Non-canonical Coordination in the Transformational Approach

Abstract

Recently introduced Transformational Semantics (TS) formalizes, restraints and makes rigorous the transformational approach epitomized by QR and Transformational Grammars: deriving a meaning (in the form of a logical formula or a logical form) by a series of transformations from a suitably abstract (tecto-) form of a sentence. TS generalizes various ‘monad’ or ‘continuation-based’ computational approaches, abstracting away irrelevant details (such as monads, etc.) while overcoming their rigidity and brittleness. Unlike QR, each transformation in TS is rigorously and precisely defined, typed, and deterministic. The restraints of TS and the sparsity of the choice points (in the order of applying the deterministic transformation steps) make it easier to derive negative predictions and control over-generation.

We apply TS to right-node raising (RNR), gapping and other instances of non-constituent coordination. Our analyses straightforwardly represent the intuition that coordinated phrases must in some sense be ‘parallel’, with a matching structure. Coordinated material is not necessarily constituent – even ‘below the surface’ – and we do not pretend it is. We answer the Kubota, Levine and Moot challenge (the KLM problem) of analyzing RNR and gapping without directional types, yet avoiding massive over-generation. We thus formalize the old idea of ‘coordination reduction’ and show how to make it work for generalized quantifiers.

1 Introduction

Non-canonical coordination – right-node raising (RNR) as in (1), argument-cluster coordination (2) and, in particular, gapping (3–7) – provides an unending stream of puzzles for the theory of semantics [8, 10]:

• (1) John likes and Mary hates Bill.

• (2) John gave a present to Robin on Thursday and to Leslie on Friday.

• (3) Mary liked Chicago and Bill Detroit.

• (4) One gave me a book and the other a CD.

• (5) Terry can go with me and Pat with you.

• (6) Mrs. J can’t live in Boston and Mr. J in LA.

• (7) Pete wasn’t called by Vanessa but rather John by Jesse.

With gapping, it is not just a simple verb that can “go missing”, as in (3). It can be a complex phrase of a verb with arguments and complements – or, as in (4), a verb and an auxiliary verb. Interactions of coordination with scope-taking are particularly challenging: a competent theory needs to handle both narrow- and wide-scope reading of “a present” in (2) and the narrow- and wide-scope coordination in (6). In (7), negation somehow scopes over the first “coordinated structure” but not over the second.

Recently in [8, 9], Kubota and Levin put forward new analyses of non-canonical coordination, applying hybrid categorial grammars they have been developing. In contrast, the analyses in [6] use plain old non-associative Lambek grammar. However, the main ideas of [6] are completely hidden behind thickets of complicated types and their interactions within a derivation. The intuition that coordinated structures must be parallel is thus lost in the details.

We present a new analysis of non-constituent coordination using the more intuitive and less round-about framework TS (formerly called AACG) [7], designed to take the ‘hacking’ out of tree-hacking. TS lets us talk about QR and other transformations towards some semantic form in a rigorous, formal, mostly deterministic way. We remind of TS in Sect. 2.

Our analyses re-expose ideas from the earlier approach of [6], but free them from the bondage of encoding. A notable feature of TS is the absence of directional types. We use it to answer the challenge posited by Kubota and Levin [10] and Moot (dubbed “the KLM problem” by Morrill): to analyze RNR within categorial-grammar–like formalisms without directional types, while avoiding massive over-generation.

One may categorize the various approaches to non-canonical coordination based on what exactly is being coordinated. Take (1), repeated below

• (1) John likes and Mary hates Bill.

which will be our running example for a while. Are the complete sentences being coordinated behind the scene, as in “John likes Bill” and “Mary hates Bill” with “Bill” being later elided? Or perhaps sentences with holes are being coordinated, as in “John likes $hyp_{obj}$”? (as done in [6, 8, 9].) Or perhaps we regard “John likes” and “Mary hates” as constituents and coordinate as such (as in CCG). In this paper we give another answer: we analyze (1) as the coordination of the complete clause “Mary hates Bill” with the cluster “John” and “likes”. The types of the cluster components and their order guide the transformation that picks the needed material from the clause “Mary hates Bill” to make the cluster the complete clause. The ‘picking transformation’ can be naturally supported within the existing setup of TS, using the same mechanism used in [7] to analyze quantification and inverse linking. The intuition of ‘picking’ is made precise and formal in Sect. 3.

The structure of the paper is as follows. Section 2 reminds TS, in a different, clearer presentation. We then describe our approach to coordination: transforming non-canonical one to the ordinary coordination of clauses. Section 4 discusses the related work that forms the context of our approach. The rigorous nature of TS makes it easier to carry analyses mechanically, by a computer. In fact, the analyses in the paper have been so programmed and executed. The implementation, in the form of a domain-specific language embedded in Haskell – ‘the semantic calculator’ – is publicly available at http://okmij.org/ftp/gengo/transformational-semantics/.

2 TS Background

Traditional Categorial Grammar approaches draw parallels between proof systems and grammars: grammaticality is identified with the existence of a derivation. It is rather challenging however to prove the absence of a derivation, and to overview the space of possible derivations in general.

TS (formerly, AACG) [7] in contrast pursues the computational approach, harking back to Transformational Generative Grammars [2] of 1960s: Rather than trying to deduce a derivation, it tries to induce the meaning (the logical formula) by applying a sequence of precisely and formally defined transformations to a suitably abstract form of a sentence. The latter abstracts away the case and the number agreement, declination, etc. The transformations are deterministic; the order of their applications is generally not. (There may still be dependencies between particular transformations imposing the order.) The transformations are partial: the failure is taken as ungrammaticality of the original sentence.

Formally, TS deals with term languages that represent typed finite trees. Each T-language is a set of well-typed terms built from typed constants (function symbols) c. Types are

The set terms d is then inductively defined as: (i) each constant c of the type $\sigma$ is a term; (ii) if c has the type $\sigma _1\rightarrow \sigma$ and d is a term of type $\sigma _1$, then $c\ d$ is a term of type $\sigma$; (iii) nothing else is a term. The set of constants and their types is a (multi-sorted) algebraic signature; A T-language is hence a term language over the signature, which defines the language.

Table 1. Signatures of various T-languages

Table 1 shows three sample languages. $T_S$ has the single base type string and numerous constants "John", "greet", "every", etc. of that type. It describes the surface, “phonetic”, form of a sentence. The constant $\text{- }\cdot \text{- }: \textsf {string}\rightarrow \textsf {string}\rightarrow \textsf {string}$ (usually written as the infix operation) signifies string concatenation. The language $T_A$ whose types are familiar categories represents the abstract form. $T_L$ is the language of formulas of predicate logic, which describe the meaning of sentences. The (infinite) sets of constants $\mathsf {var_x}\ , \mathsf {var_y}\ ,\ldots$ and the corresponding$\mathsf {U_x}\ , \ldots$ and$\mathsf {E_x}\ , \ldots$ represent (to be) bound variables and their binders. Unlike the conventional (lambda-bound) variables, they are not subject to substitution, $\alpha$-conversion or capture-avoidance. $T_L$ likewise has constants $x,y,z,\ldots$ of the type e and the corresponding sets of constants $\forall _x, \forall _y, \ldots , \exists _x,\exists _y,\ldots$ intended as binders.

As a way to introduce TS we show the quantification analysis of “John greeted every participant”. The sample sentence in the language $T_A$ has the form

$$ \mathsf {cl}\ \mathsf {john}\ (\mathsf {argp}\ \mathsf {greet}\ (\mathsf {every_x}\ \mathsf {participant}\ )) $$

to be referred to as jgep. The constant $\mathsf {cl}\$ combines an NP and a VP into a clause. (Likewise, $\mathsf {argp}$ attaches an argument to a verb and $\mathsf {ppadv}$ attaches a prepositional phrase (PP) as a VP complement.) Quantifiers are uniquely labeled by x, y, z, etc. We assume it is the job of a parser to uniquely label the quantifiers in the abstract form.

Before taking on meaning we illustrate the recovering of the surface form of jgep, by applying the following ‘phonetic’ transformation $\mathcal {L}_{syn}$.

The rules are written in the form reminiscent of top-down tree transducers. The result $\mathcal {L}\ulcorner d\urcorner$ of transforming a term d is obtained by trying to match d against the pattern in the left-hand-side of every rule. The right-hand-side of the matching rule gives the result. If no matching rule is found, the transformation is not defined (i.e., ‘fails’). The patterns may contain variables, which stand for the corresponding subterms. For example, in the first rule, $d_1$ and $d_2$ match the two children of a term whose head is $\mathsf {cl}$. The occurrences of these variables in the right-hand side of the rule are replaced by the corresponding matching branches. Intuitively, $\mathcal {L}_{sem}$ looks like a context-free-grammar of the sample sentence, with jgep being its derivation tree.

The meaning is derived by applying a sequence of transformations to a $T_A$ term. The transformation $\mathcal {L}_{Ux}$ gets rid of $\mathsf {every_x}$, introducing $\mathsf {var_x}$ and $\mathsf {U_x}$ instead. This transformation is context-sensitive. Therefore, we first define context C – a term (tree) with a hole – as follows:

$$ C = [] \mathclose {\,}\mathrel \vert \mathopen {\,}\mathsf {cl}\ C\ d \mathclose {\,}\mathrel \vert \mathopen {\,}\mathsf {cl}\ d\ C \mathclose {\,}\mathrel \vert \mathopen {\,}\mathsf {argp}\ d \ C \mathclose {\,}\mathrel \vert \mathopen {\,}\mathsf {ppadv}\ C\ d \mathclose {\,}\mathrel \vert \mathopen {\,}\mathsf {ppadv}\ d\ C $$

where the meta-variable d stands for an arbitrary term. In words: a context is the bare hole [], or a clause (the $\mathsf {cl}$ term) that contains a hole in the subject or the predicate, or a VP made of a transitive verb whose argument has a hole, or a complemented VP with the hole in the head or the complement, etc. We write C[d] for the term obtained by plugging d into the hole of C. We further distinguish two subsets of contexts $C_{cl}$ and $C_{ncl}$:

Intuitively, $C_{cl}$ is the smallest context that has a hole within a clause.

The transformation $\mathcal {L}_{Ux}$ is then stated as follows:

We now use extended top-down tree transducers, whose patterns are ‘deep’, that is, contain matching expressions within arbitrary context. As before, whenever a pattern, e.g., $C_{cl}[\mathsf {every_x}\ \ d_r]$, matches the source term, it is replaced with$\mathsf {U_x}\ d_r \ C_{cl}[\mathsf {var_x}\ ]$, and the transformation is re-applied to its subterms. That is, $C_{cl}[\mathsf {every_x}\ d_r]$ on the left hand-side of the rule matches a tree that contains, somewhere inside, a sub-expression of the form $\mathsf {every_x}\ d_r$ (a branch headed by $\mathsf {every_x}$). On the right-hand side of the rule,$C_{cl}[\mathsf {var_x}\ ]$ is the same tree in which $\mathsf {every_x}\ d_r$ subterm has been replaced with $\mathsf {var_x}$. Unlike $\mathcal {L}_{syn}$ above, the $\mathcal {L}_{Ux}$ transformation does not look like a context-free grammar. It is context-sensitive. The other difference is the presence of a default rule: if $\mathcal {L}_{Ux}\ulcorner d\urcorner$ finds no match for d, $\mathcal {L}_{Ux}$ is repeated on sub-expressions of d. In particular, $\mathcal {L}_{Ux}\ulcorner c\urcorner$ is the constant c itself (unless there is an explicit rule for that particular c). For $\mathcal {L}_{syn}$, which translates from one language, $T_A$, to another, $T_S$, the default rule does not make sense.

Our example jgep matches the left-hand side of $\mathcal {L}_{Ux}$ immediately: $d_r$ matches $\mathsf {participant}$ and $C_{cl}$ is $\mathsf {john}$ ($\mathsf {argp}$ $\mathsf {greet}$ []), The result

$$ (\mathsf {U_x}\ \ \mathsf {participant)}\ \ (\mathsf {cl}\ \ \mathsf {john}\ \ (\mathsf {argp}\ \mathsf {greet}\ \ \mathsf {var_x}\ )) $$

is in effect the Quantifier Raising (QR) of “every participant”, but in a rigorous, deterministic way. The intent of the new constants should become clear: $\mathsf {U_x}$ is to represent the raised quantifier, and $\mathsf {var_x}$ its trace. Unlike QR, the raised quantifier ($\mathsf {U_x}$$\mathsf {participant)}$ lands not just on any suitable place. $\mathcal {L}_{U}$ puts it at the closest boundary marked by the clause-forming constant $\mathsf {cl.}$ $\mathcal {L}_{U}$ is type-preserving: it maps a well-typed term to also a well-typed term. Again unlike QR, we state the correctness properties such as type-preservation. The type preservation is the necessary condition for the correctness of the transformations.

To finally obtain the meaning we apply the transformation $\mathcal {L}_{sem}$:

that produces the logical formula representing the term’s meaning. The transformation replaces $\mathsf {john}$, etc. with the corresponding logical constants and $\mathsf {U_x}$ with the universal quantifier. Since $\mathcal {L}_{sem}$ translates one language, $T_A$, into a different one, $T_L$, this transformation, like $\mathcal {L}_{syn}$, has no default rule. If the source term does not match the pattern of any $\mathcal {L}_{sem}$ rule, the transformation is undefined. In particular, applying $\mathcal {L}_{sem}$ to the original jgep term straight away is not defined because there is no rule for $\mathsf {every_x}$. The failure means that jgep cannot be given meaning – directly. However, $\mathcal {L}_{sem}\ulcorner \mathcal {L}_{Ux}\ulcorner jgep\urcorner \urcorner$ is well-defined, resulting in

3 Coordination in TS

We now apply TS to the analysis of (non-canonical) coordination. As a warm-up, we take the non-problematic “John tripped and fell,” which is an example of the conventional VP coordination. We analyze it differently, however, as ‘left-node raising’ so to speak, to introduce the technique to be later used in right-node raising (RNR), argument cluster coordination (ACC) and gapping¹.

The abstract form of our example is

$$ \mathsf {and_{S,VP}}\ (\mathsf {cl}\ \mathsf {john}\ \mathsf {tripped}\ ) \ \mathsf {fell}\ $$

The new constant $\mathsf {and_{S,VP}}$ has the type $S\rightarrow VP \rightarrow S$. As common, we assume a whole family of constants $\mathsf {and_{X,Y}}$ of different types. The constant $\mathsf {and_{S,VP}}$ – like $\mathsf {every_x}$ in the example of the previous section – is not in the domain of $\mathcal {L}_{sem}$. Therefore, to be able to derive the logical formula, we have to transform it away. The following transformation $\mathcal {L}_{a}$ does that:

The rule again is written in the form of extended top-down tree transducers: when the source term matches the rule’s pattern, it is replaced with the right-hand-side of the rule. Again, d with various subscripts are meta-variables that stand for arbitrary subterms (tree branches). Like $\mathcal {L}_{Ux}$, there is a default rule: a term that does not match the rule undergoes $\mathcal {L}_{a}$ on its subterms, if any. Applying $\mathcal {L}_{a}$ to our $T_A$ term transforms it to

$$ \mathsf {and}\ (\mathsf {cl}\ \mathsf {john}\ \mathsf {tripped}\ ) \ (\mathsf {cl}\ \mathsf {john}\ \mathsf {fell}\ ) $$

where $\mathsf {and}$ is the ordinary coordination, of the type $S\rightarrow S\rightarrow S$, which can be given the meaning of propositional disjunction and which hence is in the domain of $\mathcal {L}_{sem}$. The result is straightforward to transform to a logical formula $T_L$.

3.1 RNR in TS

Our next example is the proper RNR: “John likes and Mary hates Bill”, whose abstract form is

$$ \mathsf {and_{(NP,TV),S}}\ (\mathsf {john}\ , \mathsf {like}\ ) \ (\mathsf {cl}\ \mathsf {mary}\ (\mathsf {argp}\ \mathsf {hate}\ \mathsf {bill}\ )) $$

We have added to $T_A$ tuples (d, d) and tuple types $(\sigma ,\sigma )$. The constant $\mathsf {and_{(NP,TV),S}}$ has the type $(NP,TV) \rightarrow S \rightarrow S$. Whereas $(\mathsf {cl}\ \mathsf {mary}\$ $(\mathsf {argp}\ \mathsf {hate}\ \mathsf {bill}\ ))$ is the complete sentence,$(\mathsf {john}\ , \mathsf {like}\ )$ is certainly not. It is not even a constituent; it is just a sequence of words: a cluster. Since we added to $T_A$ tuples and new constants, we may need to extend our earlier transformation rules, specifically, $\mathcal {L}_{syn}$ for transforming into the surface form of the sentence $T_S$:

Applying $\mathcal {L}_{syn}$ to our $T_A$ clearly gives “John likes and Mary hates Bill”. This ‘phonetic’ transformation is dull and uninteresting, in contrast to the higher-order phonetics of [8].

Let us derive the meaning, the $T_L$ formula, from the same $T_A$ term. Before we can apply $\mathcal {L}_{sem}$ we need to transform away $\mathsf {and_{(NP,TV),S}}$, which is not in the domain of that transformation. We extend the $\mathcal {L}_{a}$ with a new clause:

where $d_1, d, d_5$ have to be of the type NP and $d_2$ and $d_4$ of the type TV. The transformation is context-sensitive and type-directed. It may be regarded as matching of $(d_1,d_2)$ against the complete sentence (the second argument of $\mathsf {and_{(NP,TV),S}}$). The matching is determined by the type of $\mathsf {and_{(NP,TV),S}}$. The parallel structure of the coordination is clearly visible.

Analyses of RNR without directional types (e.g., using ACG) run into trouble of over-generating “*John likes Bill and Mary hates”. Although we can write the abstract form for that sentence as well:

$$ \mathsf {and_{S,(NP,TV)}}\ (\mathsf {cl}\ \mathsf {john}\ (\mathsf {argp}\ \mathsf {like}\ \mathsf {bill}\ )) \ (\mathsf {mary,}\ \mathsf {hate}\ ) $$

we do not provide the $\mathcal {L}_{a}$ rule with the constant $\mathsf {and_{S,(NP,TV)}}$. Since it remains uneliminated, $\mathcal {L}_{sem}$ cannot be applied and the meaning cannot be derived. In TS, transformations are partial and are not guaranteed to always succeed. The original sentence is considered ungrammatical then. We discuss the choice of transformable $\mathsf {and_{XY}}$ constants in Sect. 3.4.

Let us consider another well-known troublesome example, due to P. Dekker:

In categorial grammar approaches, ‘the mother of’ and ‘John thinks that’ may be given the same type, $(S/(N\backslash S))/N$. The two phrases may hence be coordinated, over-generating (1). In TS, ‘the mother of’ cannot be given any type at all (likewise, ‘John thinks that’ is not a constituent and has no type.) We can only treat ‘the mother of’ as a cluster, of the determiner, N and the proposition. We do provide the constant $\mathsf {and_{(DET,N,POF),S}}$ with the corresponding rule

which can be used to analyze “The mother of, as well as the father of John died”. The rule does not apply to the problematic (1) since there is no similar parallel structure of the of-headed PP.

3.2 Argument Cluster Coordination and Gapping

The same transformation idea also works for argument cluster coordination (ACC) and gappping. Take for example, “Mary liked Chicago and Bill Detroit”, or, in the abstract form:

$$ \mathsf {and_{S,(NP,NP)}}\ (\mathsf {cl}\ \mathsf {mary}\ (\mathsf {argp}\ \mathsf {liked}\ \mathsf {chicago}\ ))\ (\mathsf {bill}\ , \mathsf {detroit}\ ) $$

The transformational rule for the constant $\mathsf {and_{S,(NP,NP)}}$ picks a suitable subterm that can relate two NPs from the left conjunct

It turns our $T_A$ term to

$$ \mathsf {and}\ (\mathsf {cl}\ \mathsf {mary}\ (\mathsf {argp}\ \mathsf {liked}\ \mathsf {chicago))}\ \ (\mathsf {cl}\ \mathsf {bill}\ (\mathsf {argp}\ \mathsf {liked}\ \mathsf {detroit))}\ $$

with the clear meaning. The examples (2) and (4) of Sect. 1 are dealt with similarly. One may observe that the analysis of gapping is nearly the same as that of VP coordination, used in the warm-up example.

3.3 Coordination and Scoping

The interaction of non-canonical coordination with quantification is not much different from that of the ordinary coordination of two clauses. For example, take (2) of Sect. 1, whose abstract form is

contains two components to be eliminated by transformations: $\mathsf {and_{S,(PP,PP)}}$ and the QNP$(\mathsf {a_x}\ \mathsf {present}\ )$. The latter is to be handled by $\mathcal {L}_{E}$, which is analogous to $\mathcal {L}_{U}$ but for the existential quantifier. The transformations $\mathcal {L}_{a}$ and $\mathcal {L}_{E}$ can be applied in either order, which corresponds to the wide- and narrow-scope–readings of (2). The narrow scope happens when $\mathcal {L}_{a}$ goes first, producing

The $\mathcal {L}_{Ex}$ transformation then gives

whose meaning is the conjunction of two existentially quantified formulas.

If $\mathcal {L}_{Ex}$ is applied first to the original sentence, we get

Strictly speaking, the rule analogous to $\mathcal {L}_{a}$ from Sect. 3.2 does not apply since the first conjunct now has the form $\mathsf {E_x}\ d_r \ (\mathsf {cl}\ d_1 \ d_2)$ rather than the bare $(\mathsf {cl}\ d_1 \ d_2)$. We have to hence generalize the rule to

effectively pulling out the context $C_{ncl}$ – the sequence of $\mathsf {U_x}\ d$ and $\mathsf {E_x}\ d$ quantifiers and their restrictors – and coordinating underneath. The coordination thus receives narrow scope. Such pulling of the context may seem ad hoc; however, it is this general form of $\mathcal {L}_{a}$ rules that gives the mechanism to account for the anomalous scope of negation in (7) of Sect. 1, repeated below.

The transformation involving the contrasting coordinating particle such as ‘but rather’ gets a chance to examine $C_{ncl}$ and determine if there is a negation to contrast with:

where $\mathsf {Neg}$ is the constant analogous to $\mathsf {U_x}$.

3.4 Discussion

We have presented the uniform analysis of both the canonical and non-canonical coordination, reducing the variety of coordination (VP, RNR, ACC, Gapping) to the choice of the coordinating constants $\mathsf {and_{S,X}}$ or $\mathsf {and_{X,S}}$ that adjoin material (often just a cluster of words) to a sentence. The transformation rules driven by the constants pick the pieces from the sentence to complete the material to a clause. We have thus provided a uniform mechanism of coordination. The corresponding policy is embodied in the coordinator constants like $\mathsf {and}$ hence lexicalized.

There remains a question of a general principle/pattern that governs the choice of the constants. For example, the fact that in English the coordinated sentence appears on the right for RNR but on the left for ACC and Gapping boils down to the presence of $\mathsf {and_{(NP,TV),S}}$ and $\mathsf {and_{S,(NP,NP)}}$ and the absence of $\mathsf {and_{S,(NP,TV)}}$ and $\mathsf {and_{(NP,NP),S}}$. In contrast, one may say that this fact ‘falls out’ as a consequence of like-category coordination analyses in directional categorial grammars. One may also say that the like-category coordination is itself a postulate, which does not come from any general principle, but does have significant empirical justification. Like any empirical principle, it has exceptions: unlike-category coordination, e.g., “John saw the facts and that Mary had been right”. Also, the like-category coordination leads to overgeneration, as we saw in the Dekker’s example in Sect. 3.1.

Since our TS approach is still new, we have not yet accumulated enough empirical data to discern patterns and formulate postulates that underlie the presence of coordination constants for some types and their absence for others. For now, we leave the question open.

4 Related Work

Our transformational approach is rooted in Transformational Generative Grammars [2, 3], later carried into Minimalism [4]. Our abstract form $T_A$ is similar to the spell-out of Minimalism. However, whereas the spell-out is near culmination of a syntactic derivation for Minimalists, for us, it is just the beginning. We are not interested in how structure is created through a sequence of Merges from lexical selections. Rather, we consider our abstract form as given (by a parser) and investigate its transformations into a semantic form. Our transformations are hence all covert.

Closely related to TS is the work of Butler [1], who also obtains a semantic representation as a result of a transformation from a parsed tree. Unlike us, he has applied his approach to a wealth of empirical data in many languages and has truly achieved wide coverage. His transformations are rather complex and coarse, doing many things at once, and not typed. One may view TS as an attempt to re-engineer and understand Butler’s approach and decompose his transformations into elementary steps.

We are grateful to the anonymous reviewer for pointing out the analysis of ACC and Gapping in [14].

(1) The interpretation of an elliptical construction is obtained by uniformly substituting its immediate constituents into some immediately preceding structure, and computing the interpretation of the results. [14, p. 162, (119)]

We indeed share the underlying idea of picking and substituting of ‘immediate constituents’ into the coordinated material (understood at some level as an elliptical construction). The proposal of [14] remained rather informal; the present paper may be seen as an attempt to formalize the idea, as well as to extend it to scope phenomena.

There have been other attempts to solve the KLM problem without directional types (within the ACG-like formalisms). Kanazawa [5] proposes ‘regular constraints’ to prevent over-generation (which recall structural constraints in Government and Binding). This amounts however to duplication of lexical entries. The approach [13] reins in the over-generation using subtyping. Either proposal can be classified as ‘proof search’ rather than computational like TS; in case of [13] with no guarantees that the proof search ever terminates (and, as the authors admitted, no good way to characterize the space of available derivations and detect over-generation).

5 Conclusions

We have demonstrated the transformational analyses of RNR and Gapping. The analyses make precise various eliding schemas, demanding type preservation. The asymmetry of the type of $\mathsf {and_{(NP,TV),S}}$ and similar constants is what lets us answer the Kubota, Levine and Moot challenge: how to prevent over-generation in analyses of RNR and gapping without directional types.

The idiosyncrasies of coordination are distilled to the ad hoc choice of constants $\mathsf {and_{XY}}$. There are transformations for some types XY but not for the others. There may be a pattern there. Collecting the arbitrariness in one place might make the pattern easier to find. Being able to handle the entire ellipsis part of the FraCaS corpus seems the natural first step in searching for that pattern.

It is interesting to consider interpreting the “sequence of words” as a discontinuous sentence in the sense of Morrill et al. [12].

Another future work task is to apply TS to more complicated scoping phenomena including ‘same’, ‘different’, ‘the total of’ – as well as to various wh-movement phenomena.