On Constraining Free Merge*

Jason Ginsburg Sandiway Fong**

Osaka Kyoiku University University of Arizonajginsbur@cc.osaka-kyoiku.ac.jp sandiway@email.arizona.edu

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The 43rd Meeting of the Kansai Linguistic Society

*Supported by Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research #16K02769 1

Outline

• Background• Set Merge, Pair Merge, Overgeneration

• Constraints• Extension to Other Examples• Conclusion

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Background

• In the Labeling-based work of Chomsky (2013, 2015), Merge is free.

• Merge is not feature-driven, no Edge/EPP features.

• This simplification comes at the cost of combinatorics.

• How might we mitigate and constrain the resulting massive over-generation?

• We present a promising model (with combinatorics calculated by computer) that suggests, contra expectation, Free Merge may in fact be computationally tractable, when constrained by reasonable assumptions about computational efficiency.

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Background

• Merge (Chomsky 1995, 2013, 2015, etc.)

• Set Merge• Symmetric: given two Syntactic Objects (SOs) α and β, we form the set {α, β}• If α is a (strong) head and β is an XP, then α Labels {α, β}• If both α and β are XPs, Labeling of {α, β} only occurs if

• they share prominent features, e.g. ɸ or Q features• one of the XPs α (or β) moves out, the remaining XP β (or α) will Label

• Pair Merge• Asymmetric: ordered pair <α, β>• α becomes invisible to further syntactic operations, including probing and feature

valuation, and (still visible) β Labels <α, β>

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Set Merge: Labeling

(a) Head X labels (b) Head X is too weak to label unless strengthened

(c) No label• No shared features

(d) Y labels if XP moves out• Not all copies of XP are

within this Syntactic Object • Y is not weak

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• Pair Merge• Asymmetric: ordered pair <α, β>. • α becomes invisible to further syntactic operations, including probing and

feature valuation, and (still visible) β labels <α, β>• Arc indicates Pair Merge

β is visible• If β is labeled, the label stays

the same

α is Pair Merged• α is invisible

Background

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Background

• Merge (Chomsky 1995, 2013, 2015, etc.) • Merge is free: • In {α, {β, ɣ}}, ɣ can undergo internal Set Merge or Pair Merge to form:

• {ɣ, {α, {β, ɣ}}} • <ɣ, {α, {β, ɣ}}>

• If Merge is free, then how do you block an infinite number of Merges?• In theory, any SO can undergo internal or external Set Merge or Pair Merge an

infinite number of times. • In theory, when generating a phrase, you have an infinite number of possible

derivations.

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Background

• Given 8 Merge operations, over 7 million distinct SOs (Syntactic Objects) can be formed!• We computed this via computer model• 2 separate computer models

• This overgeneration is potentially a problem.

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Simple example: the book

• Assume an undifferentiated root + categorizer analysis• Lexical Array: {d, the, n, book} (cf. Chomsky 2015)• Since Internal Merge may be iterated, we have in principle an

unbounded number of SOs scaling with Merge depth• An infinite number of unattested SOs can be generated

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Simple example: the book• Unattested SOs can be generated

• Notation: !F indicates an unvalued feature F• !case is unvalued case

• ? indicates an unlabeled structure• straight lines indicate Set Merge• arcs indicate Pair Merge

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Set MergePair Merge

Simple example: the book• More unattested Syntactic Objects

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(a) (b) (c)

(d)(e) (f) (g)

Simple example: the book

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# of possible Syntactic Objects as # Merges grows

Log-scale graph of possible SOS as # Merges grows

Free Merge must be constrained.

#Merges #SOs

1 3

2 7

3 29

4 161

5 1,423

6 18,144

7 318,480

8 7,396,976

Factors in language design

• Chomsky (2005)• Three factors in language design: (1) First Factor: genetic endowment(2) Second Factor: experience(3) Third Factor: “language-independent principles of data processing, structural architecture, and computational efficiency”

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Factors in language design

• Chomsky (2005)• (1) First Factor: Genetic Endowment• First Factor elements are (possibly) specific to the language faculty

• Labeling:• A requirement for interpretation of syntactic objects• Unlabelable structures will be rejected at the Conceptual-Intensional (CI)

Interface

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First Factor Constraints

• There are roots and categorizers• Categorizers are heads such as n, d, adj, adv, and v • Roots must be categorized• (Not all heads are categorizers or roots: e.g. T and C)

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Root book Set Merges with categorizer n • n categorizes

Root the Set Merges with the categorizer d• d categorizes

First Factor Constraints

•*pm-Root• No Pair Merge of a root• book and n are Pair Merged (PM) as <book,n>, with book as an adjunct

• book is invisible, so it can never be categorized• Since the root cannot be categorized, it can’t be labeled

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Pair Merged adjunct

!case is unvalued case

First Factor Constraints

•*pm-Cat• No Pair Merge of a categorizer

(assuming a categorizer must categorize)

• the and d are Pair Merged (PM) as <d,the>, with d as an adjunct• d is invisible, so it can not categorize

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Pair Merged adjunct

•*pm-uF• *<SO!F, α> Pair Merged SO with a uF to α• Example: <n!case, book>

• !case is an unvalued Case feature from the adjunct n • Since n!case is an adjunct, !case can never be valued• Possible exception (Chomsky, p.c.): Concord, not discussed here

Pair Merged adjunct

First Factor Constraints

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First Factor Constraints

• The Labeling Algorithm blocks unlabelable structures.• There are no shared prominent features for {book,n!case} and {the,d}• The resulting Set Merge is unlabelable (as indicated by ?)

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Unlabelable• No shared prominent features

First Factor Constraints

• First factor constraints severely constrain Pair Merge• These constraints rule out many unattested structures• Labeling Algorithm rules out unlabelable SOs

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Third-factor Constraints• *loop• The computational mechanism actively blocks (pending) repeated operations that lead to infinite

loops • not necessarily specific to language.

• Internal Set Merge (ISM) of the applies twice• ESM(the,d) -> ISM(the) -> ISM(the)

• *ISM(X) -> ISM(X) is an instance of a Third Factor constraint (*loop), as it permits Workspace enlargement without limit

formed from {the, d}by• ISM({the})• ISM({the})

Constraints• *loop• Many more complex examples are also ruled out by *loop

formed from {{the, d}, {the, {the, d}}}by• ISM{the, {the, d}}• ISM{the, {the, d}}

formed from {the, {the, d}}by• ISM({the, d})• ISM({the, d})

Constraints

• the book converges after 6 operations. • One solution found• No other possible solutions - search space is finite!

• Given constraints, we can generate the desired structure in six Merge operations. Unattested structures are ruled out. • *pm-Root, *pm-Cat, *pm-uF, *loop, Labeling Algorithm

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• the book is an NP • DP is an adjunct

• Chomsky (2007), also see Oishi (2015)• N (not D) is the head of a phrase such

as ‘the book’• determiner can be an XP

Extension to Other Examples

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Extension to Complex NPs

• We have extended this model of the NP, and explored the search space for more complex NPs such as possessives.• In my friend, me ’s is pronounced as my and functions as a DP that is

Pair Merged with friend.

25my friend

Extension to Complex NPs• Given my friend, the root friend

undergoes internal Set Merge• friend is categorized by n • Root friend is categorized twice

by different n heads• there is no restriction against

relabeling - a root can be categorized multiple times (cf. Ceccheto & Donati 2015)

• The DP {d, the} is Pair Merged• Case on me ’s pronounced as of• me ’s followed by copy of an NP

is pronounced as mine26

the friend of mine

Extension to Complex NPs• Combinatorics are given here:

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0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9

log10(#SOs

)

#Merges

LOG10(#SOs Generated) vs.#Merges

Single (correct) solution found for the friend of mine at depth 6• Nearly 1,000 SOs were considered. No other SOs

found at depths 7–9 (exhaustively explored)

See Ginsburg & Fong (2018)

Sentence: John liked the book

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Correct solution found for John liked the book at depth 11• Note that other unattested SOs are found at higher

depths

John liked the book

Example: John liked the book

spuriousanalyses

intendedanalysis

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

LOG

10(#

SYN

TACT

IC O

BJEC

TS)

# OPERATIONS

Log10(# SOs Generated) vs. # Operations

#operations = 16obtain 3 spurious analyses

#operations = 15obtain single analysis

Example Combinatorics†† naïve version

• logscale y-axis:• e.g. 6 = 106 = million• 15 operations deep:

• 25 million SOs generated

• 1 convergent SO

• 16 operations deep:• 250 million SOs• 3 spurious SOs

John liked the book

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• Following Chomsky (2015):• the book (= <{the,d},{book,n}>) undergoes Object Shift • v* transmits unvalued ɸ-features to root like• Case is checked on the book• Shared prominent ɸ-features label as <ɸ,ɸ>. • Shared ɸ-features enable the strengthened root like to label• Subject John Set Merges to the edge of v*P forming {XP,YP}

• labeled by v* because the subject raises out to the edge of T• C transmits unvalued ɸ-features to T• Unvalued ɸ on T probes and Agrees with ɸ on the subject• Prominent shared ɸ-features label as <ɸ,ɸ>• Strengthened T labels • There is a phonological operation that results in like being pronounced

at the position of v*

John liked the book

Conclusion

• This model shows that it is possible to successfully converge upon correct analyses for our test sentences given Free Merge and motivated Third Factor constraints that limit the space of possible Merge operations. • The model can generate target structures with the aid of constraints. • Constraints discussed are 1st and 3rd Factor elements. • Performance: 2nd Factor

• Our research question: are these constraints sufficient?• (A) Can the theory eliminate all unattested SOs?• (B) Is the theory constrained enough?• (C) Are other 3rd factor constraints needed?

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Implementation• This model has been implemented via 2 separate computer programs

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Sandiway Fong’s FreeMerge Machine• Calculates all possible Merges, given our

proposed constraints

References

Cecchetto, Carlo. and Caterina. Donati. 2015. (Re)labeling. Cambridge, MA: MIT Press.

Chomsky, Noam. 1995. The Minimalist Program. Cambridge, MA: MIT Press.

Chomsky, Noam. 2004. Beyond explanatory adequacy. In Belletti, A. (ed.), Structures and beyond: The cartography of syntactic structures, vol. 3, 104-131. Oxford: Oxford University Press.

Chomsky, Noam. 2005. Three factors in language design. Linguistic Inquiry 36: 1-22.

Chomsky, Noam. 2007. Approaching UG from below. In Sauerland, U. & Gartner, H.-M. (eds). Interfaces + Recursion = Language?, 1-29. New York: Mouton de Gruyter.

Chomsky, Noam. 2013. Problems of projection. Lingua 130:33-49.

Chomsky, Noam. 2015. Problems of projection: Extensions. In Domenico, E. D., Hamann, C., Matteini, S. (eds.) Structures, strategies, and beyond – studies in honor of Adrianna Belletti, 3-16. Amsterdam: John Benjamins.

Ginsburg, Jason. & Sandiway Fong. 2018. A relabeling analysis of English possessives. In Papers from the 35th Conference and the Tenth International Spring Forum of the English Linguistic Society of Japan: JELS 35. 211-216. The English Linguistic Society of Japan.

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