Recursion in Language: Is It Indirectly Constrained?

Abstract

Recursion is a key aspect of formal theories of language. In this chapter we have presented a new perspective on recursion in language, claiming that recursion in language is indirectly constrained. After discussing some aspects of the relation between linguistic theory and mathematical linguistics, we show that this perspective arises from the properties of grammars that are beyond context-free grammars with context-sensitivity that is mild in a sense and that the derivation proceeds from structures to structures. Viewing the generation process inductively, we then show that for each index of recursion, there is a form (we call this a canonical form) that can be generated. However, corresponding to the canonical form, there are variants of the form, called here as non-canonical forms that exhibit gaps in the sense that for a given index of recursion, beyond two levels of embedding, not all non-canonical forms can be generated. This property emerges from the formal architecture of the system and not from any particular linguistic or processing constraint. We call this property Indirectly Constrained Recursion (ICR), suggesting a new perspective on recursion in language. ICR is a formal structural constraint and not a linguistic or processing constraint. In linguistic theories constraints on recursion are usually expressed either in terms of some linguistic constraints or in terms of processing constraints. ICR arises from the properties of the kinds of formal systems considered here. Linguistic constraints and processing constraints are then to be considered as additional constraints. The linguistic phenomena referred to in this chapter are well-known. Hence, the main contribution of this chapter is the presentation of a new perspective on recursion as described above.

In this chapter, we have presented ICR with respect to a particular framework. However, it is expected that similar results will carry over to some of the other systems, for example, in the class of Mildly Context-Sensitive Languages and perhaps, stronger versions of ICR can be found.

Appendices

Appendices

1.1 Appendix A

1.1.1 LTAGs and MC-LTAGs

We will give a brief introduction to Lexicalized Tree Adjoining Grammars (LTAGs) and tree-local Multi-Component LTAGs (MC-LTAGs), which will be adequate to follow the derivations in Sects. 1, 2, and Appendix B.

An LTAG consists of a finite set of elementary structures (trees, directed acyclic graphs in general) where each elementary structure is associated with a lexical anchor. Each elementary structure encapsulates the syntactic/semantic arguments and dependencies (corresponding to the lexical anchor associated with the structure) such as agreement, subcategorization, filler-gap dependencies, among others. These structures are large enough to localize all these dependencies and therefore can be larger than one level trees. This property, called the extended domain of locality, is in contrast to the case of CFG’s where the dependencies are distributed over a set of rules.

The elementary trees are either initial trees or auxiliary trees. There is no recursion inside an elementary structure. Of course, an elementary structure may have the same label for the root node and the foot node in an auxiliary tree and can thus participate in a recursive structure. Since there is no recursion internal to an elementary structure, in a sense, in LTAGs recursion is factored away from the domain of dependencies. This is in contrast to CFG’s where recursion is intertwined with the domain of locality, as both may be spread over several rules.

There are two composition operations, substitution and adjoining: These operations are universal in the sense that they are the same for all LTAGs. There are no rules in the sense of the rewriting rules in a CFG.

Since all dependencies of a lexical anchor are localized we may have several elementary structures corresponding to a lexical item, depending on the structural positions each argument can occupy in different constructions associated with the lexical item. There can be many such trees but the total number is finite. One can regard the different trees associated with a lexical item as extended projections associated with that item.

Figures A.1, A.2, and A.3 show an LTAG derivation. Figure A.1 shows the collection of trees²⁰ participating in the derivation. Figure A.2 shows the composition of the elementary trees in Fig. A.1 by means of the operations of substitution (shown by a solid line) and adjoining (shown by a dotted line). Figure A.3 shows the derivation tree. The nodes are labeled by the names of trees as well as by the lexical anchors associated with the elementary trees. Solid lines represent substitution and the dotted lines represent adjoining. The lines should also be labeled by the addresses of the nodes in the host trees where substitutions and adjoinings take place. These have been omitted in order to simplify the diagram. Finally, Fig. A.4 shows the derived tree, which is the result of carrying out the compositions shown in Fig. A.3.

Fig. A.1

An LTAG derivation

Fig. A.2

An LTAG derivation

Fig. A.3

A derivation tree

Fig. A.4

A derived tree

Note that the dependency between who and e has become long distance. However, it was local in the elementary tree associated with likes in Figs. A.1 and A.2.

The derivation tree is a record of the history of composition of the elementary structures by substitution or adjoining. The derived tree is the result of carrying out all the composition operations as indicated in the derivation tree. In a sense, the derivation tree has all the information and, in fact, compositional semantics can be defined on the derivation tree (Kallmeyer and Joshi 2003).

The order of derivation in the derivation tree in Fig. A.3 is shown bottom up in a sense. This is the normal convention that is followed in a TAG derivation. However, in principle the derivation could begin with any node with the directions of the arrows reversed appropriately. The order of derivation is flexible in a sense; see also Sect. 1 and (Chomsky 2007).

1.1.2 Multi-component TAG’s (MC-TAG’s)

MC-TAG’s are like LTAG’s except that in an MC-TAG the elementary trees can be a set of trees (usually a pair of trees in the linguistic context) and at least one of the trees in the multi-component set is required to have a lexical anchor. In this sense, MC-LTAGs are lexicalized. In a tree-local MC-TAG when a multi-component tree (say with two components) composes with another tree, say t, the two components compose with the same elementary tree, t, by targeting two nodes of that elementary tree. This is why this composition is called tree-local. The two components target two nodes in the local domain of an elementary tree, preserving the notion of locality that is inherent in LTAG. Tree-local MC-LTAGs are known to be weakly equivalent to LTAGs, i.e., they generate the same set of strings but not necessarily the same set of structural descriptions. Instead of formally describing the composition operation in MC-LTAG, we will illustrate it by means of a simple example. In Fig. A.5 we have shown a simple tree-local MC-LTAG derivation.

Fig. A.5

A derivation in a tree-local MC-TAG

The derivation proceeds as follows; a3 is substituted in a2 as shown and then it is (reverse) adjoined to the b21²¹ component of b2. The multi-component tree b2 is composed with a1 as follows. b21 is substituted at the NP node of a1 and the b22 is adjoined to a1 at the root node S of a1.

In a multi-component composition in the linguistic context there is usually a constraint between the two components (besides any co-indexing across the components). For example, in our example above there is a constraint (not shown in the Fig. A.5) that the foot node of b22 immediately dominates the NP root node of b21. This constraint is obviously satisfied in the derivation in Fig. A.5.

Schabes and Shieber (1993) introduced an alternate definition of derivation for LTAG in which they allowed multiple adjoining of certain kinds of elementary trees at the same node during the derivation. They showed that multiple adjunctions of the kind described above do not affect the generative capacity of LTAGs, i.e., they are weakly equivalent to LTAGs. This notion of multiple adjunctions can be extended to the derivations in tree-local MC-LTAG where we can allow multiple adjunctions of the NP auxiliary trees that correspond to the scrambled NP’s (see Appendix B). The generative power of tree-local MC-LTAG is not affected. We use such derivations for certain scrambling patterns. In Appendix B, we will show some derivations for sentences with scrambling using tree-local MCTAG.

1.2 Appendix B

Finally, we will describe some derivations for the scrambling (permutation) patterns using LTAGs or MC-LTAGs.²² First consider

(1)	…dass	Hans	Peter	Marie	schwimmen	lassen	sah
		Hans	Peter	Marie	swim	let	saw
		NP1	NP2	NP3	V3	V2	V1

This canonical sequence and (2) below can be easily derived as shown in Figs. B.1 and B.2 above. Trees for V1, V2, and V3 are all LTAG trees (all nodes on the spine (except the lowest node) are labeled as VP nodes for convenience).

Fig. B.1

Derivation for NP1 NP2 NP3 V3 V2 V1

Fig. B.2

Derivation for NP3 NP2 NP1 V3 V2 V1

(2)

NP3

NP2

NP1

V3

V2

V1

In Fig. B.1, the V1 tree is adjoined at the root node of the V2 tree and then the derived tree is adjoined to the root node of the V3 tree. In Fig. B.2, the V1 tree is adjoined to the interior VP node of the V2 tree and the resulting tree is then adjoined into the interior VP node of the V3 tree. Finally in Fig. B.3, the V1 tree is adjoined to the interior VP node of the V2 tree and then the resulting tree is adjoined to the root node of the V3 tree.

Fig. B.3

Derivation for NP2 NP1 NP3 V3 V2 V1

Finally, in Figs. B.4 and B.5 we show some derivations for sequences with four NP’s and four VP’s. Figure B.4 shows a derivation where the V1 tree is adjoined to the interior VP node of the V2 tree and the resulting tree is adjoined to the interior VP node of the V3 tree. Finally, the resulting tree is adjoined to the interior VP node of the V4 tree.

Fig. B.4

Derivation for NP4 NP3 NP2 NP1 V4 V3 V2 V1

Fig. B.5

Derivations for two sequences NP1 NP2 NP4 NP3 V4 V3 V2 V1 and NP4 NP1 NP2 NP3 V4 V3 V2 V1

In Fig. B.5 the V1 tree is adjoined to the root node of the V2 tree and the resulting tree is adjoined to the root node of the V3 tree. The V4 tree is a multi-component tree. It has two components. The lower component of the tree has V4 and its subject argument, NP4 which has been scrambled and hence shown with a trace. The upper component has the NP4 argument which is scrambled. There is also a constraint between the two components (not shown explicitly in Fig. B.5). The VP foot node of the upper component must dominate the VP root node of the lower component when the multicomponent is composed with an elementary host tree. The two components of the multi-component tree for V4 must combine with the same elementary tree, i.e., each component must target an elementary tree in order to compose with it, either by substitution or by adjoining. This assures the locality of the composition, as required by the definition of a Tree-Local Multi-Component TAG (TL-MCTAG), which is weakly equivalent to the standard TAG. The lower component of the V4 tree is “attached” (reverse adjoined in a sense²³) to the VP foot node of the V3 tree (as VP4 is the complement argument of V3) and the upper component of the V4 tree is adjoined to the root node of the V3 tree. Note that we have multiple adjunctions at the root node of the V3 tree.²⁴ Figure B.5 thus represents two possible sequences as indicated. It can be thought of as an underspecified representation of these two sequences.

Note that in Fig. B.5, the elementary ‘tree’ for V4 is in two parts (components), the lower and the upper part. The lower part has the trace for NP4 and the upper tree has the scrambled NP4 argument. There is an implied constraint (not shown here) that after these two components of the tree for V4 compose with the tree for V3 at the nodes VP (root node) and the frontier node labeled VP* respectively, these adjoining sites are in a dominating relation. Such constraints are a part of the Tree Local MCTAG composition and further these constraints are specified for each elementary object in a TL-MCTAG.

With 4 verbs and 4 NP’s there are 24 possible sequences, permuting the NP’s and keeping the VP’s in a fixed order. It can be shown that 22 of these sequences can be generated by Tree-local MCTAG and two sequences cannot be generated. These two sequences are

(1)	NP3	NP1	NP4	NP2	V4	V3	V2	V1
(2)	NP2	NP4	NP1	NP3	V4	V3	V2	V1

Thus there is a gap of two sequences in the set of the non-canonical transforms of the canonical sentence forms with three levels of recursion. For every depth of recursion beyond three or more embeddings gaps will continue to exist, for example, with four levels of embedding there will be six gaps, etc.

In a TAG or MC-TAG the composition is always binary. However, the elementary trees are not required to be binary. In the figures above the lower components are non-binary. If one imposes the binary form restriction on the elementary trees of TAG and MC-TAG then it can be shown that the first time the gaps appear is when we have four levels of embedding.