Logical and Computational Structures for Linguistic Modeling Part 3 Mildly Context-Sensitive Formalisms

Transcription

1 Logical and Computational Structures for Linguistic Modeling Part 3 Mildly Context-Sensitive Formalisms Éric de la Clergerie <Eric.De_La_Clergerie@inria.fr> 30 Septembre 2014 Éric. de la Clergerie TAL 30/09/ / 91

2 Part I Tree Adjoining Grammars Éric. de la Clergerie TAL 30/09/ / 91

3 Extending CFG with structures CFG NP Éric. de la Clergerie TAL 30/09/ / 91

4 Extending CFG with structures CFG NP Datalog V(sing) DCG LFG HPSG S(gap(np)) λ-prolog Vocabulary Complexity unification Éric. de la Clergerie TAL 30/09/ / 91

5 Extending CFG with structures Derivation complexity combining structures RCG LCFRS TAG LIG N A N CFG NP Datalog V(sing) DCG LFG HPSG S(gap(np)) λ-prolog Vocabulary Complexity unification Éric. de la Clergerie TAL 30/09/ / 91

6 Extending CFG with structures Derivation complexity combining structures RCG LCFRS MCS TAG LIG N A N CFG NP Datalog V(sing) DCG LFG HPSG S(gap(np)) λ-prolog Vocabulary Complexity unification Éric. de la Clergerie TAL 30/09/ / 91

7 Extending CFG with structures Derivation complexity combining structures RCG LCFRS MCS TAG LIG N A N Feature TAG CFG NP Datalog V(sing) DCG LFG HPSG S(gap(np)) λ-prolog Vocabulary Complexity unification Éric. de la Clergerie TAL 30/09/ / 91

8 Outline 1 Some background about TAGs 2 Deductive chart-based TAG parsing 3 Automata-based tabular TAG parsing Éric. de la Clergerie TAL 30/09/ / 91

9 From CFG to Tree Substitution Grammars s > np vp np > pn np > det n np > np pp vp > v np vp > vp pp pp > prep np CFG productions: are too local = need decorations for info propagation are generally not lexicalized but info often propagated from words also more efficient parsing algo for lexicalized grammars Éric. de la Clergerie TAL 30/09/ / 91

10 From CFG to Tree Substitution Grammars s > np vp np > pn np > det n np > np pp vp > v np vp > vp pp pp > prep np CFG productions: are too local = need decorations for info propagation are generally not lexicalized but info often propagated from words also more efficient parsing algo for lexicalized grammars CFG productions can be grouped into trees = we get Tree Substitution Grammars (TSG) S For instance, dealing with ditransitive verb donner NP v VP NP PP manger prep NP à Éric. de la Clergerie TAL 30/09/ / 91

11 Derivation Tree vs Parse Tree TSG are strongly equivalent to CFG However, for TSG, parse trees and derivation trees are not equivalent S τ 1 S NP NP0 v NP τ 2 donne Jean VP NP NP NP1 prep a NP τ 3 une pomme PP Jean NP NP2 NP τ 4 Marie NP0 VP v NP PP donne une pomme prep à Marie τ 1 NP1 NP2 τ 2 τ 3 τ 4 Furthermore, several derivations may lead to a same parse tree Éric. de la Clergerie TAL 30/09/ / 91

12 One step farther: adjoining How to deal with: Jean donne souvent une pomme à Marie Need a way to insert the adverb somewhere in the verbal tree = adjoining operation Tree Adjoining Grammars Éric. de la Clergerie TAL 30/09/ / 91

13 TAG: a small example Tree Adjoining Grammars [TAGs] [Joshi] build parse trees from initial and auxiliary trees by using 2 tree operations: substitution and adjoining NP S subst = P John NP VP NP VP V John V sleeps sleeps Éric. de la Clergerie TAL 30/09/ / 91

14 TAG: a small example Tree Adjoining Grammars [TAGs] [Joshi] build parse trees from initial and auxiliary trees by using 2 tree operations: substitution and adjoining NP S subst = P V adj = S John NP VP NP VP V Adv NP VP V John V a lot John V sleeps sleeps V Adv sleeps a lot Éric. de la Clergerie TAL 30/09/ / 91

15 A more complex example: French comparative Paul est plus grand que lui Éric. de la Clergerie TAL 30/09/ / 91

16 A more complex example: French comparative adjp supermod supermod supermod CS adv plus adjp adjp adj grand csu que NP S NP VP pn Paul v est adjp supermod supermod CS adv adjp csu NP plus adj que pro grand lui Éric. de la Clergerie TAL 30/09/ / 91

17 TAGs: (First) Formal definition A TAG G is a tuple (N,Σ, S,I,A) where Σ a finite set of terminal symbols N a finite set of non-terminal symbols S N the axiom I and A are two finite sets of elementary trees over N Σ {ǫ} only leaf nodes ν may have a label l(ν) Σ {ǫ} the trees α in I are initial trees the trees β in A are auxiliary trees and have a unique leaf node marked ( ) as a foot f β with same label than the root node r β, i.e. l(f β ) = l(r β ) Two operations may be used to combine the elementary trees substitution of a leaf node ν of γ by some initial tree α I, (l(ν) = l(t α )) adjoining of an (internal) node ν of γ by some auxiliary tree β A G generates a tree language and a string language T(G) = {γ α = γ yield(γ) Σ α I r α = S} L(G) = {yield(γ) γ T(G)} Éric. de la Clergerie TAL 30/09/ / 91

18 Adjoining Assuming γ = (V, E) with ν V and β = (V β, E β ) with r β, f β V β, such that l(ν) = l(r β ) N with γ[adj(ν,β)] = (V, E ) V = V V β \{ν} {(x, y) E x ν y ν} E = E β {(x, r β ) (x,ν) E} {(f β, y) (ν, y) E} Note: The node sets are assumed to be renamed to avoid clashes, i.e. E E = Éric. de la Clergerie TAL 30/09/ / 91

19 Adjoining contraints A full definition of TAGs should include constraints on adjoining nodes: A TAG G is a tuple (N,Σ, S,I,A, f OA, f SA ) where, assuming V set of nodes in I A, f OA : V {0, 1} specify if adjoining on ν is obligatory (1) or not (0) f SA : V 2 A specify which auxiliary trees may be adjoined on ν note: ν becomes non-adjoinable with f SA (ν) = Adjoining constraints necessary for getting the full expressive power of TAGs but they are often implicit: no adjoining on leaf nodes (including foot nodes) explicit mandatory adjoining (MA, +) marks on some nodes explicit non adjoining (NA, ) marks on some nodes Éric. de la Clergerie TAL 30/09/ / 91

20 Derivation tree S NP NP VP S NP VP Tarzan Jane VP Adv NP VP V NP passionately Tarzan VP Adv loves V NP passionately α(loves) loves Jane subst(np 1 ) subst(np 2 ) adj(vp 1 ) α 1 (Tarzan) α 2 (Jane) β 1 (passionately) For TAGs, derivation tree not isomorphic to parse tree but close from semantic level. Éric. de la Clergerie TAL 30/09/ / 91

21 Regular Tree Languages For a TAG G, its set of derivation trees D(G) forms a regular tree language i.e., D(G) may be generated by a finite tree automaton (top-down) term rewrite rules of the form q 0 a(q 1,...,q n ), q i Q, a F may also be seen as the parse trees for some CFG G Éric. de la Clergerie TAL 30/09/ / 91

22 TAG complexity: Adjoining Root Adjunction node Spine Adjunction Foot Auxiliary Tree adjoining Input String discontinuity (hole in aux tree) crossing (both sides of the hole) Éric. de la Clergerie TAL 30/09/ / 91

23 TAG complexity: Adjoining Root Adjunction node Spine Adjunction Foot Auxiliary Tree Input String adjoining discontinuity (hole in aux tree) crossing (both sides of the hole) unbounded synchronization (both sides of spine) nested adjoining Éric. de la Clergerie TAL 30/09/ / 91

24 Expressive power of TAGs The adjoining operation extends the expressive power of TAGs w.r.t. CFGs. long distance dependencies (wh-pronoun extraction for instance) crossed dependencies as given by copy language ww or by language a n b n c n (1) omdat ik Cecillia de nijlpaarden zag voeren because I Cecilia the hippopotamuses saw feed because I saw Cecilia feed the hippopotamuses Éric. de la Clergerie TAL 30/09/ / 91

25 Expressive power (limits) TAGs can t handle the following languages: a n b m c n d m e n f m multiple copy languages w n with n > 2. Éric. de la Clergerie TAL 30/09/ / 91

26 Pumping lemma Tree Adjoining Languages satisfy a pumping lemma If L is a TAL, then there exists N, such for all w L and w > N, there exist x, y, z, v 1, v 2, w 1, w 2, w 3, w 4 Σ, such that { v1 v 2 w 1 w 2 w 3 w 4 N w 1 w 2 w 3 w 4 1 and one of the following case holds 1 w = xw 1 v 1 w 2 yw 3 v 2 w 4 z and k 0, xw k 1 v 1w k 2 ywk 3 v 2w k 4 z L 2 w = xw 1 v 1 w 2 v 2 w 3 yw 4 z and k 0, xw k+1 1 v 1 w 2 v 2 w 3 (w 2 w 4 w 3 ) k yw 4 z L 3 w = xw 1 yw 2 v 1 w 3 v 2 w 4 z and forallk 0, xw 1 y(w 2 w 1 w 3 ) k w 2 v 1 w 3 v 2 w k+1 4 L Éric. de la Clergerie TAL 30/09/ / 91

27 Closure properties of TALs As CFLs, TALs form an Abstract Family of Languages (AFL): 1 closed by intersection with regular languages 2 closed by union, concatenation, and Kleene-iteration 3 closed by homomorphism and inverse homomorphism In particular, (1) = notion of Shared Derivation Forest Éric. de la Clergerie TAL 30/09/ / 91

28 Shared Derivation Forests Formal definition in Vijay-Shanker & Weir Tarzan 1 loves 2 Jane 3 very 4 passionately 5 α 1 (Tarzan) α(loves) subst(np subst(np 1 ) 2 adj(vp ) 1 ) α 2 (Jane) β 1 (passionately) adj(adv 1 ) β 2 (very) α 1 (0, 5) α 1 (0, 1) loves(1, 2) α 2 (2, 3) β 1 (1, 5, 1, 3) β 1 (1, 5, 1, 3) β 2 (3, 5, 4, 5) passionately(4, 5) β 2 (3, 5, 4, 5) very(3, 4) α 1 (0, 1) Tarzan(0, 1) α 1 (2, 3) Janes(0, 1) More formally, use tree nodes rather than trees Space complexity in O(n 6 ) by binarization (adj on spine node ν) ν (i, j, r, s) r β (i, j, p, q) ν (p, q, r, s) Éric. de la Clergerie TAL 30/09/ / 91

29 Well formed trees Many possible ways to define elementary trees In practive, elementary trees follow some linguistic principles: lexical anchoring: at least, one non-empty lexical (frontier) node the head (or anchor) sub-categorization: a frontier node for each argument sub-categorized by the head domain of locality semantic consistency: a tree correspond to the scope of a semantic predicate with its arguments non-composition: a tree stands for a single semantic unit A few bad trees: S V S S S NP VP dort NP VP C S NP VP V que V NP donne regarde Det N le chien Éric. de la Clergerie TAL 30/09/ / 91

30 Subcategorization S NP VP S NP VP V NP PP V S donner Prep NP penser C S à que Éric. de la Clergerie TAL 30/09/ / 91

31 Feature TAGs The nodes may be decorated with a pair (top,bot) of decorations S NP VP S NP VP V S b:mode=inf V S vouloir vouloir C S b:mode=subj que When adj on ν, unification of ν.top with r β.top and ν.bot with f β.bot alternate way to express adjoining constraints Note: for flat decorations, same expressive power and complexity Éric. de la Clergerie TAL 30/09/ / 91

32 TAG families Trees derived from a canonical ones grouped into families e.g. family of transitive verbs NP0 tn0vn1 S V v mange VP NP1 twn1n0v S t.wh=+ NP1 NP0 S VP V v trn1n0v NP NP S + all other extractions (on NP 0 ) + passive + extractions on passive + ordering + multiple realizations +... que NP0 S VP V v XTAG architecture a set of trees (with anchor nodes) grouped into families a lexicon L specifying for each word w the set of families it may anchor + additional constraints Éric. de la Clergerie TAL 30/09/ / 91

33 Meta-grammars Large coverage TAG = many trees to write and maintain! Alternative: generate the trees from a higher description level: meta-grammars Abeillé, Candito hierarchy of classes, containing constraints A precedes B, A dominates B,... a class deals with a linguistic facet e.g. verb argument, refined into subject or object a class may require or provide functionalities the classes may be combined to form neutral classes the constraints of the neutral classes used to generate elementary trees = used for FRMG, a large-coverage French TAG (plus mechanisms for factorizing elementary trees) Éric. de la Clergerie TAL 30/09/ / 91

34 Long-distance dependencies (Recursive) Adjoining may replace LFG s functional uncertainty for long-distance dependencies Jean demande [quel homme Paul pense [que Marie regarde ǫ]] S NP S ( Wh) = c + = ( Focus) = ( Wh)=+ ( Focus)= (Comp) Obj Éric. de la Clergerie TAL 30/09/ / 91

35 Long-distance dependencies (TAGs) Handled through repeated adjoining S NP S quel homme NP VP S Marie v NP NP VP regarde ǫ Paul v CS pense csu que *S Éric. de la Clergerie TAL 30/09/ / 91

36 Outline 1 Some background about TAGs 2 Deductive chart-based TAG parsing 3 Automata-based tabular TAG parsing Éric. de la Clergerie TAL 30/09/ / 91

37 Deductive parsing Formalization of chart parsing Use of universe of tabulable items, representing (set of) partial parses items often build upon dotted rules A 0 A 1...A i A i+1...a n chart edges labeled by dotted rules ( items i, j, A α β ) a deductive system specifying how to derive items Éric. de la Clergerie TAL 30/09/ / 91

38 CKY as a deductive system (for CFGs) A α i, i, A α A α i (Seed) i, j, A α aβ i, j + 1, A αa β A αa β A α aβ a = a j+1 i j j + 1 (Scan) i, j, A α Bβ j, k, B γ i, k, A αb β A αb β i A α Bβ j B γ k (Complete) Éric. de la Clergerie TAL 30/09/ / 91

39 CKY for TAGs CKY algorithm for TAGs [Vijay-Shanker & Joshi 85] Presentation: Dotted trees N and N where N is a node of an elementary tree Items N, i, p, q, j and N, i, p, q, j with p, q possibly covering a foot node. N N M i N j u N v p M, i, p, q, j q p q Without adjoining: N, p,,, q With adjoining: N, u,,, v Éric. de la Clergerie TAL 30/09/ / 91

40 Rule (Adjoin) Gluing a sub-tree at a foot node. N, p, r, s, q R t, i, p, q, j N, i, r, s, j label(n) = label(r t ) (Adjoin) R t N i pq N j p rs q i rs j Éric. de la Clergerie TAL 30/09/ / 91

41 Rule (NoAdjoin) When no adjoining on a node N, p, r, s, q N, p, r, s, q (NoAdjoin) Éric. de la Clergerie TAL 30/09/ / 91

42 Rule (Complete) Gluing all node s children N i, l i, p i, q i, r i i=1,...,v N, l 1, p i, q i, r v N N 1... N v and i, l i+1 = r i (Complete) Note: At most one child (k) covers a foot node with ( p i, q i ) = (p k, q k ) N N N 1 l r 1 l k N k p k q k r k l v N v - - r v l 1 p k q k r v Éric. de la Clergerie TAL 30/09/ / 91

43 Complexity Other deductive rules needed to handle 1 substitution 2 terminal scanning + axioms Time complexity O(n max(6,1+v+2) ) with v : maximal number of children per node 2 : number of indexes to cover a possible unique foot node Normalization using binary-branching trees (v = 2) = complexity O(n 6 ) 4 indexes per item = Space complexity in O(n 4 ) for a recognizer O(n 6 ) for a parser, keeping backpointers to parents Optimal worst-case complexities but practically, even less efficient than CKY for CFGs Éric. de la Clergerie TAL 30/09/ / 91

44 Prediction, dotted trees and dotted producctions To mark prediction, new dotted trees [Shabes]: N and N Alternative: equivalence with dotted productions N N 1... N v N N 1...N v dotted tree dotted production N k, N k+1 N N 1...N k N k+1...n v R (root) R R (root) R N N N 1...N v N N N 1...N n i p j q N M N α Mβ, i, p, q, j Éric. de la Clergerie TAL 30/09/ / 91

45 Non prefix valid Earley algorithm Glue a sub-tree at foot node F t (maybe useless!) M γ, p, r, s, q R t, i, p, q, j M γ, i, r, s, j label(m) = label(r t ) (Adjoin) Advance in recognition of N s children N α Mβ, i, u, v, j M γ, j, r, s, k N αm β, i, u r, v s, k (Complete) (Adjoin) and (Complete) similar to CKY (binary form) Éric. de la Clergerie TAL 30/09/ / 91

46 Adjoining Prediction Predict adjoining at M N α Mβ, i, p, q, j label(m) = label(r t ) (CallAdj) M i pq j Éric. de la Clergerie TAL 30/09/ / 91

47 Adjoining Prediction Predict adjoining at M N α Mβ, i, p, q, j R t, j,,, j label(m) = label(r t ) (CallAdj) M i pq j t Éric. de la Clergerie TAL 30/09/ / 91

48 Foot Prediction Predict a sub-tree root at M to recognize below foot node F t F t, i,,, i label(f t ) = label(m) (CallFoot) F t Éric. de la Clergerie TAL 30/09/ / 91

49 Foot Prediction Predict a sub-tree root at M to recognize below foot node F t F t, i,,, i M γ, i,,, i label(f t ) = label(m) (CallFoot) F t M i Éric. de la Clergerie TAL 30/09/ / 91

50 Foot Prediction Predict a sub-tree root at M to recognize below foot node F t F t, i,,, i M γ, i,,, i label(f t ) = label(m) (CallFoot) F t M i The prediction of M not related to the node M having triggered the adjoining of t = Non prefix valid parsing strategy Éric. de la Clergerie TAL 30/09/ / 91

51 Complexity Space complexity remains O(n 4 ) Dotted productions = implicit binarization = time in O(n 6 ) Non prefix valid: impact difficult to evaluate in practice Note: Dotted productions also applicable to improve CKY Éric. de la Clergerie TAL 30/09/ / 91

52 Prefix valid Early [Shabes] Complexities time in O(n 9 ) and space in O(n 6 ) due to 6-index items l tl bl j fl fr Actually, tl and bl may be avoided using dotted productions Éric. de la Clergerie TAL 30/09/ / 91

53 Prefix valid Earley [Nederhof] Item with only an extra index h: h, N α β, i, p, q, j h states starting (leftmost) position of current tree A B u A B h i A j p q h, N αb β, i, p, q, j A h i j p q u, A γ, p,,, q Éric. de la Clergerie TAL 30/09/ / 91

54 Foot prediction h, N α Mβ, i, p, q, j label(f t ) = label(m) (CallFootPf) h i Éric. de la Clergerie TAL 30/09/ / 91

55 Foot prediction h, N α Mβ, i, p, q, j j, F t, k,,, k label(f t ) = label(m) (CallFootPf) h i j F t Éric. de la Clergerie TAL 30/09/ / 91

56 Foot prediction h, N α Mβ, i, p, q, j j, F t, k,,, k h, M γ, k,,, k label(f t ) = label(m) (CallFootPf) M h i j F t M k Éric. de la Clergerie TAL 30/09/ / 91

57 Adjoining return h, N α Mβ, i, u, v, j h, M γ, p, r, s, q label(m) = label(r t ) (AdjoinPf) M h i M p q Éric. de la Clergerie TAL 30/09/ / 91

58 Adjoining return h, N α Mβ, i, u, v, j j, R t, j, p, q, k h, M γ, p, r, s, q label(m) = label(r t ) (AdjoinPf) M h i R t j k M p q Éric. de la Clergerie TAL 30/09/ / 91

59 Adjoining return h, N α Mβ, i, u, v, j j, R t, j, p, q, k h, M γ, p, r, s, q h, N αm β, i, u r, v s, k label(m) = label(r t ) (AdjoinPf) M M h i R t h i k j k M p q Éric. de la Clergerie TAL 30/09/ / 91

60 Raw complexity Maximal time complexity provided by (AdjoinPf) : O(n 10 ) because of 10 indexes h, N α Mβ, i, u, v, j j, R t, j, p, q, k h, M γ, p, r, s, q h, N αm β, i, u r, v s, k label(m) = label(r t ) (AdjoinPf) But (u, v) or (r, s) equals (, ) = (Case analysis) splitting rule into 2 sub-rules = O(n 8 ) = not sufficient! Éric. de la Clergerie TAL 30/09/ / 91

61 Splitting and intermediary structures Split (AdjoinPf) into 2 successive steps with an intermediary structure [M γ, j, r, s, k] This intermediary structure combines the aux. tree with the subtree rooted at M j, R t, j, p, q, k h, M γ, p, r, s, q [M γ, j, r, s, k] (AdjoinPf-1) h, N α Mβ, i, u, v, j [M γ, j, r, s, k] h, M γ, p, r, s, q h, N αm β, i, u r, v s, k (AdjoinPf-2) Éric. de la Clergerie TAL 30/09/ / 91

62 Projection j, R t, j, p, q, k h, M γ, p, r, s, q [M γ, j, r, s, k] (AdjoinPf-1) Involves 7 indexes {j, p, q, k, h, r, s} but h not consulted h, M γ, p, r, s, q, M γ, p, r, s, q (Proj) j, R t, j, p, q, k, M γ, p, r, s, q [M γ, j, r, s, k] (AdjoinPf-1) Finally, O(n 6 ) time complexity Éric. de la Clergerie TAL 30/09/ / 91

63 Case of (AdjoinPf-2) h, N α Mβ, i, u, v, j [M γ, j, r, s, k] h, M γ, p, r, s, q h, N αm β, i, u r, v s, k (AdjoinPf-2) 10 indexes = Raw complexity in O(n 10 ) At least one pair in (u, v) or (r, s) equals (, ); Case splitting = O(n 8 ) Pair (p,q) not consulted; projection = O(n 6 ) Éric. de la Clergerie TAL 30/09/ / 91

64 Preliminary conclusion Rule splitting, intermediary structures, and projections decrease complexities but increase the number of steps To be practically validated! Designing a tabular algorithm for TAGs is complex! Designing items Understanding the invariants Formulating the deductive rules (simultaneously handling tabulation and strategy) Optimizing rules (splitting and projections) How to adapt for close formalisms such as Linear Indexed Grammars [LIG]? A 0 ([ x]) A 1 ([])...A k ([ y])...a n ([]) Éric. de la Clergerie TAL 30/09/ / 91

65 LIG Indexed Grammars: Context-Free grammars with non terminals decorated with stacks Linear Indexed Grammars: a single stack propagated per production A LIG G = (N,Σ,I, S,P) where I is a finite set of indices P is a finite set of productions of the form A[ α] A 1 []...A i [ β]...a n [] or A[] γ with γ Σ and α,β I Relationship with (linear monadic) Context-Free Tree Languages Éric. de la Clergerie TAL 30/09/ / 91

66 LIGs and TAGs LIGs and TAGs are weakly equivalent, and almost strongly equivalent TAGs may be easily encoded by LIGs, using tree nodes as non-terminals adjoining node ν in γ using aux. tree β ν[ ] r β [ ν] discharging a node ν with children ν 1,...,ν n at a foot node f β f β [ ν] ν 1 [α 1 ]...ν n [α n ] where α i = [ ] if ν i on spine, and α i = [] otherwise traversing a node ν without adjoining ν[ ] ν 1 [α 1 ]...ν n [α n ] with same conditions on α i than above Reverse way more difficult: no locality constraint between push and pop points (same aux. tree β for TAGs) Suggest using LPDAs to parse LIGs and TAGs but non efficient and non termination Éric. de la Clergerie TAL 30/09/ / 91

67 Outline 1 Some background about TAGs 2 Deductive chart-based TAG parsing 3 Automata-based tabular TAG parsing Éric. de la Clergerie TAL 30/09/ / 91

68 From formalisms to automata Methodology: Automata are operational devices used to describe the steps of Parsing Strategies Dynamic Programming interpretations of automata used to identify context-free subderivations that may be tabulated. Formalisms Automata Tabulation Notes RegExp FSA - CFG PDA O(n 3 ) Lang TAG / LIG 2-Stack Automata O(n 6 ) Becker, Clergerie & Pardo Embedded PDA O(n 6 ) Nederhof Problem: 2-stack automata (or EPDA) have the power of Turing Machine (intuition) moving left- or rightward pushing on first or second stack & popping the other one = need restrictions Éric. de la Clergerie TAL 30/09/ / 91

69 EPDA Embedded Push-Down Automata Becker are natural candidates for LIGs (and TAGs) by handling stack of stacks. Two flavors: Top-Down and Bottom-Up EPDAs WRAP(TD) UNWRAP (BU) Éric. de la Clergerie TAL 30/09/ / 91

70 2-stack automata for TAGs Solution: stack asymmetry Master Stack: to keep trace of uncompleted tree traversals Auxiliary Stack: only to keep trace of uncompleted adjunctions Adjunction info: (top-down) ν n = ν and (bottom-up) ν n = T, T, B, B : prediction and propagation info about top and bottom node decorations (Feature TAGs) Calls (top-down prediction) Returns (bottom-up propagation) ν r f ν Éric. de la Clergerie TAL 30/09/ / 91

71 2-stack automata for TAGs Solution: stack asymmetry Master Stack: to keep trace of uncompleted tree traversals Auxiliary Stack: only to keep trace of uncompleted adjunctions Adjunction info: (top-down) ν n = ν and (bottom-up) ν n = T, T, B, B : prediction and propagation info about top and bottom node decorations (Feature TAGs) Calls (top-down prediction) Returns (bottom-up propagation) ν T ν ν n transition ACALL Call Adj. ν r f ν Éric. de la Clergerie TAL 30/09/ / 91

72 2-stack automata for TAGs Solution: stack asymmetry Master Stack: to keep trace of uncompleted tree traversals Auxiliary Stack: only to keep trace of uncompleted adjunctions Adjunction info: (top-down) ν n = ν and (bottom-up) ν n = T, T, B, B : prediction and propagation info about top and bottom node decorations (Feature TAGs) Calls (top-down prediction) Returns (bottom-up propagation) ν T ν ν n transition ACALL Call Adj. ν r f ν n B+ν n f transition FCALL Call Foot f ν Éric. de la Clergerie TAL 30/09/ / 91

73 2-stack automata for TAGs Solution: stack asymmetry Master Stack: to keep trace of uncompleted tree traversals Auxiliary Stack: only to keep trace of uncompleted adjunctions Adjunction info: (top-down) ν n = ν and (bottom-up) ν n = T, T, B, B : prediction and propagation info about top and bottom node decorations (Feature TAGs) Calls (top-down prediction) Returns (bottom-up propagation) ν T ν ν n transition ACALL Call Adj. ν r f ν n B+ν n f transition FCALL Call Foot f ν Ret Foot B +ν n f f ν n transition FRET Éric. de la Clergerie TAL 30/09/ / 91

74 2-stack automata for TAGs Solution: stack asymmetry Master Stack: to keep trace of uncompleted tree traversals Auxiliary Stack: only to keep trace of uncompleted adjunctions Adjunction info: (top-down) ν n = ν and (bottom-up) ν n = T, T, B, B : prediction and propagation info about top and bottom node decorations (Feature TAGs) Calls (top-down prediction) Returns (bottom-up propagation) ν T ν ν n transition ACALL Call Adj. ν r Ret Adj. T ν ν n ν transition ARET f ν n B+ν n f transition FCALL Call Foot f ν Ret Foot B +ν n f f ν n transition FRET Éric. de la Clergerie TAL 30/09/ / 91

75 Transitions Retracing in erase mode concerns only the size of AS (not its content). Retracing possible because : WRITE transitions leave marks (PUSH, POP, NOP, NEW) in the Master Stack that can only be removed by a dual ERASE transition. Auxiliary stack PUSH POP NOP NEW WRITE transitions SWAP Transition ERASE transitions Master stack Éric. de la Clergerie TAL 30/09/ / 91

76 Context-Free derivation for PDAs Dynamic Programming : Recursive decomposition of problems into elementary subproblems that may be combined, tabulated, and reused eg the knapsack problem Éric. de la Clergerie TAL 30/09/ / 91

77 Context-Free derivation for PDAs Dynamic Programming : Recursive decomposition of problems into elementary subproblems that may be combined, tabulated, and reused eg the knapsack problem For PDAs, derivations broken into elementary Context-Free sub-derivations: A PUSH B Aσ Init Éric. de la Clergerie TAL 30/09/ / 91

78 Context-Free derivation for PDAs Dynamic Programming : Recursive decomposition of problems into elementary subproblems that may be combined, tabulated, and reused eg the knapsack problem For PDAs, derivations broken into elementary Context-Free sub-derivations: Init A PUSH B Aσ ǫ-item B, Aσ Éric. de la Clergerie TAL 30/09/ / 91

79 Context-Free derivation for PDAs Dynamic Programming : Recursive decomposition of problems into elementary subproblems that may be combined, tabulated, and reused eg the knapsack problem For PDAs, derivations broken into elementary Context-Free sub-derivations: Init A PUSH B Aσ ǫ-item B, Aσ Éric. de la Clergerie TAL 30/09/ / 91

80 Context-Free derivation for PDAs Dynamic Programming : Recursive decomposition of problems into elementary subproblems that may be combined, tabulated, and reused eg the knapsack problem For PDAs, derivations broken into elementary Context-Free sub-derivations: Init A PUSH B Aσ ǫ-item B, Aσ A is the fraction ǫ of information consulted to trigger the subderivation and not propagated to B. Éric. de la Clergerie TAL 30/09/ / 91

81 (Escaped) CF derivations for 2SA Escaped derivation A B D E C Éric. de la Clergerie TAL 30/09/ / 91

82 (Escaped) CF derivations for 2SA Escaped derivation A B D E C = 5-point xcf items AB[DE]C = ǫa ǫb, b [ ǫd, d E ] C, c [TAG] ǫa ǫb [ ǫd E ] C When no escaped part = 3-point CF items ABC = ǫa ǫb, b C (new generalization) escaped part [DE] may take place between A and B Éric. de la Clergerie TAL 30/09/ / 91

83 xcfs and TAGs Escaped derivation A B D E C A root of elementary tree B start of adjoining C current position in the tree D and E left and right borders of the foot Éric. de la Clergerie TAL 30/09/ / 91

84 Item shapes At most 5 indexes per items = Space complexity in O(n 5 ) SD-2SA restrictions & transition kinds = 6 possible item shapes A B C A B C A B C B A B C D E A B C D E A C D E Éric. de la Clergerie TAL 30/09/ / 91

85 Combining items and transitions By graphically playing with items and transitions, we find 10 composition rules with O(n 8 ) time complexity may be split into 11 rules with O(n 6 ) time complexity (Easy:) Write a POP mark: I 1 + I 2 +τ = I 3 Éric. de la Clergerie TAL 30/09/ / 91

86 Combining items and transitions By graphically playing with items and transitions, we find 10 composition rules with O(n 8 ) time complexity may be split into 11 rules with O(n 6 ) time complexity (Easy:) Write a POP mark: I 1 + I 2 +τ = I 3 I 2 τ Éric. de la Clergerie TAL 30/09/ / 91

87 Combining items and transitions By graphically playing with items and transitions, we find 10 composition rules with O(n 8 ) time complexity may be split into 11 rules with O(n 6 ) time complexity (Easy:) Write a POP mark: I 1 + I 2 +τ = I 3 I 1 I 2 τ Éric. de la Clergerie TAL 30/09/ / 91

88 Combining items and transitions By graphically playing with items and transitions, we find 10 composition rules with O(n 8 ) time complexity may be split into 11 rules with O(n 6 ) time complexity (Easy:) Write a POP mark: I 1 + I 2 +τ = I 3 I 1 I 2 τ I 3 Éric. de la Clergerie TAL 30/09/ / 91

89 Combining items and transitions By graphically playing with items and transitions, we find 10 composition rules with O(n 8 ) time complexity may be split into 11 rules with O(n 6 ) time complexity (Easy:) Write a POP mark: I 1 + I 2 +τ = I 3 I 1 I 2 τ I 3 Consultation of 3 indexes = Complexity O(n 3 ) Éric. de la Clergerie TAL 30/09/ / 91

90 Combining items and transitions (2) (complex:) Erasing a PUSH mark: I 1 + I 2 + I 3 +τ = I 4 e.g. when returning from auxiliary tree (ending adjoining) Éric. de la Clergerie TAL 30/09/ / 91

91 Combining items and transitions (2) (complex:) Erasing a PUSH mark: I 1 + I 2 + I 3 +τ = I 4 e.g. when returning from auxiliary tree (ending adjoining) τ I 2 Éric. de la Clergerie TAL 30/09/ / 91

92 Combining items and transitions (2) (complex:) Erasing a PUSH mark: I 1 + I 2 + I 3 +τ = I 4 e.g. when returning from auxiliary tree (ending adjoining) I 1 τ I 2 Éric. de la Clergerie TAL 30/09/ / 91

93 Combining items and transitions (2) (complex:) Erasing a PUSH mark: I 1 + I 2 + I 3 +τ = I 4 e.g. when returning from auxiliary tree (ending adjoining) τ I 2 I 1 I 3 Éric. de la Clergerie TAL 30/09/ / 91

94 Combining items and transitions (2) (complex:) Erasing a PUSH mark: I 1 + I 2 + I 3 +τ = I 4 e.g. when returning from auxiliary tree (ending adjoining) I 4 I 2 τ I I 3 1 Éric. de la Clergerie TAL 30/09/ / 91

95 Combining items and transitions (2) (complex:) Erasing a PUSH mark: I 1 + I 2 + I 3 +τ = I 4 e.g. when returning from auxiliary tree (ending adjoining) I 4 I 2 τ I I 3 1 Consultation of 8 indexes = Complexity O(n 8 ) need to decompose, project and use intermediary steps (as seen before) Éric. de la Clergerie TAL 30/09/ / 91

96 Cascade of partial evaluations ν f Éric. de la Clergerie TAL 30/09/ / 91

97 Cascade of partial evaluations ν CAI RAI > f I a CFI > Éric. de la Clergerie TAL 30/09/ / 91

98 Cascade of partial evaluations ν CAI RAI > CFI f RFI > I a CFI > Éric. de la Clergerie TAL 30/09/ / 91

99 Cascade of partial evaluations ν CAI RAI > CFI f RFI > I a CFI > RFI a RFI b RFI c Éric. de la Clergerie TAL 30/09/ / 91

100 Cascade of partial evaluations ν CAI RAI > CFI f RFI > I a CFI > RFI a RFI b RFI > 1 RFI c Éric. de la Clergerie TAL 30/09/ / 91

101 Cascade of partial evaluations ν CAI RAI RAI > CFI f RFI > I a CFI > RFI a RFI b RFI > 1 RFI c Éric. de la Clergerie TAL 30/09/ / 91

102 Cascade of partial evaluations ν CAI RAI RAI > 1 RAI > CFI f RFI > I a CFI > RFI a RFI b RFI > 1 RFI c Éric. de la Clergerie TAL 30/09/ / 91

103 Cascade of partial evaluations ν CAI RAI RAI > 1 RAI > CFI f RFI > I a CFI > RFI a RFI b RFI > 1 RFI c Éric. de la Clergerie TAL 30/09/ / 91

104 Cascade of partial evaluations ν RAI > CAI RAI RAI > 1 L CFI f RFI > I a CFI > RFI a RFI b RFI > 1 RFI c Éric. de la Clergerie TAL 30/09/ / 91

105 Cascade of partial evaluations ν RAI > CAI RAI RAI > 1 L CFI f RFI > I a CFI > RFI a RFI b RFI > 1 RFI c Éric. de la Clergerie TAL 30/09/ / 91

106 Simplified Cascade of partial evaluations ν R f Not the optimal worst case complexity (because yellow subtree traversed in the context of larger yellow subtree, keeping trace of unfinished adjoinings) But more efficient in practice! And suggesting extensions, based on the idea of continuation Éric. de la Clergerie TAL 30/09/ / 91

107 Simplified Cascade of partial evaluations ν R CAI f CFI > Not the optimal worst case complexity (because yellow subtree traversed in the context of larger yellow subtree, keeping trace of unfinished adjoinings) But more efficient in practice! And suggesting extensions, based on the idea of continuation Éric. de la Clergerie TAL 30/09/ / 91

108 Simplified Cascade of partial evaluations ν R CAI f CFI RFI > CFI > Not the optimal worst case complexity (because yellow subtree traversed in the context of larger yellow subtree, keeping trace of unfinished adjoinings) But more efficient in practice! And suggesting extensions, based on the idea of continuation Éric. de la Clergerie TAL 30/09/ / 91

109 Simplified Cascade of partial evaluations ν R CAI CFI f RFI > CFI > RFI RAI > Not the optimal worst case complexity (because yellow subtree traversed in the context of larger yellow subtree, keeping trace of unfinished adjoinings) But more efficient in practice! And suggesting extensions, based on the idea of continuation Éric. de la Clergerie TAL 30/09/ / 91

110 Simplified Cascade of partial evaluations ν R CAI RAI CFI f RFI > CFI > RFI RAI > Not the optimal worst case complexity (because yellow subtree traversed in the context of larger yellow subtree, keeping trace of unfinished adjoinings) But more efficient in practice! And suggesting extensions, based on the idea of continuation Éric. de la Clergerie TAL 30/09/ / 91

111 Simplified Cascade of partial evaluations ν R CAI RAI CFI f RFI > CFI > RFI RAI > Not the optimal worst case complexity (because yellow subtree traversed in the context of larger yellow subtree, keeping trace of unfinished adjoinings) But more efficient in practice! And suggesting extensions, based on the idea of continuation Éric. de la Clergerie TAL 30/09/ / 91

112 Part II MCS in general Éric. de la Clergerie TAL 30/09/ / 91

113 Outline 4 Thread Automata and MCS formalisms 5 A Dynamic Programming interpretation for TAs Éric. de la Clergerie TAL 30/09/ / 91

114 Mildly Context Sensitivity An informal notion covering formalisms such that: they are powerful enough to model crossing, such as a n b n c n they are parsable with polynomial complexity i.e. Given L, there exists k, membership w checked in O( w k ) they generate string languages satisfying the constant growth property G, G finite, n 0, w L, w > n 0 = g G, w L, w = w +g (intuition) the languages are generated by finite sets of generators Éric. de la Clergerie TAL 30/09/ / 91

115 Mildly Context Sensitivity An informal notion covering formalisms such that: they are powerful enough to model crossing, such as a n b n c n they are parsable with polynomial complexity i.e. Given L, there exists k, membership w checked in O( w k ) they generate string languages satisfying the constant growth property G, G finite, n 0, w L, w > n 0 = g G, w L, w = w +g (intuition) the languages are generated by finite sets of generators Some MCS languages: TAGs and LIGs Local Multi Component TAGs (MC-TAGs Weir) Linear Context-Free Rewriting Systems (LCFRS Weir) Simple Range Concatenation Grammars (srcg Boullier) Éric. de la Clergerie TAL 30/09/ / 91

116 Semi-linearity The Constant Growth property subsumed by stronger semi-linearity under Parikh image The Parikh image of w {a 1,...,a n } defined as p(w) = ( w a1,..., w an ) The Parikh image of L defined as p(l) = {p(w) w L} A set V of vectors over N Σ is linear is generated by a base v 0, v 1,...,v n N Σ by V = {v 0 +Σ n i=1k i v i k i N} V is semilinear if V = k i=1 V i is a finite union of linear sets V i A language L is semilinear if p(l) is semilinear (intuition) A MCS language is generated, modulo some permutations, by a finite set of generators Éric. de la Clergerie TAL 30/09/ / 91

117 MCS: discontinuity and interleaving Discontinuous interleaved constituents present in linguistic phenomena Nesting, Crossing, Topicalization, Deep extraction, Complex Word-Order... Éric. de la Clergerie TAL 30/09/ / 91

118 MCS: discontinuity and interleaving Discontinuous interleaved constituents present in linguistic phenomena Nesting, Crossing, Topicalization, Deep extraction, Complex Word-Order... A B C X1 X2 X3 X4 H X5 X6 Éric. de la Clergerie TAL 30/09/ / 91

119 MCS: discontinuity and interleaving Discontinuous interleaved constituents present in linguistic phenomena Nesting, Crossing, Topicalization, Deep extraction, Complex Word-Order... A B C X1 X2 X3 X4 H X5 X6 LFCRS: A f(b, C), f linear non erasing function on string tuples. f( x 1, x 3, x 5, x 2, x 4, x 6 ) = x 1 x 2 x 3 x 4, x 5 x 6 srcg A(x 1.x 2.x 3.x 4, x 5.x 6 ) B(x 1, x 3, x 5 ), C(x 2, x 4, x 6 ) range variables x i ; concatenation. ; holes, Éric. de la Clergerie TAL 30/09/ / 91

120 LCFRS Linear Context-Free Rewriting Systems (LCFRS), a restricted form of generalized CFGs A LCFRS is a tuple G = (N,Σ, S,P,F) where P is a finite set of productions as follows, with f F A f(a 1,...,A n ) F is a set of linear regular operations over tuples of strings in Σ f( x 1,1,...,x 1,k1,..., x n,1,...,x 1,kn ) = t 1,...t k where V = {x i,j } are variables (over Σ ) and t i (Σ V) and (regular or non-erasing) xi,j, t u, x i,j t u (linear) xi,j, x i,j t u x i,j t v = u = v Assuming arity(s) = 1, L(G) = {w S = w } where A = f() if A f() P A = f(t 1,...,t n ) if A f(a 1,...,A n ) P i, A i = t i Éric. de la Clergerie TAL 30/09/ / 91

121 RCG Range Concatenation Grammars (RCG) [Boullier] : Constraints on intervals on the input string. For language a n b n c n Z) > A(X,Y, Z). ( " a X, " b Y, " c Z) > A(X,Y, Z). ( " ", " ", " " ) >. aabbcc S(]0, 6]) aabbcc A(]0, 2],]2, 4],]4, 6]) aabbcc A(]1, 2],]3, 4],]5, 6]) aabbcc A(]2, 2],]4, 4],]6, 6]) RCG is an operational formalism for encoding linguistic formalisms where discontinuous constituents are used. RCG allow modular grammar writing concatenation Y) > G1(X),G2(Y). union G(X) > G1(X) G2(X). intersection G(X) > G1(X),G2(X). Linear non-erasing positive RCGs equivalent to LCFRS Full RCGs are PTIME (equivalent to Datalog) Éric. de la Clergerie TAL 30/09/ / 91

122 Parsing MCS MCS have theoretical polynomial complexity O(n u ) depending upon degree of discontinuity, (also fanout, arity) degree of interleaving, (also rank) But no uniform framework to express parsing strategies and tabular algorithms operational device: Deterministic Tree Walking Transducer (Weir), but no tabular algorithm operational formalism srcg with tabular algorithm (Boullier) but not for prefix-valid strategies Notion of Thread Automata to model discontinuity and interleaving through the suspension/resume of threads. Éric. de la Clergerie TAL 30/09/ / 91

123 Tree Walking Automata TWA may be used to check properties of (binary) trees (by accepting or rejecting them) A (non-deterministic) TWA is a tuple A = (Q, Σ, I, F, R, δ) where Q is a finite set of states Σ a finite set of node labels I, F, R Q the initial, accepting, rejecting states δ the finite set of transitions in Q Σ Pred Dirs Q where Pred {root, left, right,leaf} is a set of predicates for testing nodes Dirs {stay, up, left, right} a set of directions Deterministic TWA: δ : Q Σ Pred Dirs Q Given a Σ-tree τ = (V, E), a configuration is given by (ν, q) V Q Extensions: Pebble Automata (Engelfriet) Éric. de la Clergerie TAL 30/09/ / 91

124 Tree Walking Transducers Similar to TWA, but emits strings when walking over a tree (in some tree set) A deterministic TWD (Weir) is a tuple T = (Q, G,Σ O, q I, F,δ) where G = (N,Σ I, S,P) is a CFG Σ O a finite set of output symbols δ : Q (N Σ I {ǫ}) Dirs Q Σ O with Dirs = {stay, up, down 1,...,down n } A transition step given by (q,γ,ν, w) (q,γ,ν, w.v) if { (q, dir) = δ(q, label(ν)) ν = dir(ν) The language generated by T defined as L(T) = {w (q I,γ, r γ,ǫ) (q f,γ,, w)} with q f F, γ a derivation tree for G with root r γ and a virtual node parent of r γ Weir s result: L(DTWD) = LCFRL Éric. de la Clergerie TAL 30/09/ / 91

125 Thread Automata Idea: Associate a thread p per constituent and create a subthread p.u for a sub-constituent [PUSH] suspend thread at constituent discontinuity, and (resume) either the parent thread [SPOP] or some direct subthread [SPUSH] scan terminal [SWAP] delete thread after full recognition of a constituent [POP] Éric. de la Clergerie TAL 30/09/ / 91

126 Thread Automata Idea: Associate a thread p per constituent and create a subthread p.u for a sub-constituent [PUSH] suspend thread at constituent discontinuity, and (resume) either the parent thread [SPOP] or some direct subthread [SPUSH] scan terminal [SWAP] delete thread after full recognition of a constituent [POP] Recognize aaabbbccc a n b n c n ǫ 1 Éric. de la Clergerie TAL 30/09/ / 91

127 Thread Automata Idea: Associate a thread p per constituent and create a subthread p.u for a sub-constituent [PUSH] suspend thread at constituent discontinuity, and (resume) either the parent thread [SPOP] or some direct subthread [SPUSH] scan terminal [SWAP] delete thread after full recognition of a constituent [POP] Recognize aaabbbccc a n b n c n ǫ a Éric. de la Clergerie TAL 30/09/ / 91

128 Thread Automata Idea: Associate a thread p per constituent and create a subthread p.u for a sub-constituent [PUSH] suspend thread at constituent discontinuity, and (resume) either the parent thread [SPOP] or some direct subthread [SPUSH] scan terminal [SWAP] delete thread after full recognition of a constituent [POP] Recognize aaabbbccc a n b n c n ǫ a 1.1 a 1 Éric. de la Clergerie TAL 30/09/ / 91

129 Thread Automata Idea: Associate a thread p per constituent and create a subthread p.u for a sub-constituent [PUSH] suspend thread at constituent discontinuity, and (resume) either the parent thread [SPOP] or some direct subthread [SPUSH] scan terminal [SWAP] delete thread after full recognition of a constituent [POP] Recognize aaabbbccc a n b n c n ǫ a a 1.1 a 1 Éric. de la Clergerie TAL 30/09/ / 91

130 Thread Automata Idea: Associate a thread p per constituent and create a subthread p.u for a sub-constituent [PUSH] suspend thread at constituent discontinuity, and (resume) either the parent thread [SPOP] or some direct subthread [SPUSH] scan terminal [SWAP] delete thread after full recognition of a constituent [POP] Recognize aaabbbccc a n b n c n ǫ a ǫ a 1.1 a 1 Éric. de la Clergerie TAL 30/09/ / 91

131 Thread Automata Idea: Associate a thread p per constituent and create a subthread p.u for a sub-constituent [PUSH] suspend thread at constituent discontinuity, and (resume) either the parent thread [SPOP] or some direct subthread [SPUSH] scan terminal [SWAP] delete thread after full recognition of a constituent [POP] Recognize aaabbbccc a n b n c n ǫ a ǫ a 1.1 a 1 Éric. de la Clergerie TAL 30/09/ / 91

132 Thread Automata Idea: Associate a thread p per constituent and create a subthread p.u for a sub-constituent [PUSH] suspend thread at constituent discontinuity, and (resume) either the parent thread [SPOP] or some direct subthread [SPUSH] scan terminal [SWAP] delete thread after full recognition of a constituent [POP] Recognize aaabbbccc a n b n c n ǫ a ǫ a 1.1 a 1 Éric. de la Clergerie TAL 30/09/ / 91

133 Thread Automata Idea: Associate a thread p per constituent and create a subthread p.u for a sub-constituent [PUSH] suspend thread at constituent discontinuity, and (resume) either the parent thread [SPOP] or some direct subthread [SPUSH] scan terminal [SWAP] delete thread after full recognition of a constituent [POP] Recognize aaabbbccc a n b n c n ǫ a ǫ a 1.1 a 1 Éric. de la Clergerie TAL 30/09/ / 91

134 Thread Automata Idea: Associate a thread p per constituent and create a subthread p.u for a sub-constituent [PUSH] suspend thread at constituent discontinuity, and (resume) either the parent thread [SPOP] or some direct subthread [SPUSH] scan terminal [SWAP] delete thread after full recognition of a constituent [POP] Recognize aaabbbccc a n b n c n a ǫ a 1.1 a b 1 ǫ b b ǫ Éric. de la Clergerie TAL 30/09/ / 91

135 Thread Automata Idea: Associate a thread p per constituent and create a subthread p.u for a sub-constituent [PUSH] suspend thread at constituent discontinuity, and (resume) either the parent thread [SPOP] or some direct subthread [SPUSH] scan terminal [SWAP] delete thread after full recognition of a constituent [POP] Recognize aaabbbccc a n b n c n ǫ a ǫ a 1.1 a 1 b b b ǫ c c c ǫ Éric. de la Clergerie TAL 30/09/ / 91

136 Formal presentation of TA Configuration position l, active thread path p, thread store S = {p i :A i } S closed by prefix: p.u dom(s) = p dom(s) Note: stateless automata (but no problem for variants with states) Triggering function a = Φ(A) amount of information needed to trigger transitions. = useful to get linear compexity O( G ) w.r.t. grammar size G Default: Φ = Identity Driver function u δ(a) Drive thread creations and suspensions = reduce number of transitions (TA variants without δ should be possible) Éric. de la Clergerie TAL 30/09/ / 91

137 Formal presentation of TA (cont d) SWAP B α C : Changes the content of the active thread, possibly scanning a terminal. l, p,s p:b τ l + α, p,s p:c a l = α if α ǫ PUSH b [b]c : Creates a new subthread (unless present) l, p,s p:b τ l, pu,s p:b pu:c (b, u) Φδ(B) pu dom POP [B]C D : Terminates thread pu (if no existing subthreads). l, pu,s p:b pu:c τ l, p,s p:d pu dom(s) SPUSH b[c] [b]d : Resumes the subthread pu (if already created) l, p,s p:b pu:c l, pu,s p:b pu:d (b, u s ) Φδ(B) τ SPOP [B]c D[c] : Resumes the parent thread p of pu l, pu,s p:b pu:c l, p,s p:d pu:c (c, ) Φδ(C) τ Éric. de la Clergerie TAL 30/09/ / 91

138 Characterizing Thread Automata Key parameters: h maximal number of suspensions to the parent thread h finite ensures termination (of tabular parsing) d maximal number of simultaneously alive subthreads l maximal number of subthreads s maximal number of suspensions (parent + alive subthreads) s h+dh h+lh Éric. de la Clergerie TAL 30/09/ / 91

139 Characterizing Thread Automata Key parameters: h maximal number of suspensions to the parent thread h finite ensures termination (of tabular parsing) d maximal number of simultaneously alive subthreads l maximal number of subthreads s maximal number of suspensions (parent + alive subthreads) s h+dh h+lh Worst-case Complexity: } { space O(n u ) u = 2+s + x time O(n 1+u where ) x = min(s,(l d)(h + 1)) { space between O(n = 2+2s ) and [when l = d] O(n 2+s ) time between O(n 3+2s) ) and [when l = d] O(n 3+s ) Éric. de la Clergerie TAL 30/09/ / 91

140 Characterizing Thread Automata Key parameters: h maximal number of suspensions to the parent thread h finite ensures termination (of tabular parsing) d maximal number of simultaneously alive subthreads l maximal number of subthreads s maximal number of suspensions (parent + alive subthreads) s h+dh h+lh Worst-case Complexity: } { space O(n u ) u = 2+s + x time O(n 1+u where ) x = min(s,(l d)(h + 1)) { space between O(n = 2+2s ) and [when l = d] O(n 2+s ) time between O(n 3+2s) ) and [when l = d] O(n 3+s ) Push-Down Automata (PDA) for CFG TA(h=0,d=1,s=0) = space O(n 2 ) and time O(n 3 ) Éric. de la Clergerie TAL 30/09/ / 91

141 Parsing TAGs Idea: Assign a thread per elementary tree traversal (substitution or adjunction) Suspend and return to parent thread to handle a foot node Éric. de la Clergerie TAL 30/09/ / 91

142 Parsing TAGs Idea: Assign a thread per elementary tree traversal (substitution or adjunction) Suspend and return to parent thread to handle a foot node ν r ν ν β r β α r α ν ν f ν Éric. de la Clergerie TAL 30/09/ / 91

143 Parsing TAGs Idea: Assign a thread per elementary tree traversal (substitution or adjunction) Suspend and return to parent thread to handle a foot node ν r ν ν β r β f ν f ν α r α ν ν Éric. de la Clergerie TAL 30/09/ / 91

144 Parsing TAGs Idea: Assign a thread per elementary tree traversal (substitution or adjunction) Suspend and return to parent thread to handle a foot node ν r ν ν β r β f f ν f ν α r α ν ν ν Éric. de la Clergerie TAL 30/09/ / 91

145 Parsing TAGs Idea: Assign a thread per elementary tree traversal (substitution or adjunction) Suspend and return to parent thread to handle a foot node ν r ν ν β r β f f r β ν f ν α r α ν ν ν ν Éric. de la Clergerie TAL 30/09/ / 91

146 Parsing TAGs Idea: Assign a thread per elementary tree traversal (substitution or adjunction) Suspend and return to parent thread to handle a foot node ν r ν ν β r β f f r β ν f ν α r α ν ν ν ν r α Éric. de la Clergerie TAL 30/09/ / 91

147 Parsing TAGs Idea: Assign a thread per elementary tree traversal (substitution or adjunction) Suspend and return to parent thread to handle a foot node ν r ν ν ν f ν α β r α β r β ν f ν f r β ν ν r α Éric. de la Clergerie TAL 30/09/ / 91

148 Parsing TAGs Idea: Assign a thread per elementary tree traversal (substitution or adjunction) Suspend and return to parent thread to handle a foot node ν r ν ν ν f ν α β r α β r β ν f ν f r β ν ν r α Éric. de la Clergerie TAL 30/09/ / 91

149 Parsing TAGs Idea: Assign a thread per elementary tree traversal (substitution or adjunction) Suspend and return to parent thread to handle a foot node ν r ν ν ν f ν α β r α β r β ν f ν f r β ν ν r α Éric. de la Clergerie TAL 30/09/ / 91

150 Parsing TAGs Idea: Assign a thread per elementary tree traversal (substitution or adjunction) Suspend and return to parent thread to handle a foot node ν r ν ν ν f ν α β r α β r β ν f ν f r β ν ν r α One thread per tree h = 1, d = max(depth(trees)) = [s = 1+d] space O(n 4+2d ) and time O(n 5+2d ) Éric. de la Clergerie TAL 30/09/ / 91

151 Parsing TAG: an alternate parsing strategy Using more than one thread per elementary tree: 1 thread per subtree ( LIG) = implicit extraction of subtrees = implicit normal form (using a third kind of tree operation) = usual n 6 time complexity ν r ν ν ν f ν r ν ν f f ν ν Note: Similar to a TAG encoding in RCG proposed by Boullier Éric. de la Clergerie TAL 30/09/ / 91

152 Using less threads Always possible to reduce the number of live subthreads (down to 2). if a thread p has d + 1 subthreads, add a new subthread p.v that inherits d subthreads of p generally increases the number of parent suspensions h but may also exploit good topological properties, such as well-nesting (TAGs). Éric. de la Clergerie TAL 30/09/ / 91