grounded tests from Hupkes et al. (2020): systematicity , substitutivity and overgeneralisation . Since accuracy  commonly used in artificial compositionality tests  is not a suitable evaluation metric for MT, we base our evaluations on the extent to which models behave consistently , rather than correctly. In our tests for systematicity and substitutivity, we consider whether processing is local ; in our overgeneralisation test, we consider how models treat idioms that are assumed to require global processing. Our results indicate that models often do not behave compositionally under the local interpretation, but exhibit behaviour that is too local in other cases. In other words, models have the ability to process phrases both locally and globally but do not always correctly modulate between them. We further show that some inconsistencies reflect variation in natural language, whereas others are true compositional mistakes , exemplifying the need for both local and global compositionality as well as illustrating the need for tests that encompass them both. With our study, we contribute to ongoing questions about the compositional abilities of neural networks, and we provide nuance to the nature of this question when natural language is concerned: 3 E.g. SCAN's inputs are instructions (walk twice) with executions as outputs (walk walk) (Lake and Baroni, 2018). how local should the compositionality of models for natural language actually be? Aside from an empirical study, our work is also a call to action: we should rethink the evaluation of compositionality in neural networks and develop benchmarks using real data to evaluate compositionality on natural language, where composing meaning is not as straightforward as doing the math. Tests for compositional generalisation in neural networks typically assume an arithmetic-like version of compositionality, in which meaning can be computed bottom up. The compositions require only local information  they are context independent and unambiguous: walk twice after jump thrice (a fragment from SCAN by Lake and Baroni, 2018) is evaluated similarly to (2 + 1)  (4  5) . In MT, this type of compositionality would imply that a change in a word or phrase should affect only the translation of that word or phrase, or at most the smallest constituent it is a part of. For instance, the translation of the girl should not change depending on the verb phrase that follows it, and in the translation of a conjunction of two sentences, making a change in the first conjunct should not change the translation of the second. While translating in such a local way seems robust and productive, it is not always realistic  e.g. consider the translation of dates in She hated bananas and she liked dates. In linguistics and philosophy of language, the level of compositionality has been widely discussed, which led to a variety of definitions. One of the most well-known ones is from Partee (1984): The meaning of a compound expression is a function of the meanings of its parts and of the way they are syntactically combined. 4 This definition hardly places restrictions on the relationship between expressions and their parts. The type of function that relates them is unspeci-fied and could take into account the global syntactic structure or external arguments, and the parts' meanings can depend on global information. Par-tee's definition is therefore called weak , global , or open compositionality (Szabo, 2012; Garca-Ramrez, 2019). When, instead, the meaning of a compound depends only on the meanings of its largest parts, regardless of their internal structure (similar to arithmetic), that is strong , local or closed 4 This straightforwardly extends to translation, by replacing meaning with translation (Rosetta, 1994). compositionality (Jacobson, 2002; Szabo, 2012). Under the local interpretation, natural language can hardly be considered compositional  many frequent phenomena such as homonyms, idioms and scope ambiguities cannot be resolved locally (Pagin and Westerstahl, 2010; Pavlick and Callison-Burch, 2016). The global interpretation handles such cases straightforwardly but does not match up with many a person's intuitions about the compositionality of language. After all, how useful is compositionality if composing the meanings of parts requires the entire rest of the sentence? This paradox inspired debates on the compositionality of natural language and is also highly relevant in the context of evaluating compositionality in neural models. Previous compositionality tests (6) considered only the local interpretation of compositionality, but to what extent is that relevant given the type of compositionality actually required to model natural language? Here, we aim to open up the discussion about what it means for computational models of language to be compositional by considering properties that require composing meaning locally as well as globally and evaluating them in models trained on unadapted natural language corpora. We focus on English-Dutch translation, for which we can ensure good command for both languages. We train Transformer-base models (Vaswani et al., 2017) using Fairseq (Ott et al., 2019). Our training data consists of a collection of MT corpora bundled in OPUS (Tiedemann and Thottingal, 2020), of which we use the English-Dutch subset provided by Tiedemann (2020), which contains 69M sentence pairs. 5 To examine the impact of the amount 5 Visit the Tatoeba challenge for the OPUS training data. of training data  a dimension that is relevant because compositionality is hypothesised to be more important when resources are scarcer  we train one setup using the full dataset, one using 18 of the data ( medium ), and one using one million source-target pairs in the small setup. For each setup, we train models with five seeds and average the results. To evaluate our trained models, we adopt FLORES -101 (Goyal et al., 2021), which contains 3001 sentences from Wikinews, Wikijunior and WikiVoyage, translated by professional translators, split across three subsets. We train the models until convergence on the dev' set. Afterwards, we compute SacreBLEU scores on the devtest' set (Post, 2018), using beam search (beam size = 5), yielding scores of 20 . 6  . 4 , 24 . 4  . 3 and 25 . 8  . 1 for the small, medium and full datasets, respectively. 6 3.2 Evaluation data While all our models are trained on fully natural data, for evaluation we use different types of data: synthetic, semi-natural and natural data. Synthetic data For our synthetic evaluation data, we consider the data generated by Lakretz et al. (2019), previously used to probe for hierarchical structure in neural language models. This data consist of sentences with a fixed syntactic structure and diverse lexical material. We extend the vocabulary and the templates used to generate the data and generate 3000 sentences for each of the resulting 10 templates (see Table 1a). Semi-natural data In the synthetic data, we have full control over the sentence structure and lexical items, but the sentences are shorter (9 tokens vs 16 in OPUS) and simpler than typical in NMT data. To obtain more complex yet plausible test sentences, we employ a data-driven approach 6 All training details are listed in Appendix E. 4156 synthetic semi-n. to generate semi-natural data. Using the tree substitution grammar Double DOP (Van Cranenburgh et al., 2016), we obtain noun and verb phrases (NP, VP) whose structures frequently occur in OPUS. We then embed these NPs and VPs in ten synthetic templates with 3000 samples each (see Table 1b). See Appendix A for details on the data generation. Natural data Lastly, we extract natural data directly from OPUS, as detailed in the subsections of the individual tests (4). In our experiments, we consider systematicity (4.1) and substitutivity (4.2), to test for local compositionality, and idiom translation to probe for a more global type of processing (4.3). One of the most commonly tested properties of compositional generalisation is systematicity  the ability to understand novel combinations made up from known components (most famously, Lake and Baroni, 2018). In natural data, the number of potential recombinations to consider is infinite. We chose to focus on recombinations in two sentence-level context-free rules: S  NP VP and S  S CONJ S . Test design The first setup, S  NP VP , concerns recombinations of noun and verb phrases. We extract translations for input sentences from the templates from 3.2, as well as versions of them with the (1) noun (NP  NP') or (2) verb phrase (VP  VP') adapted. In (1), a noun from the NP in the subject position is replaced with a different noun while preserving number agreement with the VP. In (2), a noun in the VP is replaced. NP  NP' is applied to both synthetic and semi-natural data; VP  VP' only to synthetic data. We use 500 samples per template per condition per data type. The second setup, S  S CONJ S , involves phrases concatenated using and, and tests whether the translation of the second sentence is dependent on the first sentence. We concatenate two sentences (S 1 and S 2 ) from different templates, and we consider again two different conditions. First, in condition S 1  S (cid:48) 1 , we make a minimal change to S 1 yielding S (cid:48) 1 by changing the noun in its verb phrase. In S 1  S 3 , instead, we replace S 1 with a sentence S 3 that is sampled from a template different from S 1 . We compare the translation of S 2 in all conditions. For consistency, the first conjunct is always sampled from the synthetic data templates. The second conjunct is sampled from synthetic data, semi-natural data, or from natural sentences sampled from OPUS with similar lengths and word-frequencies as the semi-natural inputs. We use 500 samples per template per condition per data type. Figure 2 provides an illustration of the different setups experimented with. Evaluation In artificial domains, systematicity is evaluated by leaving out combinations of known components' from the training data and using them for testing purposes. The necessary familiarity of the components (the fact that they are known') is ensured by high training accuracies, and systematicity is quantified by measuring the test set accu-4157 synthetic semi natural natural 0.0 0.2 0.4 0.6 0.8 1.0 c o n s i s t e n c y f l a u t i s t m o u s t a c h e f l a u t i s t m o u s t a c h e f l a u t i s t t h e a t r e l a d y b i r d m o u s t a c h e f l a u t i s t m o u s t a c h e f l a u t i s t m o u s t a c h e a u b e r g i n e m o u s t a c h e f l a u t i s t s h o pp i n g t r o ll e y f l a u t i s t m o u s t a c h e training size small medium full racy. If the training data is a natural corpus and the model is evaluated with a measure like BLEU in MT, this strategy is not available. We observe that being systematic requires being consistent in the interpretation assigned to a (sub)expression across contexts, both in artificial and natural domains. Here, we, therefore, focus on consistency rather than accuracy, allowing us to employ a model-driven approach that evaluates the model's systematicity as the consistency of the translations when presenting words or phrases in multiple contexts. We measure consistency as the equality of two translations after accounting for anticipated changes. For instance, in the S  NP VP setup, two translations are consistent if they differ in one word only, after accounting for determiner changes in Dutch (de vs het). In the evaluation of S  S CONJ S , we measure the consistency of the translations of the second conjunct. Figure 1 shows the results for the S  NP VP and S  S CONJ S setups (numbers available in Appendix B). The average performance for the natural data closely resembles the performance on semi-natural data, suggesting that the increased degree of control did not severely impact the results obtained using this generated data. 7 In general, the consistency scores are low, illustrating that models are prone to changing their translation of a (sub)sentence after small (unrelated) adaptations to the input. It hardly matters whether that change occurs in the sentence itself ( S  NP VP ), or in the other conjunct ( S  S CONJ S ), suggesting that the processing of the models is not local as assumed in strong compositionality. Models trained on more data seem more locally compositional, a somewhat contradictory solution to achieving compositional-7 In our manual analysis (5), however, we did observe a slightly different distribution of changes between these setups. ity, which, after all, is assumed to underlie the ability to generalise usage from few examples (Lake et al., 2019). This trend is also at odds with the hypothesis that inconsistencies are a consequence of the natural variation of language, which models trained on more data are expected to better capture. Under a local interpretation of the principle of compositionality, synonym substitutions should be meaning-preserving: substituting a constituent in a complex expression with a synonym should not alter the complex expression's meaning, or, in the case of MT, its translation. Here, we test to what extent models' translations abide by this principle, by performing the substitutivity test from Hupkes et al. (2020), that measures whether the outputs remain consistent after synonym substitution. To find synonyms  source terms that translate into the same target terms  we exploit the fact that OPUS contains texts both in British and American English. Therefore, it contains synonymous terms that are spelt different  e.g. doughnut / donut  and synonymous terms with a very different form  e.g. aubergine / eggplant. We use 20 synonym pairs in total (see Figure 3b). Test design Per synonym pair, we select natural data from OPUS in which the terms appear and perform synonym substitutions. Thus, each sample has two sentences, one with the British and one with the American English term. We also insert the synonyms into the synthetic and semi-natural data using 500 samples per synonym pair per template, through subordinate clauses that modify a noun  e.g. the king that eats the doughnut . In Appendix C, Table 6, we list all clauses used. Evaluation Like systematicity, we evaluate substitutivity using the consistency score, expressing whether the model translations for a sample are identical. We report both the full sentence consistency and the consistency of the synonyms' translations only, excluding the context. Cases in which the model omits the synonym from both translations are labelled as consistent if the rest of the translation is the same for both input sequences. In Figure 3a, we summarise all substitutivity consistency scores (tables are in Appendix C). We observe trends similar to the systematicity results: models trained on larger training sets perform better and synthetic data yields more consistent translations compared to (semi-)natural data. We further observe large variations across synonyms, for which we further detail the performance aggregated across experimental setups in Figure 3b. The three lowest scoring synonyms  flautist, aubergine and ladybug  are among the least frequent synonyms (see Appendix C), which stresses the importance of frequency for the model to pick up on synonymy. In Figure 3b, we show both the regular consistency and the consistency of the synonym translations, illustrating that a substantial part of the inconsistencies are due to varying translations of the context rather than the synonym itself, stressing again the non-local processing of the models. In our final test, we focus on exceptions to compositional rules. In natural language, typical exceptions that constitute a challenge for local compositionality are idioms . For instance, the idiom raining cats and dogs should be treated globally to arrive at its meaning of heavy rainfall. A local approach would yield an overly literal, non-sensical translation (het regent katten en honden). When a model's translation is too local, we follow Hupkes et al. (2020) in saying that it overgeneralises , or, in other words, it applies a general rule to an expression that is an exception to this rule. Overgeneralisation indicates that a language learner has internalised the general rule (e.g. Penke, 2012). We select 20 English idioms for which an accurate Dutch translation differs from the literal translation from the English MAGPIE corpus (Haagsma et al., 2020). Because acquisition of idioms is dependent on their frequency in the corpus, we use idioms 1 40 80 120 160 epoch 0.0 0.2 0.4 0.6 0.8 1.0 o v e r g e n e r a li s a t i o n small 1 10 20 30 40 50 epoch medium 1 10 20 30 epoch full with at least 200 occurrences in OPUS based on exact matches, for which over 80% of the target translations does not contain a literal translation. Test design Per idiom, we extract natural sentences containing the idiom from OPUS. For the synthetic and semi-natural data types, we insert the idiom in 500 samples per idiom per template, by attaching a subordinate clause to a noun  e.g. the king that said I knew the formula by heart ' . The clauses used can be found in Appendix D, Table 7. Evaluation Per idiom, we assess how often a model overgeneralises and how often it translates the idiom globally. To do so, we identify keywords that indicate that a translation is translated locally (literal) instead of globally (idiomatic). If the key-words' literal translations are present, the translation is labelled as an overgeneralised translation. For instance, for by heart, the presence of hart (heart) suggests a literal translation. An adequate paraphrase would say uit het hoofd (from the head). See Appendix D, Table 7, for the full list of keywords. We evaluate overgeneralisation for ten intermediate training checkpoints. In Figure 4, we report our results. 8 For all evaluation data types and all training set sizes, three phases can be identified. Initially, the translations do not contain the idiom's keyword, not because the idiom's meaning is paraphrased in the translation, but because the translations consist of high-frequency words in the target language only. Afterwards, overgeneralisation peaks: the model emits a very literal translation of the idiom. Finally, the model starts to memorise the idiom's translation. This is in accordance with results from Hupkes et al. (2020), and earlier results presented in the past tense debate by  among others  Rumelhart and McClelland (1986). Although the height of the overgeneralisation peak is similar across evaluation data types and training set sizes, overgeneralisation is more prominent in converged models trained on smaller datasets than it is in models trained on the full corpus. 9 In addition to training dataset size, the type of evaluation data used also matters: there is more overgeneralisation for synthetic and semi-natural data compared to natural data, stressing the impact of the context in which an idiom is embedded. The extreme case of a context unsupportive of an idiomatic interpretation is a sequence of random words. To evaluate the hypothesis that this yields local translations, we surround the idioms with ten random words. The results (Appendix D, Table 7) indicate that, indeed, when the context provides no support at all for a global interpretation, the model provides a local translation for nearly all idioms. Overall, the results of this test provide an interesting contrast with our substitutivity and systematicity results: where in those tests, we saw processing that was less local than we expected, here, the behaviour shown by the models is instead not global enough . Our systematicity and substitutivity results demonstrate that models are not behaving compositional according to a strict definition of compositionality. However, we ourselves have argued that strict compositionality is not always appropriate to handle natural language. A reasonable question to ask is thus: are the inconsistencies we marked as non-compositional actually incorrect? Annotation setup To address this question, we perform a manual analysis. We annotate 900 inconsistent translation pairs of the systematicity and substitutivity tests to establish whether the inconsistencies are benign or concerning. We consider four different types of changes: 1. cases of rephrasing , where both translations are equally (in)correct; 2. changes reflecting different interpretations of source ambiguities ; 3. cases in which one of the two translations contains an error ; 4. formatting (mostly punctuation) changes. For substitutivity samples, we also annotate whether the changes are related to the translation of the synonym, where we distinguish cases where i. one of the synonym translations is incorrect; ii. both are incorrect but in a different manner; iii. both are correct but translated differently; iv. one synonym remains untranslated. We annotate all changes observed per pair and report the relative frequency per class. We summarise the results, aggregated over different training set sizes and the three data types, in Figure 5. For a more elaborate analysis and a breakdown per model and data type, we refer to Appendix F. Results In the systematicity test, 40% of the marked inconsistencies reflects wrongfully translated parts in one of the two sentences, whereas 38% contains examples of rephrasing, 16% reflects ambiguities in the source sentences and 6% is caused by formatting differences. For substitutivity, most inconsistencies are similar to the ones observed in systematicity: only 24% involves the synonyms' translations, where one of them being untranslated was the most frequent category. The distribution of these types of inconsistencies differ strongly per training data type. For models trained on less data, inconsistencies are more likely to represent errors, whereas models trained on more data rephrase more often. This result emphasises that 4160 for lower-resource settings, being compositional is particularly relevant. Another demonstration of this relevance comes from the observation that although models can emit correct translations for nearly all synonyms, 10 they do not always do so, depending on the context. To give a peculiar example: in The child admires the king that eats the { doughnut, donut } , the snack was occasionally translated as ezel (donkey). Robustness and predictability Finally, we would like to stress that while rephrasing often might seem benign rather than concerning from the perspective of emitting adequate translations, its harmlessness still deserves some thought. There is a fine line between rephrasing and mistranslating: whether the single largest business establishment is referred to as de grootste (the largest) or de enige grootste (the only largest) may make or break a translation. Furthermore, if changes are unrelated to the contextual change (e.g. replacing soccer with football), this can be undesirable from a robustness and reliability perspective. This point becomes even more pronounced in cases where both translations are correct but have a different meaning. To analyse the extent to which inconsistencies are actually unmotivated, we investigated if we could trace them back to the contextual change, in particular focusing on whether changing synonyms from British to American spelling or vice versa might trigger a change in style or tone. We could not find evidence of such motivations, indicating that even correct cases of rephrasing were not caused by contextual changes that were necessary to take into account. In previous work, a variety of artificial tasks have been proposed to evaluate compositional generalisation using non-i.i.d. test sets that are designed to assess a specific characteristic of compositional behaviour. Examples are systematicity (Lake and Baroni, 2018; Bastings et al., 2018; Hupkes et al., 2020), substitutivity (Mul and Zuidema, 2019; Hupkes et al., 2020), localism (Hupkes et al., 2020; Saphra and Lopez, 2020), productivity (Lake and Baroni, 2018) or overgeneralisation (Korrel et al., 2019; Hupkes et al., 2020; Dankers et al., 2021). Generally, neural models struggle to generalise in such evaluation setups. There are also studies that consider compositional generalisation on more natural data. Such studies typically focus on either MT (Lake and Baroni, 2018; Raunak et al., 2019; Li et al., 2021) or semantic parsing (Finegan-Dollak et al., 2018; Keysers et al., 2019; Kim and Linzen, 2020; Shaw et al., 2021). Most of these studies consider small and highly controlled subsets of natural language. Instead, we focus on models trained on fully natural MT datasets, which we believe to be the setup for compositionality evaluation that does most jus-tice to the complexity of natural language: contrary to semantic parsing, where the outputs are structures created by expert annotators, in translation both inputs and outputs are fully-fledged natural language sentences. To the best of our knowledge, the only attempt to explicitly measure compositional generalisation of NMT models trained on large natural MT corpora is the study presented by Raunak et al. (2019). They measure productivity  generalisation to longer sentence lengths  of an LSTM-based NMT model trained on a full-size, natural MT dataset. Other studies using NMT, instead, consider toy datasets generated via tem-plating (Lake and Baroni, 2018) or focus on short sentences excluding more complex constructions that contribute to the complexity of natural language for compositional generalisation, such as polysemous words or metaphors (Li et al., 2021). Whether neural networks can generalise compositionally is often studied using artificial tasks that assume strictly local interpretations of compositionality. We argued that such interpretations exclude large parts of language and that to move towards human-like productive usage of language, tests are needed that assess how compositional models trained on natural data are. 11 We laid out reformulations of three compositional generalisation tests  systematicity, substitutivity and overgeneralisation  for NMT models trained on natural corpora, and assessed models trained on different amounts of data. Our work provides an empirical contribution but also highlights vital hurdles to overcome when considering what it means for models of natural language to be compositional. Below, we reflect on these hurdles and our results. The proxy-to-meaning problem Compositionality is a property of the mapping between the form and meaning of an expression. Since translation is a meaning-preserving mapping from form in one language to form in another, it is an attractive task to evaluate compositionality: the translation of its sentence can be seen as a proxy to its meaning. However, while expressions are assumed to have only one meaning, translation is a many-to-many mapping: the same sentence can have multiple correct translations. This does not only complicate evaluation  MT systems are typically evaluated with BLEU because accuracy is not a suitable option  it also raises questions about how compositional the desired behaviour of an MT model should be. On the one hand, one could argue that for optimal generalisation, robustness, and accountability, we like models to behave systematically and consistently: we expect the translations of expressions to be independent of unrelated contextual changes that do not affect their meaning (e.g. swapping out a synonym in a nearby sentence). Additionally, model performance could be improved if small changes do not introduce errors in unrelated parts of the translation. On the other hand, non-compositional behaviour is not always incorrect  it is one of the main arguments in our plead to test compositionality in the wild'  and we observe that indeed, not all non-compositional changes alter the correctness of the resulting translations. Changing a translation from atleet (athlete) to sporter (sportsman) based on an unrelated word somewhat far away may not be (locally) compositional, but is it a problem? And how do we separate such harmful' mistakes from helpful ones? The locality problem Inextricably linked to the proxy-to-meaning problem is the locality problem. In our tests we see that small, local source changes elicit global changes in translations . For instance, in our systematicity tests, changing one noun in a sentence elicited changes in the translation of a sentence that it was conjoined with. In our substitutivity test, even synonyms that merely differed in spelling (e.g. doughnut and donut) elicited changes to the remainder of the sentence. This counters the idea of compositionality as a means of productively reusing language: if a phrase's translation depends on (unrelated) context that is not in its direct vicinity, this suggests that more evidence is required to acquire the translation of this phrase. Tests involving synthetic data present the models with sentences in which maximally local behaviour is possible, and we argue that it is, therefore, also desirable. Our experiments show that even in such setups, models do not translate in a local fashion: with varying degrees of correctness, they frequently change their translation when we slightly adapt the input. On the one hand, this well-known volatility (see also Fadaee and Monz, 2020) might be essential for coping with ambiguities for which meanings are context-dependent. On the other hand, our manual analysis shows that the observed non-compositional behaviour does not reflect the incorporation of necessary contextual information and that oftentimes it is even altering the correctness of the translations. Furthermore, this erratic behaviour highlights a lack of default reasoning, which can, in some cases, be problematic or even harmful, especially if faithfulness (Parthasarathi et al., 2021) or consistency is important. In linguistics, it has been discussed how to extend the syntax and semantics such that problem cases' can be a part of a compositional language (Westerstahl, 2002; Pagin and Westerstahl, 2010). In such formalisations, global information is used to disambiguate the problem cases, while other parts of the language are still treated locally. In our models, global behaviour appears in situations where a local treatment would be perfectly suitable and where there is no clear evidence for ambiguity. We follow Baggio (2021) in suggesting that we should learn from strategies employed by humans, who can assign compositional interpretations to expressions but can for some inputs also derive non-compositional meanings. For human-like linguistic generalisation, it is vital to investigate how models can represent both these types of processing, providing a locally compositional treatment when possible and deviating from that when needed. Conclusion In conclusion, with this work, we contribute to the question of how compositional models trained on natural data are, and we argue that MT is a suitable and relevant testing ground to ask this question. Focusing on the balance between local and global forms of compositionality, we formulate three different compositionality tests and discuss the issues and considerations that come up when considering compositionality in the context of natural data. Our tests indicate that models show both local and global processing, but not necessarily for the right samples. Furthermore, they underscore the difficulty of separating helpful and harmful types of non-compositionality, stressing the need to rethink the evaluation of compositionality using natural language, where composing meaning is not as straightforward as doing the math. We thank Sebastian Riedel, Douwe Kiela, Thomas Wolf, Khalil Sima'an, Marzieh Fadaee, Marco Baroni, Brenden Lake and Adina Williams for providing feedback on this draft and our work in several different stages of it. We thank Michiel van der Meer for contributing to the initial experiments that led to this paper. A special thanks goes to Angela Fan, who assisted us at several points to get the ins and outs of training large MT models and double-checked several steps of our pipeline and to our ARR reviewers, who provided amazingly high quality feedback. VD is supported by the UKRI Centre for Doctoral Training in Natural Language Processing, funded by the UKRI (grant EP/S022481/1) and the University of Edinburgh.
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