Foundations of XML Types: Tree Automata

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1 1 / 43 Foundtions of XML Types: Tree Automt Pierre Genevès CNRS (slides mostly sed on slides y W. Mrtens nd T. Schwentick) University of Grenole Alpes,

2 2 / 43 Why Tree Automt? Foundtions of XML type lnguges (DTD, XML Schem, Relx NG...) Provide generl frmework for XML type lnguges A tool to define regulr tree lnguges with n opertionl semntics Provide lgorithms for efficient vlidtion Bsic tool for sttic nlysis (proofs, decision procedures in logic)

3 3 / 43 Prelude: Word Automt strt even odd Trnsitions even odd odd even...

4 4 / 43 From Words to Trees: Binry Trees Binry trees with n even numer of s How to write trnsitions?

5 4 / 43 From Words to Trees: Binry Trees Binry trees with n even numer of s even even odd even odd even even How to write trnsitions?

6 4 / 43 From Words to Trees: Binry Trees Binry trees with n even numer of s even even odd even odd even even How to write trnsitions? (even, odd) even (even, even) odd etc.

7 5 / 43 Rnked Trees? They come from prse trees of dt (or progrms)... A function cll f (, ) is rnked tree f

8 5 / 43 Rnked Trees? They come from prse trees of dt (or progrms)... A function cll f (g(,, c), h(i)) is rnked tree f g h c i

9 6 / 43 Rnked Alphet A rnked lphet symol is: formlistion of function cll symol with n integer rity() rity() indictes the numer of children of Nottion (k) : symol with rity() = k

10 7 / 43 Exmple Alphet: { (2), (2), c (3), # (0) } Possile tree: c # # # # # # # # # #

11 8 / 43 Rnked Tree Automt A rnked tree utomton A consists in: Alphet(A): Sttes(A): Rules(A): Finl(A): where : finite lphet of symols finite set of sttes finite set of trnsition rules finite set of finl sttes ( Sttes(A)) Rules(A) re of the form (q 1,..., q k ) (k) q if k = 0, we write ɛ (0) q

12 8 / 43 Rnked Tree Automt A rnked tree utomton A consists in: Alphet(A): Sttes(A): Rules(A): Finl(A): where : finite lphet of symols finite set of sttes finite set of trnsition rules finite set of finl sttes ( Sttes(A)) Rules(A) re of the form (q 1,..., q k ) (k) q if k = 0, we write ɛ (0) q

13 9 / 43 Exmple: Boolen Expressions How do they work? Principle Alphet(A) = {,, 0, 1} Sttes(A) = {q 0, q 1} 1 ccepting stte t the root: Finl(A) = {q 1} Rules(A) ɛ 0 q 0 ɛ 1 q 1 (q 1, q 1) q 1 (q 0, q 1) q 1 (q 0, q 1) q 0 (q 1, q 0) q 1 (q 1, q 0) q 0 (q 1, q 1) q 1 (q 0, q 0) q 0 (q 0, q 0) q 0

14 9 / 43 Exmple: Boolen Expressions How do they work? 0 q 0 1 q 1 1 q 1 0 q 0 0 q 0 1 q 1 1 q 1 1 q 1 Principle Alphet(A) = {,, 0, 1} Sttes(A) = {q 0, q 1} 1 ccepting stte t the root: Finl(A) = {q 1} Rules(A) ɛ 0 q 0 ɛ 1 q 1 (q 1, q 1) q 1 (q 0, q 1) q 1 (q 0, q 1) q 0 (q 1, q 0) q 1 (q 1, q 0) q 0 (q 1, q 1) q 1 (q 0, q 0) q 0 (q 0, q 0) q 0

15 9 / 43 Exmple: Boolen Expressions How do they work? q 1 q 1 q 1 q 0 q 1 q 1 q 1 0 q 0 1 q 1 1 q 1 0 q 0 0 q 0 1 q 1 1 q 1 1 q 1 Principle Alphet(A) = {,, 0, 1} Sttes(A) = {q 0, q 1} 1 ccepting stte t the root: Finl(A) = {q 1} Rules(A) ɛ 0 q 0 ɛ 1 q 1 (q 1, q 1) q 1 (q 0, q 1) q 1 (q 0, q 1) q 0 (q 1, q 0) q 1 (q 1, q 0) q 0 (q 1, q 1) q 1 (q 0, q 0) q 0 (q 0, q 0) q 0

16 10 / 43 Terminology Lnguge(A) : set of trees ccepted y A For tree utomton A, Lnguge(A) is regulr tree lnguge [Thtcher nd Wright, 1968, Doner, 1970]

17 11 / 43 Exmple Tree utomton A over { (2), (2), # (0) } which recognizes trees with n even numer of s Alphet(A) : {,, #} Sttes(A) : Finl(A) : Rules(A) :

18 11 / 43 Exmple Tree utomton A over { (2), (2), # (0) } which recognizes trees with n even numer of s Alphet(A) : {,, #} Sttes(A) : {even, odd} Finl(A) : {even} Rules(A) : (even, even) odd (even, odd) even (odd, even) even (odd, odd) odd ɛ # even (even, even) even (even, odd) odd (odd, even) odd (odd, odd) even

19 12 / 43 Outline 1. Cn we implement tree utomton efficiently? (notion of determinism) 2. Are tree utomt closed under set-theoretic opertions? 3. Cn we check type inclusion? 4. Cn we uild equivlent top-down tree utomt? 5. Nice theory. But... wht should I do with my unrnked XML trees? Cn we pply this for XML type-checking?

20 13 / 43 Deterministic Tree Automt Deterministic does not hve two rules of the form: for two different sttes q nd q (q 1,..., q k ) (k) q (q 1,..., q k ) (k) q Intuition At most one possile trnsition t given node implementtion...

21 14 / 43 Cn we Mke Tree Automton Deterministic? Theorem (determiniztion) From given non-deterministic (ottom-up) tree utomton we cn uild deterministic tree utomton Corollry Non-deterministic nd deterministic (ottom-up) tree utomt recognize the sme lnguges. Complexity EXPTIME ( Sttes(A det ) = 2 Sttes(A) )

22 15 / 43 Implementing Vlidtion Memership Checking Given tree utomton A nd tree t, does t Lnguge(A)? Remrk We cn implement even if A is non-deterministic... Exmple Automton with Finl(A) = {q f } nd : ɛ c q q q q q q q f (q, q) q c c

23 15 / 43 Implementing Vlidtion Memership Checking Given tree utomton A nd tree t, does t Lnguge(A)? Remrk We cn implement even if A is non-deterministic... Exmple Automton with Finl(A) = {q f } nd : ɛ c q q q q q q q f (q, q) q {q} c c {q}

24 15 / 43 Implementing Vlidtion Memership Checking Given tree utomton A nd tree t, does t Lnguge(A)? Remrk We cn implement even if A is non-deterministic... Exmple Automton with Finl(A) = {q f } nd : ɛ c q q q q q q q f (q, q) q {q, q } {q, q } {q} c c {q}

25 15 / 43 Implementing Vlidtion Memership Checking Given tree utomton A nd tree t, does t Lnguge(A)? Remrk We cn implement even if A is non-deterministic... Exmple Automton with Finl(A) = {q f } nd : ɛ c q q q q q q q f (q, q) q {q} {q, q } {q, q } {q} c c {q}

26 15 / 43 Implementing Vlidtion Memership Checking Given tree utomton A nd tree t, does t Lnguge(A)? Remrk We cn implement even if A is non-deterministic... Exmple Automton with Finl(A) = {q f } nd : ɛ c q q q q q q q f (q, q) q {q, q } {q} {q, q } {q, q } {q} c c {q}

27 15 / 43 Implementing Vlidtion Memership Checking Given tree utomton A nd tree t, does t Lnguge(A)? Remrk We cn implement even if A is non-deterministic... Exmple Automton with Finl(A) = {q f } nd : ɛ c q q q q q q q f (q, q) q {q, q, q f } {q, q } {q} {q, q } {q, q } {q} c c {q}

28 15 / 43 Implementing Vlidtion Memership Checking Given tree utomton A nd tree t, does t Lnguge(A)? Remrk We cn implement even if A is non-deterministic... Exmple Automton with Finl(A) = {q f } nd : ɛ c q q q q q q q f (q, q) q {q, q, q f } {q, q, q f } {q, q } {q} {q, q } {q, q } {q} c c {q}

29 15 / 43 Implementing Vlidtion Memership Checking Given tree utomton A nd tree t, does t Lnguge(A)? Remrk We cn implement even if A is non-deterministic... Exmple Automton with Finl(A) = {q f } nd : ɛ c q q q q q q q f (q, q) q {q, q, q f } {q, q, q f } {q, q } {q} {q, q } {q, q } {q} c c {q} Complexity Memership-Checking is in PTIME (time liner in the size of the tree)

30 16 / 43 Set-Theoretic Opertions Recll We hve seen tht neither locl tree grmmrs nor single-type tree grmmrs re closed under oolen opertions (e.g. union) Wht out tree utomt?

31 17 / 43 Exmple Closure under Union nd Intersection... Automton A: even numer of s (even, even) odd Automton B: even numer of s (even, even) even

32 17 / 43 Exmple Closure under Union nd Intersection... Automton A: even numer of s (even, even) odd Automton B: even numer of s (even, even) even odd even even even even even

33 17 / 43 Exmple Closure under Union nd Intersection... Automton A: even numer of s (even, even) odd Automton B: even numer of s (even, even) even odd even even even even even ((even, even), (even, even)) (odd, even)

34 18 / 43 Product Construction Given A nd B, uild A B Alphet(A B) = Alphet(A) Alphet(B) Sttes(A B) = Sttes(A) Sttes(B) Finl(A B) = {(q, q ) q Finl(A) q Finl(B)} Rules(A B) = ( (q 1, q), 1..., (q k, q k ) ) (k) (q, q ) q 1,..., q k (k) q Rules(A) q, 1..., q k (k) q Rules(B)

35 19 / 43 Closure under Union Given A nd B, uild A B Alphet(A B) = Alphet(A) Alphet(B) Sttes(A B) = Sttes(A) Sttes(B) Rules(A B) = ( (q 1, q), 1..., (q k, q k ) ) (k) (q, q ) q 1,..., q k (k) q Rules(A) q, 1..., q k (k) q Rules(B) Finl(A B) = {(q, q ) q Finl(A) q Finl(B)}

36 20 / 43 Closure under intersection Given A nd B, uild A B Alphet(A B) = Alphet(A) Alphet(B) Sttes(A B) = Sttes(A) Sttes(B) Rules(A B) = ( (q 1, q), 1..., (q k, q k ) ) (k) (q, q ) q 1,..., q k (k) q Rules(A) q, 1..., q k (k) q Rules(B) Finl(A B) = {(q, q ) q Finl(A) q Finl(B)}

37 21 / 43 Complexity of the Product Size of the Result Automton Sttes(A B) = Sttes(A) Sttes(B) Rules(A B) Rules(A) Rules(B) Qudrtic increse in size

38 22 / 43 Closure under Negtion: Completion Definition : Complete Tree Automton For ech (k) Alphet(A) nd q 1,..., q k Sttes(A), there exists rule with some q (q 1,..., q k ) q Intuition At lest one trnsition t given node...

39 23 / 43 Exmple Incomplete (deterministic) tree utomton Tree utomton A for {(, )} : ɛ q (q, q ) q with Finl(A) = {q } Completion of A Add sink stte q p ɛ q ɛ q p (q, q ) q (q, q ) q p (q, q ) q p (q, q ) q p (q, q ) q p (q, q ) q p (q, q ) q p (q, q ) q p (q p, q) σ q p for ll q {q, q, q p} nd σ {, } (q, q p) σ q p for ll q {q, q, q p} nd σ {, } with Finl(A) = {q }

40 23 / 43 Exmple Incomplete (deterministic) tree utomton Tree utomton A for {(, )} : ɛ q (q, q ) q with Finl(A) = {q } Completion of A, Complementtion of A Add sink stte q p ɛ q ɛ q p (q, q ) q (q, q ) q p (q, q ) q p (q, q ) q p (q, q ) q p (q, q ) q p (q, q ) q p (q, q ) q p (q p, q) σ q p for ll q {q, q, q p} nd σ {, } (q, q p) σ q p for ll q {q, q, q p} nd σ {, } with Finl(A) = {q, q p}

41 24 / 43 Closure under Negtion: Summry Building the Complement of A Mke A deterministic Complete the result Switch finl non-finl sttes Complexity Determiniztion of A : exponentil explosion (sttes: 2 Sttes(A) ) Completion of the result: exponentil explosion of the numer of rules: ( ) k Alphet(A) 2 Sttes(A) where k is the mximl rnk Switching finl non-finl sttes: liner Totl: exponentil explosion

42 25 / 43 Emptiness Test Given tree utomton A, is Lnguge(A)? Principle Compute the set of rechle sttes nd then see if ny of them re in the finl set Complexity PTIME (time proportionl to A )

43 26 / 43 Appliction for Checking Type Inclusion Type Inclusion Given two tree utomt A 1 nd A 2, is Lnguge(A 1) Lnguge(A 2)? Theorem Continment for non-deterministic tree utomt cn e decided in exponentil time Principle Lnguge(A 1 A 2)? = For this purpose, we must mke A 2 deterministic (size: O(2 A 2 )) EXPTIME Essentilly no etter solution [Seidl, 1990]

44 26 / 43 Appliction for Checking Type Inclusion Type Inclusion Given two tree utomt A 1 nd A 2, is Lnguge(A 1) Lnguge(A 2)? Theorem Continment for non-deterministic tree utomt cn e decided in exponentil time Principle Lnguge(A 1 A 2)? = For this purpose, we must mke A 2 deterministic (size: O(2 A 2 )) EXPTIME Essentilly no etter solution [Seidl, 1990]

45 27 / 43 Top-Down Tree Automt Is tht useful?... Exmple: Connection with Strings c cd = ((c(d))) = d Reding strings from left to right = reding trees top-down ( e.g. streming vlidtion...)

46 Top-Down Tree Automt: Exmple Principle strting from the root, guess correct vlues check t leves 3 sttes: q 0, q 1, cc initil stte t the root: q 1 ccepting if ll leves leled cc Trnsitions q 1 (q 1, q 1) q 1 (q 0, q 1) q 0 (q 0, q 1) q 1 (q 1, q 0) q 0 (q 1, q 0) q 1 (q 1, q 1) q 0 (q 0, q 0) q 0 (q 0, q 0) q 1 1 cc q 0 0 cc 28 / 43

47 Top-Down Tree Automt: Exmple q Principle strting from the root, guess correct vlues check t leves 3 sttes: q 0, q 1, cc initil stte t the root: q 1 ccepting if ll leves leled cc Trnsitions q 1 (q 1, q 1) q 1 (q 0, q 1) q 0 (q 0, q 1) q 1 (q 1, q 0) q 0 (q 1, q 0) q 1 (q 1, q 1) q 0 (q 0, q 0) q 0 (q 0, q 0) q 1 1 cc q 0 0 cc 28 / 43

48 Top-Down Tree Automt: Exmple q 1 q 1 q 1 q 0 q 1 q 1 q 1 0 q 0 1 q 1 1 q 1 0 q 0 0 q 0 1 q 1 1 q 1 1 q 1 cc cc cc cc cc cc cc cc Principle strting from the root, guess correct vlues check t leves 3 sttes: q 0, q 1, cc initil stte t the root: q 1 ccepting if ll leves leled cc Trnsitions q 1 (q 1, q 1) q 1 (q 0, q 1) q 0 (q 0, q 1) q 1 (q 1, q 0) q 0 (q 1, q 0) q 1 (q 1, q 1) q 0 (q 0, q 0) q 0 (q 0, q 0) q 1 1 cc q 0 0 cc 28 / 43

49 29 / 43 Top-Down Tree Automt A top-down tree utomton A consists in: Alphet(A): Sttes(A): Rules(A): Initil(A): where: finite lphet of symols finite set of sttes finite set of trnsition rules finite set of initil sttes ( Sttes(A)) Rules(A) re of the form q (k) (q 1,..., q k ) Top-down tree utomt lso recognize ll regulr tree lnguges

50 29 / 43 Top-Down Tree Automt A top-down tree utomton A consists in: Alphet(A): Sttes(A): Rules(A): Initil(A): where: finite lphet of symols finite set of sttes finite set of trnsition rules finite set of initil sttes ( Sttes(A)) Rules(A) re of the form q (k) (q 1,..., q k ) Top-down tree utomt lso recognize ll regulr tree lnguges

51 30 / 43 Top-down Determinism Deterministic Top-Down Tree Automton for ech q Sttes(A) nd Alphet(A) there is t most one rule there is t most one initil stte q (k) (q 1,..., q k )

52 30 / 43 Top-down Determinism Deterministic Top-Down Tree Automton for ech q Sttes(A) nd Alphet(A) there is t most one rule there is t most one initil stte q (k) (q 1,..., q k ) Cn We Mke Top-Down Automt Deterministic? () Yes () No (c) Mye...

53 31 / 43 Cn We Mke Top-Down Automt Deterministic? Mye!

54 31 / 43 Cn We Mke Top-Down Automt Deterministic? Mye! Deterministic top-down tree utomt do not recognize ll regulr tree lnguges Exmple c c

55 31 / 43 Cn We Mke Top-Down Automt Deterministic? Mye! Deterministic top-down tree utomt do not recognize ll regulr tree lnguges Exmple c c Initil(A) = q 0

56 31 / 43 Cn We Mke Top-Down Automt Deterministic? Mye! Deterministic top-down tree utomt do not recognize ll regulr tree lnguges Exmple c c Initil(A) = q 0 (q, q) q 0

57 31 / 43 Cn We Mke Top-Down Automt Deterministic? Mye! Deterministic top-down tree utomt do not recognize ll regulr tree lnguges Exmple c c Initil(A) = q 0 (q, q) q 0 q ɛ

58 31 / 43 Cn We Mke Top-Down Automt Deterministic? Mye! Deterministic top-down tree utomt do not recognize ll regulr tree lnguges Exmple c c Initil(A) = q 0 (q, q) q 0 q ɛ q c ɛ

59 31 / 43 Cn We Mke Top-Down Automt Deterministic? Mye! Deterministic top-down tree utomt do not recognize ll regulr tree lnguges Exmple c c Initil(A) = q 0 (q, q) q 0 q ɛ q c ɛ lso ccepts... c c

60 32 / 43 Expressive Power of Tree Automt: Summry Theorem The following properties re equivlent for tree lnguge L: () L is recognized y ottom-up non-deterministic tree utomton () Lis recognized y ottom-up deterministic tree utomton (c) L is recognized y top-down non-deterministic tree utomton (d) L is generted y regulr tree grmmr Proof Ide () (): determiniztion (see [Comon et l., 1997]) () (c): sme thing seen from 2 different wys (d) () :? (horizontl recursion?)

61 33 / 43 Unrnked Trees String s Tree c Rnked Tree c c c Unrnked Tree e c d... c d... c c e e c e Unrnked Tree Automt? 1. either we dpt rnked tree utomt 2. or we encode unrnked trees re rnked trees...

62 34 / 43 Unrnked Tree Automt Rnked Trees Trnsitions cn e descried y finite sets: δ(σ, q) = {(q 1, q 2), (q 3, q 4),...} q σ σ 1 σ 2 q 1 q 2 Unrnked Trees q σ σ 1 σ 2... σ n q 1 q 2... q n δ(σ, q)? δ(σ, q) For unrnked trees, δ(σ, q) is regulr tree lnguge δ(σ, q) my e specified y regulr expression or y finite word utomton [Murt, 1999]

63 35 / 43 Second Option Cn we encode unrnked trees s rnked trees? e c d... c d... c c e? e c e

64 36 / 43 Encoding Unrnked Trees As Binry Trees Bijective Encoding first child; next siling encoding Allows to focus on inry trees without loss of generlity Results for rnked trees hold for unrnked trees s well

65 37 / 43 Tree Automt: Summry Definition A tree lnguge is regulr iff it is recognized y non-deterministic tree utomton Advntges Closure, decidle opertions Generl tool (theoreticl nd lgorithmic) Limittions n n

66 38 / 43 Appliction for Sttic Type-Checking Recll: The XML Sttic Type-Checking Prolem Given the following: Source code of some progrm f (in e.g. XQuery) A type T in for input documents A type T out for expected output documents Prolem: Does f (t) T out for ll t T in?

67 39 / 43 Sttic Type-Checking for XQuery: Principles <doc> { FOR $uthor in /ooks/ook/uthor[...] RETURN <uthor>... { /ooks/ook/title } </uthor> } </doc> f () t T in f (t) T inf? T out Approch in two steps: 1. Compute T inf = {f (t) t T in } 2. Check whether T inf T out holds In cse T inf T out holds, then we know tht for ny t T in, f (t) T out

68 40 / 43 Sttic Type-Checking Using Tree Automt Mny reserch ppers XQuery s stndrd type system inspired from dvnces in PLs, e.g.: XDuce [Hosoy nd Pierce, 2003]: domin specific lnguge Sound sttic type system sed on n implementtion of tree utomt continment optimized for prcticl cses Use pttern-mtching for selection (no support for XPth) XQuery s stndrd type system As defined in the spec: Sound PTIME Sound nd more precise type systems in EXPTIME

69 41 / 43 XQuery nd Sttic Type-Checking Open Issues Type inference requires lnce etween precision nd complexity Sound pproximtions cn e more or less imprecise Poor precision (e.g. type pproximtion) in type inference yields more flse lrms XQuery s stndrd type system is limited: Currently: no or poor support of XPth (e.g. only child nd descendnt xes re poorly supported) Support for typing XPth hs een open for yers (in prticulr for ckwrd xes) Cn you guess why is tht difficult? 2015 ICFP pper with solution for this issue (sed on dvnces in tree logics)

70 42 / 43 Concluding Remrks on Tree Automt Tree utomt re prt of the theoreticl tools tht provide the underlying guiding principles for XML (like the reltionl lger provides the underlying principles for reltionl dtses) Importnt reserch chllenges still remin

71 43 / 43 </session> For those who wnt to lern more Pointers to relted reserch results: [1] XQuery nd Sttic Typing: Tckling the Prolem of Bckwrd Axes. Pierre Genevès nd Nils Gesert. In ICFP 15: Proceedings of the ACM SIGPLAN Interntionl Conference on Functionl Progrmming (ICFP), Sept

72 Comon, H., Duchet, M., Gilleron, R., Jcquemrd, F., Lugiez, D., Tison, S., nd Tommsi, M. (1997). Tree utomt techniques nd pplictions. Aville on: relese Octoer, 1st Doner, J. (1970). Tree cceptors nd some of their pplictions. Journl of Computer nd System Sciences, 4: Hosoy, H. nd Pierce, B. C. (2003). XDuce: A stticlly typed XML processing lnguge. ACM Trns. Inter. Tech., 3(2): Murt, M. (1999). Hedge utomt: forml model for XML schemt. Seidl, H. (1990). Deciding equivlence of finite tree utomt. SIAM J. Comput., 19(3): Thtcher, J. W. nd Wright, J. B. (1968). 43 / 43

73 Generlized finite utomt theory with n ppliction to decision prolem of second-order logic. Mthemticl Systems Theory, 2(1): / 43

Chapter Five: Nondeterministic Finite Automata. Formal Language, chapter 5, slide 1

Chapter Five: Nondeterministic Finite Automata. Formal Language, chapter 5, slide 1 Chpter Five: Nondeterministic Finite Automt Forml Lnguge, chpter 5, slide 1 1 A DFA hs exctly one trnsition from every stte on every symol in the lphet. By relxing this requirement we get relted ut more