Automata theory and its applications

Transcription

1 Automata theory and its applications Lecture 12-14: Automata over ranked finite trees Zhilin Wu State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences January 16, 2013 Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

2 Ranked and unranked alphabets Ranked alphabet: A finite set pσ, arityq with arity : Σ Ñ N. Every node labeled by σ have aritypσq children Motivation: Terms, e.g. p5, p3, 4qq A tree domain D: A subset of N satisfying D is prefix-closed, P D, it holds x P P D, xj P D for every j : 0 j i. A tree t over a ranked alphabet pσ, arityq is a tupe pd, Lq where D is a tree domain, L : D Ñ Σ P D, the number of children of x is arityplpxqq. Let T Σ denote the set of ranked trees over the alphabet Σ. Unranked alphabet: A finite set Σ Every node labeled by σ can have an arbitrary number of children Motivation: XML documents Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

3 Outline 1 Bottom-up versus top-down 2 Determinization and closure properties 3 Pumping lemma 4 Regular tree grammars 5 Relationship with CFL 6 Regular tree expressions 7 Equivalence with MSO 8 Minimization of tree automata 9 Decision problems 10 Tree-walking automata Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

4 Bottom-up finite tree automata (BUTA) A BUTA A is a tuple pq, Σ, δ, F q, where δ includes the tuples of the following form pq 1,..., q k, a, qq s.t. k aritypaq. In particular, if aritypaq 0, then δ includes the tuples pa, qq. Runs of BUTA: Let t pd, Lq be a tree and A pq, Σ, δ, q 0, F q be a BUTA, a run of A over t is a tree r A,t pd, L 1 q, where L 1 : D Ñ Q P D with children x 1,..., x k, pqpx 1 q,..., qpx k q, a, qpxqq P δ. A run r A,t pd, L 1 q is accepting if L 1 pεq P F. Deterministic BUTA (DBUTA): for every pq 1,..., q k, a, qq, pq 1,..., q k, a, q 1 q P δ, it holds q q 1. Example: Boolean expressions that evaluate to true. A ptq 0, q 1 u, t^, _, 0, 1u, δ, tq 1 uq where δ is defined as follows, p0, q 0 q, p1, q 1 q P δ, pq 0, q 0, _, q 0 q, pq 1, q 0, _, q 1 q, pq 0, q 1, _, q 1 q, pq 1, q 1, _, q 1 q P δ, pq 0, q 0, ^, q 0 q, pq 1, q 0, ^, q 0 q, pq 0, q 1, ^, q 0 q, pq 1, q 1, ^, q 1 q P δ. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

5 Top-down finite tree automata (TDTA) A TDTA A is a tuple pq, Σ, δ, Iq, where δ includes the tuples of the following form pq, a, q 1,..., q k q s.t. k aritypaq. In particular, if aritypaq 0, then δ includes the tuples pq, aq. Runs of TDTA: Similar to those of BUTA. A run pd, L 1 q is accepting if L 1 pεq P I. Deterministic TDTA (DTDTA): I is a singleton, for every pq, a, q 1,..., q k q, pq, a, q1, 1..., qk 1 q P δ, it holds q 1 q1, 1..., q k qk 1. Proposition. BUTA TDTA. pq 1,..., q k, a, qq ðñ pq, a, q 1,..., q k q Regular languages over ranked trees: Tree languages defined by BUTAs. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

6 Outline 1 Bottom-up versus top-down 2 Determinization and closure properties 3 Pumping lemma 4 Regular tree grammars 5 Relationship with CFL 6 Regular tree expressions 7 Equivalence with MSO 8 Minimization of tree automata 9 Decision problems 10 Tree-walking automata Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

7 BUTA DBUTA Let A pq, Σ, δ, F q be BUTA. Construct an equivalent DBUTA A 1 pq 1, Σ, δ 1, F 1 q. Idea: Extension of subset construction. Q 1 2 Q, ps 1,..., S k, a, Sq P δ 1, where S tq Dq 1,..., q k.pq 1,..., q k, a, qq P i P S i u, F 1 ts S X F Hu. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

8 TDTA DTDTA Proposition. Let A pq, Σ, δ, q 0, F q be a DTDTA. Suppose t apt 1,..., t k q, t 1 apt 1 1,..., t 1 kq P LpAq, then t 2 apt 2 1,..., t 2 k q P LpAq, t2 i t i or t 2 i t1 i. Proof. Let x (resp. x 1 ) be the root of t (resp. t 1 ), x 1,..., x k (resp. x 1 1,..., x 1 k ) be the children of x (resp. x 1 ). A is deterministic and the run r A,t, r A,t 1 are accepting ñ r A,t pxq r A,t 1px 1 q q 0 P I, r A,t 1px i q r A,t 1px 1 i q. Thus, r A,t 2, a composition of r A,t Dptiq s or r A,t 1 Dpt 1 i q s, is also accepting. q 0 a q 1 t 1... t k q k q t... 1 t 1 q k k q 0 a... Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

9 TDTA DTDTA Proposition. Let A pq, Σ, δ, q 0, F q be a DTDTA. Suppose t apt 1,..., t k q, t 1 apt 1 1,..., t 1 kq P LpAq, then t 2 apt 2 1,..., t 2 k q P LpAq, t2 i t i or t 2 i t1 i. Corollary. TDTA DTDTA. Proof. Let L: The Boolean expressions that evaluate to true. Claim. L cannot be defined by DTDTA. To the contrary, suppose that L is recognized by a DTDTA. Then _p0, 1q, _p1, 0q P L implies that _p0, 0q P L, a contradiction. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

10 Closure properties Theorem. Regular tree languages are closed under union, intersection and complementation. Union: Suppose A 1 pq 1, Σ, δ 1, F 1 q and A 2 pq 2, Σ, δ 2, F 2 q. Then A pq 1 Y Q 2, Σ, δ 1 Y δ 2, F 1 Y F 2 q defines LpA 1 q Y LpA 2 q. Intersection: Suppose A 1 pq 1, Σ, δ 1, F 1 q and A 2 pq 2, Σ, δ 2, F 2 q. Then A pq 1 Q 2, Σ, δ, F 1 F 2 q defines LpA 1 q X LpA 2 q, where ppq 1, q 1 1q,..., pq k, q 1 k q, a, pq, q1 qq P δ iff pq 1,..., q k, a, qq P δ 1 and pq 1 1,..., q 1 k, a, q1 q P δ 2. Complementation: Let L T Σ be defined by a DBUTA A pq, Σ, δ, F q. Then pq, Σ, δ, QzF q defines T Σ zl. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

11 Closure properties Proposition. Regular tree languages are closed under letter projection (θ : Σ Ñ Σ 1 ). Proof. Let L T Σ be defined by a BUTA A pq, Σ, δ, F q. Then A 1 pq, Σ 1, δ 1, F q s.t. pq 1,..., q k, a 1, qq P δ 1 iff Da P Σ.pq 1,..., q k, a, qq P δ and θpaq a 1. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

12 Outline 1 Bottom-up versus top-down 2 Determinization and closure properties 3 Pumping lemma 4 Regular tree grammars 5 Relationship with CFL 6 Regular tree expressions 7 Equivalence with MSO 8 Minimization of tree automata 9 Decision problems 10 Tree-walking automata Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

13 Pumping lemma Contexts over Σ: A ranked tree over the alphabet Σ Y t u s.t. the arity of is zero. Let C Σ,n denote the set of all contexts over Σ with n occurrences of. By convention, C Σ,0 T Σ. We also use C Σ to denote C Σ,1. Let C P C Σ,n and C 1 P C k1,..., C n P C kn. Then CrC 1,..., C n s is the context obtained from C by replacing the i-th (from left to right) occurrence of with C i for every i : 1 i n. C... C 1... C k Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

14 Pumping lemma Pumping lemma. Let L T Σ be a regular tree language. Then there exists k P N s.t. if the depth of a tree t P L is k (the root is of depth 0), then DC, C 1 P C Σ and t 1 P T Σ s.t. t CrC 1 rt 1 ss and for all n P N, CrC 1n rt 1 ss P L. Proof. Let L defined by a BUTA A pq, Σ, δ, F q and k Q. q C C q t... Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

15 Pumping lemma Pumping lemma. Let L T Σ be a regular tree language. Then there exists k P N s.t. if the depth of a tree t P L is k (the root is of depth 0), then DC, C 1 P C Σ and t 1 P T Σ s.t. t CrC 1 rt 1 ss and for all n P N, CrC 1n rt 1 ss P L. Proof. Let L defined by a BUTA A pq, Σ, δ, F q and k Q. Application of pumping lemma: The tree language fpg i paq, g i paqq i P N ( is not regular. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

16 Outline 1 Bottom-up versus top-down 2 Determinization and closure properties 3 Pumping lemma 4 Regular tree grammars 5 Relationship with CFL 6 Regular tree expressions 7 Equivalence with MSO 8 Minimization of tree automata 9 Decision problems 10 Tree-walking automata Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

17 Regular tree grammar (RTG) A regular tree grammar: G pn, Σ, S, Rq, where N nonterminals, Σ: terminals, S P N : Start symbol, R contains rules of the form A Ñ α, where α P T ΣYN Symbols in N has arity zero. Derivation relation s Ñ G t: DC P C ΣYN, A P N, and A Ñ α P R s.t. s CrAs, t Crαs. The language generated by G (denoted LpGq): LpGq tt P T Σ S Ñ G tu. Example: S Ñ ^ps, Sq, S Ñ _ps, Sq, S Ñ 0, S Ñ 1. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

18 BUTA RTG RTG ñ Reduced RTG Reachable nonterminals (RpGq): N P N is reachable from S iff Dt P T ΣYN, S Ñ G t and N occurs in t. Productive nonterminals (P pgq): N P N is productive iff Dt P T Σ s.t. N Ñ G t. Reduced regular tree grammars: A RTG whose nonterminals are all reachable and productive. Computation of RpGq (resp. P pgq): Computation of a fixpoint. R 0 pgq tsu (resp. P 0 pgq tn P N Dα P T Σ. N Ñ αu), R i 1 pgq R i pgq Y tn P N DN 1 P R i pgq. N 1 Ñ α, N occurs in αu (resp. P i 1 pgq P i pgq Y N P N Dα P T ΣYPipGq. N Ñ α ( ). Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

19 BUTA RTG Reduced RTG ñ Normalized RTG Normalized RTG: Reduced RTG whose rules are of the form A Ñ apa 1,..., A k q or A Ñ a (where a P Σ, A 1,..., A k P N ). Repeat the following rule until a normalized RTG is obtained: Select a rule A Ñ α not of the desired form, let α apα 1,..., α k q, introduce new nonterminals A 1,..., A k, replace A Ñ α by A Ñ apa 1,..., A k q, A 1 Ñ α 1,..., A k Ñ α k. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

20 BUTA RTG Reduced RTG ñ Normalized RTG Normalized RTG: Reduced RTG whose rules are of the form A Ñ apa 1,..., A k q or A Ñ a (where a P Σ, A 1,..., A k P N ). Repeat the following rule until a normalized RTG is obtained: Select a rule A Ñ α not of the desired form, let α apα 1,..., α k q, introduce new nonterminals A 1,..., A k, replace A Ñ α by A Ñ apa 1,..., A k q, A 1 Ñ α 1,..., A k Ñ α k. Normalized RTG ñ TDTA TDTA ñ RTG Idea: Nonterminals as states, A Ñ apa 1,..., A k q ñ pa, a, A 1,..., A k q P δ, A Ñ a ñ pa, aq P δ. Idea: States as nonterminals, pq, a, q 1,..., q k q P δ ñ q Ñ apq 1,..., q k q. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

21 Outline 1 Bottom-up versus top-down 2 Determinization and closure properties 3 Pumping lemma 4 Regular tree grammars 5 Relationship with CFL 6 Regular tree expressions 7 Equivalence with MSO 8 Minimization of tree automata 9 Decision problems 10 Tree-walking automata Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

22 Yield operator Compute a word from a tree by concatenating the leaves of the tree from the left to the right. Yieldpaq a if a P Σ and aritypaq 0, Yieldpapt 1,..., t n qq Yieldpt 1 q... Yieldpt n q. E E id ( E ) id + id Yieldptq pid idq id. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

23 Ranked derivation trees of CFG Let G pn, Σ, S, Rq be a CFG. For each A P N s.t. there is a rule A Ñ α P R with α k, introduce a new nonterminal pa, kq. A ranked derivation tree is obtained from a derivation tree as follows: For every node x labeled by A with k-children, replace A by pa, kq. Let DT R pgq denote the set of ranked derivation trees of words in LpGq. E : id E E E (E, 3) E peq (E, 3) (E, 1) ( (E, 3) ) id (E, 1) + (E, 1) id id Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

24 CFL and Regular tree languages Theorem. The following fact hold. 1 For each CFG G, DT R pgq is a regular tree language. 2 For each regular tree language L T Σ, YieldpLq is a CFL. 3 There exists a regular tree language which is not the set of derivation trees of any CFG. Proof. 1. CFG G pn, Σ, P, Sq ñ RTG G 1 pn, Σ 1, S, Rq Σ 1 Σ Y tεu Y tpa, pq DA Ñ α s.t. α pu. R: A Ñ a 1... a p ñ A Ñ pa, pqpa 1,..., a p q, A Ñ ε ñ A Ñ pa, 0qpεq. 2. Let L generated by a normalized RTG G pn, Σ, S, Rq. Then the CFG pn, Σ, P, Sq generates YieldpLq, where P : A Ñ apa 1,..., A k q ñ A Ñ A 1... A k. 3. The regular tree language L tfpgpaq, gpbqqu is not the derivation tree of any CFG. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

25 Outline 1 Bottom-up versus top-down 2 Determinization and closure properties 3 Pumping lemma 4 Regular tree grammars 5 Relationship with CFL 6 Regular tree expressions 7 Equivalence with MSO 8 Minimization of tree automata 9 Decision problems 10 Tree-walking automata Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

26 Syntax of regular tree expressions Regular expressions: Regular tree expressions: r : a r 1 Y r 2 r 1 r 2 r 1 How to define concatenation and star operator over ranked trees? Main idea: Use a finite set of ports P t 1,..., k u to denote the positions of trees where concatenations happen Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

27 Syntax of regular tree expressions Regular expressions: Regular tree expressions: r : a r 1 Y r 2 r 1 r 2 r 1 How to define concatenation and star operator over ranked trees? Main idea: Use a finite set of ports P t 1,..., k u to denote the positions of trees where concatenations happen Regular tree expressions over Σ and P (denoted by RegpΣ, Pq): where r : a fpr 1,..., r n q r 1 Y r 2 r 1 i r 2 pr 1 q, i, a P Σ Y P such that aritypaq 0 (where ports in P have arity 0), f P Σ such that aritypfq n, i : 1 i k and i P P. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

28 Semantics of regular tree expressions r : a fpr 1,..., r n q r 1 Y r 2 r 1 i r 2 pr 1 q, i a tau, fpr 1,..., r n q tfps 1,..., s n q s 1 P r 1,..., s n P r n u, r 1 Y r 2 r 1 Y r 2, r 1 i r 2 r 1 i r 2, pr 1 q, i r 1, i. Suppose L, L 1 T ΣYP, then L i L 1 tr i Ð L 1 s, where tr i Ð L 1 s is defined as follows: Suppose i occurs n times in t, then tr i Ð L 1 s is the set of trees obtained from t by choosing n trees s 1,..., s n P L 1 and replacing the j-th (left-to-right) occurrence of i by s j. Let L T ΣYP, then tpl L 0, i t i u, and L n 1, i L i L n, i. In addition, let L, i L n, i. npn Example: pfp 1, 0qq, Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

29 BUTA Regular tree expressions From regular tree expressions to BUTA By induction on the structure of regular tree expressions, show that for every regular tree expression r, r is a regular tree language. Proposition. If L and L 1 are regular tree languages, then L i L 1 and L, i are also regular tree languages. L i L 1 : Suppose A pq, Σ Y P, δ, F q and A 1 pq 1, Σ Y P, δ 1, F 1 q are two BUTAs defining L and L 1 such that Q X Q 1 H. Then L i L 1 is defined by B pq 2, Σ Y P, δ 2, F 2 q, where Q 2 Q Y Q 1, F 2 F, δ 2 δ Y δ 1 Y tpq 1 1,..., q 1 n, σ, qq p i, qq P δ, Dq 1 P F 1.pq 1 1,..., q 1 n, σ, q 1 q P δ 1 u. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

30 BUTA Regular tree expressions From regular tree expressions to BUTA By induction on the structure of regular tree expressions, show that for every regular tree expression r, r is a regular tree language. Proposition. If L and L 1 are regular tree languages, then L i L 1 and L, i are also regular tree languages. L, i : Suppose A pq, Σ Y P, δ, F q is a BUTA defining L such that if p i, qq P δ, then there are no transitions pq 1,..., q n, σ, qq with σ P Σ. The BUTA A 1 pq, Σ Y P, δ 1, F 1 q defines L, i, where δ 1 δ Y tpq 1,..., q n, σ, qq p i, qq P δ, Dq 1 P F. pq 1,..., q n, σ, q 1 q P δu, F 1 tq p i, qq P δu. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

31 BUTA Regular tree expressions From BUTA to regular tree expressions Let A pq, Σ, δ, F q be a BUTA such that Q tq 1,..., q n u. The goal: Define a regular tree expression r A over Σ and Q for LpAq. The idea: Notation T pi, j, Kq s.t. 1 i n, 0 j n, and K Q. The set of trees t pd, Lq satisfying that D a run of A over t, say r A,t, s.t. r A,t pεq q i, for every leaf x in t, Lpxq P K or Lpxq a P Σ s.t. aritypaq 0, and for all the non-leaf nodes x ε in t, r A,t pxq P tq 1,..., q j u. LpAq T pi, n, Hq. q i PF Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

32 BUTA Regular tree expressions From BUTA to regular tree expressions Let A pq, Σ, δ, F q be a BUTA such that Q tq 1,..., q n u. The goal: Define a regular tree expression r A over Σ and Q for LpAq. The idea: Notation T pi, j, Kq s.t. 1 i n, 0 j n, and K Q. Induction on j. j 0: T pi, 0, Kq is the set of trees q i P K, or σ P Σ s.t. aritypσq 0 and pσ, q i q P δ, or trees σpθ 1,..., θ k q s.t. j 0: aritypσq : 1 j k, θ j P K or θ j P Σ s.t. aritypθ j q 0, there are q 1 1,..., q 1 k P Q : 1 j k, either q1 j θ j or pθ j, q 1 j q P δ, and pq 1 1,..., q 1 k, σ, q iq P δ. T pi, j, Kq T pi, j 1, Kq Y T pi, j 1, K Y tq juq qj T pj, j 1, K Y tq juq,q j qj T pj, j 1, Kq. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

33 From BUTA to regular tree expressions: An example Example: Boolean expressions that evaluate to true. BUTA A ptq 1, q 2 u, t^, _, 0, 1u, δ, tq 2 uq, where δ is defined as follows, p0, q 1 q, pq 1, q 1, _, q 1 q, pq 1, q 1, ^, q 1 q, pq 2, q 1, ^, q 1 q, pq 1, q 2, ^, q 1 q P δ, p1, q 2 q, pq 2, q 1, _, q 2 q, pq 1, q 2, _, q 2 q, pq 2, q 2, _, q 2 q, pq 2, q 2, ^, q 2 q P δ. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

34 From BUTA to regular tree expressions: An example Example: Boolean expressions that evaluate to true. BUTA A ptq 1, q 2 u, t^, _, 0, 1u, δ, tq 2 uq, where δ is defined as follows, p0, q 1 q, pq 1, q 1, _, q 1 q, pq 1, q 1, ^, q 1 q, pq 2, q 1, ^, q 1 q, pq 1, q 2, ^, q 1 q P δ, p1, q 2 q, pq 2, q 1, _, q 2 q, pq 1, q 2, _, q 2 q, pq 2, q 2, _, q 2 q, pq 2, q 2, ^, q 2 q P δ. Then T p2, 2, Hq T p2, 1, Hq Y T p2, 1, 2q q2 T p2, 1, 2q,q 2 q2 T p2, 1, Hq. T p2, 1, Hq T p2, 0, Hq Y T p2, 0, 1q q1 T p1, 0, 1q,q 1 q1 T p1, 0, Hq. T p2, 1, 2q T p2, 0, 2q Y T p2, 0, 1q q1 T p1, 0, t1, 2uq,q 1 q1 T p1, 0, 2q. T p2, 0, Hq 1 Y _p0, 1q Y _p1, 0q Y _p1, 1q Y ^p1, 1q, T p1, 0, Hq 0 Y _p0, 0q Y ^p0, 0q Y ^p1, 0q Y ^p0, 1q, T p2, 0, 1q T p2, 0, Hq Y _p1, q 1q Y _pq 1, 1q, T p1, 0, 1q T p2, 0, 2q T p1, 0, Hq Y _pq 1, 0q Y _p0, q 1q Y _pq 1, q 1q Y ^ p0, q 1q Y ^pq 1, 0q Y ^pq 1, 1q Y ^p1, q 1q Y ^pq 1, q 1q, T p2, 0, Hq Y _p0, q 2q Y _pq 2, 0q Y _ pq 2, q 2q Y ^pq 2, 1q Y ^p1, q 2q Y ^pq 2, q 2q, T p1, 0, 2q T p1, 0, Hq Y ^p0, q 2q Y ^pq 2, 0q, T p1, 0, t1, 2uq T p1, 0, 1q Y T p1, 0, 2q Y ^pq 1, q 2q Y ^pq 2, q 1q. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

35 Outline 1 Bottom-up versus top-down 2 Determinization and closure properties 3 Pumping lemma 4 Regular tree grammars 5 Relationship with CFL 6 Regular tree expressions 7 Equivalence with MSO 8 Minimization of tree automata 9 Decision problems 10 Tree-walking automata Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

36 MSO over ranked trees Let Σ be a ranked alphabet and k maxtaritypσq σ P Σu. Syntax of MSOrpP σ q σpσ, psuc i q 1 i k s ϕ : P σ pxq x y suc i px, yq x y Xpxq ϕ 1 _ϕ 2 ϕ 1 Dxϕ 1 DXϕ 1 Semantics Let t P T Σ, then t pd, Lq can be seen as a relational structure pd, pp σ q σpσ, psuc i q 1 i k, q s.t. P σ tx P D Lpxq σu, suc i tpx, xpi 1qq x, xpi 1q P Du and is the transitive closure of suc i. i Then MSO formulas are interpreted over t P T Σ in a natural way. Example: There is a path of even length in the tree DX 1 X 2 pdxpϕ ε pxq ^ X 1 pxqq ^ ϕ path px 1, X 2 q ^ ϕ even px 1, X 2 qq, where ϕ path 1pxq _ X 2pxqq ^ px 1pyq _ X 2pyqq Ñ px y _ x y _ y xqq, ϕ even : rpxq ^ ϕ leaf pxqq Ñ _ isuc ipx, yq ^ X 3 rpyqq ^ Dxpϕ leaf pxq ^ X 2pxqq ϕ εpxq y _ x yq, ϕ leaf pxq : Dypx yq. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40,

37 MSO BUTA From MSO to BUTA: Normal form for MSOrpP σ q σpσ, psuc i q 1 i k s formulas: ϕ : X P σ X Y SingpXq suc i px, Y q ϕ 1 _ ϕ 2 ϕ 1 DXϕ 1 Induction on the structure of MSO formulas, utilizing the closure of BUTAs under projection and Boolean operations. From BUTA to MSO: Let A pq, Σ, δ, F q be a BUTA such that Q tq 1,..., q n u. Then where ϕ init : ϕ trans : ϕ : Dq 1... q n pϕ init ^ ϕ trans ^ ϕ acc q, aritypσq0 pq 1,...,q r,σq ϕ acc Dxpϕ ε pxq ^ š leaf pxq ^ P σ pxqq Ñ š pσ,qqpδ qpxqq, ppσ pxq ^ ^i suc š i px, y i q ^ ^i q i py i 1... y r Ñ qpxq, qpf pq 1,...,q r,σ,qqpδ Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

38 Outline 1 Bottom-up versus top-down 2 Determinization and closure properties 3 Pumping lemma 4 Regular tree grammars 5 Relationship with CFL 6 Regular tree expressions 7 Equivalence with MSO 8 Minimization of tree automata 9 Decision problems 10 Tree-walking automata Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

39 Myhill-Nerode theorem for tree languages Let L T Σ. Define a congruence L on T Σ as follows: t L t 1 iff for all contexts C, Crts P L ô Crt 1 s P L. Proposition. L is a congruence, that is, if t L t 1, then for all contexts C, Crts L Crt 1 s. Proof. For all contexts C 1, if C 1 rcrtss P L, then rc 1 rcssrts P L. From the fact that t L t 1, we have rc 1 rcssrt 1 s P L, that is, C 1 rcrtss P L. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

40 Myhill-Nerode theorem for tree languages Let L T Σ. Define a congruence L on T Σ as follows: t L t 1 iff for all contexts C, Crts P L ô Crt 1 s P L. Theorem. Let L T Σ. Then L is regular iff L is of finite index. Proof. Only if direction: Suppose L is recognized by a deterministic BUTA A pq, Σ, δ, F q. Define A as follows: t A t 1 iff r A,t pεq r A,t 1pεq. Evidently, A is of finite index, since Q is finite. It remains to show A L. If t A t 1, then r A,t pεq r A,t 1pεq. context C, r A,Crts pεq r A,Crt 1 s pεq. Thus, for every context C, Crts P L iff Crt 1 s P L, we have t L t 1. If direction: Suppose L is of finite index. Define A L pq L, Σ, δ L, F L q as follows. Q L is the set of equivalence classes of L, F L trts t P Lu, for every σ P Σ s.t. aritypσq k and rt 1 s,..., rt k s P Q, prt 1 s,..., rt k s, σ, rσpt 1,..., t k qsq P δ L. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

41 Myhill-Nerode theorem for tree languages Let L T Σ. Define a congruence L on T Σ as follows: t L t 1 iff for all contexts C, Crts P L ô Crt 1 s P L. Theorem. Let L T Σ. Then L is regular iff L is of finite index. Corollary. For every regular tree language L T Σ, there is a unique deterministic BUTA of the minimum size defining L. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

42 Minimization of deterministic tree automata Let A pq, Σ, δ, F q be a deterministic BUTA defining L. Compute inductively an equivalence relation A over Q as follows, until i A i 1 A. q 0 A q1 iff q P F ô q 1 P F, q i 1 A q1 iff q i A q1 and for every σ P Σ s.t. aritypσq k, q 1,..., q j 1, q j 1,..., q k P Q, δpq 1,..., q j 1, q, q j 1,..., q k, σq i A δpq 1,..., q j 1, q 1, q j 1,..., q k, σq. t 1 P T Σ, t L t 1 iff r A,t pεq A r A,t 1pεq. Lemma. The following two facts hold. 1 For every t, t 1 P T Σ and context C, r A,t pεq A r A,t 1pεq implies r A,Crts pεq A r A,Crt 1 s pεq. 2 t L t 1 implies r A,t pεq A r A,t 1pεq. Proof. 1. Induction on the depth of in C. 2. By an induction on i, prove that t L t 1 implies r A,t pεq i A r A,t 1pεq. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

43 Outline 1 Bottom-up versus top-down 2 Determinization and closure properties 3 Pumping lemma 4 Regular tree grammars 5 Relationship with CFL 6 Regular tree expressions 7 Equivalence with MSO 8 Minimization of tree automata 9 Decision problems 10 Tree-walking automata Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

44 Membership and emptiness Membership problem: Given t P T Σ and a BUTA A pq, Σ, δ, F q, is t P LpAq? Nonemptiness problem: Given a BUTA A pq, Σ, δ, F q, is LpAq H? Theorem. The membership and nonemptiness problem can be solved in polynomial time. Proof. Membership: A bottom-up computation of the reachable states pr x q xpt. Let t pd, Lq P T Σ and A pq, Σ, δ, F q be a BUTA. for every leaf x of t, R x tq plpxq, qq P δu, for every non-leaf node x labeled by Lpxq σ of arity k, R x tq Dq 1 P R x0,..., q k P R xpk 1q. pq 1,..., q k, σ, qq P δu. t P LpAq iff R ε X F H Nonemptiness: Bottom-up computation of the set of reachable states R. R 0 tq pσ, qq P δu, R i 1 R i Y tq Dσ P Σ, Dq 1,..., q k P R i. pq 1,..., q k, σ, qq P δu. LpAq H iff R X F H Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

45 Language inclusion Language inclusion: Given two BUTAs A 1 pq 1, Σ, δ 1, F 1 q and A 2 pq 2, Σ, δ 2, F 2 q, is LpA 1 q LpA 2 q? Universality: Given a BUTA A pq, Σ, δ, F q, is LpAq T Σ? Theorem. The language inclusion and universality problem are EXPTIME-complete. Proof. Upper bound: Construct a BUTA A 1 2 defining the complement of LpA 2 q, decide whether LpA 1 q X LpA 1 2q H. Lower bound (Universality): Reduction from polynomial space alternating Turing machines. Use a BUTA to describe the unsuccessful computations. C ε C 0 C 1 i C 00 C01 C10 C i Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

46 Outline 1 Bottom-up versus top-down 2 Determinization and closure properties 3 Pumping lemma 4 Regular tree grammars 5 Relationship with CFL 6 Regular tree expressions 7 Equivalence with MSO 8 Minimization of tree automata 9 Decision problems 10 Tree-walking automata Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

47 Definition We restrict our attention to binary trees. Let T ypes tr, 0, 1u tl, iu. Let t pd, Lq be a binary tree and x P D, then the type of x (denoted by typepxq), is an element of T ypes s.t. the first component of typepxq: r: the root, 0: the left-child, 1: the right-child, the second component of typepxq: l: a leaf, i: an internal node. A tree walking automaton (TWA) A is a tuple pq, Σ, I, F, δq, where Q: the set of states, I : the set of initial states, F : the set of accepting states, δ Q T ypes Σ tò, 0, 1u Q s.t. for every pq, pb, lq, σ, act, q 1 q P δ, we have act Ò. A deterministic TWA is a TWA A pq, Σ, I, F, δq such that I is a τ, σq P Q T ypes Σ, δpq, τ, σq is a singleton. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

48 Definition: continued Let A be a TWA and t be a binary tree. A configuration of A over t is a pair pq, xq s.t. q P Q and x is a node in t. An initial configuration is a configuration pq, εq with q P I. A run of A over t is a sequence pq 0, x 0 qpq 1, x 1 q... pq n, x n q such that pq 0, x 0q is an initial : 0 j n, Dact s.t. pq j, typepx jq, Lpx jq, act, q j 1q P δ, and if act Ò and typepx j q P tp0, lq, p0, iqu, then x j x j 1 0, if act Ò and typepx j q P tp1, lq, p1, iqu, then x j x j 1 1, if act 0 and typepx j q P tr, 0, 1u tiu, then x j 1 x j 0, if act 1 and typepx j q P tr, 0, 1u tiu, then x j 1 x j 1. A run pq 0, x 0 qpq 1, x 1 q... pq n, x n q is accepting iff q n P F. A tree t is accepted by A if there is an accepting run of A over t. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

49 TWA: Example Depth-first search of trees Three states tq, q left, q right u pq, pb, iq, a, 0, qq P δ, where b r, 0, 1, pq, xq s.t. x is not a leaf ñ goes to the left child of x and the state is changed to q. pq, p0, lq, a, Ò, q left q P δ (resp. pq, p1, lq, a, Ò, q right q P δ), pq, xq s.t. x is a leaf and the left (resp. right) child of its parent ñ goes to the parent of x and the state is changed to q left (resp. q right ). Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

50 TWA: Example Depth-first search of trees Three states tq, q left, q right u pq, pb, iq, a, 0, qq P δ, where b r, 0, 1, pq, xq s.t. x is not a leaf ñ goes to the left child of x and the state is changed to q. pq, p0, lq, a, Ò, q left q P δ (resp. pq, p1, lq, a, Ò, q right q P δ), pq, xq s.t. x is a leaf and the left (resp. right) child of its parent ñ goes to the parent of x and the state is changed to q left (resp. q right ). pq left, pb, iq, a, 1, qq P δ, pq left, xq s.t. x is a not a leaf ñ goes to the right child of x and the state is changed to q. pq right, p0, lq, a, Ò, q left q, pq right, p0, iq, a, Ò, q left q P δ, pq right, xq s.t. x is the left child of its parent ñ goes to the parent of x and the state is changed to q left. pq right, p1, lq, a, Ò, q right q, pq right, p1, iq, a, Ò, q right q P δ, pq right, xq s.t. x is the right child of its parent ñ goes to the parent of x and the state is changed to q right. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

51 TWA BUTA Theorem. From every TWA, an equivalent BUTA can be constructed. Let A pq, Σ, I, F, δq be a TWA. W.l.o.g. assume that for every tree t, every accepting run of A over t stops at the root of t. The idea: For each non-leaf node x in t pd, Lq, define Rpxq Q Q as follows. pp, qq P Rpxq iff D a partial run of A over t, say pq 1, x 1 q... pq n, x n q, s.t. x 1 x, q 1 : 1 i n, x i P t x, x n is the parent of x, q n q, and pq n 1, typepxq, Lpxq, q n, Òq P δ. Observation. Rpxq only depends on t x. p R(x) x0 p q q x q p x1 Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

52 TWA BUTA Theorem. From every TWA, an equivalent BUTA can be constructed. Let A pq, Σ, I, F, δq be a TWA. W.l.o.g. assume that for every tree t, every accepting run of A over t stops at the root of t. For every leaf node x, Rpxq tpp, qq pp, typepxq, Lpxq, q, Òq P δu. For every non-leaf node x, Rpxq is computed from Rpx0q, Rpx1q as follows. If there exist p 1,..., p k, q 1,..., q k 1 P Q, b 1,..., b k P t0, 1u : 1 i k. pq i, typepxq, Lpxq, p i, b i q P δ, pp i, q i 1 q P Rpxb i q, p q 1, pq k 1, typepxq, Lpxq, q, Òq P δ, then pp, qq P Rpxq. There is an accepting run of A over t iff Rpεq X pi F q H. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

53 TWA BUTA Let Σ ta, b, cu such that aritypaq aritypcq 0 and aritypbq 1. Theorem. The language K T Σ defined in the following cannot be defined by TWAs. Let t P T Σ and x be a non-leaf node in t. Then x is balanced if both t x0 and t x1 contain a leaf labeled by a. Define K T Σ as follows: t P K iff for every leaf x labeled by a in t, there are an even number of balanced proper ancestors of x. Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40

54 Proof. Emptiness of TWA Theorem. The emptiness of TWAs is EXPTIME-complete. Upper bound: Translation to BUTAs. Lower bound: Reduction from Polynomial space alternating Turing machines. Encode the successful computations of PSPACE ATMs by binary trees and check the consistency of adjacent configurations using TWAs. C ε i C 0 C 1 i C 00 C01 C10 C Zhilin Wu (SKLCS) Automata over ranked finite trees January 16, / 40