The Combinatorics of Merge*and Workspace Right-Sizing
Sandiway Fong

Robert Berwick

Jason Ginsburg

University of Arizona

Massachusetts Institute ofTechnology

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

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Being human: our brains• Popular conception, e.g., Gazzaniga(2008), pg. 8: we congratulate
ourselves on our brain power:• "I want to begin by simply
recognizing the huge differences
between the human mind andbrain and other minds and brains[…]"• We posit:• The marvel we call the human brain isactually the weak link in our cognitive
apparatus.
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Sensory Apparatus• Our sensory apparatus provides way too much information for thebrain to process. There is ample biological evidence for this:• Our eyes:• have sensitivity down to the single photon level (Tinsley et al. 2016)• have peak acuity of 77 cycles/degree (Curcio et al. 1990)• Our nose:• sensitivity can be of the order of parts per billion (ppb) (Wackermannová et al., 2016).

• Our ears:• eardrums can detect vibrations smaller than the diameter of a hydrogen atom (Fletcher& Munson, 1933).

• All completely unnecessary for survival.

• [Dynamic range is huge: yet neurons can fire…]
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Why such sensitivity?
• Evolutionary pressure unknown:• could be a simple matter of chemistry• Anyway:• It's clear, in case after case, the brain does not make use of the full resolutionof our sensory inputs.

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Throwing away information• Since we cannot cope with the sensory deluge, the brain tosses out
most of the sensory information:• (attention puzzle) how to sieve the sensory information?
• (dynamic range puzzle) yet we can report on single photons (above chance)• Our hypothesis:• Same goes for language, the brain (must) economize where it can:
• Chomsky (2005) terms the pressure for computational efficiency a ThirdFactor consideration.
• This applies to the reduction of Merge to simplest (binary) Merge (cf. Komachiet al., 2019).
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Resolution in Language• Simplest Merge may follow directly from Workspace (WS) sizing
constraints.
• Although simplest Merge (by itself) has demonstrably undesirable
combinatorics (e.g. from an initial WS of just two lexical items, about
8 million distinct sets can be formed in just 8 Merges), language does
not make full use of this resolution.

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Workspace Size
• 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.

*From Fong and Ginsburg (2018)7

Simple example: the book
#Merges

#SOs

132

7

3

29

4

161

51,423

6

18,144

7

318,480

87,396,976# of possible Syntactic Objects as# Merges grows*From Fong and Ginsburg (2018)

Log-scale graph of possible SOS as # Merges grows

Free Merge must be constrained.
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Workspace Right-Sizing• Accessible terms in a workspace (Noam Chomsky, p.c.; Chomsky 2017Reading lecture)• WS = Workspace• SO = Syntactic Object• Define WS Size as # SOs + # Accessible terms• # accessible terms of a SO = number of proper subsets of the SO + lexicalitems• Subject to Minimal Search (lower copies not considered accessible)

9Workspace Right-Sizing• {a, {b, c}} is a Syntactic Object SO• d and e are SOs
Accessible terms = 4
SOs = 3
Workspaced#SOs = 3
#acc. terms = 4
WS Size = #SOs + #acc. terms
WS Size = 3 + 4
WS Size = 7e10

Workspace Right-Sizing• External Merge (EM) of d and {a, {b, c}}#SOs = 3#acc. terms = 4
WS Size = 7#SOs = 2#acc. terms = 6

Accessible terms = 4
SOs = 3dee#SOs = 2#acc. terms = 6
WS Size = 2 + 6
WS Size = 8
External Merge (EM)decreases the # of SOs by 1,but increases the numberof accessible terms by 2.
EM increases WS Size by 1

11Workspace Right-Sizing• Internal Merge of b and {a, {b, c}}#SOs = 3#acc terms = 4
WS Size = 7

d# acc. terms = 6#SOs = 3#acc. terms = 6
WS Size = 3 + 6
WS Size = 9SOs = 3ede

Internal Merge (IM) keeps the# of SOs the same, andincreases the number ofaccessible terms by 2
IM increases WS Size by 2

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Countercyclic Merge
John has eaten dinner.
l Assume Perfective Merges counter-cyclically with T{a, {b, c}}[{a, {b, c}},d] à [{a, c}, {a, {b, c}},d] (Huijbregts 2019){a, c}

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Countercyclic Merge#SOs = 2# acc. terms = 4
WS Size = 6SOsacc. terms#SOs = 3#acc. terms = 4
WS Size = 3 +4
WS Size = 7dd

WS Size increases by 1
WS Size increase is okay
Is there any way to blockthis?
a is 1 acc. Termc is 1 acc term

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Proposal 1: Merge cannot decrease WS Size.
• You canʼt remove anything from the WS.
Proposal 2: Copies are not accessible terms.

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Workspace Right-Sizing
Minimal Search: exclude lower copies from set of accessible terms• Copy of b is indicated as b

• Internal Merge of b and {a, {b, c}}# acc. terms#SOs = 3#acc. terms = 4
WS Size = 7

d#SOs = 3
#acc. terms = 4
WS Size = 3 + 4
WS Size = 7SOse

IM keeps the # of SOs andaccessible terms the same.

denot accessible

IM does not change WS Size

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Countercyclic Merge
John has eaten dinner.
l Assume Perfective Merges counter-cyclically with T{a, {b, c}}[{a, {b, c}},d] à [{a, c}, {a, {b, c}},d] (Huijbregts 2019){a, c}

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Countercyclic Merge#SOs = 2# acc. terms = 4
WS Size = 6#SOs = 3

not accessible

dd#SOs = 3#acc. terms = 2
WS Size = #SOs + #Acc.
WS Size = 5

WS Size decreases
This isn’t allowed

acc. terms

18Sideward Movement[{a,b}, {c,d}] —> [{a,c},{a,b}, {c,d}] (Huijbregts 2019)• Not sure if examples like this exist, but they still need to be blocked.
#SOs = 2# acc. terms = 4
WS Size = 6

not accessible#SOs = 3#acc. terms = 2
WS Size = #SOs + #Acc.
WS Size = 5

WS Size decreases
This isn’t allowed#acc. terms = 2

19Sideward Movement• Relative Clause adjunction (following Nunes 2001)• Which claim that John made was he willing to discuss? (Chomsky 1993: 36, Nunes 2001:316){a, b}[{a, b}, c, d] à [{a, c}, {a, b}, d] (Huijbregts 2019)[CP was he willing to discuss [which claim]1]{a, c}[CP [which claim]1 [OP2 that John made Op2]]

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Workspace Right-Sizing• Sideward Merge#SOs = 3# acc. terms = 2
WS Size = 5#SOs = 3# acc. terms = 2
Not accessible because it is an SO
WS Size = 5dcd
not accessible

Give Example phrase

WS Size remains the same
Similar to IM• Note that this may be a problem• The WS Size doesn’t decrease.
• This is a problem if we want to bansideward movement
21Conclusion• Unconstrained Simplest Merge leads to an undesirable combinatorial explosion in thenumber of possible Syntactic Objects.
• Research Question: what are the well-motivated constraints that don't involve newmachinery (a problem for evolution)?
• Possible Answer: there is no new machinery, only the simplest possible bound on WS
Size, i.e., WS Size must not decrease and WS Size must not expand by more than 1.
Further consequences (not discussed here):• Can replace other constraints such as on vacuous movement (e.g. iterated IM) – now canbe blocked by WS Size narrow limits – normally would require loop detection machinery.
• Can also explain why language cannot count despite the free availability of IM, i.e. thesuccessor function is simply a loop
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References
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Chomsky, Noam. (2017) Lecture at University of Reading.
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Fletcher, H. and Munson, W. A. (1933). Loudness, its definition, measurement and calculation. The Journal of the Acoustical Society of America, 5, 82–108.
Gazzaniga, M. (2008). Human: The Science Behind What Makes Us Unique. New York, Harper Collins.
Huijbregts, Riny. (2019) Special and general theory of Merge with special and general thanks to Noam. Syntax Interface Lectures 2019, Utrecht.
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Tinsley, J. N., Molodstov, M. U., Prevedel, R., Wartmann, D., Espigulé-Pons, J., Lauwers, M., & Vaziri, A. (2016). Direct detection of a single photon by humans. NatureCommunications, 7, 12172.
Wackermannová, M., Pinc, L., & Jebavý, L. (2016). Olfactory sensitivity in mammalian species. Physiological Research, 65, 369-390.

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