The relationship between recent CG analyses of ellipsis and HPSG counterparts seems to be similar to the situation with competing analyses on coordination. Both Barker (2013) and Kubota & Levine (2017) exploit hypothetical reasoning to treat the antecedent of an elided material as a "constituent" with fullfledged semantic interpretation at an abstract combinatoric component of syntax. The anaphoric mechanism can then refer to both the syntactic and semantic information of the antecedent expression to capture syntactic sensitivity observed in ellipsis phenomena, without the need to posit hierarchical representations at the ellipsis site. Due to its surface-oriented nature, HPSG is not equipped with an analogous abstract combinatoric component that assigns "constituent" status to expressions that do not (in any obvious sense) correspond to constituents in the surface representation. In HPSG, the major work in restricting the possible form of ellipsis is instead taken over by constructional schemata, which can encode syntactic information of the antecedent to capture connectivity effects, as is done, for example, with the use of the SAL-UTT feature in Ginzburg & Sag's (2000: Chapter 8) analysis of sluicing (cf. Nykiel & Kim 2021, Chapter 19 of this volume).

Kubota & Levine (2020: Chapter 8) extend Kubota & Levine's (2017) approach further to the treatment of interactions between VP ellipsis and extraction, which has often been invoked in the earlier literature (in particular, Kennedy 2003) as providing crucial evidence for covert structure analysis of ellipsis phenomena (see also Jacobson 2018 for a related proposal, cast in a variant of CCG). At least some of the counterproposals that Kubota & Levine formulate in their argument against the covert structure analysis seem to be directly compatible with the HPSG approach to ellipsis, but (so far as I am aware) no concrete analysis of extraction/ellipsis interaction currently exists in the HPSG literature.

### **4.2.4 Mismatches in right-node raising**

While right-node raising (RNR) has mostly been discussed in connection to coordination in the literature, it is well-known that RNR is not necessarily restricted to coordination environments (see, for example, Wilder 2018 for a recent overview). Moreover, it has recently been pointed out by Abeillé et al. (2016) and Shiraïshi et al. (2019) that RNR admits certain types of syntactic mismatch between the RNR'ed material and the selecting head in a non-adjacent conjunct.

### Yusuke Kubota

The current literature seems to agree that RNR is not a unitary phenomenon, and that at least some type of RNR should be treated via a mechanism of surface ellipsis, which could be modeled as deletion of syntactic (or prosodic) objects or via some sort of anaphoric mechanism (cf. Nykiel & Kim 2021: Section 6.2, Chapter 19 of this volume, Chaves 2014, Shiraïshi et al. 2019; see also Kubota & Levine 2017: footnote 15).

One point that is worth emphasizing in this connection is that while the "NCC as constituent coordination" analysis of RNR in CG discussed in Section 2.4.1 (major evidence for which comes from the interactions between various sorts of scopal operators and RNR as noted in Section 4.2.1) is well-known, neither CCG nor TLCG is by any means committed to the idea that *all* instances of RNR should be analyzed this way. In fact, given the extensive evidence for the non-unitary nature of RNR reviewed in Chaves (2014) and the syntactic mismatch data from French offered by Abeillé et al. (2016) and Shiraïshi et al. (2019), it seems that a comprehensive account of RNR in CG (or, for that matter, in any other theory) would need to recognize the non-unitary nature of the phenomenon, along lines similar to Chaves's (2014) recent proposal in HPSG. While there is currently no detailed comprehensive account of RNR along these lines in the CG literature, there does not seem to be any inherent obstacle for formulating such an account.

# **4.3 Binding**

Empirical phenomena that have traditionally been analyzed by means of Binding Theory (both in the transformational and the non-transformational literature; cf. Müller 2021a, Chapter 20 of this volume) potentially pose a major challenge to the "non-representational" view of the syntax-semantics interface common to most variants of CG. The HPSG Binding Theory in Pollard & Sag (1992; 1994: Chapter 6) captures Principles A and B at the level of argument structure, while Principle C makes reference to the configurational structure (i.e. the featurestructure encoding of the constituent geometry). The status of Principle C itself is controversial to begin with, but if this condition needs to be stated in the syntax, it would possibly constitute one of the greatest challenges to CG-based theories of syntax, since, unlike phrase structure trees, the proof trees in CG are not objects that a principle of grammar can directly refer to.

While there seems to be no consensus in the current CG literature on how the standard facts about binding theory are to be accounted for, there are some important ideas and proposals in the wider literature of CG-based syntax (broadly construed to include work in the Montague Grammar tradition). First, as for Principle A, there is a recurrent suggestion in the literature that these effects

#### 29 HPSG and Categorial Grammar

can (and should) be captured simply via strictly lexical properties of reflexive pronouns (e.g. Keenan 1988; Szabolcsi 1992; see Büring 2005: 43–44 for a concise summary). For example, for a reflexive in the direct object position of a transitive verb bound by the subject NP, the following type assignment (where the reflexive pronoun first takes a transitive verb and then the subject NP as arguments) suffices to capture its bound status:

# (48) himself; *.*() (); ((NP\S)/NP)\NP\S

This approach is attractively simple, but there are at least two things to keep in mind, in order to make it a complete analysis of Principle A in CG. First, while this lexical treatment of reflexive binding may at first sight appear to capture the locality of binding quite nicely, CG's flexible syntax potentially overgenerates unacceptable long-distance binding readings for (English) reflexives. Since RNR can take place across clause boundaries, it seems necessary to assume that hypothetical reasoning for the Lambek-slash (or a chain of Function Composition that has the same effect in CCG) can generally take place across clause boundaries. But then, expressions such as *thinks Bill hates* can be assigned the same syntactic type (i.e. (NP\S)/NP) as lexical transitive verbs, overgenerating non-local binding of a reflexive from a subject NP in the upstairs clause (\* *John thinks Bill hates himself* ).

In order to prevent this situation while still retaining the lexical analysis of reflexivization sketched above, some kind of restriction needs to be imposed as to the way in which reflexives combine with other linguistic expressions. One possibility would be to distinguish between lexical transitive verbs and derived transitive verb-like expressions by positing different "modes of composition" in the two cases in a "multi-modal" version of CG.

The other issue is that the lexical entry in (48) needs to be generalized to cover all cases in which a reflexive is bound by an argument that is higher in the obliqueness hierarchy. This amounts to positing a polymorphic lexical entry for the reflexive. The use of polymorphism is not itself a problem, since it is needed in other places in the grammar (such as coordination) anyway. But this account would amount to capturing the Principle A effects purely in terms of the specific lexical encoding for reflexive pronouns (unlike the treatment in HPSG which explicitly refers to the obliqueness hierarchy).

While Principle A effects are in essence amenable to a relatively simple lexical treatment along lines sketched above, Principle B turns out to be considerably more challenging for CG. To see this point, note that the lexical analysis of reflexives sketched above crucially relies on the fact that the constraint associ-

### Yusuke Kubota

ated with reflexives corresponds to a straightforward semantic effect of variable binding. Pronouns instead require *disjointness* of reference from less oblique coarguments, but such an effect cannot be captured by simply specifying some appropriate lambda term as the semantic translation for the pronoun.

To date, the most detailed treatment of Principle B effects in CG that explicitly addresses this difficulty is the proposal by Jacobson (2008), formulated in a version of CCG (Steedman 1996 proposes a different approach to binding, which will be briefly discussed at the end of this section). The key idea of Jacobson's account of Principle B effects is that NPs are divided by a binary-valued feature ±p, with pronouns marked NP[+p] and all other NPs NP[−p]. In all lexical entries of the form in (49), all NP (and PP) arguments in any realization of /\$ are specified as [−p].26

(49) *k*; ; VP/\$

The effect of this restriction is to rule out pronouns from argument positions of verbs with ordinary semantic denotations. On this approach, the only way a lexically specified functional category can take [+p] arguments is via the application of the following irreflexive operator:27

	- b. \* John talked to Mary about her .
	- c. \* John explained himself to him .

Cases such as the following also call for an extension (also a relatively straightforward one):

26Here, /\$ is an abbreviation of a sequence of argument categories sought via /. Thus, VP/\$ can be instantiated as VP/NP, VP/NP/NP, VP/PP/NP, etc.

27For expository purposes, I state the operator in (50) in its most restricted form, dealing with only the case where there is a single syntactic argument apart from the subject. A much broader coverage is of course necessary in order to handle cases like the following:

What is needed in effect is a schematic type specification that applies to a pronoun in any or all argument positions, i.e., stated on an input of the form VP/\$/XP[−p]/\$ to yield an output of the form VP/\$/XP[+p]/\$. To ensure the correct implementation of this extension, some version of the "wrapping" analysis needs to be assumed (cf. Jacobson 2008: 194), so that the order of the arguments in verbs' lexical entries is isomorphic to the obliqueness hierarchy (of the sort discussed by Pollard & Sag 1992).

(ii) \* John is proud of him .

By assuming (following Jacobson 2008) that the [±p] feature percolates from NPs to PPs and by generalizing the irreflexive operator still further so that it applies not just to VP/XP[−p] but to AP/XP[−p] as well, the ungrammaticality of (ii) follows straightforwardly.

29 HPSG and Categorial Grammar

# (50) φ*.*φ;  *.* () ()*,* ≠ ; (VP/NP[+p])↾(VP/NP[−p])

The greyed-in part ≠ separated from the truth conditional meaning by a comma is a presupposition introduced by the pronoun-seeking variant of the predicate. It says that the subject and object arguments are forced to pick out different objects in the model. For the semantics of pronouns themselves, one can assume, following the standard practice, that free (i.e. unbound) pronouns are simply translated as arbitrary variables (cf. Cooper 1979).

Crucially, the operator in (50) is restricted in its domain of application to the set of signs which are specified in the lexicon. I notate this restriction by using the dashed line notion in what follows. Then (51) will be derived as in (52).

(51) John praises him.

$$\begin{array}{lcl} \text{(52)} & \lambda \upmu\text{.q};\\ & \lambda f \lambda \ulu\&\nu.f(u)(v), u \neq v; & \text{pairs};\\ & \text{(VP/NP[+p])} \{\text{(VP/NP[-p])} & \text{paraise;VP/NP[-p]}\\ & \text{pairs}; \lambda u \lon.\text{pairs}(u)(v), u \neq v; \text{VP/NP[+p]} & \text{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\textit{\langle>}}}}}}}}}}}}}}}}}}}}\\ \text{\textit{\textit{\textit{\textit{\textit{\cdot}}}}}}}, \text{)} & \text{pairs}; \lambda u \lon.\text{pairs}(u)(v), u \neq v; \text{VP/NP[+p]} & \text{\textit{\textit{\dot{\textit{\langle\textit{\langle\langle\mathbf{\langle\langle\mathbf{\cdot{\langle\langle\mathbf{\boldsymbol{\langle\langle\mathbf{\boldsymbol{\langle\langle\mathbf{\boldsymbol{\langle\langle\mathbf{\boldsymbol{\langle\langle\mathbf{\cdot{\langle\langle\mathbf{\boldsymbol{\langle\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\mathbf{\langle\mathbf{\boldsymbol{\langle\mathbf{\boldsymbol{\langle\$$

The presupposition ≠ **j** ensures that the referent of the pronoun is different from John.

Thus, Jacobson's approach captures the relevant conditions on the interpretation of pronouns essentially as a type of lexical presupposition tied to the denotation of the pronoun-taking verb, and the syntactic feature [±p] mediates the distributional correlation between the pronoun and the verb that subcategorizes for it. The idea is essentially the same as in the HPSG Binding Theory, except that the relevant condition is directly encoded as a restriction on the denotation itself, since the standard CG syntax-semantics interface does not admit of syntactic indices of the sort assumed in HPSG.

Unlike Jacobson's proposal outlined above, Steedman's (1996: Chapter 2) analysis of binding conditions in CCG recognizes the syntactic forms of the logical language that is used to write the denotations of linguistic expressions as the "level" at which binding conditions are stated. This approach can be thought of as a "compromise" which enables a straightforward encoding of the HPSG-style Binding Conditions by (slightly) deviating from the CG doctrine of not admitting any representational object at the syntax-semantics interface (see Dowty 1997 for a critique of the approach to binding by Steedman 1996 discussing this issue clearly).

### Yusuke Kubota

Steedman's approach can be best illustrated by taking a look at the analysis of (53).28

(53) \* Every student praised him .

According to Steedman, pronouns receive translations of the form **pro**(), where **pro** is effectively a term that marks the presence of (the translation of) a pronoun at some particular syntactic position in the logical formula that represents the meaning of the sentence.

With this assumption, the translation for (53) that needs to be ruled out (via Principle B) is as follows:

(54) ∀ [**student**() → **praise**(**pro**()) ()]

And this is where the CCG Binding Theory kicks in. The relevant part of the structure of the logical formula in (54) can be more perspicuously written as a tree as in Figure 2, which makes clear the hierarchical relation between sub-terms.29 Principle B states that pronouns need to be locally free. Figure 2 violates this

Figure 2: Logical formula as a tree

28At the same time that he formulates an essentially syntactic account of Principle B via the term **pro** in the translation language, Steedman (1996: 29) briefly speculates on the (somewhat radical) possibility of relegating Principle B entirely to the pragmatic component of pronominal anaphora resolution. However, the relevant discussion is rather sketchy, and the details of such a pragmatic alternative are not entirely clear.

29Since binding conditions are stated at the level of the translation language, this approach raises the issue of whether it can correctly capture the binding relations in constructions in which there is a mismatch between the surface argument structure and the underlying semantics, such as in subject-to-object raising constructions (*John believes himself to be a descendant of Beethoven*). Steedman (1996) does not contain an explicit discussion on this type of data, but it seems likely that one will need to assume a particular syntax for the translation language in order to accommodate this type of data in his approach.

#### 29 HPSG and Categorial Grammar

condition since there is a locally c-commanding term that binds **pro**() (where a term binds term when they are semantically bound by the same operator).

Principles A and C are formulated similarly by making crucial reference to the structures of the terms that represent the semantic translations of sentences.

What one can see from the comparison of different approaches to binding in CG and the treatment of binding in HPSG is that although HPSG and CG are both lexicalist theories of syntax, and there is a general consensus that binding conditions are to be formulated lexically rather than configurationally, there are important differences in the actual implementations of the conditions between approaches that stick to the classical Montagovian tradition (embodying the tenet of "direct compositionality" in Jacobson's terms) and those that make use of (analogues of) representational devices more liberally.

Finally, some comments are in order regarding the status of Principle C, the part of Binding Theory that is supposed to rule out examples such as the following:

	- b. \* He talked to John's brother.

The formulation of Principle C has always been a problem in lexicalist theories of syntax. While Principles A and B can be stated by just making reference to the local argument structure of a predicate in the lexicon, the global nature of Principle C seems to require looking at the whole configurational structure of the sentence in which the referring term appears (but see Branco (2002) for an alternative view; see also Müller (2021a), Chapter 20 of this volume). In fact, Pollard & Sag (1992; 1994: Chapter 6) opt for this solution, and their definition of the Principle C has a somewhat exceptional status within the whole theory (which otherwise adheres to strict locality conditions) in directly referring to the configurational structure.

Essentially the same problem arises in CG. Steedman's (1996) formulation of Principle C can be thought of as an analog of Pollard & Sag's (1992; 1994) proposal, where global reference to hierarchical structure is made not at the level of phrase structure, but instead at the level of "logical structure", that is, in the syntactic structure of the logical language used for writing the meanings of natural language expressions. As already noted above, if one takes the Montagovian, or "direct compositional", view of the syntax-semantics interface that is more traditional/standard in CG research, this option is unavailable.

Thus, Principle C has a somewhat cumbersome place within lexicalist theories in general. However, unlike Principles A and B, the status of Principle C in the

### Yusuke Kubota

grammar is still considerably unclear and controversial to begin with (see Büring 2005: 122–124 for some discussion on this point). In particular, it has been noted in the literature (Lasnik 1986) that there are languages such as Thai and Vietnamese that do not show Principle C effects. If, as suggested by some authors (cf., e.g., Levinson 1987; 1991), the effects of Principle C can be accounted for by pragmatic principles, that would remove one major sticking point in both HPSG and CG formulations of the Binding Theory.

# **5 A brief note on processing and implementation**

The discussion above has mostly focused on linguistic analysis. In this final section, I will briefly comment on implications for psycholinguistics and computational linguistics research.

As should already be clear from the above discussion, different variants of both HPSG and CG make different assumptions about the relationship between the competence grammar and theories of performance. To make things even more complicated, such assumptions are often implicit. As a first approximation, it is probably fair to say that HPSG (at least the "bare-bones" version of it) and CCG are more similar to each other than they are to TLCG in being surfaceoriented. TLCG makes heavy use of hypothetical reasoning in the analyses of certain linguistic phenomena, and, as should already be clear at this point, the role it plays in the grammar is much like the role of movement operations in Mainstream Generative Grammar.

As repeatedly emphasized by practitioners of HPSG and CCG (see, for example, Sag & Wasow 2011, Steedman 2012: Section 13.7 and Wasow 2021, Chapter 24 of this volume), all other things being equal, it is more preferable to make the relationship between the competence grammar and the model of performance as transparent as possible. It is unlikely that any reasonable researcher would deny such a claim, but it begs one big question: how exactly are we to understand the qualification "all other things being equal"? Practitioners of TLCG in general seem to have a somewhat more detached take on the relationship between competence and performance, and I believe the consensus there is more in line with (what seems to be) the spirit of Mainstream Generative grammar: the goal is to clarify the most fundamental principles of grammar and state them in the simplest form possible. TLCG subscribes to the thesis that (a certain variety of) logic is indeed the underlying principle of grammar of natural language. This is an attractive view, but at the same time language exhibits phenomena that suggest that pushing this perspective to the limit is unlikely to be the most fruitful

#### 29 HPSG and Categorial Grammar

research strategy. The right approach is probably one that combines the insights of both surface-oriented approaches (such as HPSG and CCG) and more abstract approaches (such as TLCG and Mainstream Generative Grammar).

At a more specific level, one attractive feature of CCG (but not CG in general), when viewed as an integrated model of the competence grammar and human sentence processing, is that it enables surface-oriented, incremental analyses of strings from left to right. This aspect was emphasized in the early literature of CCG (Ades & Steedman 1982; Crain & Steedman 1985), but it does not seem to have had much impact on psycholinguistic research in general since then. A notable exception is the work by Pickering & Barry (1991; 1993) in the early 90s. There is also some work on the relationship between processing and TLCG (see Morrill 2011: Chapters 9 and 10, and references therein). In any event, a serious investigation of the relationship between competence grammar and human sentence processing from a CG perspective (either CCG or TLCG) is a research topic that is waiting to be explored, much like the situation with HPSG (see Wasow 2021, Chapter 24 of this volume).

As for connections to computational linguistics (CL)/natural language processing (NLP) research, like HPSG (cf. Bender & Emerson 2021, Chapter 25 of this volume), large-scale computational implementation has been an important research agenda for CCG (see, for example, White & Baldridge 2003; Clark & Curran 2007). I refer the reader to Steedman (2012: Chapter 13) for an excellent summary on this subject (this chapter contains a discussion of human sentence processing as well). Together with work on linguistically informed parsing in HPSG, CCG parsers seem to be attracting some renewed interest in CL/NLP research recently, due to the new trend of combining the insights of statistical approaches and linguistically-informed approaches. In particular, the straightforward syntax-semantics interface of (C)CG is an attractive feature in building CL/NLP systems that have an explicit logical representation of meaning. See, for example, Lewis & Steedman (2013) and Mineshima et al. (2016) for this type of work. TLCG research has traditionally been less directly related to CL/NLP research. But there are recent attempts at constructing large-scale treebanks (Moot 2015) and combining TLCG frameworks with more mainstream approaches in NLP research such as distributional semantics (Moot 2018).

# **6 Conclusion**

As should be clear from the above discussion, HPSG and CG share many important similarities, mainly due to the fact that they are both variants of lexicalist

### Yusuke Kubota

syntactic theories. This is particularly clear in the analyses of local dependencies in terms of lexically encoded argument structure information. Important differences emerge once one turns one's attention to less canonical types of phenomena, such as atypical types of coordination (nonconstituent coordination, Gapping) and the treatment of "constructional" patterns that are not easily lexicalizable. In general, HPSG has a richer and more comprehensive treatment of various empirical phenomena, whereas CG has a lot to offer to grammatical theory (perhaps somewhat paradoxically) due to the very fact that the potentials of the logic-based perspective it embodies has not yet been explored in full detail. It is more likely than not that the two will continue to develop as distinct theories of natural language syntax (and semantics). I hope that the discussion in the present chapter has made it clear that there are still many occasions for fruitful interactions between the two approaches both at the level of analytic ideas for specific empirical phenomena and at the more general, foundational level pertaining to the overall architecture of grammatical theory.

# **Acknowledgments**

I'd like to thank Jean-Pierre Koenig, Bob Levine and Stefan Müller for comments. This work is supported by the NINJAL collaborative research project "Crosslinguistic Studies of Japanese Prosody and Grammar".

# **References**

29 HPSG and Categorial Grammar

### Yusuke Kubota

Publications. http : / / csli - publications . stanford . edu / HPSG / 2004 / beavers sag.pdf (10 February, 2021).

29 HPSG and Categorial Grammar

### Yusuke Kubota

#### 29 HPSG and Categorial Grammar

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*and norms in science* (Synthese Library 260), 347–368. Dordrecht: Springer Verlag. DOI: 10.1007/978-94-017-0538-7\_21.

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### Yusuke Kubota

Lexical-Functional Grammar: A formal system for grammatical representation. In Mary Dalrymple, Ronald M. Kaplan, John T. Maxwell III & Annie Zaenen (eds.), *Formal issues in Lexical-Functional Grammar* (CSLI Lecture Notes 47), 29–130. Stanford, CA: CSLI Publications, 1995.

29 HPSG and Categorial Grammar

### Yusuke Kubota

Culicover & Paul M. Postal (eds.), *Parasitic gaps* (Current Studies in Linguistics 35), 181–222. Cambridge, MA: MIT Press.

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### Yusuke Kubota

#### 29 HPSG and Categorial Grammar

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*plicit models of grammar: A guide to current models*, 225–267. Oxford: Wiley-Blackwell. DOI: 10.1002/9781444395037.

29 HPSG and Categorial Grammar

### Yusuke Kubota

29 HPSG and Categorial Grammar

#### Yusuke Kubota

Zaenen, Annie. 1983. On syntactic binding. *Linguistic Inquiry* 14(3). 469–504.

# **Chapter 30**

# **HPSG and Lexical Functional Grammar**

Stephen Wechsler The University of Texas

# Ash Asudeh

University of Rochester & Carleton University

> This chapter compares two closely related grammatical frameworks, Head-Driven Phrase Structure Grammar (HPSG) and Lexical Functional Grammar (LFG). Among the similarities: both frameworks draw a lexicalist distinction between morphology and syntax, both associate certain words with lexical argument structures, both employ semantic theories based on underspecification, and both are fully explicit and computationally implemented. The two frameworks make available many of the same representational resources. Typical differences between the analyses proffered under the two frameworks can often be traced to concomitant differences of emphasis in the design orientations of their founding formulations: while HPSG's origins emphasized the formal representation of syntactic locality conditions, those of LFG emphasized the formal representation of functional equivalence classes across grammatical structures. Our comparison of the two theories includes a point by point syntactic comparison, after which we turn to an exposition of Glue Semantics, a theory of semantic composition closely associated with LFG.

# **1 Introduction**

Head-Driven Phrase Structure Grammar is similar in many respects to its sister framework, Lexical Functional Grammar or LFG (Bresnan et al. 2016; Dalrymple et al. 2019). Both HPSG and LFG are lexicalist frameworks in the sense that they distinguish between the morphological system that creates words and the syntax proper that combines those fully inflected words into phrases and sentences.

Stephen Wechsler & Ash Asudeh. 2021. HPSG and Lexical Functional Grammar. In Stefan Müller, Anne Abeillé, Robert D. Borsley & Jean- Pierre Koenig (eds.), *Head-Driven Phrase Structure Grammar: The handbook*, 1395–1446. Berlin: Language Science Press. DOI: 10.5281/zenodo.5599878

#### Stephen Wechsler & Ash Asudeh

Both frameworks assume a lexical theory of argument structure (Müller & Wechsler 2014; compare also Davis, Koenig & Wechsler 2021, Chapter 9 of this volume) in which verbs and other predicators come equipped with valence structures indicating the kinds of complements that the word is to be combined with. Both theories treat certain instances of control (equi) and raising as lexical properties of control or raising predicates (on control and raising in HPSG see Abeillé 2021, Chapter 12 of this volume). Both theories allow phonologically empty nodes in the constituent structure, although researchers in both theories tend to avoid them unless they are well-motivated (Sag & Fodor 1995; Berman 1997; Dalrymple et al. 2019: 734–742). Both frameworks use recursively embedded attribute-value matrices (AVMs). These structures directly model linguistic expressions in LFG, but are understood in HPSG as grammatical descriptions satisfied by the directed acyclic graphs that model utterances.1

There are also some differences in representational resources, especially in the original formulations of the two frameworks. But each framework now exists in many variants, and features that were originally exclusive to one framework can often now be found in some variant of the other. HPSG's valence lists are ordered, while those of LFG usually are not, butAndrews & Manning (1999) use an ordered list of terms (subject and objects) in LFG. LFG represents grammatical relations in a *functional structure* or f-structure that is autonomous from the constituent structure, while HPSG usually lacks anything like a functional structure. But in Bender's (2008) version of HPSG, the COMPS list functions very much like LFG's f-structure (see Section 10 below). This chapter explores the utility of various formal devices for the description of natural language grammars, but since those devices are not exclusively intrinsic to one framework, this discussion does not bear on the comparative utility of the frameworks themselves. For a comparison of the representational architectures and formal assumptions of the two theories, see Przepiórkowski (2021), which complements this chapter.

We start with a look at the design considerations guiding the development of the two theories, followed by point by point comparisons of syntactic issues organized by grammatical topic. Then we turn to the semantic system of LFG, beginning with a brief history by way of explaining its motivations, and continuing with a presentation of the semantic theory itself. A comparison with HPSG semantics is impractical and will not be attempted here, but see Koenig & Richter (2021), Chapter 22 of this volume for an overview of HPSG semantics.

1Both frameworks are historically related to unification grammar (Kay 1984). Unification is defined as an operation on feature structures: the result of unifying two mutually consistent feature structures is a feature structure containing all and only the information in the original two feature structures. Neither framework actually employs a unification operation, but unification can produce structures resembling the ones in use in LFG and HPSG.

30 HPSG and Lexical Functional Grammar

# **2 Design principles in the origins of HPSG and LFG**

The HPSG and LFG frameworks were originally motivated by rather different design considerations. In order to make a meaningful comparison between the frameworks, it is helpful to understand those differences.

HPSG grew out of the tradition, established by Chomsky (1957), of studying the computational properties of natural language syntax as a window onto the capabilities of human cognitive processes. The advent of Generalized Phrase Structure Grammar (Gazdar et al. 1985) was an important milestone in that tradition, bringing as it did the surprising prospect that the entire range of syntactic phenomena known to exist at the time could be described with a context-free grammar (CFG). If this could be maintained, it would mean that natural language is context-free in the sense of the Chomsky Hierarchy (Chomsky 1957), which is an answer to the question raised by Chomsky that was very different from, and more interesting than, Chomsky's own answer. Then Shieber (1985) and Culy (1985) showed that certain phenomena such as recursive cross-serial dependencies in Swiss German and Bambara exceeded the generative capacity of a context-free grammar.

Despite that development, it nonetheless appeared that languages for the most part hewed rather closely to the context-free design. Thus began the search for more powerful grammatical formalisms that would preserve the insight of a context-free grammar while allowing for certain phenomena that exceed that generative capacity.2 HPSG grew out of this search. As Sag et al. (2003: 83) observe, HPSG "is still closely related to standard CFG". In fact, an HPSG grammar largely consists of constraints on local sub-trees (i.e., trees consisting of a node and its immediate daughters), which would make it a context-free grammar were it not that the nodes themselves are complex, recursively-defined feature structures. This CFG-like character of HPSG means that the framework itself has the potential to embody an interesting theory of locality. At the same time, the original theory also allowed for the description of non-local relations, and new non-local devices were added in versions of the theory developed later (a proposal to add functional structure was mentioned in Section 1, and others will be described below). The flexibility of HPSG thus provides for the study of locality and non-locality, while also allowing for grammatical description and theory construction by syntacticians with no interest in locality.

The architecture of LFG was originally motivated by rather different concerns: an interest in typological variation from a broadly functionalist perspective, from

2There were also some problems with the GPSG theory of the lexicon, in which complement selection was assimilated to the phrase structure grammar. For discussion see Müller & Wechsler (2014: Section 4.1).

#### Stephen Wechsler & Ash Asudeh

which one studies cross-linguistic variation in the expression of functionally equivalent elements of grammar. For this reason two levels of representation are discerned: a functional structure or *f-structure* representing the internal grammatical relations in a sentence that are largely invariant across languages, and a categorial constituent structure or *c-structure* representing the external morphological and syntactic expression of those relations, which vary, often rather dramatically, across different typological varieties of language. For example, probably all (or nearly all) languages have subjects and objects, hence those relations are represented in f-structure. But languages vary as to the mechanisms for signaling subjecthood and objecthood, the three main mechanisms being word order, head-marking, and dependent-marking (Nichols 1986), hence those mechanisms are distinguished at c-structure. The word *functional* in the name of the LFG framework is a three-way pun, referring to the grammatical *functions* that play such an important role in the framework, the mathematical *functions* that are the basis for the representational formalism, and the generally *functionalist*friendly nature of the LFG approach.

Despite these differing design motivations, there is no dichotomy between the frameworks with respect to the actual research undertaken within the two research communities. Typological variation within almost every area of grammar has been studied in HPSG, and locality is studied within LFG by developing theories of the mapping between c-structure and f-structure (see Bresnan et al. 2016: 88–128). In the remainder of this chapter we will survey various phenomena and compare HPSG and LFG approaches.

# **3 Phrases and endocentricity**

A phrasal node shares certain grammatical features with specific daughters. In HPSG, this is accomplished by means of structure-sharing (reentrancies) in the immediate dominance schemata and other constraints on local sub-trees such as the Head Feature Principle. LFG employs essentially the same mechanism for feature sharing in a local sub-tree but implements it slightly differently, so as to better address the design motivations of the theory. Each node in a phrase structure is paired with an f-structure, which is formally a set of attribute-value pairs. It is through the f-structure that the nodes of the phrase structure share features. The phrase structure is referred to as *c-structure*, for categorial or constituent structure, in order to distinguish it from f-structure. Context-free phrase structure rules license c-structures, and the c-structure elements are annotated with *functional equations* which describe the corresponding f-structure. The correspondence function from c-structures to f-structures, , defines and constrains the f-structure on the basis of the equations collected from the c-structure anno-

#### 30 HPSG and Lexical Functional Grammar

tations and from lexical entries of the terminal nodes.3 For example, the phrase structure grammar in (1) and lexicon in (2) license the tree in Figure 1. 4*,*5

$$\begin{array}{ccccc} \text{(1)} & \text{a. } \text{S} & \rightarrow & \text{NP} & \text{VP} \\ & & & \text{(\uparrow\\_sub)} = \downarrow & \uparrow = \downarrow \\ & \text{b. } \text{NP} & \rightarrow & \begin{pmatrix} \text{Det} \\ \uparrow = \downarrow \end{pmatrix} & \uparrow = \downarrow \\ & \text{c. } \text{VP} & \rightarrow & \text{V} \\ & & \uparrow = \downarrow & \binom{\uparrow}{\uparrow} \text{onj} \langle \rangle = \downarrow \end{array}$$

$$\begin{array}{ccccc} \text{(2)} & \text{a. } \text{this:} \text{Det} & \quad \text{(\uparrow\\_\text{PRO})} = \downarrow \\ & \text{(\uparrow\\_\text{NUM})} = \text{sc} \\ & \text{b. } \text{ lion: } \text{N} & \quad \text{(\uparrow\\_\text{PRED}) = \langle \text{lion'} \\ & \text{(\uparrow\\_\text{PRED}) = \langle \text{so} \\ & \text{c. } \text{ roor: V} & \quad \langle \uparrow \text{PRED} \rangle = \text{'roor} \\ & \text{-s: } \text{infl} & \quad \langle \uparrow \text{TENSE} \rangle = \text{PRS} \\ & \text{(\uparrow\\_\text{SUB})} = \downarrow \\ & \text{(\downarrow\\_\text{PERS}) = 3 \\ & \text{(\downarrow\\_\text{NUM})} = \text{sg} \end{array}$$

Each node in the c-structure maps to an f-structure, that is, to a set of attributevalue pairs. Within the equations, the up and down arrows are metavariables over f-structure labels, interpreted as follows: the up arrow refers to the f-structure to which the c-structure mother node maps, and the down arrow refers to the f-structure that its own c-structure node maps to. To derive the f-structure from Figure 1, we instantiate the metavariables to specific function names and solve for the f-structure associated with the root node (here, S). In Figure 2, the f-structure labels 1 , 2, etc. are subscripted to the node labels. The arrows have been replaced with those labels.

Collecting all the equations from this tree and solving for 1 , we arrive at the f-structure in (3):

3Taken together, the set of equations in a c-structure is called the functional description or *f-description*.

4 In (2c) the verb is broken down into the root *roar* and third person singular suffix *-s* to show the functional equations contributed by each morpheme. However, LFG does not require that words be analyzed into morphemes. It is compatible with morpheme-based (e.g., Ishikawa 1985; Bresnan et al. 2016: 384–385, 395–396) or realizational morphology (Dalrymple et al. 2019: Chapter 12) or any other theory of morphology that associates word forms with grammatical features.

5For simplicity's sake, in this basic example we assume an NP analysis rather than a DP one (Brame 1982). However, much recent LFG work assumes DP; see Bresnan et al. (2016) and Dalrymple et al. (2019) for further discussion.

#### 30 HPSG and Lexical Functional Grammar

$$\begin{array}{c} \text{(3)} \quad f\_{\text{i}} \begin{bmatrix} \text{PRED \\_\text{'Iion'}}\\ \text{SUBJ} \\ \text{PERS 3}\\ \text{PERS} + \\ \text{PRED} \end{bmatrix} \end{array} \Bigg| \begin{array}{c} \text{(PRED \\_\text{'Iion'})}\\ \text{PERS} \end{array} \right] \Bigg| \begin{array}{c} \text{(}\text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \\ \text{PRED} \quad \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \\ \text{TENSE \\_\text{'PRES}} \end{array} \Bigg) \Bigg| \begin{array}{c} \text{(}\text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'S'} \text{'S'} \\ \text{TENSE \\_\text{'PRES}} \end{array} \Bigg) \end{array} \Bigg) \Bigg) \begin{array}{c} \text{(}\text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'S'} \text{'S'} \text{'S'} \text{'S'} \text{'S'} \text{'S'} \text{'S'} \text{'S'} \text{'S'} \text{'S'} \text{'S'} \end{array} \right) \Bigg) \begin{array}{c} \text{(}\text{'Ion'} \text{'Ion'} \text{'Ion'} \text{'S'} \text{'S'} \text{'S'} \text{'S'} \text{'S'} \text{'S'} \text{'S'}$$

F-structures are subject to three general well-formedness conditions/principles:6

Completeness and Coherence together play a similar role to the Valence Principle in HPSG (Pollard & Sag 1994: 348) in that they guarantee that only exactly the right number of syntactic dependents are in the structure. Uniqueness means that an f-structure has to be consistent in its feature values7 and also means that the f-structure is a function in the mathematical sense, a set of ordered pairs such that no two pairs have the same first member. Moreover, each PRED value is assumed to bear a unique index (normally suppressed, as we do here), so that even two instances of apparently the "same" PRED value cannot be identical; this plays an important role in LFG's analysis of pronominal affixes and agreement (see Section 7).

Since the up and down arrows refer to nodes of the local sub-tree, LFG annotated phrase structure rules like those in (1) can often be directly translated into HPSG immediate dominance schemata and principles constraining local sub-trees. By way of illustration, let FS (for *f-structure*) be an HPSG attribute corresponding to the f-structure projection function. Then the LFG rule in (7a) (repeated from (1a) above) is equivalent to the HPSG rule in (7b):

$$\begin{array}{ccccc} \text{(7)} & \text{a. } \text{S} & \rightarrow & \text{NP} & \text{VP} \\ & & & \text{(\uparrow} \text{ sUB} \text{j}) = \downarrow & \uparrow = \downarrow \\ \text{b. } & \text{S[\_{Fs}[\square]} & \rightarrow & \text{NP[\_{Fs}[\square]]} & \text{VP[\_{Fs}[\square][\text{sUB} \text{j}]]} \\ \end{array}$$

Let us compare the two representations with respect to heads and dependents.

Taking heads first, the VP node annotated with ↑ = ↓ is an *f-structure head*, meaning that the features of the VP are identified with those of the mother S. This

6Here we state them informally, just to capture the key intuitions, but see Kaplan & Bresnan (1982: 37 (reprint pagination)) or Dalrymple et al. (2019: 52–53) for precise definitions.

7 In fact, another name for the uniqueness principle is *Consistency* (Dalrymple et al. 2019: 53).

#### Stephen Wechsler & Ash Asudeh

effect is equivalent to the tag 1 in (7b). Hence ↑ = ↓ has an effect similar to HPSG's Head Feature Principle. However, in LFG the part of speech categories and their projections such as N, V, Det, NP, VP, DP, etc. belong to the c-structure and not the f-structure. As a consequence, those features are not subject to sharing, and any principled correlations between such categories, such as the fact that N is the head of NP, V the head of VP, C the head of CP, and so on, are instead captured in an explicit version of (extended) X-bar theory applying exclusively to the c-structure (Grimshaw 1998). The version of extended X-bar theory in Bresnan et al. (2016: Chapter 6) assumes that all nodes on the right side of the arrow of the phrase structure rule are optional, with many unacceptable partial structures ruled out in the f-structure instead. Also, not all structures need to be endocentric (i.e., not all structures have a head daughter in c-structure). The LFG category S shown in (7a) is inherently exocentric, lacking a c-structure head whose c-structure category could influence its external syntax (the f-structure head of S is the daughter with the ↑ = ↓ annotation, here the VP). (English is also assumed to have endocentric clauses of category IP, where an auxiliary verb of category I (for Inflection) serves as the c-structure head.) S is used for copulaless clauses and also for the flat structures of nonconfigurational clauses in languages such as Warlpiri (see Section 10 below).

Functional projections like DP, IP, and CP are typically assumed to form a "shell" over the lexical projections NP, VP, AP, and PP (plus CP can appear over S). While this assumption is widespread in transformational approaches, its origins can be found in non-transformational research, including early LFG: CP was proposed in Fassi-Fehri's (1981: 141) LFG treatment of Arabic, and IP (the idea of the sentence as functional projection) is found in Falk's (1983) LFG analysis of the English auxiliary system (Falk called it "MP" instead of "IP").8

Extended projections are formally implemented in LFG by having the functional head (such as D) and its lexical complement (such as NP) be f-structure co-heads. See for example the DP *this lion* in Figure 3, where D, NP, and N are all annotated with ↑ = ↓, hence the DP, D, NP, and N nodes all map to the same f-structure. What makes this identity possible is that function words lack a PRED feature that would otherwise indicate a semantic form.9 Content words such as *lion* have such a feature ([PRED 'lion']), and so if the D had one as well, then they would clash in the f-structure. Note more generally that the f-structure flattens out much of the hierarchical structure of the corresponding c-structure.

8For more on the origins of extended projections see Bresnan et al. (2016: 124–125).

9The attribute PRED ostensibly stands for "predicate", but really it means something more like "has semantic content", as there are lexical items, such as proper names, which have PRED features but are not predicates under standard assumptions.

#### 30 HPSG and Lexical Functional Grammar

Figure 3: Functional heads in LFG

Complementation works a little differently in LFG from HPSG. Note that the LFG rule (7a) indicates the SUBJ grammatical function on the subject NP node, while the pseudo-HPSG rule (7b) indicates the SUBJ function on the VP functor selecting the subject. A consequence of the use of functional equations in LFG is that a grammatical relation such as SUBJ can be locally associated with its formal exponents, whether a configurational position in phrase structure (as in Figure 1), head-marking (agreement, see (2c)), or dependent marking (case). A nominative case affix specialized for exclusively marking subjects can introduce a so-called "inside-out" functional designator, (SUBJ ↑), which requires that the f-structure of the NP or DP bearing that case ending be the value of a SUBJ attribute (Nordlinger 1998).10 Other argument cases effectively resolve an annotation on the NP or DP node to the appropriate grammatical function. In all of these situations, the attribute encoding a grammatical function, such as SUBJ or OBJ, is directly associated with an element filling that function. This aspect of LFG representations makes it convenient for functionalist and typological work on grammatical relations.

10Inside-out designators can be identified by their special syntax: the attribute symbol (here SUBJ) precedes instead of following the function symbol (here the metavariable ↑). They are defined as follows: for function *f*, attribute *a*, and value *v*, () = iff () = . If *f* is the f-structure representing a nominal in nominative case, then (SUBJ *f* ) refers to the f-structure of the clause whose subject is that nominal.

Stephen Wechsler & Ash Asudeh

# **4 Grammatical functions and valence**

*Grammatical functions* (or *grammatical relations*) like subject and object play an important role in LFG theory, and it is worthwhile to compare the status of grammatical functions in LFG and HPSG. Grammatical functions in LFG are best understood in the context of the break with Transformational Grammar that led to LFG and other alternative frameworks. Chomsky (1965: 68–74) argued that while grammatical functions clearly play a role in grammar, they need not be explicitly incorporated into the grammar as such. Instead he proposed to define them in phrase structural terms: the "subject" is the NP immediately dominated by S, the "object" is the NP immediately dominated by VP, and so on. This theoretical assumption necessitated the use of transformations and a profusion of certain null elements: an affixal subject, for example, would have to be inserted under an NP node that is the daughter of S, and then moved to its surface position as affix; a subject that is phonologically null but anaphorically active would have to be generated in that position as well, hence the need for "null *pro*". Early alternatives to Transformational Grammar such as Relational Grammar (Perlmutter 1983) and LFG also sought to capture the equivalence across different alternative expressions of a single argument, but rejected the transformational model. Instead the grammar licenses an abstract representation of the "subject", for example, essentially as a set of features. The language-specific grammar maps a predicator's semantic roles to these abstract representations (named "subject", "object", and so on), and also maps those representations onto the expressions in the language.

As a non-transformational theory, HPSG breaks down the relation between semantic roles and their grammatical expressions into two distinct mappings, in roughly the same way that LFG does. The intermediate representations of arguments consists of sets of features, just like in LFG. However, while LFG provides a consistent cross-linguistic representation of each grammatical function type, many variants of the HPSG framework allow the representation to vary from language to language. For example, LFG identifies subjects in every language with the attribute SUBJ, while HPSG identifies the subject, if at all, as a specific element of the ARG-ST list, but which element it is can vary with the language. An English subject is usually assumed to be the first element in the list, or ARG-ST|FIRST, while the definition of a German subject necessarily involves the CASE feature (Reis 1982). Note that it would not work to represent German subjects as ARG-ST|FIRST, because in German subjectless sentences with arguments (such as datives), the first item in the ARG-ST list is a non-subject. Also, the HPSG valence feature SUBJ (or SPR "specifier") does not represent the subject grammatical

#### 30 HPSG and Lexical Functional Grammar

function in the LFG sense, but is instead closer to Chomsky's (1965: 68–74) definition of subject as an NP in a particular phrase structural position. However, nothing precludes an HPSG practitioner from adopting LFG-style grammatical function features within HPSG, and many have done so, such as the qVAL feature of Kropp Dakubu et al. (2007) or the ERG feature proposed by Pollard (1994). Similarly, Andrews & Manning (1999) present a version of LFG that incorporates an HPSG-style valence list.

This illustrates once again LFG's orientation towards cross-linguistic grammar comparison. HPSG practitioners who do not adopt an LFG-style SUBJ feature can still formulate theories of "subjects", comparing subjects in English, German, Icelandic, and so on, but they lack a formal correlate of the notion "subject" comparable to LFG's SUBJ feature. Meanwhile from the perspective of a single grammar, the original motivation for adopting grammatical functions, in reaction to the pan-phrase-structural view of the transformational approach, informed both LFG and HPSG in the same way.

In LFG, a lexical predicator such as a verb selects its complements via grammatical functions, which are native to f-structure, rather than using c-structure categories. A transitive verb selects a SUBJ and OBJ, which are features of f-structure, but it cannot select for the category DP because such part of speech categories belong only to c-structure. For example, the verb stem *roar* in (2c) has a PRED feature whose value contains (↑ SUBJ), which has the effect of requiring a SUBJ function in the f-structure. The f-structure (shown in (3)) is built using the defining equations, as described above. Then that f-structure is checked against any *existential constraints* such as the expression (↑ SUBJ), which requires that the f-structure contain a SUBJ feature. That constraint is satisfied, as shown in (3). Moreover, the fact that (↑ SUBJ) appears in the angled brackets means that it expresses a semantic role of the 'roar' relation, hence the SUBJ value is required to contain a PRED feature, which is satisfied by the feature [PRED 'lion'] in (3).

Selection for grammatical relations instead of formal categories enables LFG to capture the flexibility in the expression of a given grammatical relation described at the end of the previous section. As noted there, in many languages the subject can be expressed either as an independent NP/DP phrase as in English, or as a pronominal affix on the verb. As long as the affix introduces a PRED feature and is designated by the grammar as filling the SUBJ relation, then it satisfies the subcategorization requirements imposed by a verb. A more subtle example of flexible expression of grammatical functions can be seen in English constructions where an argument can in principle take the form of either a DP (as in (8a)) or a clause (as in (8b)) (the examples in (8) are from Bresnan et al. 2016: 11–12).

#### Stephen Wechsler & Ash Asudeh

	- b. That he was sick, we talked about for days.
	- c. We talked about that problem for days.
	- d. \* We talked about that he was sick for days.

The preposition *about* selects neither a DP nor a CP *per se*, but rather selects the grammatical function OBJ.

(9) *about*: P (↑ PRED) = 'abouth(↑ OBJ)i'

It is not the preposition but the local c-structure environment that conditions the category of that argument: the canonical prepositional object position rightadjacent to *about* can only house a DP (compare (8c) and (8d)), while the topic position allows either DP or CP (compare (8a) and (8b)). In LFG, the grammatical functions such as SUBJ and OBJ represent equivalence classes across various modes of c-structure expression.

Some HPSG approaches to filler-gap mismatches are presented in Borsley & Crysmann (2021: Section 9), Chapter 13 of this volume. They are essentially similar to the LFG account just presented, in that they work by allowing the preposed clause and the gap to differ in certain features. The main difference between research on this problem in the two frameworks is that in LFG the bifurcation between matching and non-matching features of filler/gap is built into the framework in the separation between f-structure and c-structure.11

# **5 Head mobility**

The lexical head of a phrase can sometimes appear in an alternative position apparently outside of what would normally be its phrasal projection. Assuming that an English finite auxiliary verb is the (category I) head of its (IP) clause, then that auxiliary appears outside its clause in a yes/no question:

(10) a. [IP she is mad]

b. Is [IP she \_ mad]?

Transformational grammars capture the systematic relation between these two structures with a head-movement transformation that leaves the source IP structure intact, with a trace replacing the moved lexical head of the clause. The

11Note that c-structure and f-structure are autonomous but not independent. To constrain the relation between the c-structure category and f-structure, one can use either metacategories (Dalrymple et al. 2019: 691–698) or the CAT function (Dalrymple et al. 2019: 265).

#### 30 HPSG and Lexical Functional Grammar

landing site of the moved clausal head is often assumed to be C, the complementizer position, as motivated by complementarity between the fronted verb and a lexical complementizer. This complementarity is observed most strikingly in German verb-second versus verb-final alternations, but is also found in other languages, including some English constructions such as the following:

	- b. I wonder, is [IP she \_ mad]?
	- c. \* I wonder whether is she mad.

Non-transformational frameworks like HPSG and LFG offer two alternative approaches to head mobility, described in the HPSG context in Müller (2021: Section 5), Chapter 10 of this volume. Let us consider these in turn.

In the constructional approach, the sentences in (11) have been treated as displaying two distinct structures licensed by the grammar (Sag et al. 2020 and other references in Müller 2021: Section 5, Chapter 10 of this volume). For example, assuming ternary branching for the sentence in (10b), then the subject DP *she* and predicate AP *mad* would normally be assumed to be sisters of the fronted auxiliary *is*. On that analysis, the phrase structure is flattened out so that *she mad* is not a constituent. In fact, for English the fronting of *is* can even be seen as a consequence of that flattening: English is a head-initial language, so the two dependents*she* and *mad* are expected to follow their selecting head *is*. This analysis is common in HPSG, and it could be cast within the LFG framework as well.

The second approach is closer in spirit to the head movement posited in Transformational Grammar. It is found in both HPSG and LFG, but the formal implementation is rather different in the two frameworks. The HPSG "head movement" account due to Borsley (1989) posits a phonologically empty element that can function as the verb in its canonical position (such as (10a)), and that empty element is structure-shared with the verb that is fronted. This treats the variation in head position similarly to non-local dependencies involving phrases. See Müller (2021: Section 5.1), Chapter 10 of this volume.

The LFG version of "head movement" takes advantage of the separation of autonomous c- and f-structures. Recall from the above discussion of the DP in Figure 3 that functional heads such as determiners, auxiliaries, and complementizers do not introduce new f-structures, but rather map to the same f-structure as their complement phrases. The finite auxiliary can therefore appear in either the I or C position without this difference in position affecting the f-structure, as we will see presently. Recall also that c-structure nodes are optional and can be omitted as long as a well-formed f-structure is generated. Comparing the non-

terminal structures of Figure 4 and Figure 5, the I preterminal node is omitted from the latter structure, but otherwise they are identical. (The lexical equations in Figure 5 are the same as the ones in Figure 4 but are omitted for clarity.) Given the many ↑ = ↓ annotations, the C, I, and AP nodes (as well as IP and CP) all map to the same f-structure, namely the one shown in (12).

$$\begin{array}{c} \text{(12)} \quad f \begin{bmatrix} \text{PRED \\_} \text{pro} \\ \text{SUBJ} \\ \text{PERS \ 3} \\ \text{GEND FEM} \end{bmatrix} \end{array} \Bigg| \begin{array}{c} \text{(12)} \quad f \end{array} \right]$$

$$\begin{array}{c} \text{(12)} \quad \text{(12)} \quad f \text{(12)} \\ \begin{bmatrix} \text{PRED \\_} \text{mod} \langle \langle f \text{ \\_sup} \rangle \rangle \\ \text{FIN} \end{bmatrix} \end{array}$$

The C and I positions are appropriate for markers of clausal grammatical features such as finiteness ([FIN ±]), encoded either by auxiliary verbs like finite *is* or complementizers like finite *that* and infinitival *for*: *I said that/\*for she is present* vs. *I asked for/\*that her to be present*. English has a specialized class of auxiliary verbs

#### 30 HPSG and Lexical Functional Grammar

Figure 5: Head mobility in LFG

for marking finiteness from the C position, while in languages like German all finite verbs, including main verbs, can appear in a C position that is unoccupied by a lexical complementizer. Summarizing, the LFG framework enables a theory of head mobility based on the intuition that a clause has multiple head positions where inflectional features of the clause are encoded.

# **6 Agreement, case, and constraining equations**

The basic theory of agreement is the same in LFG and HPSG (see Wechsler 2021, Chapter 6 of this volume): agreement occurs when multiple feature sets arising from distinct elements of a sentence specify information about a single abstract object, so that the information must be mutually consistent (Kay 1984). The two forms are said to agree when the values imposed by the two constraints are compatible, while ungrammaticality results when they are incompatible. An LFG example is seen in Figure 1 above, where the noun, determiner, and verbal suffix each specify person and/or number features of the same SUBJ value.

The basic mechanism for case marking works in essentially the same way as agreement in both frameworks: in case marking, distinct elements of a sentence specify case information about a single abstract object, hence that information must be compatible. To account for the contrast in (13a), nominative CASE equa-

#### Stephen Wechsler & Ash Asudeh

tions are associated with the pronoun *she* and added to the entry for the verbal agreement suffix *-s*:

### (13) a. She/\*Her/\*You rules.

b. *she*: D (↑ PRED) = 'pro' (↑ CASE) = NOM (↑ PERS) = 3 (↑ NUM) = SG (↑ GEND) = FEM c. *her*: D (↑ PRED) = 'pro' (↑ CASE) = ACC (↑ PERS) = 3 (↑ NUM) = SG (↑ GEND) = FEM d. *you*: D (↑ PRED) = 'pro' (↑ PERS) = 2 e. -*s*: *infl* (↑ TENSE) = PRES (↑ SUBJ) = ↓ (↓ PERS) = 3 (↓ NUM) = SG (↓ CASE) = NOM

The variant of (13a) with *her* as subject is ruled out due to a clash of CASE within the SUBJ f-structure. The variant with *you* as subject is ruled out due to a clash of PERS features. This mechanism is essentially the same as in HPSG, where it operates via the valence features.

This account allows for underspecification of both the case assigner and the case-bearing element, and of both the trigger and target of agreement. In English, for example, gender is marked on some pronouns but not on a verbal affix, and (nominative) case is not marked on nominals, with the exception of pronouns, but is governed by the finite verb. But certain case and agreement phenomena do not tolerate underspecification, and for those phenomena LFG offers an account using a *constraining equation*, a mechanism absent from HPSG and indeed ruled out by current foundational assumptions of HPSG theory (Richter 2021, Chapter 3 of this volume). (Some early precursors to HPSG included a special feature value called ANY that functioned much like an LFG constraining equation, e.g. Shieber 1986: 36–37, but that device has been eliminated from HPSG.) The functional equations described so far in this chapter work by building the f-structure,

#### 30 HPSG and Lexical Functional Grammar

as illustrated in Figure 1 and (3) above; such equations are called *defining equations*. A constraining equation has the same syntax as a defining equation, but it functions by checking the completed f-structure for the presence of a feature. An f-structure lacking the feature designated by the constraining equation is illformed.

The following lexical entry for *she* is identical to the one in (13b) above, except that the CASE equation has been replaced with a constraining equation, notated with = .

(14) *she*: D (↑ PRED) = 'pro' (↑ CASE) = NOM (↑ PERS) = 3 (↑ NUM) = SG (↑ GEND) = FEM

The f-structure is built from the defining equations, after which the SUBJ field is checked for the presence of the [CASE NOM] feature, as indicated by the constraining equation. If this feature has been contributed by the finite verb, as in (13), then the sentence is predicted to be grammatical; if there is no finite verb (and there is no other source of nominative case) then it is ruled out. This predicts the following grammaticality pattern:

	- a. She did! / \*Her did!
	- b. \* She! / Her!

English nominative pronouns require the presence of a finite verb, here the finite auxiliary *did*. Constraining equations operate as output filters on f-structures and are the primary way to grammatically specify the obligatoriness of a form, especially under the assumption that all daughter nodes are optional in the phrase structure. As described in Section 4 above, obligatory dependents are specified in the lexical form of a predicator using existential constraints like (↑ SUBJ) or (↑ OBJ). These are equivalent to constraining equations in which the particular value is unspecified, but some value must appear in order for the f-structure to be well-formed.

A constraining equation for case introduced by the case-*assigner*, rather than the case-bearing element, predicts that the appropriate case-bearing element must appear. A striking example from Serbo-Croatian is described by Wechsler & Zlatić (2003: 134), who give this descriptive generalization:

#### Stephen Wechsler & Ash Asudeh

(16) Serbo-Croatian Dative/Instrumental Case Realization Condition: If a verb or noun assigns dative or instrumental case to an NP, then that case must be morphologically realized by some element within the NP.

In Serbo-Croatian most common nouns, proper nouns, adjectives, and determiners are inflected for case. An NP in a dative position must contain at least one such item morphologically inflected for dative case, and similarly for instrumental case. The verb *pokloniti* 'give' governs a dative object, such as *ovom studentu* in (17a). But a quantified NP like *ovih pet studenata* 'these five students' has invariant case, namely genitive on the determiner and noun, and an undeclinable numeral *pet* 'five'. Such a quantified NP can appear in any case position, except when it fails to satisfy the condition in (16), such as this dative position (Wechsler & Zlatić 2003: 125):

	- b. \* pokloniti give.INF knjige books.ACC [ovih this.GEN.PL pet five studenata] student.GEN.PL intended: 'to give books to these five students'

Similarly, certain foreign names such as *Miki* and loanwords such as *braon* 'brown', 'brunette' are undeclinable, and can appear in any case position, except those ruled out by (16). Thus example (18a) is unacceptable, while the inflected possessive adjective *mojoj* 'my' saves it, as shown in (18b). When the possessive adjective realizes the case feature, it is acceptable. In (18c) we contrast the undeclined loan word *braon* 'brown' with the inflected form *lepoj* 'beautiful'. The example is acceptable only with the inflected adjective (Wechsler & Zlatić 2003: 134).

c. Divim admire.1SG se REFL \*braon brown / lepoj beautiful.DAT.SG Miki. Miki intended: 'I admire brunette / beautiful Miki.'

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This complex distribution is captured simply by positing that the dative (and instrumental) case assigning equations on verbs and nouns, such as the verbs *pokloniti* and *divim* in the above examples, are constraining equations:

(19) (↑ OBL CASE) = DAT

Any item in dative form within the NP, such as *ovom* or *studentu* in (17a) or *mojoj* or *lepoj* in (18b,c), could introduce the [CASE DAT] feature that satisfies this equation, but if none appears then the sentence fails. In contrast, other case-assigning equations (e.g., for nominative, accusative, or genitive case, or for cases assigned by prepositions) are defining equations, which therefore allow the undeclined NPs to appear. This sort of phenomenon is easy to capture using an output filter such as a constraining equation, but rather difficult otherwise. See Wechsler & Zlatić (2001) for further examples and discussion.

# **7 Agreement and affixal pronouns**

Agreement inflections that include the person feature derive historically from incorporated pronominal affixes. Distinguishing between agreement markers and affixal pronouns can be a subtle and controversial matter. LFG provides a particular formal device for representing this distinction within the f-structure: a true pronoun, whether affixal or free, introduces a semantic form (formally a PRED feature) with the value 'pro', while an agreement inflection does not. For example, Bresnan & Mchombo (1987) argue that the Chicheŵa (Bantu) object marker (OM) is an incorporated pronoun, while the subject marker (SM) alternates between agreement and incorporated pronoun, as in this example:

(20) Njûchi 10.bee zi-ná-wá-lum-a 10.SM-PST-2.OM-bite-FV a-lenje. 2-hunter (Chicheŵa) 'The bees bit them, the hunters.'

According to Bresnan & Mchombo (1987: 745), the class 2 object marker *wá-* is a pronoun, so the phrase *alenje* 'the hunters' is not the true object, but rather a postposed topic cataphorically linked to the object marker, with which it agrees in noun class (a case of *anaphoric agreement*). Meanwhile, the class 10 subject marker *zi-* alternates: when an associated subject NP (*njûchi* 'bees' in (20)) appears, then it is a grammatical agreement marker, but when no subject NP appears, then it functions as a pronoun. This is captured in LFG with the simplified lexical entries in (21):

#### Stephen Wechsler & Ash Asudeh

The PRED feature in (21b) is obligatory while that of (21c) is optional, as indicated by the parentheses around the latter. These entries interact with the grammar in the following manner. The two grammatical functions governed by the verb in (21a) are SUBJ and OBJ (the *governed* functions are the ones designated in the predicate argument structure of a predicator). According to the Principle of Completeness, a PRED feature must appear in the f-structure for each governed grammatical function that appears within the angle brackets of a predicator (indicating assignment of a semantic role). By the uniqueness condition it follows that there must be *exactly one* PRED feature, since a second such feature would cause a clash of values.12

The OM *wá-* introduces the [PRED 'pro'] into the object field of the f-structure of this sentence; the word *alenje* 'hunters' introduces its own PRED feature with value 'hunter', so it cannot be the true object, and instead is assumed to be in a topic position. Bresnan & Mchombo (1987: 744–745) note that the OM can be omitted from the sentence, in which case the phrasal object (here, *alenje*) is fixed in the immediately post-verbal position, while that phrase can alternatively be preposed when the OM appears. This is explained by assuming that the postverbal position is an OBJ position, while the adjoined TOPIC position is more flexible.

The subject *njûchi* can be omitted from (20), yielding a grammatical sentence meaning 'They (some class 10 plural entity) bit them, the hunters.' The optional PRED feature equation in (21c) captures this pro-drop property: when the equation appears, then a phrase such as *njûchi* cannot appear in the subject position, since this would lead to a clash of PRED values ('pro' versus 'bee'); but when the equation is not selected, then *njûchi* must appear in the subject position in order for the f-structure to be complete.

The diachronic process in which a pronominal affix is reanalyzed as agreement has been modeled in LFG as the historic loss of the PRED feature, along with the retention of the pronoun's person, number, and gender features (Coppock & Wechsler 2010). The anaphoric agreement of the older pronoun with its antecedent then becomes reanalyzed as grammatical agreement of the inflected verb with an external nominal. Finer transition states can also be modeled in terms of selective feature loss. Clitic doubling can be modeled as optional loss

12Recall that each PRED value is assumed to be unique, so that two 'pro' values cannot unify.

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of the PRED feature with retention of some semantic vestiges of the pronominal, such as specificity of reference.

The LFG analysis of affixal pronouns and agreement inflections can be translated into HPSG by associating the former but not the latter with pronominal semantics in the CONTENT field. The complementarity between the appearance of a pronominal inflection and an analytic phrase filling the same grammatical relation would be modeled as an exclusive disjunction between those two options, which captures the effects of the uniqueness of PRED values in LFG.

# **8 Lexical mapping**

LFG and HPSG both adopt *lexical approaches to argument structure* in the sense of Müller & Wechsler (2014): a verb or other predicator is equipped with a valence structure indicating the grammatical expression of its semantic arguments as syntactic dependents. Both frameworks have complex systems for mapping semantic arguments to syntactic dependents that are designed to capture prevailing semantic regularities within a language and across languages. The respective systems differ greatly in their notation and formal properties, but it is unclear whether there are any theoretically interesting differences, such as types of analysis that are available in one but not the other. This section identifies some of the most important analogues across the two systems, namely LFG's Lexical Mapping Theory (LMT; Bresnan et al. 2016: Chapter 14) and the theory of macroroles proposed by Davis and Koenig for HPSG (Davis 1996; Davis & Koenig 2000; see Davis, Koenig & Wechsler 2021: Section 4, Chapter 9 of this volume).13

In LMT, the argument structure is a list of a verb's argument slots, each labeled with a thematic role type such as Agent, Instrument, Recipient, Patient, Location, and so on, in the tradition of Charles Fillmore's *Deep Cases* (Fillmore 1968; 1977) and Pāṇini's *kārakas* (Kiparsky & Staal 1969). The ordering is determined by a thematic hierarchy that reflects priority for subject selection.14 The thematic role type influences a further classification by the features [±] and [±] that

13A recent alternative to LMT based on Glue Semantics has been developed by Asudeh & Giorgolo (2012) and Asudeh et al. (2014), incorporating the formal mapping theory of Findlay (2016), which in turn is based on Kibort (2007). It preserves the monotonicity of LMT, discussed below, but uses Glue Semantics to model argument realization and valence alternations. Müller (2018) discusses this Glue approach in contrast to HPSG approaches in light of lexicalism and argument structure. Note, though, that despite what might be implied by the title of the Müller volume, the Asudeh et al. treatment is not *necessarily* "phrasal" (i.e., non-lexicalist). The Glue framework assumed by Asudeh et al. can accommodate either a lexicalist or non-lexicalist position. It is a theoretical matter as to which is correct.

14The particular ordering proposed in Bresnan & Kanerva 1989: 23 and Bresnan et al. 2016: 329 is the following: *agent beneficiary experiencer/goal instrument patient/theme locative*.

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conditions the mapping to syntactic functions (this version is from Bresnan et al. 2016: 331):

(22) Semantic classification of argument structure roles for function: patientlike roles:

[−] secondary patientlike roles: [+] other semantic roles: [−]

The features [±] (thematically *restricted*) and [±] (*objective*) cross-classify grammatical functions: subject is [−*,* −], object is [−*,* +], obliques are [+*,* −] and restricted objects are [+*,* +]. A monotonic derivation (where feature values cannot be changed) starts from the argument list with the Intrinsic Classification (I.C. in example (23) below), then morpholexical operations such as passivization can suppress a role (not shown), then the thematically highest role (such as the Agent), if [−], is selected as Subject, and then any remaining features receive positive values by default.

In the macro-role theory formulated for HPSG, the analogues of [−] and [−] are the macro-roles Actor (ACT) and Undergoer (UND), respectively. The names of these features reflect large general groupings of semantic role types, but there is not a unique semantic entailment such as "agency" or "affectedness" associated with each of them. Actor and Undergoer name whatever semantic roles map to the subject and object, respectively, of a transitive verb.15 On the semantic side they are disjunctively defined: *x* is the Actor and *y* is the Undergoer iff "*x* causes a change in *y*, or *x* has a notion of *y*, or …" (quoted from Davis, Koenig & Wechsler 2021: 328, Chapter 9 of this volume). Such disjunctive definitions are the

15Note, for example, that within this system the "Undergoer" argument of the English verb *undergo*, as in *John underwent an operation*, is the object, and not the subject as one might expect if being an Undergoer involved actually undergoing something.

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HPSG analogues of the LMT "semantic classifications" shown in (22) above. In the HPSG macro-role system, linking constraints dictate that the ACT argument maps to the first element of ARG-ST, and that the UND argument maps to some nominal element of ARG-ST; (24) and (25) are from Davis, Koenig & Wechsler (2021: 329), Chapter 9 of this volume (each set of dots in the list value of ARG-ST represents zero or more list items):

$$\begin{array}{ll} \text{(24)} & \begin{bmatrix} \text{CONTENT} [\text{KEY} \quad \text{[ACT [\text{]}]} \\\\ \text{ARG-ST} & \begin{Bmatrix} \text{NP} [\text{]}, \dots \end{Bmatrix} \end{bmatrix} \\\\ \text{(25)} & \begin{bmatrix} \text{CONTENT} [\text{KEV} \left[ \text{UND} \left[ \text{]} \right] \\\\ \text{ARG-ST} & \begin{Bmatrix} \text{...}, \text{NP} [\text{[ $\mathbb{Z}$ ]}, \dots \end{Bmatrix} \end{bmatrix} \end{array} \end{array}$$

The first element of ARG-ST maps to the subject of an active voice verb, so (24)– (25) imply that if there is an ACT, then that ACT is the subject, and otherwise the UND is the subject (in the latter case NP 2 is the initial item in the list, given that the ellipsis represents zero or more items). Similarly, in LMT as described above, the subject is the [−] highest argument, if there is one, and otherwise it is the [−] argument. In this simple example we can see how the two systems accomplish exactly the same thing. A careful examination of more complex examples might point up theoretical differences, but it seems more likely that the two systems can express virtually the same set of mappings.

In LFG the *argument structure* (or *a-structure*) contains the predicator and its argument roles classified and ordered by thematic role type and further classified by Intrinsic Classification. It is considered a distinct level of representation, along with c-structure and f-structure. As a consequence, the grammar can make reference to the initial item in a-structure, such as the agent in (23), which is considered the "most prominent" role and often called the *a-subject* ("argument structure subject") in LFG parlance. To derive the passive voice mapping, the a-subject is suppressed in a morpholexical operation that crucially takes place before the subject is selected:

(26) Derivation of *eaten* as in *A yam was eaten* (*by Pam*).:

#### Stephen Wechsler & Ash Asudeh

(The optional *by*-phrase is considered to be an adjunct realizing the passivized a-subject.) Note that the passive alternation is *not* captured by a procedural rule that replaces one grammatical relation (such as OBJ) with another (such as SUBJ). The mapping from word strings to f-structures in LFG is monotonic, in the sense that information cannot be destroyed or changed. As a result, the mapping between internal and external structures is said to be transparent in the sense that the grammatical relations of parts of the sentence are preserved in the whole (for discussion of this point, see Bresnan et al. 2016: Chapter 5). In early versions of LFG, monotonicity was assumed for the syntax proper, while destructive procedures were permitted in the lexicon. This was canonized in the *Principle of Direct Syntactic Encoding*, according to which all grammatical relation changes are lexical (Kaplan & Bresnan 1982: 180; Bresnan et al. 2016: 77). At that time, an LFG passive rule operated on fully specified predicate argument structures, replacing OBJ with SUBJ, and SUBJ with an OBL*by* or an existentially bound variable. The advent of LMT brought monotonicity to the lexicon as well. The HPSG lexicon is also monotonic, if lexical rules are formulated as unary branching rules (see Davis & Koenig 2021: Section 5.1, Chapter 4 of this volume).

# **9 Long distance dependencies**

In LFG a long distance dependency is modeled as a reentrancy in the f-structure. The HPSG theory of long distance dependencies is based on that of GPSG (Gazdar 1981) and uses the percolation of a SLASH feature through the constituent structure. But LFG and HPSG accounts are essentially very similar, both working by decomposing a long distance dependency into a series of local dependencies. As we will see, there are nevertheless some minor differences with respect to what hypothetical extraction patterns can be expressed.

Both frameworks allow accounts either with or without gaps: regarding LFG see Bresnan et al. (2016: Chapter 9) for gaps and Dalrymple et al. (2019: Chapter 17) for gapless; regarding HPSG see Pollard & Sag (1994: Chapter 4) for gaps and Sag et al. (2003: Chapter 14) for gapless. Gaps have been motivated by the (controversial) claim that the linear position of an empty category matters for the purpose of weak crossover and other binding phenomena (Bresnan et al. 2016: 210–223). In this section we compare gapless accounts.

LFG has two grammaticalized discourse functions, TOP (topic) and FOC (focus). A sentence with a left-adjoined topic position is depicted in Figure 6. The topic phrase *Ann* serves as the object of the verb *like* within the clausal complement of *think*. This dependency is encoded in the second equation annotating the topic node, where the variable ranges over strings of attributes representing

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grammatical functions such as SUBJ, OBJ, OBL, or COMP. These strings describe paths through the f-structure. In this example is resolved to the string COMP OBJ, so this equation has the effect of adding to the f-structure in (27) the curved line representing token identity.

Figure 6: Long distance dependencies in LFG

HPSG accounts are broadly similar. One HPSG version relaxes the requirement that the arguments specified in the lexical entry of a verb or other predicator must all appear in its valence lists. Arguments are represented by elements of the ARG-ST list, so the list for the verb *like* contains two NPs, one each for the subject and object. In a sentence with no extraction, those ARG-ST list items map to the valence lists, the first item appearing in SUBJ and any remaining ones in

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COMPS. To allow for extraction, one of those ARG-ST list items is permitted to appear on the SLASH list instead. The SLASH list item is then passed up the tree by means of strictly local constraints, until it is bound by the topicalized phrase (see Borsley & Crysmann 2021, Chapter 13 of this volume and Bouma et al. 2001).

The LFG dependency is expressed in the f-structure, not the c-structure. Bresnan et al. (2016: Chapter 2) note that this allows for category mismatches between the phrases serving as filler and those in the canonical, unextracted position. This was discussed in Section 4 and illustrated with example (8) above.

Constraints on extraction such as accessibility conditions and islandisland constraint constraints can be captured in LFG by placing constraints on the attribute string (Dalrymple et al. 2019: Chapter 17).16 If subjects are not accessible to extraction, then we stipulate that SUBJ cannot be the final attribute in ; if subjects are islands, then we stipulate that SUBJ cannot be a non-final attribute in . If the f-structure is the only place such constraints are stated, then this makes the interesting (but unfortunately false; see presently) prediction that the theory of extraction cannot distinguish between constituents that map to the same f-structure. For example, as noted in Section 5, function words like determiners and their contentful sisters like NP are usually assumed to be f-structure co-heads, so the DP *the lion* maps to the same f-structure as its daughter *lion* (see Figure 3). This predicts that if the DP can be extracted, then so can the NP, but of course that is not true:

	- b. \* Lion, I think she saw the.

These two sentences have exactly the same f-structures, so any explanation for the contrast in acceptability must involve some other level. For example, one could posit that the phrase structure rules can introduce some items obligatorily (see Snijders 2015: 239), such as an obligatory sister of the determiner *the*. 17

# **10 Nonconfigurationality**

Some languages make heavy use of case and agreement morphology to indicate the grammatical relations, while allowing very free ordering of words within the clause. Such radically nonconfigurational syntax receives a straightforward anal-

16Asudeh (2012) shows that LFG's *off-path constraints* (Dalrymple et al. 2019: 225–230) can even capture quite strict locality conditions on extraction, in the spirit of *successive cyclicity* in movement-based accounts (Chomsky 1973; 1977), but without movement/transformations.

17This would depart from the assumption that all nodes are optional, adopted in Bresnan et al. (2016).

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ysis in LFG, due to the autonomy of functional structure from constituent structure. Indeed, the notion that phrasal position and word-internal morphology can be functionally equivalent is a foundational motivation for that separation of structure and function, as noted in Section 2 above. As Bresnan et al. (2016: 5) observe, "The idea that words and phrases are alternative means of expressing the same grammatical relations underlies the design of LFG, and distinguishes it from other formal syntactic frameworks."

The LFG treatment of nonconfigurationality will be illustrated with a simplified analysis, from Bresnan et al. (2016: 352–353), of the clausal syntax of Warlpiri, a Pama-Nyungan language of northern Australia. The following example gives three of the many possible grammatical permutations of words, all expressing the same truth-conditional content.

	- b. Kurdu-jarra-rlu child-DUAL-ERG ka-pala PRES-DUAL maliki dog.ABS wajilipi-nyi chase-NONPAST wita-jarra-rlu. small-DUAL-ERG 'The two small children are chasing the dog.'
	- c. Maliki dog.ABS ka-pala PRES-DUAL kurdu-jarra-rlu child-DUAL-ERG wajilipi-nyi chase-NONPAST wita-jarra-rlu. small-DUAL-ERG 'The two small children are chasing the dog.'

The main constraint on word order is that the auxiliary (here, the word *kapala*) must immediately follow the first daughter of S, where that first daughter can be any other word in the sentence, or else a multi-word NP as in (29a). Apart from that constraint, all word orders are possible. Any word or phrase in the clause can precede the auxiliary, and the words following the auxiliary can appear in any order.

The LFG analysis of these sentences works by directly specifying the auxiliarysecond constraint within the c-structure rule in (30a).18 Then the lexical entries directly specify the case, number, and other grammatical features of the word

18The c-structure is slightly simplified for illustrative purposes. In the actual c-structure proposed for Warlpiri, the second position auxiliary is the c-structure head (of IP) taking a flat S as its right sister (see Austin & Bresnan 1996: 225). Because the IP functional projection and its complement S map to the same f-structure (as discussed in Section 5 above), the analysis presented here works in exactly the same way regardless of whether this extra structure appears.

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forms, including case-assignment properties of the verb (see (32)). The framework does the rest, licensing all and only grammatical word orderings and generating an appropriate f-structure (see (33)).19

19The value of LFG's ADJ feature is a set of f-structures, as there can be multiple adjuncts, in fact indefinitely many. We use the set membership symbol as an attribute (Dalrymple et al. 2019: 229–230), which results in the f-structure for 'small' being in a set.

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The functional annotations on the NP nodes (see (31)) can vary as long as they secure Completeness and Coherence, given the governing predicate. In this case the main verb is transitive, so SUBJ and OBJ must be realized, each with exactly one PRED value.20 The noun *wita* 'small' is of category N, as Warlpiri does not distinguish between nouns and adjectives; but it differs functionally from the nouns for 'child' and 'dog' in that it modifies another noun. This is indicated by embedding its PRED feature under the ADJ ("adjunct") attribute (see the first equation in (32c)).

Comparing HPSG accounts of nonconfigurationality is instructive. Two HPSG approaches are described in Müller (2021), Chapter 10 of this volume: the *order domain* approach (Donohue & Sag 1999) and the *non-cancellation* approach (Bender 2008). In the order domain approach, the words of Warlpiri are combined into constituent structures resembling those of a configurational language like English. For example, in an order domain analysis of the Warlpiri sentences in (29), as well as all other acceptable permutations of those word forms, the words for 'two small children' together form an NP constituent. However, a domain feature DOM lists the phonological forms of the words in each constituent, and allows that list order to vary freely relative to the order of the daughters. This effective shuffling of the DOM list applies recursively on the nodes of the tree, up to the clausal node. It is the DOM list order that determines the order of words for pronunciation of the sentence. That function of the DOM feature is carried out in LFG by the c-structure.

The *non-cancellation* approach effectively introduces into HPSG correlates of c-structure and f-structure. In essence, an f-structure is added to HPSG in the form of a feature, much like the FS feature in the pseudo-HPSG rule in (7b) above. Instead of FS, that feature is called COMPS and has a list value. Unlike the valence feature normally called COMPS, items of this COMPS feature are not canceled from the list (and unlike ARG-ST, this feature is shared between a phrase and its head daughter, so it appears on non-terminal nodes). The items of that list are referenced by their order in the list. Special phrasal types define grammatical relations between a non-head daughter and an item in the COMPS list of the head daughter. These phrasal types are equivalent to LFG annotated phrase structure rules. For example, suppose the second item in COMPS (*2nd-comp*) is the object. Then *head-2nd-comp-phrase* in Bender (2008: 12) is equivalent to an LFG rule where the non-head daughter has the OBJ annotation (see (31a)). Since the list item is not canceled from the list, it remains available for other items to com-

20By the Principle of Completeness, both SUBJ and OBJ appear in the f-structure; by the Principle of Coherence, no other governable grammatical function appears in the f-structure.

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bine and to modify the object, using a different phrasal type. Non-cancellation mechanisms bring HPSG closer to LFG by relying on a level of structure that is autonomous from the constituent structure responsible for the grouping and ordering of words for pronunciation. See also Müller (2008) for a non-cancellation approach to depictive predicates in English and German.

# **11 Raising and control**

Raising and control (equi) words can be treated in virtually the same way in LFG and HPSG. Taking raising first, a subject raising word (such as *seem* in (34)) specifies that its subject is (also) the subject of its predicate complement (see Abeillé (2021), Chapter 12 of this volume).

(34) Pam seems to visit Fred.

a. *seem*: (↑ PRED) = 'seemh(↑ XCOMP)i (↑ SUBJ)' (↑ SUBJ) = (↑ XCOMP SUBJ) b. *seem*: [ ARG-ST h 1 NP, VP[*inf*, SUBJ h 1 i]i]

The LFG entry for *seem* in (34a) contains the grammatical function XCOMP ("open complement"), the function reserved for predicate complements such as the infinitival phrase *to visit Fred*. The functional control equation specifies that its subject is identical to the subject of the verb *seem*; the tag 1 plays the same role in the simplified HPSG entry in (34b).

The f-structure for (34) is shown here (with simplified structures for *Pam* and *Fred*):

Turning next to equi, similar proposals have been made in both frameworks, such that it is the referential index of the controller and the controllee that are identical:

(36) Pam hopes to visit Fred.

a. *hope*: (↑ PRED) = 'hopeh(↑ SUBJ) (↑ COMP)i' (↑ COMP SUBJ PRED) = 'pro' (↑ SUBJ INDEX) = (↑ COMP SUBJ INDEX) b. *hope*: [ ARG-ST h NP 1 , VP[*inf*, SUBJ h NP 1 i]i]

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The LFG entry for *hope* in (36a) is adapted from Dalrymple et al. (2019: 572). It states that the subject of the controlled infinitival is a pronominal that is coindexed with the subject of the control verb. Similarly, the subscripted tags in (36b) represent coindexing between the subject of the control verb and the controlled subject of the complement.

One interesting difference between the two frameworks concerns the representation of restrictions on the grammatical function of the target of control or raising. The basic HPSG theory of control and raising (for example, the one presented in Pollard & Sag 1994: 132–145) allows only for control (or raising) of subjects and not complements. More precisely, it allows for control/raising of the outermost or final dependent to be combined with the verbal projection that is a complement of the control verb. This is because of the list cancellation regime that operates with valence lists (on non-cancellation theories, see Section 10). The expression VP in (34b/36b) represents an item with an empty COMPS list. In a simple English clause, the verb combines with its complement phrases to form a VP constituent, with which the subject is then combined to form a clause. Assuming the same order of combination in the control or raising structure, it is not possible to raise or control the complement of a structure that contains a structural subject, as in (37a):

	- b. Fred seems to be visited by Pam.

The intended meaning would be that of (37b). The passive voice is needed in order to make *Fred* the subject of *visit* and thus available to be raised. This restriction that only subjects can be raised follows from the basic HPSG theory of Pollard & Sag (1994: 132-145), while in LFG it follows only if raising equations like the one in (38) are systematically excluded.

$$(\text{38}) \quad (\uparrow \text{ звед}) = (\uparrow \text{ хсомр овд})$$

At the same time, the HPSG framework allows for mechanisms that can be used to violate the restriction to subjects, and such mechanisms have been proposed, including the adoption of something similar to an f-structure in HPSG (this is the non-cancellation theory described in Section 10). This illustrates the point made in Section 2 above, that the framework was originally designed to capture locality conditions, but is flexible enough to capture non-local relations as well.

This raises the question of whether the restriction to subject controllees is universal. In fact, it appears that some languages allow the control of non-subjects, but it is still unclear whether these control relations are established via the grammatical relations and therefore justify equations such as (38). For instance, Kroeger (1993) shows that Tagalog has two types of control relation. In the more

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specialized type, which occurs only with a small set of verbs or in a special construction in which the downstairs verb appears in non-volitive mood, both the controller and controllee must be subjects. Kroeger analyzes this type using a functional control equation like the one in (34a). In the more common type of Tagalog control, the controllee must be the Actor argument, while the grammatical relations of controllee and controller are not restricted. (Tagalog has a rich voice system, often called a focus marking system, regulating which argument of a verb is selected as its subject.) This latter type of Tagalog control is defined on argument structure (Actors, etc.), so a-structure rather than f-structure is appropriate for representing the control relations.

# **12 Semantics**

HPSG was conceived from the start as a theory of the *sign* (de Saussure 1916), wherein each constituent is a pairing of form and meaning. So semantic representation and composition were built into HPSG (and the related framework of Sign-Based Construction Grammar; Boas & Sag 2012), as reflected in the title of the first HPSG book (Pollard & Sag 1987), *Information-Based Syntax and Semantics*. LFG was not founded as a theory that included semantics, but a semantic component was developed for LFG shortly after its foundation (Halvorsen 1983). The direction of semantics for LFG changed some ten years later and the dominant tradition is now Glue Semantics (Dalrymple et al. 1993; Dalrymple 1999; 2001; Asudeh 2012; Dalrymple et al. 2019).

This section presents a basic introduction to Glue Semantics (Glue); this is necessary to fully understand a not insignificant portion of LFG literature of the past fifteen years, which interleaves LFG syntactic analysis with Glue semantic analysis. The section is not meant as a direct comparison of LFG and HPSG semantics, for two reasons. First, as explained in the previous paragraph, HPSG is inherently a theory that integrates syntax and semantics, but LFG is not; the semantic module that Glue provides for LFG can easily be pulled out, leaving the syntactic component exactly the same.21 Second, as will become clear in the next section, at a suitable level of abstraction, Glue offers an underspecified theory of semantic composition, in particular scope relations, which is also the goal of an influential HPSG semantic approach, Minimal Recursion Semantics (MRS; Copestake et al. 2005). But beyond observing this big-picture commonality, comparison of Glue and MRS would require a chapter in its own right. Our goal is to present enough of Glue Semantics for readers to grasp the main intuitions behind it, without presupposing much knowledge of formal semantic theory. The

21On the relation between the PRED feature and the semantic component, see footnote 24 below.

#### 30 HPSG and Lexical Functional Grammar

references listed at the end of the previous paragraph (especially Dalrymple et al. 2019) are good places to find additional discussion and references.

The rest of this section is organized as follows. In Section 12.1 we present some more historical background on semantics for LFG and HPSG. In Section 12.2, we present Glue Semantics, as a general compositional system in its own right. Then, in Section 12.3, we look at the syntax–semantics interface with specific reference to an LFG syntax. For further details on semantic composition and the syntax– semantics interface in constraint-based theories of syntax, see Koenig & Richter (2021), Chapter 22 of this volume for semantics for HPSG and Asudeh (2021) for Glue Semantics for LFG.

# **12.1 Brief history of semantics for LFG and HPSG**

Various theories of semantic representation have been adopted by the different non-transformational syntactic frameworks over the years. The precursor to HPSG, GPSG (Gazdar et al. 1985), was paired by its designers with a then fairly standard static Montogovian semantics (Montague 1973), but GPSG itself was subsequently adopted as the syntactic framework used by Kamp & Reyle (1993: 9) for Discourse Representation Theory, a dynamic theory of semantics. Initial work on semantics for LFG also assumed a Montogovian semantics (Halvorsen 1983; Halvorsen & Kaplan 1988). But with the increasing interest in Situation Semantics (Barwise & Perry 1983) in the 1980s at Stanford University and environs (particularly SRI International and Xerox PARC), the sites of the foundational work on both HPSG and LFG, both frameworks incorporated a Situation Semantics component (on LFG see Fenstad et al. 1987). Interest in the use of Situation Semantics did not last as long in LFG as it did in HPSG, where Situation Semantics was carried over into the second main HPSG book (Pollard & Sag 1994) and beyond (Ginzburg & Sag 2000).

Beginning in the nineties, the focus subsequently shifted in new directions due to a new interest in computationally tractable theories of the syntax–semantics interface, to support efforts at large-scale grammar development, such as the Par-Gram project for LFG (Butt et al. 1999; 2002) and the LinGO/Grammar Matrix and CoreGram projects for HPSG (Flickinger 2000; Bender et al. 2002; 2010; Müller 2015).22 This naturally led to an interest in underspecified semantic representa-

22Readers can explore the current incarnations of these projects at the following links (checked 2021-04-30):

#### Stephen Wechsler & Ash Asudeh

tions, so that semantic ambiguities such as scope ambiguity could be compactly encoded without the need for full enumeration of all scope possibilities. Two examples for HPSG are *Lexical Resource Semantics* (Richter 2004; Penn & Richter 2004) and *Minimal Recursion Semantics* (Copestake et al. 2005). Similarly, focus in semantics for LFG shifted to ways of encoding semantic ambiguity compactly and efficiently. This led to the development of Glue Semantics.

# **12.2 General Glue Semantics**

In this section, we briefly review Glue Semantics itself, without reference to a particular syntactic framework. Glue Semantics is a general framework for semantic composition that requires *some* independent syntactic framework but does not presuppose anything about syntax except headedness, which is an uncontroversial assumption across frameworks. This makes the system flexible and adaptable, and it has been paired not just with LFG, but also with Lexicalized Tree-Adjoining Grammar (Frank & van Genabith 2001), HPSG (Asudeh & Crouch 2002b), Minimalism (Gotham 2018), and Universal Dependencies (Gotham & Haug 2018).

In Glue Semantics, meaningful linguistic expressions—including lexical items but possibly also particular syntactic configurations—are associated with *meaning constructors* of the following form:23

# (39) M :

M is an expression from a *meaning language* which can be anything that supports the lambda calculus; is an expression of *linear logic* (Girard 1987), which specifies the semantic composition (it "glues meanings together"), based on a syntactic parse. By convention a colon separates them. Glue Semantics is related to (Type-Logical) Categorial Grammar (Carpenter 1998; Morrill 1994; 2011; Moortgat 1997), but the terms of the linear logic specify just semantic composition without regard to word order (see Asudeh 2012 for further discussion). Glue Semantics is therefore useful in helping us focus on semantic composition in its own right.

The principal compositional rules for Glue Semantics are those for the linear implication connective, ⊸, which are here presented in a natural deduction format, in which each connective has an elimination rule (⊸E, in this case) and an introduction rule (⊸I, in this case).

23It is in principle possible for a linguistic expression to have a phonology and syntax but not contribute to interpretation, such as the expletives *there* and *it* or the *do*-support auxiliary in English; see Asudeh (2012: 113) for some discussion of expletive pronouns in the context of Glue.

30 HPSG and Lexical Functional Grammar

(40) Functional application : Implication elimination (modus ponens)

$$\frac{f:A \multimap B \quad \quad a:A}{f(a):B} \multimap\_{\mathcal{E}}^{\mathcal{E}}$$

(41) Functional abstraction : Implication introduction (hypothetical reasoning)

$$\frac{[a:A]^1}{\vdots}$$

$$\frac{f:B}{\lambda a.f:A \multimap B} \multimap\_{I,1}$$

Focusing first on the right-hand, linear logic side, the implication elimination rule is just standard modus ponens. The implication introduction rule is hypothetical reasoning. A hypothesis is made in the first line as an assumption, indicated by presenting it in square brackets with an index that flags the particular hypothesis/assumption. Given this hypothesis, if through some series of proof steps, indicated by the vertical ellipsis, we derive a term, then we are entitled to discharge the assumption, using its flag to indicate that it is this particular assumption that has been discharged, and conclude that the hypothesis implies the term so-derived. In each of these rules, the inference over the linear logic term corresponds to an operation on the meaning term, via the Curry-Howard Isomorphism between formulas and types (Curry & Feys 1958; Howard 1980). The rule for eliminating the linear implication corresponds to functional application. The rule for introducing the linear implication corresponds to functional abstraction. These rules will be seen in action shortly.

In general, given some head *ℎ* and some arguments of the head *1 , . . . ,* , an implicational term like the following models consumption of the arguments to yield the saturated meaning of the head: *1* ⊸ *. . .* ⊸ ⊸*ℎ*. For example, let us assume the following meaning constructor for the verb *likes* in the sentence *Max likes Sam*:

# (42) *..***like**() () : ⊸ ⊸

Let's also assume that is mnemonic for the semantic correspondent of the (single word) phrase *Sam*, similarly mnemonic for *Max*, and for *likes*. In other words, the meaning constructor for *likes* would be associated with the lexical entry for the verb and specified in some general form such that it can be instantiated by the syntax (we will see an LFG example shortly); here we are assuming that the instantiation has given us the meaning constructor in (42).

#### Stephen Wechsler & Ash Asudeh

Given this separate level of syntax, the glue logic does not have to worry about word order and is permitted to be commutative (unlike the logic of Categorial Grammar, see also Kubota 2021, Chapter 29 of this volume on Categorial Grammar and Müller 2021: 379, Chapter 10 of this volume on HPSG approaches allowing saturation of elements from the valence lists in arbitrary order). We could therefore freely reorder the arguments for *likes* in (42) above, as in (43) below, such that we instead first compose with the subject and then the object, but still yield the meaning appropriate for the intended sentence *Max likes Sam* (rather than for *Sam likes Max*):

# (43) *..***like**() () : ⊸ ⊸

As we will see below, the commutativity of the glue logic yields a simple and elegant treatment of quantifiers in non-subject positions, which are challenging for other frameworks (see, for example, the careful pedagogical presentation of the issue in Jacobson 2014: 244–263).

First, though, let us see how this argument reordering, otherwise known as Currying or Schönfinkelization, works in a proof, which also demonstrates the rules of implication elimination and introduction:

$$\begin{array}{c} \text{(44)} \quad \frac{\lambda y.\lambda x.f(y)(x):a\multimap b\multimap c}{\lambda x.f(v)(x):b\multimap c}^{1}\multimap\_{\mathcal{E}} & \multimap\_{\mathcal{E}} & [u:b]^{2} \\ \hline & & & & [u:b]^{2} & \multimap c \\ & & & & \overline{\lambda v.f(v)(u):a\multimap c}^{\sim T\_{1},1} \\ & & & \overline{\lambda y.f(y)(u):a\multimap c}^{\sim T\_{1},1} & \multimap\_{a} \\ & & & \overline{\lambda u.\lambda y.f(y)(u):b\multimap a\multimap c}^{\sim T\_{1},2} & \multimap\_{a} \\ \hline & & & \overline{\lambda x.\lambda y.f(y)(x):b\multimap a\multimap c}^{\sim T\_{1},2} \\ \end{array}$$

The general structure of the proof is as follows. First, an assumption (hypothesis) is formed for each argument, in the order in which they originally occur, corresponding to a variable in the meaning language. Each assumed argument is then allowed to combine with the implicational term by implication elimination. Once the implicational term has been entirely reduced, the assumptions are then discharged in the same order that they were made, through iterations of implication introduction. The result is the original term in curried form, such that the order of arguments has been reversed but without any change in meaning. The two steps of -equivalence, notated ⇒ , are of course not strictly necessary, but have been added for exposition.

This presentation has been purposefully abstract to highlight what is intrinsic to the glue logic, but we need to see how this works with a syntactic framework to

#### 30 HPSG and Lexical Functional Grammar

see how Glue Semantics actually handles semantic composition and the syntax– semantics interface. So next, in Section 12.3, we will review LFG+Glue.

# **12.3 Glue Semantics for LFG**

Glue for LFG will be demonstrated by analyses of the following three examples:

(45) a. Blake called Alex.

Example (45a) is a simple case of a transitive verb with two proper name arguments, but is sufficient to demonstrate the basics of the syntax–semantics interface in LFG+Glue. Example (45b) is a case of a quantifier in object position, which is challenging to compositionality because there is a type clash between the simplest type we can assign to the verb, h*,* h*,* ii, and the simplest type that would be assigned to the quantifier, hh*,* i*,* i. In other theories, this necessitates either a syntactic operation which is undermotivated from a purely syntactic perspective, e.g. Quantifier Raising (QR) in interpretive theories of composition, such as Logical Form semantics (May 1977; 1985; Heim & Kratzer 1998), or a type shifting operation of some kind in directly compositional approaches, as in categorial or type-logical frameworks; see Jacobson (2014: Chapter 14) for further discussion and references. Example (45c) also demonstrates this point, but it more importantly demonstrates that quantifier scope ambiguity can be handled in Glue without positing an undermotivated syntactic ambiguity, but nevertheless while maintaining the simplest types for both quantifiers.

The relevant aspects of the lexical entries involved are shown in Table 30.1. Other syntactic aspects of the lexical items, such as the fact that *called* has a SUBJ and an OBJ, are specified in its meaning constructor. Minimal f-structures are provided below for each example. The subscript indicates the semantic structure that corresponds to the annotated f-structure term. The types for the lexical items are the minimal types that would be expected. Note that in Glue these are normally associated directly with the semantic structures, for example ↑ and (↑ OBJ) ⊸ (↑ SUBJ) ⊸ ↑ , but they have been presented separately for better exposition; see Dalrymple et al. (2019: 299–305) for further discussion. We do not show semantic structures here, as they are not necessary for this simple demonstration.

The functions associated with *everybody* and *somebody* are, respectively, **every** and **some** in the meaning language, where these are the standard quantificational determiners from generalized quantifier theory (Montague 1973; Barwise

#### Stephen Wechsler & Ash Asudeh

& Cooper 1981; Keenan & Faltz 1985). The function **every** returns true iff the set characterized by its restriction is a subset of the set characterized by its scope. The function **some** returns true iff the intersection of the set characterized by its restriction and the set characterized by its scope is non-empty. The universal quantification symbol ∀ in the glue logic/linear logic terms for the quantifiers ranges over semantic structures of type . It is unrelated to the meaning language functions **every** and **some**. Hence even the existential word *somebody* has the universal ∀ in its linear logic glue term. The ∀-terms thus effectively say that *any* type semantic structure that can be found by application of proof rules such that the quantifier's semantic structure implies can serve as the scope of the quantifier; see Asudeh (2005: 393–394) for basic discussion of the interpretation of ∀ in linear logic. This will become clearer when quantifier scope is demonstrated shortly.

Table 30.1: Relevant lexical details for the three examples in (45)

Let us assume the following f-structure for (45a):

$$\begin{array}{ll} \text{(46)} & c \begin{bmatrix} \text{pRED \\_call'}\\ \text{subJ} & b \begin{bmatrix} \text{pRED \\_Blueke'}\\ \text{OBJ} & a \begin{bmatrix} \text{pRED \\_Alex'} \end{bmatrix} \end{bmatrix} \end{array} \end{array}$$

Note that here, unlike in previous sections, the PRED value for the verb does not list its subcategorization information. This is because we've made the move that is standard in much Glue work to suppress this information.24 The f-structures are named mnemonically by the first character of their PRED value. All other f-structural information has been suppressed for simplicity. Based on these f-

24Indeed, one could go further and argue that PRED values do not list subcategorization at all, in which case the move is not just notational, and that the Principles of Completeness and Coherence instead follow from the resource-sensitivity of Glue Semantics; for some discussion, see Asudeh (2012: 112–114) and Dalrymple et al. (2019: 299–301).

#### 30 HPSG and Lexical Functional Grammar

structure labels, the relevant meaning constructors in the lexicon in Table 30.1 are instantiated as follows ( subscripts suppressed):

(47) Instantiated meaning constructors: **blake** : **alex** : *..***call**() () : ⊸ ⊸

These meaning constructors yield the following proof, which is the only available normal form proof for the sentence, where ⇒ indicates -equivalence:25

(48) Proof:

$$\begin{array}{c} \lambda y.\lambda \text{x.call}(y)(\mathbf{x}): a \multicolumn{3}{c}{\multicolumn{1}{c}{\mathbf{a}}}(y)(\mathbf{x}): a \multicolumn{3}{c}{\multicolumn{1}{c}{\mathbf{a}}}(y)(\mathbf{x}): a \multicolumn{3}{c}{\mathbf{a}}}{\text{ \multicolumn{1}{c}{\lambda x.\text{call}(\mathbf{a}\mathbf{e}\mathbf{x})(\mathbf{x}) \,(\mathbf{a}\mathbf{e}\mathbf{x}): b \multicolumn{3}{c}{\mathbf{a}}} \multicolumn{3}{c}{\mathbf{a}}} \multicolumn{3}{c}{\multicolumn{1}{c}{\mathbf{a}}} \multicolumn{3}{c}{\mathbf{a}}}{\begin{array}{c} \lambda \mathbf{x}.\text{call}(\mathbf{a}\mathbf{e}\mathbf{x})(\mathbf{x}): b \multicolumn{3}{c}{\mathbf{a}} \multicolumn{3}{c}{\mathbf{a}} \multicolumn{3}{c}{\mathbf{b}} \text{l} \&\mathbf{a}\mathbf{e}: b} \multicolumn{3}{c}{\mathbf{a}} \multicolumn{3}{c}{\mathbf{a}} \multicolumn{3}{c}{\mathbf{b}} \multicolumn{3}{c}{\mathbf{a}} \multicolumn{3}{c}{\mathbf{b}} \multicolumn{3}{c}{\mathbf{b}} \multicolumn{3}{c}{\mathbf{a}} \multicolumn{3}{c}{\mathbf{b}} \multicolumn{3}{c}{\mathbf{b}}} \multicolumn{3}{c}{c}{\mathbf{a}} \multicolumn{3}{c}{\mathbf{b}} \multicolumn{3}{c}{\mathbf{b}} \multicolumn{3}{c}{\mathbf{c}} \end{array}$$

The final meaning language expression, **call**(**alex**) (**blake**), gives the correct truth conditions for *Blake called Alex*, based on a standard model theory.

Let us next assume the following f-structure for (45b):

$$\text{(49)} \quad c \begin{bmatrix} \text{pRED \text{\textquotedblleft}call\text{\textquotedblright}}\\ \text{subj} \text{ b} \begin{bmatrix} \text{pRED \text{\textquotedblleft}Blue\text{\textquotedblright}}\\ e \text{\textquotedblleft} \text{pRED \textquotedblright} \end{bmatrix} \end{bmatrix}$$

Based on these f-structure labels, the meaning constructors in the lexicon are instantiated as follows ( subscripts again suppressed):

(50) Instantiated meaning constructors: *..***call**() () : ⊸ ⊸ *.***every**(**person***,* ) : ∀*.*( ⊸ ) ⊸ **blake** :

These meaning constructors yield the following proof, which is again the only available normal form proof:26

25The reader can think of the normal form proof as the minimal proof that yields the conclusion, without unnecessary steps of introducing and discharging assumptions; see Asudeh & Crouch (2002a) for some basic discussion.

26We have not presented the proof rule for Universal Elimination, ∀E, but it is trivial; see Asudeh (2012: 396).

#### Stephen Wechsler & Ash Asudeh

(51) Proof:

The final meaning language expression, **every**(**person***, .***call**() (**blake**)), again gives the correct truth conditions for *Blake called everybody*, based on a standard model theory with generalized quantifiers.

Notice that the quantifier does not move in the syntax, contra QR analyses; see Gotham (2018) for contrastive discussion. The quantifier is just an OBJ in fstructure, and no special type shifting was necessary. This is because the proof rules allow us to temporarily fill the position of the object quantifier with a hypothetical meaning constructor that consists of a type variable paired with the linear logic term for the object; this assumption is then discharged to return the scope of the quantifier, ⊸ , and the corresponding variable is bound, to yield the function that maps individuals called by Blake to a truth value. In other words, we have demonstrated that this approach scopes the quantifier without positing an *ad hoc* syntactic operation and without complicating the type of the object quantifier or the transitive verb. This is ultimately due to the commutativity of the glue logic, linear logic, since the proof does not have to deal with the elements of composition (words) in their syntactic order, because the syntax is separately represented by c-structure (not shown here) and f-structure.

Lastly, let us assume the following f-structure for (45c), *Everybody called somebody*:

(52) PRED 'call' SUBJ - PRED 'everybody' OBJ - PRED 'somebody'

Based on these f-structure labels, the meaning constructors in the lexicon are instantiated as follows:

(53) Instantiated meaning constructors: *..***call**() () : ⊸ ⊸ *.***some**(**person***,* ) : ∀*.*( ⊸ ) ⊸ *.***every**(**person***,* ) : ∀*.*( ⊸ ) ⊸

#### 30 HPSG and Lexical Functional Grammar

These meaning constructors yield the following proofs, which are the only available normal form proofs, but there are two distinct proofs, because of the scope ambiguity:27

(54) Proof 1 (subject wide scope):

(55) Proof 2 (object wide scope):

*.***some**(**person***,* ) : ∀*.*( ⊸ ) ⊸ *.***every**(**person***,* ) : ∀*.*( ⊸ ) ⊸ *..***call**() () : ⊸ ⊸ [ : ] 1 ⊸E*,* ⇒ *.***call**() () : ⊸ ⊸E*,* ∀E [/]*,* ⇒ **every**(**person***, .***call**() ()) : ⊸I*,*1 *.***every**(**person***, .***call**() ()) : ⊸ ⊸E*,* ∀E [/]*,* ⇒ **some**(**person***, .***every**(**person***, .***call**() ())) : ⇒ **some**(**person***, .***every**(**person***, .***call**() ())) :

The final meaning language expressions in (54) and (55) give the two possible readings for the scope ambiguity, again assuming a standard model theory with generalized quantifiers. Once more, notice that neither quantifier moves in the syntax (again, contra QR analyses): they are respectively just a SUBJ and an OBJ in f-structure. And, once more, no special type shifting is necessary. It is a key strength of this approach that even quantifier scope ambiguity can be captured without positing *ad hoc* syntactic operations (and, again, without complicating the type of the object quantifier or the transitive verb). Again, this is ultimately due to the commutativity of the glue logic.

# **13 Conclusion**

HPSG and LFG are rather similar syntactic frameworks, both of them important declarative lexicalist alternatives to Transformational Grammar. They allow for

27We have made the typical move in Glue work of not showing the trivial universal elimination step this time.

#### Stephen Wechsler & Ash Asudeh

the expression of roughly the same set of substantive analyses, where analyses are individuated in terms of deeper theoretical content rather than superficial properties. The same sort of analytical options can be compared under both systems, answers to questions such as whether a phenomenon is to be captured on a lexical level or in the syntax, whether a given word string is a constituent or not, the proper treatment of complex predicates, and so on. Analyses in one framework can often be translated into the other, preserving the underlying intuition of the account.

Against the backdrop of a general expressive similarity, we have pointed out a few specific places where one framework makes certain modes of analysis available that are not found in the other. The main thesis of this chapter is that the differences between the frameworks stem from different design motivations, reflecting subtly different methodological outlooks. HPSG is historically rooted in context-free grammars and an interest in the study of locality. LFG is based on the notion of functional similarity or equivalence between what are externally rather different structures. For example, fixed phrasal positions, case markers, and agreement inflections can all function similarly in signaling grammatical relations. LFG makes this functional similarity highly explicit.

# **Abbreviations**

FV final vowel

# **Acknowledgments**

We would like to thank the editors of the volume, Anne Abeillé, Bob Borsley, Jean-Pierre Koenig, and Stefan Müller. JP and Stefan provided particularly close reads of the paper that greatly improved it. We would also like to thank the anonymous reviewers for their insightful comments. Any remaining errors are our own.

# **References**

Abeillé, Anne. 2021. Control and Raising. In Stefan Müller, Anne Abeillé, Robert D. Borsley & Jean- Pierre Koenig (eds.), *Head-Driven Phrase Structure Grammar: The handbook* (Empirically Oriented Theoretical Morphology and Syntax), 489–535. Berlin: Language Science Press. DOI: 10.5281/zenodo.5599840.

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30 HPSG and Lexical Functional Grammar

Stephen Wechsler & Ash Asudeh

30 HPSG and Lexical Functional Grammar

Stephen Wechsler & Ash Asudeh

*the LFG '18 conference, University of Vienna*, 208–226. Stanford, CA: CSLI Publications. http://csli-publications.stanford.edu/LFG/LFG-2018/ (10 February, 2021).

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In Mary Dalrymple, Ronald M. Kaplan, John T. Maxwell III & Annie Zaenen (eds.), *Formal issues in Lexical-Functional Grammar* (CSLI Lecture Notes 47), 29–130. Stanford, CA: CSLI Publications, 1995.

Stephen Wechsler & Ash Asudeh

30 HPSG and Lexical Functional Grammar

*Conference on Head-Driven Phrase Structure Grammar, Center for Computational Linguistics, Katholieke Universiteit Leuven*, 423–443. Stanford, CA: CSLI Publications. http://csli-publications.stanford.edu/HPSG/2004/ (10 February, 2021).

Stephen Wechsler & Ash Asudeh

# **Chapter 31**

# **HPSG and Dependency Grammar**

# Richard Hudson

University College London

HPSG assumes Phrase Structure (PS), a partonomy, in contrast with Dependency Grammar (DG), which recognises Dependency Structure (DS), with direct relations between individual words and no multi-word phrases. The chapter presents a brief history of the two approaches, showing that DG matured in the late nineteenth century, long before the influential work by Tesnière, while Phrase Structure Grammar (PSG) started somewhat later with Bloomfield's enthusiastic adoption of Wundt's ideas. Since DG embraces almost as wide a range of approaches as PSG, the rest of the chapter focuses on one version of DG, Word Grammar. The chapter argues that classical DG needs to be enriched in ways that bring it closer to PSG: each dependent actually adds an extra node to the head, but the nodes thus created form a taxonomy, not a partonomy; coordination requires strings; and in some languages the syntactic analysis needs to indicate phrase boundaries. Another proposed extension to bare DG is a separate system of relations for controlling word order, which is reminiscent of the PSG distinction between dominance and precedence. The "head-driven" part of HPSG corresponds in Word Grammar to a taxonomy of dependencies which distinguishes grammatical functions, with complex combinations similar to HPSG's re-entrancy. The chapter reviews and rejects the evidence for headless phrases, and ends with the suggestion that HPSG could easily move from PS to DG.

# **1 Introduction**