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The 43rd Meeting of theKansai Linguistic Society*On Constraining Free Merge
Jason Ginsburg
Osaka Kyoiku University

Sandiway Fong**
University of Arizonajginsbur@cc.osaka-kyoiku.ac.jp

sandiway@email.arizona.edu*Supported by Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research #16K02769
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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 overgeneration?
• We present a promising model (with combinatorics calculated bycomputer) that suggests, contra expectation, Free Merge may in fact
be computationally tractable, when constrained by reasonableassumptions 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>4Set Merge: Labeling(a) Head X labels (b) Head X is too weak tolabel unless strengthened(c) No label• No shared features(d) Y labels if XP moves out• Not all copies of XP arewithin this Syntactic Object• Y is not weak5

Background• Pair Merge• Asymmetric: ordered pair <α, β>.
• α becomes invisible to further syntactic operations, including probing andfeature valuation, and (still visible) β labels <α, β>• Arc indicates Pair Merge
α is Pair Merged• α is invisible

β is visible• If β is labeled, the label stays
the same6

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 ofMerges?

• In theory, any SO can undergo internal or external Set Merge or Pair Merge aninfinite 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 (SyntacticObjects) 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 anunbounded number of SOs scaling with Merge depth• An infinite number of unattested SOs can be generated

9Simple example: the book
• Unattested SOs can be generated• Notation: !F indicates an unvalued feature F• !case is unvalued case•••? indicates an unlabeled structurestraight lines indicate Set Merge
arcs indicate Pair Merge

Set Merge

Pair Merge

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Simple example: the book
• More unattested Syntactic Objects
(a)

(d)

(b)

(e)

(c)

(f)

(g)

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Simple example: the book
#Merges

#SOs

132

7

3

29

4

161

5

1,423

6

18,144

7318,480

87,396,976# of possible Syntactic Objects as #Merges grows

Log-scale graph of possible SOS as # Merges grows

Free Merge must be constrained.
12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”

13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)
Root book Set Merges with categorizer n• n categorizes

Root the Set Merges with the categorizer d• d categorizes

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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
Pair Merged adjunct!case is unvalued case16

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
Pair Merged adjunct17

First Factor Constraints• *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 adjunct18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 ?)Unlabelable• No shared prominent features19

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 infiniteloops• 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 Workspaceenlargement 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, {the, d}}by• ISM({the, d})• ISM({the, d})formed from {{the, d}, {the, {the, d}}}by
• ISM{the, {the, d}}• ISM{the, {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• the book is an NP• DP is an adjunct• Chomsky (2007), also see Oishi (2015)• N (not D) is the head of a phrase suchas ‘the book’• determiner can be an XP23

Extension to Other Examples

24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 isPair Merged with friend.

my friend

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Extension to Complex NPs• Given my friend, the root friend
undergoes internal Set Merge• friend is categorized by n• Root friend is categorized twiceby 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 minethe friend of mine

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Extension to Complex NPs• Combinatorics are given here:
6LOG10(#SOs Generated) vs. # Merges

5log10(#SOs)

4

32

1

0
1

2

3

4

5
# Merges67

89Single (correct) solution found for the friend ofmine at depth 6• Nearly 1,000 SOs were considered. No other SOs
found at depths 7–9 (exhaustively explored)
See Ginsburg & Fong (2018)27

Sentence: John liked the book

Correct solution found for John liked the book at depth
11• Note that other unattested SOs are found at higher
depthsJohn liked the book28

Example: John liked the bookspuriousanalysesintended
analysisExample Combinatorics†† naïve version• logscale y-axis:

Log10(# SOs Generated) vs. # Operations98

LOG10(#SYNTACTIC OBJECTS)• e.g. 6 = 106 = million• 15 operations deep:#operations = 15
obtain single analysis

7• 25 million SOsgenerated• 1 convergent SO

654• 16 operations deep:#operations = 16
obtain 3 spuriousanalyses

32• 250 million SOs• 3 spurious SOs

1
0
0

1

2

3

4

5

6

7

8

9

# OPERATIONS

10

11

12

13

14

1516

John liked the book•
•
•
•
•
•
•
•
•
•
•
•

Following Chomsky (2015):
the book (= <{the,d},{book,n}>) undergoes Object Shiftv* 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 book31

Conclusion• This model shows that it is possible to successfully converge uponcorrect analyses for our test sentences given Free Merge andmotivated 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?

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

Sandiway Fong’s FreeMerge Machine• Calculates all possible Merges, given ourproposed constraints33

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, andbeyond – 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 TenthInternational Spring Forum of the English Linguistic Society of Japan: JELS 35. 211-216. The English Linguistic Society of Japan.

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