Logical Transductions for the Typology of Ditransitive Prosody (2022)

Abstract: Given the empirical landscape of possible prosodic parses, this paper examines the computations required to formalize the mapping from syntactic structure to prosodic structure. In particular, we use logical tree transductions to define the prosodic mapping of ditransitive verb phrases in SVO languages, building off of the typology described in Kalivoda (2018). Explicit formalization of syntax-prosody mapping revealed a number of unanswered questions relating to the fine details of theoretical assumptions behind prosodic mapping.

## introduction within computational and mathematical phonology, there is ample work on formalizing segmental and suprasegmental phonological processes that are word-bounded, such as by using finite state acceptors (fsas) and transducers (fsts) @xcite @xcite @xcite , or using equivalent logical transductions @xcite @xcite @xcite . until recently however, there was little work on the computational machinery required by sentence-level or phrase-level phonology (prosodic phonology). this gap may be because early work on prosodic phonology found that some common aspects of prosody were computationally regular over strings, and can be formalized with fsas @xcite . however, the abstract representations that are the target of prosodic processes are subject to extensive debates in the linguistic literature, and they play a crucial role for questions about the nature of the linguistic phenomena at the phonology-syntax interface @xcite @xcite . it is an established fact that phonological processes can refer to domains larger than a word. these domains form hierarchical layers: the prosodic word (w or pw), the prosodic phrase (p or pph) foot_0 , and the intonational phrase (i or ip). these prosodic constituents show systemic relations with syntactic constituents. however, such relations have been argued not to be strictly isomorphic -that is, prosodic constituency cannot be read directly from syntactic constituency. the characteristics of the mapping between syntactic structure and prosodic structure are important to theoretical approaches that consider prosodic constituency to be relevant for phonological generalizations. in this sense, distransitive constructions -verbs with multiple internal arguments (e.g. gave mary books) -are a core example of prosodic-syntax mismatches cross-linguistically. building on the systematic report of such mismatches in svo languages provided by @xcite , this paper works out a formalization of the typology of attested syntax-prosodic mappings for ditransitive constructions in terms logical transductions @xcite . in other linguistic domains, the rigor provided by computational/mathematical formalization has helped researchers commit to details of their theoretical assumptions, and fully understand the impact of particular representational choices. in line with this observation, this paper contributes to recent work laying the ground for mathematical investigations of the syntax-prosody interface @xcite @xcite . these first steps already shed light on how a variety of theoretical details often unspecified in the literature need further clarification before extensive logical formalization of the syntax-prosody interface can be achieved. the paper is organized as follows. section 2 goes over the basic empirical typology of ditransitive prosody. section 3 presents the formal preliminaries for the logical notation. section 4 formally defines the bulk of syntactic information relevant for ditransitive prosody. section 5 shows how such information can be used to formally define the mapping from syntax to prosody. we then discuss ( §6) and conclude ( §7). ## typology of ditransitive prosody in prosodic phonology, syntactic constituents (e.g. xp's) are said to map onto prosodic constituents (e.g. prosodic phrases). these two types of constituents are often mis-aligned, meaning that an xp can be larger or smaller than its corresponding prosodic phrase. unsurprisingly, different languages have different rules for how xps are mapped. in this paper, we focus on a formal exploration of the prosody of ditransitive sentences in svo languages, given that there is data available on their typology @xcite . ## logical tree transductions in this section, we illustrate the use of monadic second order (mso) logic to define tree-to-tree transductions. mso transductions are equivalent to regular functions @xcite , and have been commonly employed to model both segmental and autosegmental phonological processes @xcite @xcite . for the current discussion, we assume familiarity with logic (boolean connectives, first-order quantification, etc.) and set notation on the reader's part. with logical transductions, the input tree model is defined in terms of a signature ⟨d,r⟩. the segments are defined in terms of a set of domain elements d taken from the set of positive integers. for tree models, the common practice is to use gorn-addresses. the domain elements satisfy a set of relations r which can be unary or binary. unary relations designate the labels l of these domain elements, e.g. the label v(x) designates domain elements which are nodes labeled v (for verb). domain elements are connected via b c a a a b b b c a a a b as a toy example, take a tree transduction that changes root nodes that are labeled a into root nodes that are labeled b (figure 2 ). we first illustrate the logical definition of a tree for the input tree in this example transduction, with more extended illustration of each logical statement in equation 2 in figure 3 . the model definition first establishes the domain of the structure, here using gorn addresses. each unary relation corresponds to labels and is the set of nodes for whom that label applies. for instance, the set for a(x) are the nodes which are labeled a: these are the nodes with gorn addreses ε,0,1,00,01,10,11, as can also be seen in figure 3 . each binary relation is a set of pairs for which the binary relation holds: equation 2 thus states that the dominance relation ¡ holds for nodes ϵ and 0, meaning that the node with addres ϵ dominates the node with address 0, and so on. proper dominance (¡ + ) is defined as the transitive closure of immediate dominance (¡). in order to transform input trees into output trees, mso logical transductions define a copy set c of some fixed size k. the k members of the copy set act as indexes for copies of the input. if the output structure needs less than or equal nodes as the input, then a copy set of size 1 is sufficient: |c| = 1. if the output has a larger number of nodes than the input, then a larger copy set is needed. output functions define segments in the output copies in terms of the input segments. the apostrophe marks output elements. we mark these functions using ϕthis font. for example, to change a root node a to b, we need a transduction with a copy set of size 1, since the output tree has the same number of nodes as the input tree. in order to make the transduction easier to read, we define the root a segment with the predicate in (1) as a shorthand, using this font. crucially, every pair of segments has the same dominance relation in the output as in the input (2). nodes in the output are labeled a if they are labeled a in the input and they are not the root (3). the label b is generated for all underlying b's and for underlying root a's (4). nodes labeled c in the input stay c in the output (5). we visualize an example of this transduction in figure 2 . for representational ease, in what follows we use simple integers like {1,2,3,...} as numeral shorthands for gorn addresses. ## formalizing core syntactic information in ditransitives, prosodic phrasing is sensitive to some but not all aspects of the syntactic structure @xcite @xcite @xcite @xcite @xcite . these aspects are overtness, headedness, tree geometry, arguments, and linearity. it ignores category labels. in this section, we define predicates that pick out these aspects of syntactic structure. these predicates will be later used to define the logical mappings from syntax to prosody. note that existent prosodic mapping studies have not directly addressed adjunction, namely the nature of the prosodic mapping when an unbounded number of adjoining phrases are added to the sentence. additionally, unbounded adjunction introduces non-locality between a head and its argument. because of the lack of data and these non-trivial open issues related to adjunction, we set it aside in our preliminary formalization. ## discussion in this paper, we used logical tree transductions to characterize mappings between syntactic and prosodic structure in ditransitive constructions. based on the cross-linguistic typology of prosodic mappings reported in @xcite , we showed that logical transductions seem appropriate to derive the alignment mismatches between syntactic and prosodic constituents. in doing so, we highlighted how details of prosodic and syntactic structures often left unspecified in the linguistic literature become fundamental in deciding the linguistic naturalness of such mappings. these results then provide a baseline for future, extensive formalization of syntax-prosody mismatches and open the way for a vast array of computationally informed questions and computationally-driven empirical predictions. ## conclusion this paper offers a contribution to the scarce existing literature on the formal characterization of prosodic processes, and their relation to syntactic representations. while much work remains to be done, our results further show how careful mathematical formalization can help up refine long-standing theoretical questions, suggest the need for more and different types of data, and make us more critical of theoretical assumptions about linguistic representations across subdomains.