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Tree Automata Foundations of XML Types: Tree Automata Pierre Genevès CNRS (slides mostly based on slides by W. Martens and T. Schwentick) University Grenoble Alpes, 2020–2021 1 / 43 Why Tree Automata? • Foundations of XML type languages (DTD, XML Schema, Relax NG...) • Provide a general framework for XML type languages • A tool to define regular tree languages with an operational semantics • Provide algorithms for efficient validation • Basic tool for static analysis (proofs, decision procedures in logic) 2 / 43 Prelude: Word Automata b b a start even odd a Transitions even !a odd odd !a even ... 3 / 43 From Words to Trees: Binary Trees Binary trees with an even number of a’s a a a b a b b How to write transitions? 4 / 43 From Words to Trees: Binary Trees Binary trees with an even number of a’s a even a even a odd b even a odd b even b even How to write transitions? 4 / 43 From Words to Trees: Binary Trees Binary trees with an even number of a’s a even a even a odd b even a odd b even b even How to write transitions? (even; odd) !a even (even; even) !a odd etc: 4 / 43 Ranked Trees? They come from parse trees of data (or programs)... A function call f (a; b) is a ranked tree f a b 5 / 43 Ranked Trees? They come from parse trees of data (or programs)... A function call f (g(a; b; c); h(i)) is a ranked tree f g h a b c i 5 / 43 Ranked Alphabet A ranked alphabet symbol is: • a formalisation of a function call • a symbol a with an integer arity(a) • arity(a) indicates the number of children of a Notation a(k): symbol a with arity(a)= k 6 / 43 Example Alphabet: fa(2); b(2); c(3); #(0)g Possible tree: a b a a c # # # # a # b # # a # # # 7 / 43 Ranked Tree Automata A ranked tree automaton A consists in: Alphabet(A): finite alphabet of symbols States(A): finite set of states Rules(A): finite set of transition rules Final(A): finite set of final states (⊆ States(A)) where : a(k) Rules(A) are of the form (q1; :::; qk ) ! q (0) if k = 0, we write a! q 8 / 43 Ranked Tree Automata A ranked tree automaton A consists in: Alphabet(A): finite alphabet of symbols States(A): finite set of states Rules(A): finite set of transition rules Final(A): finite set of final states (⊆ States(A)) where : a(k) Rules(A) are of the form (q1; :::; qk ) ! q (0) if k = 0, we write a! q 8 / 43 How do they work? Example: Boolean Expressions ^ _ ^ ^ _ _ ^ 0 1 1 0 0 1 1 1 Rules(A) Principle 0 1 ! q0 ! q1 ^ _ • Alphabet(A)= f^; _; 0; 1g (q1; q1) ! q1 (q0; q1) ! q1 ^ _ • States(A)= fq0; q1g (q0; q1) ! q0 (q1; q0) ! q1 ^ _ • 1 accepting state at the root: (q1; q0) ! q0 (q1; q1) ! q1 ^ _ Final(A)= fq1g (q0; q0) ! q0 (q0; q0) ! q0 9 / 43 How do they work? Example: Boolean Expressions ^ _ ^ ^ _ _ ^ 0 q0 1 q1 1 q1 0 q0 0 q0 1 q1 1 q1 1 q1 Rules(A) Principle 0 1 ! q0 ! q1 ^ _ • Alphabet(A)= f^; _; 0; 1g (q1; q1) ! q1 (q0; q1) ! q1 ^ _ • States(A)= fq0; q1g (q0; q1) ! q0 (q1; q0) ! q1 ^ _ • 1 accepting state at the root: (q1; q0) ! q0 (q1; q1) ! q1 ^ _ Final(A)= fq1g (q0; q0) ! q0 (q0; q0) ! q0 9 / 43 How do they work? Example: Boolean Expressions ^ q1 _ q1 ^ q1 ^q0 _q1 _q1 ^q1 0 q0 1 q1 1 q1 0 q0 0 q0 1 q1 1 q1 1 q1 Rules(A) Principle 0 1 ! q0 ! q1 ^ _ • Alphabet(A)= f^; _; 0; 1g (q1; q1) ! q1 (q0; q1) ! q1 ^ _ • States(A)= fq0; q1g (q0; q1) ! q0 (q1; q0) ! q1 ^ _ • 1 accepting state at the root: (q1; q0) ! q0 (q1; q1) ! q1 ^ _ Final(A)= fq1g (q0; q0) ! q0 (q0; q0) ! q0 9 / 43 Terminology • Language(A) : set of trees accepted by A • For a tree automaton A, Language(A) is a regular tree language [Thatcher and Wright, 1968, Doner, 1970] 10 / 43 Example Tree automaton A over fa(2); b(2); #(0)g which recognizes trees with an even number of a’s Alphabet(A): fa; b; #g States(A): Final(A): Rules(A): 11 / 43 Example Tree automaton A over fa(2); b(2); #(0)g which recognizes trees with an even number of a’s Alphabet(A): fa; b; #g States(A): feven; oddg Final(A): feveng Rules(A): (even; even) !a odd (even; even) !b even (even; odd) !a even (even; odd) !b odd (odd; even) !a even (odd; even) !b odd (odd; odd) !a odd (odd; odd) !b even # ! even 11 / 43 Outline 1. Can we implement a tree automaton efficiently? (notion of determinism) 2. Are tree automata closed under set-theoretic operations? 3. Can we check type inclusion? 4. Can we build equivalent top-down tree automata? 5. Nice theory. But... what should I do with my unranked XML trees? ! Can we apply this for XML type-checking? 12 / 43 Deterministic Tree Automata Deterministic does not have two rules of the form: a(k) (q1; :::; qk ) ! q (k) a 0 (q1; :::; qk ) ! q for two different states q and q0 Intuition At most one possible transition at a given node ! implementation... 13 / 43 Can we Make a Tree Automaton Deterministic? Theorem (determinization) From a given non-deterministic (bottom-up) tree automaton we can build a deterministic tree automaton Corollary Non-deterministic and deterministic (bottom-up) tree automata recognize the same languages. Complexity jStates(A)j EXPTIME (jStates(Adet)j = 2 ) 14 / 43 Implementing Validation Membership Checking Given a tree automaton A and a tree t, does t 2 Language(A)? Remark b We can implement even if A is b non-deterministic... b Example a Automaton with Final(A)= fqf g and : c b b ! q q ! qb q ! q b b b a qb ! qf (q; q) ! q c c 15 / 43 Implementing Validation Membership Checking Given a tree automaton A and a tree t, does t 2 Language(A)? Remark b We can implement even if A is b non-deterministic... b Example a Automaton with Final(A)= fqf g and : c b b ! q q ! qb q ! q b b b a qb ! qf (q; q) ! q fqg c c fqg 15 / 43 Implementing Validation Membership Checking Given a tree automaton A and a tree t, does t 2 Language(A)? Remark b We can implement even if A is b non-deterministic... b Example a Automaton with Final(A)= fqf g and : c b b ! q q ! qb q ! q fq; qbg b b fq; qbg b a qb ! qf (q; q) ! q fqg c c fqg 15 / 43 Implementing Validation Membership Checking Given a tree automaton A and a tree t, does t 2 Language(A)? Remark b We can implement even if A is b non-deterministic... b Example a fqg Automaton with Final(A)= fqf g and : c b b ! q q ! qb q ! q fq; qbg b b fq; qbg b a qb ! qf (q; q) ! q fqg c c fqg 15 / 43 Implementing Validation Membership Checking Given a tree automaton A and a tree t, does t 2 Language(A)? Remark b We can implement even if A is b non-deterministic... b fq; qbg Example a fqg Automaton with Final(A)= fqf g and : c b b ! q q ! qb q ! q fq; qbg b b fq; qbg b a qb ! qf (q; q) ! q fqg c c fqg 15 / 43 Implementing Validation Membership Checking Given a tree automaton A and a tree t, does t 2 Language(A)? Remark b We can implement even if A is b fq; qb; qf g non-deterministic... b fq; qbg Example a fqg Automaton with Final(A)= fqf g and : c b b ! q q ! qb q ! q fq; qbg b b fq; qbg b a qb ! qf (q; q) ! q fqg c c fqg 15 / 43 Implementing Validation Membership Checking Given a tree automaton A and a tree t, does t 2 Language(A)? Remark b fq; qb; qf g We can implement even if A is b fq; qb; qf g non-deterministic... b fq; qbg Example a fqg Automaton with Final(A)= fqf g and : c b b ! q q ! qb q ! q fq; qbg b b fq; qbg b a qb ! qf (q; q) ! q fqg c c fqg 15 / 43 Implementing Validation Membership Checking Given a tree automaton A and a tree t, does t 2 Language(A)? Remark b fq; qb; qf g We can implement even if A is b fq; qb; qf g non-deterministic... b fq; qbg Example a fqg Automaton with Final(A)= fqf g and : c b b ! q q ! qb q ! q fq; qbg b b fq; qbg b a qb ! qf (q; q) ! q fqg c c fqg Complexity Membership-Checking is in PTIME (time linear in the size of the tree) 15 / 43 Set-Theoretic Operations Recall • We have seen that neither local tree grammars nor single-type tree grammars are closed under boolean operations (e.g. union) • What about tree automata? 16 / 43 Closure under Union and Intersection... Example • Automaton A: even number of a’s a • (even; even) ! odd • Automaton B: even number of b’s a • (even; even) ! even 17 / 43 Closure under Union and Intersection..
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