CCG Cross-Serial Dependencies

CCG derivations for Dutch cross-serial dependencies, in the rule-restricted (classical) model CCG.Classical ([Ste00a]). Cross-serial dependencies are the classic witness for CCG's super-context-free power; the derivations thread a verb's argument slots through the cluster via generalized composition.

The derivation pattern

For "Jan Piet zag zwemmen" (Jan saw Piet swim), with the verb-raising category InfSubj = (S\NP)/NP for zwemmen:

• zag >B zwemmen : (S\NP)/(S\NP) ∘ (S\NP)/NP = (S\NP)/NP (forward composition)
• (zag zwemmen) Piet : (S\NP)/NP NP = S\NP (forward application)
• Jan (zag zwemmen Piet) : NP S\NP = S (backward application)

Every step has primary target S, so each is a valid rule instance of CCG.Classical.

Implementation notes

Two constructions are given. The verb-raising derivations (rightward /NP slots, harmonic B/B²) encode the cross-serial binding pattern but their yield does not match Dutch surface order — see jan_zag_zwemmen_piet_yield. The surface-faithful derivations (leftward \NP slots, forward crossed composition fcompX1, following the book's own Dutch fragment) get both the binding and the attested word order — see two_np_sub_yield / three_np_sub_yield.

Categories for Dutch verb clusters

def CCG.CrossSerial.VP : Cat

Verb phrase (infinitival).

Equations

• CCG.CrossSerial.VP = CCG.S.lslash CCG.NP

def CCG.CrossSerial.ControlV : Cat

Control verb: takes VP, gives VP (e.g. "helpen" in 2-verb clusters).

Equations

• CCG.CrossSerial.ControlV = CCG.CrossSerial.VP.rslash CCG.CrossSerial.VP

def CCG.CrossSerial.PercV : Cat

Perception verb: (S\NP)/(S\NP) (e.g. "zag" = saw).

Equations

• CCG.CrossSerial.PercV = (CCG.S.lslash CCG.NP).rslash CCG.CrossSerial.VP

def CCG.CrossSerial.InfSubj : Cat

Infinitival verb needing its (raised) subject: (S\NP)/NP. In Dutch verb-raising the infinitive's subject surfaces in an object-like position, picked up via composition.

Equations

• CCG.CrossSerial.InfSubj = (CCG.S.lslash CCG.NP).rslash CCG.NP

def CCG.CrossSerial.ControlVR : Cat

Verb-raising control verb ((S\NP)/NP)/(S\NP): each restructuring verb provides an extra /NP slot for its own raised subject, in addition to its VP complement. This is what threads multiple argument slots through a 3+-verb cluster:

• zwemmen : (S\NP)/NP — base: needs subject
• helpen : ((S\NP)/NP)/(S\NP) — VR: needs complement, passes an /NP
• zag : (S\NP)/(S\NP) — matrix: standard perception verb

Equations

• CCG.CrossSerial.ControlVR = ((CCG.S.lslash CCG.NP).rslash CCG.NP).rslash (CCG.S.lslash CCG.NP)

def CCG.CrossSerial.PercVSub : Cat

Subordinate-clause perception verb ((S\NP)\NP)/VP: infinitival complement to the right, object and subject NPs to the left (book: zag := ((S₊SUB\NP)\NP)/VP₋SUB; the toy Cat drops the features). Sub = subordinate-clause head — contrast InfSubj, whose /NP is a raised-subject slot.

Equations

• CCG.CrossSerial.PercVSub = ((CCG.S.lslash CCG.NP).lslash CCG.NP).rslash CCG.CrossSerial.VP

def CCG.CrossSerial.InfHeadSub : Cat

Infinitival head with a raised object, (VP\NP)/VP (book: zien := (VP\NP)/VP₋SUB).

Equations

• CCG.CrossSerial.InfHeadSub = (CCG.CrossSerial.VP.lslash CCG.NP).rslash CCG.CrossSerial.VP

def CCG.CrossSerial.dutchLexicon : List LexEntry

The Dutch lexicon (reference; the derivations below select specific entries).

Equations

• One or more equations did not get rendered due to their size.

Lexical entries

def CCG.CrossSerial.jan_lex : Classical.Derivation

Equations

• CCG.CrossSerial.jan_lex = CCG.Classical.Derivation.lex CCG.NP "Jan"

def CCG.CrossSerial.piet_lex : Classical.Derivation

Equations

• CCG.CrossSerial.piet_lex = CCG.Classical.Derivation.lex CCG.NP "Piet"

def CCG.CrossSerial.marie_lex : Classical.Derivation

Equations

• CCG.CrossSerial.marie_lex = CCG.Classical.Derivation.lex CCG.NP "Marie"

def CCG.CrossSerial.zag_lex : Classical.Derivation

Equations

• CCG.CrossSerial.zag_lex = CCG.Classical.Derivation.lex CCG.CrossSerial.PercV "zag"

def CCG.CrossSerial.zwemmen_vr : Classical.Derivation

zwemmen in its verb-raising category (S\NP)/NP.

Equations

• CCG.CrossSerial.zwemmen_vr = CCG.Classical.Derivation.lex CCG.CrossSerial.InfSubj "zwemmen"

def CCG.CrossSerial.helpen_vr : Classical.Derivation

helpen in its verb-raising category ((S\NP)/NP)/(S\NP).

Equations

• CCG.CrossSerial.helpen_vr = CCG.Classical.Derivation.lex CCG.CrossSerial.ControlVR "helpen"

Derivation: "Jan Piet zag zwemmen" (2 NPs, 2 Vs)

def CCG.CrossSerial.zag_comp_zwemmen : Classical.Derivation

zag >B zwemmen: (S\NP)/(S\NP) ∘ (S\NP)/NP = (S\NP)/NP.

Equations

• CCG.CrossSerial.zag_comp_zwemmen = CCG.CrossSerial.zag_lex.fc1 CCG.CrossSerial.zwemmen_vr

def CCG.CrossSerial.zag_zwemmen_piet : Classical.Derivation

(zag zwemmen) Piet: (S\NP)/NP NP = S\NP.

Equations

• CCG.CrossSerial.zag_zwemmen_piet = CCG.CrossSerial.zag_comp_zwemmen.fa CCG.CrossSerial.piet_lex

def CCG.CrossSerial.jan_zag_zwemmen_piet : Classical.Derivation

Jan (zag zwemmen Piet): NP S\NP = S.

Equations

• CCG.CrossSerial.jan_zag_zwemmen_piet = CCG.CrossSerial.jan_lex.ba CCG.CrossSerial.zag_zwemmen_piet

Derivation: "Jan Piet Marie zag helpen zwemmen" (3 NPs, 3 Vs)

The cross-serial bindings are Jan→zag, Piet→helpen, Marie→zwemmen. helpen_vr passes an /NP slot (for Piet) through while taking its VP complement, so B/B² thread both Piet's and Marie's slots through the cluster into ((S\NP)/NP)/NP.

def CCG.CrossSerial.helpen_comp_zwemmen : Classical.Derivation

helpen >B zwemmen: ((S\NP)/NP)/(S\NP) ∘ (S\NP)/NP = ((S\NP)/NP)/NP.

Equations

• CCG.CrossSerial.helpen_comp_zwemmen = CCG.CrossSerial.helpen_vr.fc1 CCG.CrossSerial.zwemmen_vr

def CCG.CrossSerial.zag_comp2_helpen_zwemmen : Classical.Derivation

zag >B² (helpen zwemmen): (S\NP)/(S\NP) ∘² ((S\NP)/NP)/NP = ((S\NP)/NP)/NP.

Equations

• CCG.CrossSerial.zag_comp2_helpen_zwemmen = CCG.CrossSerial.zag_lex.fc2 CCG.CrossSerial.helpen_comp_zwemmen

def CCG.CrossSerial.verbs_marie : Classical.Derivation

verb cluster + Marie: ((S\NP)/NP)/NP NP = (S\NP)/NP.

Equations

• CCG.CrossSerial.verbs_marie = CCG.CrossSerial.zag_comp2_helpen_zwemmen.fa CCG.CrossSerial.marie_lex

def CCG.CrossSerial.verbs_marie_piet : Classical.Derivation

• Piet: (S\NP)/NP NP = S\NP.

Equations

• CCG.CrossSerial.verbs_marie_piet = CCG.CrossSerial.verbs_marie.fa CCG.CrossSerial.piet_lex

def CCG.CrossSerial.jan_piet_marie_zag_helpen_zwemmen_deriv : Classical.Derivation

• Jan: NP S\NP = S.

Equations

• CCG.CrossSerial.jan_piet_marie_zag_helpen_zwemmen_deriv = CCG.CrossSerial.jan_lex.ba CCG.CrossSerial.verbs_marie_piet

Verification

theorem CCG.CrossSerial.two_np_derives_S : jan_zag_zwemmen_piet.cat = some S

The 2-NP cross-serial derivation yields category S (under the target restriction).

theorem CCG.CrossSerial.three_np_derives_S : jan_piet_marie_zag_helpen_zwemmen_deriv.cat = some S

The 3-NP cross-serial derivation yields category S, threading three argument slots through the verb cluster via B and B².

theorem CCG.CrossSerial.verb_cluster_cat : zag_comp2_helpen_zwemmen.cat = some (((S.lslash NP).rslash NP).rslash NP)

The verb cluster composes into ((S\NP)/NP)/NP — a 3-place predicate wanting Jan (\NP), Piet (/NP) and Marie (/NP).

theorem CCG.CrossSerial.jan_zag_zwemmen_piet_yield : jan_zag_zwemmen_piet.yield = ["Jan", "zag", "zwemmen", "Piet"]

The 2-verb derivation's surface yield.

This does not match Dutch surface order ("Jan Piet zag zwemmen"): the verb-cluster category (S\NP)/NP looks rightward for its arguments, so the derivation tree spells out "Jan zag zwemmen Piet". The categories here capture the cross-serial binding pattern but not the linear order; the surface-faithful derivations below get both.

Surface-faithful derivations (leftward argument categories)

[Ste00a]'s own analysis (ch. 6; appendix summary of the Dutch fragment) gives subordinate-clause cluster verbs leftward NP slots and composes the cluster by forward crossed composition (fcompX1), so the NPs precede the whole cluster and the yield is the attested "Jan Piet (Marie) zag (helpen) zwemmen".

def CCG.CrossSerial.zag_sub : Classical.Derivation

Equations

• CCG.CrossSerial.zag_sub = CCG.Classical.Derivation.lex CCG.CrossSerial.PercVSub "zag"

def CCG.CrossSerial.helpen_sub : Classical.Derivation

Equations

• CCG.CrossSerial.helpen_sub = CCG.Classical.Derivation.lex CCG.CrossSerial.InfHeadSub "helpen"

def CCG.CrossSerial.zwemmen_bare : Classical.Derivation

Equations

• CCG.CrossSerial.zwemmen_bare = CCG.Classical.Derivation.lex CCG.CrossSerial.VP "zwemmen"

def CCG.CrossSerial.jan_piet_zag_zwemmen_sub : Classical.Derivation

"(dat) Jan Piet zag zwemmen": zag applies to bare zwemmen and the NPs attach leftward — the 2-verb cluster needs no composition.

Equations

• CCG.CrossSerial.jan_piet_zag_zwemmen_sub = CCG.CrossSerial.jan_lex.ba (CCG.CrossSerial.piet_lex.ba (CCG.CrossSerial.zag_sub.fa CCG.CrossSerial.zwemmen_bare))

def CCG.CrossSerial.jan_piet_marie_zag_helpen_zwemmen_sub : Classical.Derivation

"(dat) Jan Piet Marie zag helpen zwemmen": helpen zwemmen : VP\NP forms by application, zag >B× (helpen zwemmen) crosses the rightward /VP into a leftward-seeking cluster ((S\NP)\NP)\NP, and the three NPs attach leftward — Marie to helpen's slot, Piet to zag's object slot, Jan as subject: the cross-serial binding falls out of the category threading.

Equations

• One or more equations did not get rendered due to their size.

theorem CCG.CrossSerial.two_np_sub_derives_S : jan_piet_zag_zwemmen_sub.cat = some S

theorem CCG.CrossSerial.three_np_sub_derives_S : jan_piet_marie_zag_helpen_zwemmen_sub.cat = some S

theorem CCG.CrossSerial.crossed_cluster_cat : (zag_sub.fcx1 (helpen_sub.fa zwemmen_bare)).cat = some (((S.lslash NP).lslash NP).lslash NP)

The crossed cluster is a leftward-seeking 3-place predicate.

theorem CCG.CrossSerial.two_np_sub_yield : jan_piet_zag_zwemmen_sub.yield = ["Jan", "Piet", "zag", "zwemmen"]

The surface-faithful derivations spell out the attested word order (contrast jan_zag_zwemmen_piet_yield).

theorem CCG.CrossSerial.three_np_sub_yield : jan_piet_marie_zag_helpen_zwemmen_sub.yield = ["Jan", "Piet", "Marie", "zag", "helpen", "zwemmen"]