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An Introduction to Minimalist Grammars: Complexity of the Shortest Move Constraint (July 22, 2009) Gregory Kobele Jens Michaelis Humboldt Universit Universit at zu Berlin at Bielefeld kobele@rz.hu-berlin.de

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SLIDE 1

An Introduction to Minimalist Grammars: Complexity of the Shortest Move Constraint

(July 22, 2009) Gregory Kobele Humboldt Universit¨ at zu Berlin

kobele@rz.hu-berlin.de

Jens Michaelis Universit¨ at Bielefeld

jens.michaelis@uni-bielefeld.de

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SLIDE 2

Head movement constraint (HMC)

(Stabler 1997)

  • The implementation of

head movement in MGs is in accordance with the HMC – demanding a moving head not to pass over the closest c-commanding head. To put it differently, whenever we are concerned with a case of successive head movement, i.e. recursive adjunction of a (complex) head to a higher head, it obeys strict cyclicity.

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SLIDE 3

Successive cyclic left head adjunction

. . . . . . X’ WP X W X W’ VP

t W

Y’ XP Y X Y W X X’ WP

t X

W’ VP

t W

Z’ YP Z Y Z X Y X W Y’ XP

t Y

X’ WP

t X

W’ VP

t W

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SLIDE 4

Shortest movement condition (SMC)

(Stabler 1997, 1999)

  • The number of competing licensee features triggering a

movement is (finitely) bounded by n. In the strictest version n = 1, i.e., there is at most one maximal projection displaying a matching licensee feature:

∗

< +f

  • +f ...
  • f ...
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SLIDE 5

Specifier island condition (SPIC)

(Stabler 1999)

  • Proper “extraction” from specifiers is blocked.

∧

∗

< +f

  • +f ...

>

specifier

  • f ...
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SLIDE 6

SMC and SPIC — restricting the move-operator domain MG

– SMC , – SPIC + SMC , – SPIC – SMC , + SPIC + SMC , + SPIC

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SLIDE 7

SMC and SPIC — restricting the move-operator domain MG

– SMC , – SPIC + SMC , – SPIC – SMC , + SPIC + SMC , + SPIC

(Michaelis 1998, 2001; Harkema 2001)

LCFRS ⊆ LCFRS (Michaelis 2001, 2002)

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SLIDE 8

+ SMC , – SPIC — generative capacity

  • The crucial methods, in particular,
  • developed to prove that MGs provide a weakly equivalent

subclass of LCFRSs (cf. Michaelis 1998), and

  • leading to the succinct, chain-based MG-reformulation

presented in Stabler & Keenan 2000 [2003] — reducing “classical” MGs to their “bare essentials:”

  • Defining a finite partition on the “relevant” MG-tree set,

– giving rise to a finite set of nonterminals in LCFRS-terms, – deriving all possible “terminal yields.”

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SLIDE 9

Reducing an MG(+SMC,-/+SPIC)

Let G = Features , Lexicon , Ω , c be an MG A minimal expression τ ∈ Closure(G) is relevant :⇐ ⇒ for each licensee -x , there is at most one maximal projection in τ that displays -x .

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SLIDE 10

Reducing an MG(+SMC,-/+SPIC)

Let G = Features , Lexicon , Ω , c be an MG A minimal expression τ ∈ Closure(G) is relevant :⇐ ⇒ for each licensee -x , there is at most one maximal projection in τ that displays -x .

  • In fact, this kind of structure is characteristic of each expression

τ ∈ Closure(G) involved in creating a complete expression in G due to the SMC.

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SLIDE 11

A finite partition of set of relevant expressions

Basic idea : consider relevant τ ∈ Closure(G)

  • Reduce τ to a tuple such that for each maximal projection

displaying an unchecked syntactic feature, there is exactly one component of the tuple consisting of the projection’s head-label, but with the suffix of non-syntactic features truncated.

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SLIDE 12

A finite partition of set of relevant expressions

Basic idea : consider relevant τ ∈ Closure(G)

  • Reduce τ to a tuple such that for each maximal projection

displaying an unchecked syntactic feature, there is exactly one component of the tuple consisting of the projection’s head-label, but with the suffix of non-syntactic features truncated.

  • nly finitely many equivalence classes

Relevance : The resulting tuple has at most m+1 components , m = |Licensees| . Structure building by cancellation of features : Each tuple component is the suffix of the syntactic prefix of the label

  • f a lexical item.
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SLIDE 13

A finite partition of set of relevant expressions

Basic idea : consider relevant τ ∈ Closure(G)

  • Reduce τ to a tuple such that for each maximal projection

displaying an unchecked syntactic feature, there is exactly one component of the tuple consisting of the projection’s head-label, but with the suffix of non-syntactic features truncated.

  • nly finitely many equivalence classes

Relevance : The resulting tuple has at most m+1 components , m = |Licensees| . Structure building by cancellation of features : Each tuple component is the suffix of the syntactic prefix of the label

  • f a lexical item.
  • regarding the partition, applications of ‘merge’ and ‘move’

do not depend on the chosen representatives

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SLIDE 14

Reducing an MG(+SMC,-/+SPIC)

> < . w 1 . w 2 < σ0 . w 0 > . w 3 < σ4 . w 4 > < σ5 . w 5 . w 6 . w 7

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SLIDE 15

Reducing an MG(+SMC,-/+SPIC)

> < . w 1 . w 2 < σ0 . w 0 > . w 3 < σ4 . w 4 > < σ5 . w 5 . w 6 . w 7

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SLIDE 16

Reducing an MG(+SMC,-/+SPIC)

> < . w 1 . w 2 < σ0 . w 0 > . w 3 < σ4 . w 4 > < σ5 . w 5 . w 6 . w 7

  • σ0 . w 1w 2w 0 , σ4 . w 3w 4w 7 , σ5 . w 5w 6
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SLIDE 17

Reducing an MG(+SMC,-/+SPIC)

> < . w 1 . w 2 < σ0 . w 0 > . w 3 < σ4 . w 4 > < σ5 . w 5 . w 6 . w 7

  • σ0

, σ4 , σ5

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SLIDE 18

Reducing an MG(+SMC,-/+SPIC)

> < . w 1 . w 2 < σ0 . w 0 > . w 3 < σ4 . w 4 > < σ5 . w 5 . w 6 . w 7

  • σ0

, σ4 , σ5

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SLIDE 19

Reducing an MG(+SMC,-/+SPIC)

> < . w 1 . w 2 < σ0 . w 0 > . w 3 < σ4 . w 4 > < σ5 . w 5 . w 6 . w 7

  • σ0

, σ4 , σ5

  • =

⇒

  • w 1w 2w 0 , w 3w 4w 7 , w 5w 6
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SLIDE 20

MG-example 2

(α0)

=t .c .that

(α5)

v .laugh

(α1)

=t .+wh .c . ∅

(α6)

=n .d .-k .the

(α2)

=˜ v .+k .t . ∅

(α7)

=n .d .-k .-wh .which

(α3)

=v .=d .˜ v . ∅

(α8)

n .king

(α4)

=d .+k .v .eat

(α9)

n .pie

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SLIDE 21

MG-example 2 =n .d .-k .-wh .which n .pie

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SLIDE 22

MG-example 2

:: = simple , : = complex

=n .d .-k .-wh .which

=n .d .-k .-wh .which , ::

n .pie

n .pie , ::

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SLIDE 23

MG-example 2

:: = simple , : = complex

=n .d .-k .-wh .which

=n .d .-k .-wh .which , ::

n .pie

n .pie , :: <

d .-k .-wh .which pie

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SLIDE 24

MG-example 2

:: = simple , : = complex

=n .d .-k .-wh .which

=n .d .-k .-wh .which , ::

n .pie

n .pie , :: <

d .-k .-wh .which pie

d .-k .-wh .which pie , :

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SLIDE 25

MG-example 2

<

+k .v .eat

<

  • k .-wh .which

pie

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SLIDE 26

MG-example 2

:: = simple , : = complex <

+k .v .eat

<

  • k .-wh .which

pie

+k.v .eat ,-k .-wh .which pie , :

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SLIDE 27

MG-example 2

> <

  • k .the

king

<

˜ v . ∅

> <

  • wh .which

pie

<

eat

ε

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SLIDE 28

MG-example 2

:: = simple , : = complex > <

  • k .the

king

<

˜ v . ∅

> <

  • wh .which

pie

<

eat

ε ˜

v .eat ,-wh .which pie ,-k .the king , :

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SLIDE 29

SMC and SPIC — restricting the move-operator domain MG

– SMC , – SPIC + SMC , – SPIC – SMC , + SPIC + SMC , + SPIC

(Michaelis 1998, 2001; Harkema 2001)

LCFRS ⊆ LCFRS (Michaelis 2001, 2002) ?

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SLIDE 30

– SMC , + SPIC — generative capacity

  • Gärtner & Michaelis 2005 shows that MG(–SMC,+SPIC)s allow

derivation of non-mildly context-sensitive languages.

  • Kobele & Michaelis 2005 shows that, in fact, every recursively

enumerable language can be derived by an MG(–SMC,+SPIC). This is true for essentially two reasons:

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SLIDE 31

– SMC , + SPIC — generative capacity

  • Because of the SPIC, movement of a constituent α into a specifier

position freezes every proper subconstituent β within α.

  • Without the SMC, therefore, the complement line of a tree can

technically be used as two independent counters, or, as a queue.

∧

<

α

β < complement line

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SLIDE 32

MG-example — complexity results concerning LCs

  • An example of a non-mildly context-sensitive MG(–SMC,+SPIC)

deriving a language without constant growth property, namely,

  • a2n | n ≥ 0
  • =
  • a , a a , a a a a , a a a a a a a a , . . .
  • 1

2 4 8 . . .

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SLIDE 33

MG-example — complexity results concerning LCs w.-m =w.x.-l =x.+m.y.-m =y.+l.z.-l =z.y.-l =z.x.-l =x.+m.c =c.+l.c.a

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SLIDE 34

MG-example — complexity results concerning LCs

licensee -m “marks” end/start of “outer” cycle “initialize”

w.-m =w.x.-l =x.+m.y.-m ∧

“outer” cycle end “outer” cycle “appropriately:” check licensee -m start new “outer” cycle: introduce new licensee -m

∧ ∧ =y.+l.z.-l =z.y.-l ∧

“inner” cycle “reintroduce” and “double” the just checked licensee -l

∧ =z.x.-l =x.+m.c =c.+l.c.a

“finalize” leave final cycle “appropriately:” check licensee -m check successively licensee -l, each time introducing an a

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SLIDE 35

MG-example — complexity results concerning LCs

end k-th “inner” cycle YP(-l) start k-th “inner” cycle ZP(

+l,-l)

YP(-l) ZP(

+l,-l)

end 2j-j -th “inner” cycle start 2j-j -th “inner” cycle end j-th / start j+1-th “outer” cycle: check and “reintroduce” -m YP(

+m,-m)

“double” last checked -l XP(-l) check and “reintroduce” -l ZP(

+l,-l)

end i-th “inner” cycle: “double” last checked -l start i-th “inner” cycle: check and “reintroduce” -l YP(-l) ZP(

+l,-l)

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SLIDE 36

MG-example — complexity results concerning LCs

YP(-l) ZP(-l) YP(-l) ZP(-l) YP(-m) XP(-l) ZP(-l) YP(-l) ZP(-l)

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SLIDE 37

MG-example — complexity results concerning LCs

  • Starting the “outer” cycle, the currently derived tree shows 2n

successively embedded complements on the complement line each with an unchecked instance of -l, and a lowest one with an unchecked instance of -m.

  • Going through the cycle provides a successive “roll-up” of those

complements in order to check the displayed features. Thereby, 2n+1 successively embedded complements on the complement line are created, again, all displaying feature -l and a lowest one displaying feature -m.

  • Leaving the cycle procedure after a cycle has been completed,

leads to a final checking of the displayed licensees, where for each instance of -l an instance of a is introduced in the structure.

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SLIDE 38

+ SMC , + SPIC — generative capacity

  • In contrast to the – SMC , + SPIC - case,

adding the SPIC to the SMC has a restrictive effect (Michaelis 2005).

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SLIDE 39

+ SMC , + SPIC — generative capacity MG

– SMC , – SPIC + SMC , – SPIC – SMC , + SPIC + SMC , + SPIC

(Michaelis 1998, 2001; Harkema 2001)

LCFRS LCFRS(1,2) (Michaelis 2001, 2002) type 0

(Kobele & Michaelis 2005)

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SLIDE 40

LCFRS(1,2) — a restricted LCFRS-normal form

An LCFRS G = N , T , F , R , S is an LCFRS(1, 2) iff

  • each nonterminating rule is of the form A → f( B ) or A → f( B , C ),
  • if A → f( B , C ), nonterminal B derives only simple terminal strings.
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SLIDE 41

LCFRS(1,2) — a restricted LCFRS-normal form

An LCFRS G = N , T , F , R , S is an LCFRS(1, 2) iff

  • each nonterminating rule is of the form A → f( B ) or A → f( B , C ),
  • if A → f( B , C ), nonterminal B derives only simple terminal strings.
  • Excludes a non-indexed, but LCFRS-string language such as:
  • w1· · · wn zn wn· · · z1 w1 z0 wnR· · · w1R
  • wi ∈ {a , b}+, zn· · · z0 Dyck word
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SLIDE 42

LCFRS(1,2) — a restricted LCFRS-normal form

Indexed Grammar LCFRS

✛

Lexample LCFRS(1,2)

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SLIDE 43

SMC and SPIC — restricting the move-operator domain MG

– SMC , – SPIC + SMC , – SPIC – SMC , + SPIC + SMC , + SPIC

↓ ↓ ↓ ↓ ↓ ↓

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SLIDE 44

SMC and SPIC — restricting the move-operator domain MG

– SMC , – SPIC + SMC , – SPIC – SMC , + SPIC + SMC , + SPIC

LCFRS type 0 MELL-proof-search (Salvati 2008)

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SLIDE 45

SMC and SPIC — restricting the move-operator domain MG

– SMC , – SPIC + SMC , – SPIC – SMC , + SPIC + SMC , + SPIC

↓ ↓ ↓ ↓ ↓ ↓

? ? ?

↓ ↓ ↓

? ? ?

↑ ↑ ↑

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SLIDE 46

A further extension — multiple wh-movement and the SMC

  • A potential objection against MG(+SMC)’s : you cannot deal with

multiple wh-movement. /* example from Bulgarian */ koji kogoj kakvok ti e pital tj tk

who whom what

AUX

ask

Recall the SMC-implementation in MGs: the number of competing

licensee features triggering a movement is (finitely) bounded.

Answer : we can, if we implement the wh-cluster hypothesis going

back to Rudin (1988) such that we introduce two new syntactic feature types and a corresponding operator.

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SLIDE 47

A further extension — multiple wh-movement and the SMC

  • c(luster)-licensees:

c(luster)-licensors:

△x , △y , △z , . . . ▽x , ▽y , ▽z , . . .

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SLIDE 48

Structure building functions

cluster : Trees − →

part 2Trees

  • φ ∈ Domain(cluster) :⇐

⇒

  • The highest specifier χ of φ displays c-licensor ▽x
  • there is a ( unique [ SMC ] ) maximal projection ψ within φ that

displays the corresponding c-licensee △x

  • cluster( φ )

=

> < χ[ . . . ] ψ[ . . . ] φ{ ψ[ △x . . . ] − → ε }

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SLIDE 49

Structure building functions

cluster : Trees − →

part 2Trees

> φ χ

▽x...

ψ

△x...

  • >

φ′ < χ′

▽x ... ▽x

ψ′

△x ... △x

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SLIDE 50

A further extension — multiple wh-movement and the SMC

  • In order to outline the general case, we next sketch derivations for

wh-clustering with two wh-phrases: crucially exactly one -wh licensee is necessary for deriving a well-formed cluster, and no more than one △wh is displayed at any derivation step.

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SLIDE 51

Wh-clustering, n = 2, crucial step 1

< +wh ... >

▽wh . -wh ... △wh ...

  • <

+wh ... > <

  • wh ...

△wh ... △wh

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SLIDE 52

Wh-clustering, n = 2, crucial step 2

> <

▽wh ... ▽wh△wh ... △wh

< +wh ...+wh > ε

  • <

+wh ... > <

  • wh ...

△wh ... △wh