Intensionalizing Abstract Meaning Representations: Non-Veridicality and Scope

Abstract Meaning Representation (AMR) is a graphical meaning representation language designed to represent propositional information about argument structure. However, at present it is unable to satisfyingly represent non-veridical intensional contexts, often licensing inappropriate inferences. In this paper, we show how to resolve the problem of non-veridicality without appealing to layered graphs through a mapping from AMRs into Simply-Typed Lambda Calculus (STLC). At least for some cases, this requires the introduction of a new role :content which functions as an intensional operator. The translation proposed is inspired by the formal linguistics literature on the event semantics of attitude reports. Next, we address the interaction of quantifier scope and intensional operators in so-called de re/de dicto ambiguities. We adopt a scope node from the literature and provide an explicit multidimensional semantics utilizing Cooper storage which allows us to derive the de re and de dicto scope readings as well as intermediate scope readings which prove difficult for accounts without a scope node.


Introduction
Abstract Meaning Representation (AMR) is a graphical meaning representation in which graphs are rooted, directed, and acyclic (Banarescu et al., 2013). Non-terminal nodes are assigned variable IDs, terminal nodes are sense concepts (e.g., believe-01, boy, etc.) or constants (e.g., the polarity attribute -, cardinals, names, etc.), and labelled edges represent semantic relations between * Omitted from LAW-DMR'21 submission due to error.
(1) a. The boy hugged the dog. into logical forms. For instance, AMR graphical representations have been enriched to represent Tense and Aspect (Donatelli et al., 2018(Donatelli et al., , 2019Van Gysel et al., 2021), quantifier scope (Pustejovsky et al., 2019;Van Gysel et al., 2021), semantic number (Stabler, 2017), and speech acts (Bonial et al., 2020), while translations into first and higher-order logics have been proposed as a means of capturing coreference (Artzi et al., 2015) and quantifier scope (Bos, 2016(Bos, , 2020Stabler, 2017;Lai et al., 2020). Despite these advances in theoretical work, AMR encounters issues when it comes to the semantics of intensional contexts. For instance,  note that an AMR like (3-b), represented as a conjunction of logical triples, would permit an inference to 'The girl is sick' by conjunction elimination.
(3) a. The boy believes that the girl is sick.
b. (b / believe-01 :ARG0 (b2 / boy) :ARG1 (s / sick-05 :ARG1 (g / girl))) An even more striking consequence of this nonveridical problem is demonstrated by the following examples. 1 (4) a. The boy believes he is sick. b. (b / believe-01 :ARG0 (b2 / boy) :ARG1 (s / sick-05 :ARG1 b2)) c. (b / believe-01 :ARG0 (b2 / boy :ARG1-of (s / sick-05)) :ARG1 s)) At first glance, it might appear that the predicate sick-05 is in a non-veridical context in (4-b) and a veridical context in (4-c). This is because the grammatical subject position of the verb believe is a veridical environment, while its sentential complement is not. However, these graphs are in fact logically equivalent because for any AMR relation R, R(x)(y) ⇔ R-of(y)(x). Consequently, (4-b) and (4-c) depict the same conjunction of logical triples once inverse relations are normalized.  provide a solution to the problem of non-veridicality in a graph-based representation by making use of a Graphic Knowledge Representation (GKR) (Kalouli and Crouch, 2018), a layered graph in which identifiers, or names, are assigned to sub-graphs (Carroll et al., 2005). A GKR separates conceptual structure (i.e., predicate-argument structure) and contextual structure into two sub-graphs. The graph in (5) is a simplified GKR of (3-a). The lower context bel represents the intensional context of the boy's belief and is non-veridical (or averidical) with respect to the upper context top. Consequently, the GKR does not permit the inference from 'The boy believes the girl is sick' to 'The girl is sick.' Another problem for AMR's representation of attitude reports such as those containing the verb 'believe' is that PropBank argument structures often do not distinguish between propositional and non-propositional arguments. For instance, the PropBank argument structure for believe-01 assigns both nominal and clausal arguments the same argument role, :ARG1. 2 Despite this, the PropBank frame includes a note that 'believe' can have both a theme and a propositional argument simultaneously (e.g., 'Mary believed John that he didn't eat the last piece of pie'). 3 Given this lack of distinction between different types of arguments, it is not sufficient to lexically specify that the :ARG1 of believe-01 is always a non-veridical environment. While this might work for (3-b), it would also mean that we could not infer the existence of a girl from (6). In what follows, we propose the introduction of an intensional relation :content responsible for introducing propositional arguments. Replacing the :ARG1 of believe-01 in (4) with the new :content role ensures firstly that (4-b) contains a non-veridical environment and secondly that (4-c) is not a representation for a coherent natural language sentence. Crucially, the addition of :content and the translation function proposed for AMRs into logical forms offers a satisfying representation for intensional contexts without the need for additional graph structure. Finally, we show how our logical forms interact with scope taking elements to derive attested interpretations of attitude reports with quantifier phrases (QPs).

Extensional Semantics for AMRs
We start by defining a simple translation for basic AMRs (without intensional operators or quantifiers) into the Simply-Typed Lambda Calculus (STLC). Following Bos (2016) we define the syntax of AMRs recursively. A simplex AMR is a constant c, variable x, or an instance assignment (x / P). Complex graphs are defined recursively as one or more subgraphs {A 1 , . . . , A n } connected to an instance assignment by n relations.
Our semantics for AMRs is a departure from that of Bos (2016) and Lai et al. (2020). For now, we assume that the interpretation function . compositionally maps AMRs to a simple first-order calculus embedded in the STLC. 4 AMR constants and variables are mapped to STLC constants of type e, AMR predicates are mapped to STLC constants of type e → t, and AMR roles are mapped to STLC constants of type e → e → t. 5 Simple instance assignments are mapped to functional applications: To state a semantics for complex AMRs, we first state a semantics for a role assignment in (10), consisting of a role and an embedded AMR, via pattern matching on the embedded AMR. A sequence of role assignments ρ 1 . . . ρ n is interpreted via iterated conjunction (11), and finally a complex AMR is interpreted by saturating a role sequence with the main variable of the AMR (12).
Basic AMRs are thereby translated into simple conjunctive first-order formulae.
(13) The boy admires himself. a. (a / admire-01 Finally, we declare an operation close (version 1) which, applied to a STLC expression ϕ, introduces an existential quantifier which unselectively binds variables in the set Free(ϕ).
Crucially for what follows, Free(ϕ) does not correspond to the classical notion of free variables in ϕ, but rather should be defined to ensure that recurrent variables are not bound by close prior to instance assignment. We define Free as follows: 6 Applying close to the example above returns an existential statement.
Putting re-entrant nodes to one side, it is harmless to defer existential binding of free variables, (ψ)). This does away with the need to use continuationpassing style (c.f., Bos 2016;Lai et al. 2020). Cases involving re-entrant nodes such as the dog scratched itself are handled straightforwardly via matching variables.

An Intensional Semantics
Next, we systematically intensionalize the interpretation in a standard way by replacing the propositional type t with s → t (the type of a function from worlds to truth values; Gallin 1975). Our existing interpretation procedure for basic AMRs remains largely intact, although we tweak the definitions of conjunction and existential quantification in order to accommodate the presence of additional world arguments.
To provide a semantics for intensional operators such as attitude predicates in AMR, we adopt a variant of Kratzer's (2006) Davidsonian event semantics for attitude verbs which has undergone a number of refinements (e.g., Moulton, 2009;Elliott, 2016Elliott, , 2020. More specifically, we propose a translation of AMRs of attitude reports into logical forms in which attitude events are associated with propositional content via a dedicated modal operator cont. In order to achieve this, we increase the AMR inventory of semantic roles with a :content role which is interpreted as a relation of type e → (s → t) → (s → t).
To see how this resolves the problem of nonveridicality consider again the two AMRs in (4), the first of which is repeated in (19-a) with the :ARG1 role changed to :content. The underlined argument shows that the world of evaluation for the translation of the :content argument is shifted to w 2 , and thus we cannot infer that the boy is sick in w as desired.
(20) (b / believe-01 :ARG0 (b2 / boy :ARG1-of (s / sick-05)) :content s)) The semantic rule for :content is only welldefined if the embedded AMR is an instance assignment, due to the type of cont. It follows that the interpretation of (20), in which :content embeds a recurrent variable, is undefined. For this reason, AMRs in which :content embeds a constant or variable should be avoided by annotators. The issue here is that reentrant nodes are used to model a diverse range of linguistic phenomena (Szubert et al., 2020), but are inappropriate for modelling anaphora to a proposition, such as 'The boy who is sick believes it'. 7 4 Content and Quantifier Scope: de re and de dicto Readings Another property of intensional operators is their ability to interact scopally with other operators such as quantifier phrases (QPs) (see Keshet and Schwarz 2019 for a recent overview). For example, the sentence in (21) could be paraphrased as in (21-a) which can be analyzed as an existential QP 'a violin' taking scope over the attitude predicate 'hope', in which case the restrictor argument of the QP is evaluated in the actual world (i.e., there is an actual violin that the boy wants). In contrast, the reading in (21-b) can be analyzed as the existential being within the scope of the attitude (i.e., the boy hopes to be a violin-owner, but is not necessarily concerned about owning any particular violin).
(21) The boy hopes to buy a violin. a. There is a violin the boy hopes to buy.
De re b. The boy hopes to be a violin-owner.

De dicto
To capture these readings, we develop a semantics for AMRs enriched with additional graph structure for modelling scope (Pustejovsky et al., 2019) based on the mechanism of Cooper storage (Cooper, 1983;Kobele, 2018).

Scope Semantics
Before discussing the scope interaction of quantifiers and cont, let us first develop a semantics for scope in non-intensional contexts. Following Pustejovsky et al. (2019) and Van Gysel et al. (2021) we make use of a scope node, which has a predicative argument representing the core argument structure and reentrant variables to represent the order of quantifier scope. For example, in (22-b), the QP of :ARG0 scopes over that of :ARG1. Next, we define an explicit interpretation of scope nodes using Cooper storage (Cooper, 1983;Kobele, 2018). A store is an assignment of variables to STLC expressions of type (e → t) → t. 8 (23) {(x, every(boy)), (y, some(girl)} Instead of simply mapping AMRs to expressions of STLC, we map them to a pair consisting of a store s and an ordinary semantic value (i.e., an STLC expression). We assume the following notational conventions.
In this system, AMRs like those we have considered so far update the store vacuously (25-a) while retaining their ordinary semantic value (25-b).
In order to illustrate how :quant is interpreted, we begin by stating a semantics for an instance assignment decorated with a single :quant role. 9 We assume that AMR determiner constants such as every are mapped to STLC constants of type (e → t) → (e → t) → t, such as the quantificational determiner every (e.g., Barwise and Cooper, 1981). The store is updated with a generalized quantifier constructed from the determiner and the property in the instance assignment (26-a). The ordinary semantic value, on the other hand, is the truth constant , the addition of which is redundant in a string of conjunctions (26-b). On this semantics, basic AMRs have a non-trivial ordinary value but perform a vacuous store update. Conversely, QPs perform a non-trivial store update but have a redundant ordinary semantic value. Crucially, however, QPs still contribute one or more variables to the ordinary value of a role assignment via pattern matching, as in (27) below.
We revise our translation function to ensure that the store gets passed up during the derivation.

(27) Role assignment (revised)
: In combination with (26-a), this ensures that the store of a complex AMR will contain the indexquantifier pairs added to the store by its subgraphs, as shown in the following example.
In order to retrieve the quantifier from the store, we declare an operation pop x which, given a variable x, store s, and logical form ϕ retrieves the expression in s paired with x, and applies it to λx . ϕ. We write s x for the expression in s paired with x.
We also restate our close operation which now existentially binds any free variables which are not associated with an index in the store.
As mentioned above, scope nodes are decorated with roles embedding reentrant nodes indicating scope-takers (:ARGn), and a role to indicate the scope site (:pred). We state the semantics for a complex AMR headed by a scope node syncategorematically: a scope node with arguments x 1 ,. . . ,x n induces evaluation of the quantifiers stored at x 1 ,. . . ,x n . 10,11 (33) Interpreting scope nodes (s / scope :ARG0 x 1 ... : Consider again the example 'every boy danced', but now with a scope node.

Deriving the de re and de dicto readings
Now we reintroduce world variables to see how this interpretation function can translate both the de re and de dicto readings of (21) above, starting with the de re reading. In (35-a), the variable v is the :ARG0 of the scope node and consequently the QP a violin takes scope over the attitude verb.
(35) a. (s / scope :ARG0 v :pred (h / hope-01 :ARG0 (b / boy) :content (b2 / buy-01 Here, there is a specific violin in the world of evaluation w which the boy hopes to buy. For the de dicto reading, we do not need to do anything special. However, to close off the interpretation we can either embed the entire AMR under a scope node or use the close operation to bind any free variables. (36) a. (h / hope-01 :ARG0 (b / boy) :content (b2 / buy-01 :ARG0 b :ARG1 (v / violin)))) b. close ( (36-a) Here, the existential is within the scope of the lambda operator λw 2 and the restrictor argument violin is evaluated in the boy's hope worlds w 2 .

Intermediate de dicto reading
Although Pustejovsky et al. (2019) frame their proposal as "embed[ding an AMR] under a scope graph", our implementation also permits the embedding of a scope node within an AMR. Doing so allows us to derive intermediate scope readings. Consider the following example.
(37) The boy thinks the girl hopes to buy a violin. a. The boy thinks there is a violin that the girl hopes to buy.
In this intermediate reading, the intensional object might not be a violin in the actual world, nor in the girl's desire worlds, rather it is a violin in the boys belief worlds. Stabler (2017) notes that accounts such as Bos (2016) :content (b / buy-01 :ARG0 g :ARG1 (v / violin :quant a)))))

Problematic Scope
Besides the de re and de dicto readings, Fodor (1970) considers two further readings in which the restrictor argument of the QP is interpreted separately from the scopal position of the quantifier.
(39) The boy hopes to buy a violin. a. There are things which are violins and the boy hopes to buy one of them. Non-specific de re b. #There is a thing the boy hopes to buy which he believes is a violin.

Specific de dicto
Although there is a general consensus in the literature that (39-b) is not a possible reading of (39), the reading in (39-a) is possible. At present, we cannot derive the reading in (39-a). While it is possible to enrich AMRs further to accommodate such interpretations, it remains to be seen whether such an effort is worthwhile. Although such interpretations are attested and theoretically significant, they are not common, and accounting for such interpretations would likely involve enriching the graphical representation of AMRs in ways which would make them far less tractable for annotators and parsers. We leave it to future research to determine whether we can accommodate this third reading without unintentionally complicating the AMRs in undesirable ways.

Discussion
We have proposed to extend the expressive power of AMR in two respects. Firstly, we have enriched the graphical form of AMRs by increasing the inventory of AMR roles with the role :content. We did so in order to distinguish between intensional and non-intensional arguments of modal operators such as attitude predicates. Secondly, we provided a translation of AMRs into logical forms which allowed us to solve the problem of nonveridicality as well as complex scope interactions between quantifiers and intenstional operators. The addition of :content has ramifications for AMR annotation as well as backwards compatibility. We opted to avoid proposing the addition of new numbered argument roles for predicate like believe-01 since this would involve a wide-scale revision of AMR's frameset as well as complicating any effort to convert between existing corpora and an enriched corpus, as this would involve a many-to-many mapping. Instead, we proposed a semantically motivated intensional relation :content which will reduce the complexity of any conversion effort, requiring only a many-to-one mapping.
When designing meaning representations, there is inevitably a trade-off between how adequate the representation is (e.g., how much semantic information is present) and how tractable the representations are for large scale annotation projects. Although we believe that :content should not be any more difficult to annotate than non-core roles in AMR, it is likely that annotators may struggle to resolve and represent quantifier scope and de re/de dicto ambiguities. Thankfully, in addition to being representationally adequate, the scope node approach adopted from Pustejovsky et al. (2019) and Van Gysel et al. (2021) is both intuitive and transparent. Future research could aim to gauge the level inter-annotator agreement when representing such phenomena in a small scale annotation task.
Beyond AMR, the enriched graphical language Uniform Meaning Representation (UMR) (Van Gysel et al., 2021) also utilizes scope nodes to capture quantifier scope relations. The :content role and translation function proposed here can be adopted wholesale for UMR, and thus will prove useful in future annotation projects for this more expressive annotation scheme.
Finally, the :content role and its intensional translation may also facilitate downstream NLP tasks. Specifically, different attitude predicates trigger different lexical inferences regarding the truth of their complement depending on whether they are factive (e.g., know-01), counterfactive (e.g., pretend-01) or non-veridical (e.g., believe-01). A number of rich resources exist in this domain including the MegaVeridicality data sets  which contain factuality judgments for a comprehensive list of English verbs that embed finite clauses as well as a variety of predicates which embed non-finite clauses. Resources such as these may be deployed alongside AMR for NLI, since the logical forms which our interpretation function produces represent the scope of the attitude verb, unlike in a flat list of logical triples.
A python script for translating AMR into STLC is available at https://github.com/ emorynlp/Intensionalizing-AMR, and we are currently working on developing a script to convert intensional uses of numbered arguments into the new :content role which we also plan to make publicly available.

Conclusion
Abstract Meaning Representations (AMRs) are unable to represent non-veridical environments in a semantically satisfying way. When an AMR is translated into a conjunctions of logical triples, it permits spurious inferences to the truth of any of its subgraphs via conjunction elimination. We proposed to rectify this through the introduction of a novel AMR role :content, before providing an intensional interpretation for AMRs which correctly invalidates such inferences. We then showed how such a semantics can be combined with a means of modelling scope to derive de re and de dicto readings of natural language sentences with intensional operators and quantifiers. We concluded that the inclusion of a scope node (Pustejovsky et al., 2019;Van Gysel et al., 2021) is necessary in order to capture intermediate scope readings, and we provided a translation function from AMRs into STLC permitting the derivation of complex interactions of natural language quantifiers with intensional operators. This work is part of the concerted effort to increase the expressive power of AMRs while also maintaining tractable representations ensuring that large scale annotation projects can be performed with minimal instruction.