Applications of Tree Automata Theory Lecture I: Tree Automata

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1 Applications of Tree Automata Theory Lecture I: Tree Automata Andreas Maletti Institute of Computer Science Universität Leipzig, Germany on leave from: Institute for Natural Language Processing Universität Stuttgart, Germany Yekaterinburg August 23, 204 Lecture I: Tree Automata A. Maletti

2 Roadmap Theory of Tree Automata 2 Parsing Basics and Evaluation 3 Parsing Advanced Topics 4 Machine Translation Basics and Evaluation 5 Theory of Tree Transducers 6 Machine Translation Advanced Topics Always ask questions right away! Lecture I: Tree Automata A. Maletti 2

3 Trees Motivation and Notation Lecture I: Tree Automata A. Maletti 3

4 Trees Parses We must bear in mind the Community as a whole S PRP MD We must VB PP bear IN PP in NN DT NN IN mind the Community as DT NN a whole Lecture I: Tree Automata A. Maletti 4

5 Trees Parses È âû äåéñòâèòåëüíî ñìîæåòå èçìåíèòü ñâîþ æèçíü, âîñïîëüçîâàâøèñü ïðèíöèïîì 80 ê 20 conj èçìåíèòü subj È âû ñìîæåòå adv auxs acc äåéñòâèòåëüíî æèçíü âîñïîëüçîâàâøèñü adj ñâîþ adv misc ins, ïðèíöèïîì gen card gen 80 ê 20 Lecture I: Tree Automata A. Maletti 4

6 Trees XML <library> <book> <title>the Golden Ticket</title> <author>lance Fortnow</author> <publisher>princeton Univ. Press</publisher> </book><book> <title>anna Karenina</title> <author>leo Tolstoy</author> <publisher>the Russian Messenger</publisher> </book> </library> library book book title author publisher title author publisher The Golden Ticket Lance Fortnow Princeton Univ. Press Anna Karenina Leo Tolstoy The Russian Messenger Lecture I: Tree Automata A. Maletti 5

7 Trees Sets Σ and Q Definition Set T Σ (Q) of Σ-trees indexed by Q is smallest T q T for all q Q σ(t,..., t k ) T for all k N, σ Σ, and t,..., t k T We assume Σ Q = and write σ instead of σ() Lecture I: Tree Automata A. Maletti 6

8 Trees Example Σ = {σ, α} and Q = {q, p} σ(σ(p, q), σ(σ(q, σ))) T Σ (Q) σ σ σ p q σ q σ Lecture I: Tree Automata A. Maletti 7

9 Trees Example Σ = {σ, α} and Q = {q, p} σ(σ(p, q), σ(σ(q, σ))) T Σ (Q) σ σ σ p q σ q σ Notes obvious recursion & induction principle Lecture I: Tree Automata A. Maletti 7

10 Trees Recursion Definition (Gorn address) Mapping pos: T Σ (Q) 2 N assigning positions pos(q) = {ε} pos(σ(t,..., t k )) = {ε} {i.w i k, w pos(t i )} Lecture I: Tree Automata A. Maletti 8

11 Trees Recursion Definition (Gorn address) Mapping pos: T Σ (Q) 2 N assigning positions pos(q) = {ε} pos(σ(t,..., t k )) = {ε} {i.w i k, w pos(t i )} Definition Leaves of t T Σ (Q): leaves(t) = {w pos(t) w. / pos(t)} Lecture I: Tree Automata A. Maletti 8

12 Trees Recursion S PRP MD We must VB PP bear IN PP in NN DT NN IN mind the Community as DT NN a whole Leaves marked in red Lecture I: Tree Automata A. Maletti 9

13 Trees Recursion S 2 PRP We MD must VB bear PP IN in 2 2 NN 3 2 mind the 2 DT NN Community 2 IN as PP DT NN a 2 2 whole Lecture I: Tree Automata A. Maletti 9

14 Trees Recursion S 2 PRP We MD must VB bear PP IN in 2 2 NN 3 2 mind the 2 DT NN Community 2 IN as PP DT NN a 2 2 whole Address of marked DT : Lecture I: Tree Automata A. Maletti 9

15 Trees Trees t, u T Σ (Q) and position w pos(t) in t Notation t(w) = label of t at position w t w = subtree rooted in w t[u] w = tree obtained by replacing the subtree at w in t by u Lecture I: Tree Automata A. Maletti 0

16 Trees S 2 PRP We MD must VB PP bear IN in NN DT NN mind the 2 Community 2 IN as PP DT NN a 2 2 whole t(2.2..) = bear Lecture I: Tree Automata A. Maletti

17 Trees S 2 PRP We MD must VB PP bear IN in NN DT NN mind the 2 Community 2 IN as PP DT NN a 2 2 whole t(2.2..) = bear t = (DT(the), NN(Community)) Lecture I: Tree Automata A. Maletti

18 Trees S 2 PRP We MD must VB PP bear IN in NN DT NN mind the 2 Community 2 IN as PP DT NN a 2 2 whole t(2.2..) = bear t = (DT(the), NN(Community)) t = t [(DT(the), NN(Community))] Lecture I: Tree Automata A. Maletti

19 Trees S 2 PRP We MD must VB PP bear IN in NN DT NN mind the 2 Community 2 IN as PP DT NN a 2 2 whole t(2.2..) = bear t = (DT(the), NN(Community)) t = t [(DT(the), NN(Community))] Lecture I: Tree Automata A. Maletti

20 Tree Language Representations Lecture I: Tree Automata A. Maletti 2

21 Tree Language Tree language (or forest) = subset of T Σ (Q) Motivation parses of a sentence translations of an input sentence valid XML documents... parse forest translation forest XML schema Lecture I: Tree Automata A. Maletti 3

22 Tree Language How to represent a set of trees? enumerate them Lecture I: Tree Automata A. Maletti 4

23 Tree Language How to represent a set of trees? enumerate them enumerate them cleverly (e.g., add sharing) Lecture I: Tree Automata A. Maletti 4

24 Tree Language L = {σ(σ(p, q), σ(σ(q, σ))), α(σ(q, σ), α)} Enumeration of L: σ σ p q σ σ q σ α σ q σ α Subtree-shared enumeration of L: (packed forest) σ σ p q σ σ σ α α Lecture I: Tree Automata A. Maletti 5

25 Tree Language How to represent a set of trees? enumerate them enumerate them cleverly (packed forest) Lecture I: Tree Automata A. Maletti 6

26 Tree Language How to represent a set of trees? enumerate them enumerate them cleverly parse forest of a CFG (packed forest) Lecture I: Tree Automata A. Maletti 6

27 Parse Forest of a CFG Example S MD PP VB PP MD must... Lecture I: Tree Automata A. Maletti 7

28 Parse Forest of a CFG Example S MD PP VB PP MD must... S PRP MD We must VB PP bear IN PP in NN DT NN IN mind the Community as DT NN a whole Lecture I: Tree Automata A. Maletti 7

29 Parse Forest of a CFG Example S MD PP VB PP MD must... S PRP MD We must VB PP bear IN PP in NN DT NN IN mind the Community as DT NN a whole Lecture I: Tree Automata A. Maletti 7

30 Parse Forest of a CFG Example S MD PP VB PP MD must... S PRP MD We must VB PP bear IN PP in NN DT NN IN mind the Community as DT NN a whole Lecture I: Tree Automata A. Maletti 7

31 Parse Forest of a CFG Example S MD PP VB PP MD must... S PRP MD We must VB PP bear IN PP in NN DT NN IN mind the Community as DT NN a whole Lecture I: Tree Automata A. Maletti 7

32 Parse Forest of a CFG Example S MD PP VB PP MD must... S PRP MD We must VB PP bear IN PP in NN DT NN IN mind the Community as DT NN a whole Lecture I: Tree Automata A. Maletti 7

33 Parse Forest of a CFG Example S MD PP VB PP MD must... S PRP MD We must VB PP bear IN PP in NN DT NN IN mind the Community as DT NN a whole Lecture I: Tree Automata A. Maletti 7

34 Local Tree Grammar Definition A local tree grammar (LTG) is a CFG G = (N, Q, S, P) finite set N nonterminals finite set Q terminals S N start nonterminals finite set P N (N Q) productions It will compute the derivation trees of the CFG Lecture I: Tree Automata A. Maletti 8

35 Local Tree Grammar LTG G = (N, Q, S, P) Definition (Generated tree language) L(G) contains exactly the trees t T N (Q) t(ε) S root label in S t(w) t(w.) t(w.k) P for every internal node w pos(t) with {i N w.i pos(t)} = {,..., k} t(w) Q for all w leaves(t) label child labels is a production of G leaves labeled by Q Lecture I: Tree Automata A. Maletti 9

36 Local Tree Grammar Observation LTGs generate exactly the parse forests of CFGs Lecture I: Tree Automata A. Maletti 20

37 Local Tree Grammar Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 20

38 Local Tree Grammar Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 20

39 Local Tree Grammar Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 20

40 Local Tree Grammar Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 20

41 Local Tree Grammar Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 20

42 Local Tree Grammar Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 20

43 Local Tree Grammar Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 20

44 Local Tree Grammar Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 20

45 Local Tree Grammar Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 20

46 Local Tree Grammar Observation LTGs generate exactly the parse forests of CFGs Properties simple no ambiguity (unique explanation for each generated tree) not closed under (most) BOOLEAN operations (union/intersection/complement: / / ) not closed under (non-injective) relabelings... Lecture I: Tree Automata A. Maletti 20

47 Local Tree Grammar LTG G = (N, Q, S, P) No ambiguity S PRP$ My NN dog VBZ sleeps is in L(G) if and only if S is a start nonterminal 2 all the productions in it are in P 3 all leaves are labeled by elements of Q Lecture I: Tree Automata A. Maletti 2

48 Local Tree Grammar Observation Local tree languages are not closed under union Proof. The following single-element tree languages are local: PRP$ My NN dog S VBZ sleeps PRP I S VBD scored AD RB well But their union is not local as it must also generate trees for My dog scored well and *I sleeps Lecture I: Tree Automata A. Maletti 22

49 Tree Language How to represent a set of trees? enumerate them enumerate them cleverly parse forest of a CFG (packed forest) (local tree languages) Lecture I: Tree Automata A. Maletti 23

50 Tree Language How to represent a set of trees? enumerate them enumerate them cleverly parse forest of a CFG (packed forest) (local tree languages) Lecture I: Tree Automata A. Maletti 23

51 Tree Language How to represent a set of trees? enumerate them enumerate them cleverly parse forest of a CFG tree substitution grammar (packed forest) (local tree languages) Lecture I: Tree Automata A. Maletti 23

52 Local Tree Grammar CFG production L R R 2 R 3 represented by tree L R R 2 R 3 Glue fragments together to obtain larger trees: S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 24

53 Local Tree Grammar CFG production L R R 2 R 3 represented by tree L R R 2 R 3 Glue fragments together to obtain larger trees: S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 24

54 Local Tree Grammar CFG production L R R 2 R 3 represented by tree L R R 2 R 3 Glue fragments together to obtain larger trees: S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 24

55 Local Tree Grammar CFG production L R R 2 R 3 represented by tree L R R 2 R 3 Glue fragments together to obtain larger trees: S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 24

56 Local Tree Grammar CFG production L R R 2 R 3 represented by tree L R R 2 R 3 Glue fragments together to obtain larger trees: S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 24

57 Local Tree Grammar CFG production L R R 2 R 3 represented by tree L R R 2 R 3 Glue fragments together to obtain larger trees: S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 24

58 Local Tree Grammar CFG production L R R 2 R 3 represented by tree L R R 2 R 3 Glue fragments together to obtain larger trees: S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 24

59 Local Tree Grammar CFG production L R R 2 R 3 represented by tree L R R 2 R 3 Glue fragments together to obtain larger trees: S PRP MD We must VB PP But why only small tree fragments? Lecture I: Tree Automata A. Maletti 24

60 Tree Substitution Grammar Definition (JOSHI 969) A tree substitution grammar (TSG) is a tuple (N, Q, S, P) finite set N nonterminals finite set Q terminals S N start nonterminals finite set P T N (Q) \ Q tree fragments Lecture I: Tree Automata A. Maletti 25

61 Tree Substitution Grammar Definition (JOSHI 969) A tree substitution grammar (TSG) is a tuple (N, Q, S, P) finite set N nonterminals finite set Q terminals S N start nonterminals finite set P T N (Q) \ Q tree fragments Typical fragments [POST 20] S S VBD PP CD PRP TO Lecture I: Tree Automata A. Maletti 25

62 Tree Substitution Grammar TSG G = (N, Q, S, P) and sentential forms ξ, ζ T N (Q) Definition ξ G ζ if there exist a tree fragment t P and a position w leaves(ξ) such that ξ = ξ[t(ε)] w and ζ = ξ[t] w Lecture I: Tree Automata A. Maletti 26

63 Tree Substitution Grammar TSG G = (N, Q, S, P) and sentential forms ξ, ζ T N (Q) Definition ξ G ζ if there exist a tree fragment t P and a position w leaves(ξ) such that ξ = ξ[t(ε)] w and ζ = ξ[t] w Intuition In a derivation step: Find a leaf labeled by n N 2 Find a fragment t P such that t(ε) = n 3 Replace selected n by t Lecture I: Tree Automata A. Maletti 26

64 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

65 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

66 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

67 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

68 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

69 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

70 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

71 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

72 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

73 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

74 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

75 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

76 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

77 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

78 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

79 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

80 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

81 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

82 Tree Substitution Grammar Example Fragments: S(, ) (MD, ) (VB, PP, ) (PRP) PRP(We) MD(must) S PRP We MD must VB PP Lecture I: Tree Automata A. Maletti 27

83 Tree Substitution Grammar TSG G = (N, Q, S, P) Definition for all n N: L(G, n) = {t T N (Q) w leaves(t): t(w) Q, n G t} L(G) = s S L(G, s) Lecture I: Tree Automata A. Maletti 28

84 Tree Substitution Grammar TSG G = (N, Q, S, P) and TSL = {L(G) G TSG} Definition for all n N: L(G, n) = {t T N (Q) w leaves(t): t(w) Q, n G t} L(G) = s S L(G, s) Theorem FIN TSL all finite languages are TSL Lecture I: Tree Automata A. Maletti 28

85 Tree Substitution Grammar TSG G = (N, Q, S, P) and TSL = {L(G) G TSG} Definition for all n N: L(G, n) = {t T N (Q) w leaves(t): t(w) Q, n G t} L(G) = s S L(G, s) Theorem FIN TSL all finite languages are TSL 2 LTL TSL all local tree languages are TSL Lecture I: Tree Automata A. Maletti 28

86 Tree Substitution Grammar Theorem Tree substitution languages (TSL) have the following properties: closed under CFL LTL TSL union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 29

87 Tree Substitution Grammar Theorem Tree substitution languages (TSL) have the following properties: closed under CFL LTL TSL union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 29

88 Tree Substitution Grammar Theorem Tree substitution languages (TSL) have the following properties: closed under CFL LTL TSL union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 29

89 Tree Substitution Grammar Theorem Tree substitution languages (TSL) have the following properties: closed under CFL LTL TSL union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 29

90 Tree Substitution Grammar Theorem Tree substitution languages (TSL) have the following properties: closed under CFL LTL TSL union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 29

91 Tree Substitution Grammar Properties simple more expressive than local tree grammars ambiguity (several explanations for a generated tree) not closed under BOOLEAN operations (union/intersection/complement: / / ) not closed under (non-injective) relabelings... Lecture I: Tree Automata A. Maletti 30

92 Tree Substitution Grammar Theorem Tree substitution languages are not closed under union Proof. Counterexample must be infinite artificial example S S C a C b C C a b L = {S(C n (a), a) n N} L 2 = {S(C n (b), b) n N} Their union is not a tree substitution language Lecture I: Tree Automata A. Maletti 3

93 Tree Substitution Grammar Theorem Tree substitution languages are not closed under intersection Proof. Ideas? Lecture I: Tree Automata A. Maletti 32

94 Tree Language How to represent a set of trees? enumerate them enumerate them cleverly parse forest of a CFG tree substitution grammar regular tree grammar (packed forest) (local tree languages) Lecture I: Tree Automata A. Maletti 33

95 Regular Tree Grammar Definition (BRAINERD, 969) A regular tree grammar (RTG) is a tuple G = (N, Σ, S, P) with finite set N nonterminals finite set Σ terminals S N start nonterminals finite set P N T Σ (N) productions Remark Instead of (n, t) we write n t Lecture I: Tree Automata A. Maletti 34

96 Regular Tree Grammar Example N = {n 0, n, n 2, n 3, n 4, n 5, n 6 } Σ = {,, S,... } S = {n 0 } and the following productions: n 4 n 5 n 2 n 3 n 0 S n 4 n 0 n n 6 S n 2 n 4 Lecture I: Tree Automata A. Maletti 35

97 Regular Tree Grammar RTG G = (N, Σ, S, P) and sentential forms ξ, ζ T Σ (N) Definition (Derivation Semantics) ξ G ζ if there exist a production n t P and a position w leaves(ξ) such that ξ = ξ[n] w and ζ = ξ[t] w Lecture I: Tree Automata A. Maletti 36

98 Regular Tree Grammar RTG G = (N, Σ, S, P) and sentential forms ξ, ζ T Σ (N) Definition (Derivation Semantics) ξ G ζ if there exist a production n t P and a position w leaves(ξ) such that ξ = ξ[n] w and ζ = ξ[t] w Definition (Recognized tree language) L(G) = {t T Σ s S : s G t} Lecture I: Tree Automata A. Maletti 36

99 Regular Tree Grammar Productions S S n 4 n 5 n 2 n 3 n 0 n n 4 n 0 n 6 n 2 n 4 Derivation: n 0 G S n n 4 G n S n 5 n 2 n 3 Lecture I: Tree Automata A. Maletti 37

100 Regular Tree Grammar Productions S S n 4 n 5 n 2 n 3 n 0 n n 4 n 0 n 6 n 2 n 4 Derivation: n 0 G S n n 4 G n S n 5 n 2 n 3 Lecture I: Tree Automata A. Maletti 37

101 Regular Tree Grammar Productions S S n 4 n 5 n 2 n 3 n 0 n n 4 n 0 n 6 n 2 n 4 Derivation: n 0 G S n n 4 G n S n 5 n 2 n 3 Lecture I: Tree Automata A. Maletti 37

102 Regular Tree Grammar regular tree languages RTL = {L(G) G RTG} Theorem tree substitution languages regular tree languages Proof. We can express the union counterexample easily Lecture I: Tree Automata A. Maletti 38

103 Regular Tree Grammar Theorem Regular tree languages (RTL) have the following properties: closed under CFL LTL TSL RTL union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 39

104 Regular Tree Grammar Theorem Regular tree languages (RTL) have the following properties: closed under CFL LTL TSL RTL union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 39

105 Regular Tree Grammar Theorem Regular tree languages (RTL) have the following properties: closed under CFL LTL TSL RTL union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 39

106 Regular Tree Grammar Theorem Regular tree languages (RTL) have the following properties: closed under CFL LTL TSL RTL union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 39

107 Regular Tree Grammar Theorem Regular tree languages (RTL) have the following properties: closed under CFL LTL TSL RTL union intersection (rank-bounded) complement relabeling Lecture I: Tree Automata A. Maletti 39

108 Regular Tree Grammar Properties simple more expressive than tree substitution grammars ambiguity (several explanations for a generated tree) closed under all BOOLEAN operations (union/intersection/complement: / / ) closed under (non-injective) relabelings... Lecture I: Tree Automata A. Maletti 40

109 Regular Tree Grammar RTG G = (N, Σ, S, P) Definition (BRAINERD, 969) G is in normal form if t = σ(n,..., n k ) with σ Σ and n,..., n k N for all n t P Lecture I: Tree Automata A. Maletti 4

110 Regular Tree Grammar RTG G = (N, Σ, S, P) Definition (BRAINERD, 969) G is in normal form if t = σ(n,..., n k ) with σ Σ and n,..., n k N for all n t P Example productions S S n 4 n 5 n 2 n 3 n 0 n n 4 n 0 n 6 n 2 n 4 Lecture I: Tree Automata A. Maletti 4

111 Regular Tree Grammar Theorem (BRAINERD, 969) Every RTG is equivalent to an RTG in normal form Proof. Simply cut large rules introducing new states n 0 n 6 S n 2 n 4 n 0 S n 6 n n n 2 n 4 Lecture I: Tree Automata A. Maletti 42

112 Standard Representation Tree Automata Lecture I: Tree Automata A. Maletti 43

113 Tree Automaton Definition (THATCHER, 970; ROUNDS, 970) A tree automaton (TA) is an RTG in normal form Lecture I: Tree Automata A. Maletti 44

114 Tree Automaton Definition (THATCHER, 970; ROUNDS, 970) A tree automaton (TA) is an RTG in normal form Remarks bottom-up: rules written as σ(n,..., n k ) n top-down: rules written as n σ(n,..., n k ) Lecture I: Tree Automata A. Maletti 44

115 Tree Automaton Definition top-down deterministic if n N, k N, σ Σ at most one n,..., n k N : n σ(n,..., n k ) P and S = bottom-up deterministic if k N, σ Σ, n,..., n k N at most one n N : σ(n,..., n k ) n P (red determines blue) Lecture I: Tree Automata A. Maletti 45

116 Tree Automaton Theorem (THATCHER, WRIGHT, 968; DONER, 970) top-down det. RTL bottom-up det. RTL = RTL Lecture I: Tree Automata A. Maletti 46

117 Tree Automaton Theorem (THATCHER, WRIGHT, 968; DONER, 970) top-down det. RTL bottom-up det. RTL = RTL Proof. Let G = (N, Σ, S, P) be a tree automaton. We construct bottom-up det. TA G = (2 N, Σ, S, P ) with S = {N N N S } contains start nonterminal for each σ Σ, N,..., N k N, and k rk P (σ) {n n σ(n,..., n k ) P, n i N i } σ(n,..., N k ) P no other productions are in P Lecture I: Tree Automata A. Maletti 46

118 Tree Automaton Theorem (THATCHER, WRIGHT, 968; DONER, 970) top-down det. RTL bottom-up det. RTL = RTL Proof. Let G = (N, Σ, S, P) be a tree automaton. We construct bottom-up det. TA G = (2 N, Σ, S, P ) with S = {N N N S } contains start nonterminal for each σ Σ, N,..., N k N, and k rk P (σ) {n n σ(n,..., n k ) P, n i N i } σ(n,..., N k ) P no other productions are in P Strictness by the tree language L = {σ(α, β), σ(β, α)} Lecture I: Tree Automata A. Maletti 46

119 Tree Automaton RTL top-down det. RTL TSL bottom-up det. RTL LTL FIN Lecture I: Tree Automata A. Maletti 47

120 Tree Automaton RTL top-down det. RTL TSL bottom-up det. RTL LTL FIN Lecture I: Tree Automata A. Maletti 47

121 Tree Automaton RTL top-down det. RTL TSL bottom-up det. RTL LTL FIN Remark finite tree languages top-down deterministic RTL Lecture I: Tree Automata A. Maletti 47

122 Tree Automaton Theorem Regular tree languages are closed under all BOOLEAN operations substitution (quotients) and iteration (non-deterministic) relabelings linear homomorphisms inverse homomorphisms Lecture I: Tree Automata A. Maletti 48

123 Tree Automaton Theorem Regular tree languages are closed under substitution Definition L, L T Σ tree languages and α Σ L[α L ] contains all trees obtained from a tree of L by replacing each leaf labeled α by a tree of L different occurrences can be differently replaced Lecture I: Tree Automata A. Maletti 49

124 Tree Automaton Theorem Regular tree languages are closed under substitution L[α L ] t t t 2 t 3 t L t, t 2, t 3 L Lecture I: Tree Automata A. Maletti 49

125 Tree Automaton DTA = bottom-up deterministic tree automaton Definition A TA is minimal in C if all equivalent TAs of C have at least as many states Theorem Complexity of minimization problems: outp. C \ inp. model DTA TA DTA PTime ExpTime TA PSpace-hard ExpTime in ExpTime Lecture I: Tree Automata A. Maletti 50

126 Tree Automaton Definition Height of a tree t T Σ (Q): height(q) = 0 height(σ(t,..., t k )) = + max {height(t i ) i k} Lecture I: Tree Automata A. Maletti 5

127 Tree Automaton Definition Height of a tree t T Σ (Q): height(q) = 0 height(σ(t,..., t k )) = + max {height(t i ) i k} Intuition Height of t = number of edges in a maximal path from the root to a leaf Lecture I: Tree Automata A. Maletti 5

128 Tree Automaton Theorem (Pumping Lemma) For every regular tree language L T Σ there exists n N such that for every t L with height(t) n there exist contexts c, c C Σ and t T Σ such that t = c[c [t ]] c c[c [c c [t ] ]] L for any number of c c c c c. t c t Lecture I: Tree Automata A. Maletti 52

129 Tree Automaton Problem Complexity DTA TA Emptiness PTime PTime Finiteness PTime PTime Universality PTime ExpTime Inclusion PTime ExpTime Equivalence PTime ExpTime Membership in logdcfl logcfl Lecture I: Tree Automata A. Maletti 53

130 Literature Textbooks COMON, DAUCHET et al. Tree Automata Techniques and Applications GÉCSEG, STEINBY Tree Languages Handbook of Formal Languages, vol. 3, Springer, 997 GÉCSEG, STEINBY Tree Automata Akadémiai Kiadó, 984 Lecture I: Tree Automata A. Maletti 54