Structural Relations

1. Structural Relations The mathematical properties of phrase structure trees

2. Important!! • This is the most mathematical and technical topic we will cover. • Even if you have trouble with the formal definitions. Try to understand the INTUITIVE idea behind them. Don’t get lost in the details of the formalism.

3. Structural Relations • Structural relations: the formal relationships between items of a tree • Why should we care? We want to be able to talk about specific relationships in terms of structures. • Structural relationships are actually very simple! Don’t let the formalism scare you!

4. M Root node Branches Non-terminal nodes N O D E F G H J Terminal nodes Some basic terms Labels: M,N,O,D,E,F,G,H,J Node: Any point with a label

5. A [A B C [D E F G]] contained inside [A ] B C D E F G Dominance • Intuitively: this is containment. If a node contains another, then it dominates it: A dominates B,C,D,E,F,G D dominates E,F,G

6. Dominance • Other intuitive definitions: • “On top of”, • “Can follow a branch downwards to”.

7. Dominance • A slightly more formal definition: • Dominance: Node A dominates node B if and only if A is higher up in the tree than B and if you can trace a line from A to B going only downwards.

8. A B C D E F G Immediate Dominance • Node A immediately dominates node B if there is no intervening node G which is dominated by A, but dominates B • (in other words, A is the first node that dominates B) A dominates B,C,D,E,F,G but A immediately dominates only B,C,D

9. A * A B B C C Properties of domination • x ≤x(every linguistic element dominates itself) • if x ≤ y ≤ z then x ≤ z(the dominance relation is transitive) • if x ≤ y ≤ x then, x = y(the dominance relation is unidirectional) • if x ≤ z and y ≤ z then either x ≤ y or y ≤ x (or both if x =y =z) (multiple mothers are ruled out)

10. Exhaustive Domination • Node A exhaustively dominates a SET of nodes {B,C,…,D}, • provided it immediately dominate all the members of the set (so that there is no member of the set that is not dominated by A) • AND there is no node G immediately dominated by A that is not a member of the set.

11. F A G B C D E H I Exhaustive Domination A exhaustively dominates the set {B,C,D,E} A does NOT exhaustively dominate the set {B,C,D} A does NOT exhaustively dominate the set {B,C,D, E, H}

12. F A G B C D E H I A formal definition of Constituency • Constituent: A set of nodes exhaustively dominated by a single node. {E, H} are NOT a constituent

13. Constituent vs Constituent of • Constituent of does NOT mean the same thing as constituent. • Essentially constituent of is the opposite of domination. • A dominates B, then we say B is a constituent of A. • immediate constituent of is the opposite of immediate domination.

14. Some Informal Terms • Mother: the node that immediately dominates another. • Daughter: the node that is immediately dominated by another (is an immediate constituent of another). • Sisters: two nodes that share the same mother.

15. Root Node S NP VP Terminal Nodes D N V the platypus laughed Root and Terminal Nodes • Root node: A node with no mother • Terminal node: A node with no daughters

16. The intuitive part Precedence • Precedence: Node A precedes node B if and only if • (1) A is to the left of B • (2) and neither A dominates B nor B dominates A • (3) and every node dominating A either precedes B or dominates B why do we need clauses (2) and (3)?

17. Breaking down the definition of precedence • (2) and neither A dominates B nor B dominates A Is the ball to the left or right of the box? Neither! You can’t precede or follow something that dominates (contains) you or you dominate (contain).

18. S NP VP D N NP the clown D N V the donkey kissed Breaking down the definition of precedence • (3) and every node dominating A either precedes B or dominates B. Does kiss precede clown? Obviously not! “and every node dominating kissed either precedes clown or dominates clown” Not true

19. No Crossing Branches Constraint • If one node X precedes another node Y then X and all nodes dominated by X must precede Y and all nodes dominated by Y. * M N O P R Q S

20. Properties of precedence • if xy, then NOT(yx) (You can’t both precede and follow) • if xy z, then xz (precedence is transitive) • if xy or yx, then NOT(x≤y) and NOT (y≤x) (precedence and dominance are mutually exclusive)there is a mistake in the textbook on this one. • xy iff for all terminals u, v, x ≤ u and y ≤ v jointly imply uv (don’t cross lines) A B C

21. Immediate Precedence • Immediate Precedence: • A immediately precedes B if there is no node G which follows A but precedes B. A B G A G B

22. A c-commands C,D,E,F,G,H M A C D E F G H C-command • Intuitively: The relationship between a node and its sister, and all the daughters of its sister Note: D does NOT c–command A

23. you can’t command something you dominate Sisterhood C-command • Node A c-commands node B if every branching node dominating A also dominates B, and A does not itself dominate B.

24. M A B D E F G H Symmetric C-command • A symmetrically c-commands B, if A c-commands B AND B c-commands A • SAME THING AS SISTERHOOD A & B symmetrically c-command one another A does NOT symmetrically c-command D

25. M A C B E F G H Asymmetric C-command • A asymmetrically c-commands B, if A c-commands B but B does NOT c-command A. • (intuitively -- A is B’s aunt)

26. Grammatical Relations • Subject: NP daughter of S • Object: NP daughter of VP • Object of a Preposition: NP daughter of PP

27. Summary • Grammatical Relations: relationship between nodes. • Dominance (=containment) • immediate dominance (=motherhood) • exhaustive dominance (=constituent) • Precedence (=to the left) • immediate precedence (=adjacent & to the left)

28. Summary • C-command: Sisters & nieces • Symmetric C-command: sisters • Asymmetric C-command: Aunt asymm. c-commands nieces • Grammatical Relations • Subject: NP daughter of S • Object: NP daughter of VP • Object of P: NP daughter of PP