Intensionalizing A bstract M eaning R epresentations: Non-Veridicality and Scope (2021)

Abstract: 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 @xcite . 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 nodes. the inventory of amr disambiguated predicate senses (e.g., believe-01) are based on propbank argument structure frames @xcite @xcite . the graph in (1-b) is an amr for the sentence in (1-a). more commonly, however, amrs are represented in penman notation @xcite as in (1-c) or occasionally as a conjunction of logical triples (1-d). recent developments have seen the expressive power of amr improved both in terms of its graphic representation as well as its translation into logical forms. for instance, amr graphical representations have been enriched to represent tense and aspect @xcite @xcite , quantifier scope @xcite , semantic number @xcite , and speech acts @xcite , while translations into first and higher-order logics have been proposed as a means of capturing coreference @xcite and quantifier scope @xcite @xcite . despite these advances in theoretical work, amr encounters issues when it comes to the semantics of intensional contexts. for instance, @xcite 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. 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. 1 these examples are based on examples in the freakshow section of the amr guidelines, available at: https: //github.com/amrisi/amr-guidelines/blob/ master/amr.md#amr-freak-show @xcite provide a solution to the problem of non-veridicality in a graph-based representation by making use of a graphic knowledge representation (gkr) @xcite , a layered graph in which identifiers, or names, are assigned to sub-graphs @xcite . 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. foot_0 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 @xcite and @xcite . for now, we assume that the interpretation function . compositionally maps amrs to a simple first-order calculus embedded in the stlc. 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. 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: 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, since ψ ∧ ∃x(ϕ) ⇔ ∃x(ψ ∧ ϕ) (if x / ∈ free(ψ)). this does away with the need to use continuationpassing style (c.f., bos 2016; @xcite ). 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 @xmath0; 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 @xcite davidsonian event semantics for attitude verbs which has undergone a number of refinements (e.g., @xcite @xcite . 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 @xmath1. 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. 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 @xcite , 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 @xcite 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). ## 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 @xcite 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.