Streamed Validation of XML Documents

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1 Preliminries DTD Document Type Definition References Jnury 29, 2009

2 Preliminries DTD Document Type Definition References Structure Preliminries Unrnked Trees Recognizble Lnguges DTD Document Type Definition simple DTDs Specilized DTDs Strong Vlidtion Vlidting well-formed XML Documents References

3 Preliminries DTD Document Type Definition References Unrnked Trees From XML to unrnked Trees <b o o k C o l l e c t i o n> <book> < t i t l e>the Lord o f the Rings</ t i t l e> </ book> <book> <r e l t e d> < t i t l e>the Lord o f the Rings</ t i t l e> </ r e l t e d> < t i t l e>the H i s t o r y o f Middle e r t h</ t i t l e> </ book> </ b o o k C o l l e c t i o n>

4 Preliminries DTD Document Type Definition References Unrnked Trees From XML to unrnked Trees bookcollection book book title relted title title

5 Preliminries DTD Document Type Definition References Unrnked Trees From XML to unrnked Trees r c b c c

6 Preliminries DTD Document Type Definition References Unrnked Trees From XML to unrnked Trees r c b c c Forml representtion: Σ = {r,, b, c} r((c()), (b(c()), c())) = t T Σ

7 Preliminries DTD Document Type Definition References Unrnked Trees From XML to unrnked Trees r c b c c Forml representtion: Σ = {r,, b, c} r((c()), (b(c()), c())) = t T Σ String representtion: rccbccbccr = [t] [T Σ ]

8 Preliminries DTD Document Type Definition References Recognizble Lnguges Recognizble Lnguges Myhill-Nerode Theorem Let L be lnguge over n lphbet Σ. We define the Nerode reltion Σ Σ s follows: for every u, v Σ : u v w Σ : uw L vw L The Nerode reltion prtitions Σ in equivlence clsses. Theorem (Myhill-Nerode Theorem) A lnguge L is recognizble iff the Nerode reltion prtitions Σ in finitely mny equivlence clsses. [Bder, 2007]

9 Preliminries DTD Document Type Definition References simple DTDs DTD Document Type Definition Definition A DTD is tuple (Σ, r, P) where Σ is n lphbet, r Σ is clled the root lbel, nd P { R Σ, R Reg Σ } is finite set of so-clled productions. Nottion: D d... set of trees stisfying DTD d L(d) = [D d ]... set of string representtions of the trees in D d

10 Preliminries DTD Document Type Definition References simple DTDs DTD Document Type Definition Exmple A DTD which is stisfied by the tree c r b c c cn be: d = (Σ, r, P) where Σ = {r,, b, c} nd P = {r, bc + c, b c, c ε} So L(d) = {r} {cc, bccbcc} {r}.

11 Preliminries DTD Document Type Definition References Specilized DTDs Specilized DTDs Definition (specilized DTD) A specilized DTD over Σ is tuple d = (Σ, Σ, d, µ) where Σ nd Σ re lphbets, d is DTD over Σ, nd µ: Σ Σ is mpping.

12 Preliminries DTD Document Type Definition References Specilized DTDs Specilized DTDs Exmple Specilized DTD which is only stisfied by the tree d = (Σ, Σ, d, µ) where c r b c : c Σ = {r,, b, c}, Σ = {r, x, y, b, c}, d = (Σ, r, P), P = {r xy, x c, y bc, b c, c ε}, { α Σ if α {x, y}, : µ(α) = α otherwise.

13 Preliminries DTD Document Type Definition References Specilized DTDs Specilized DTDs Exmple Σ = {r,, b, c}, Σ = {r, x, y, b, c}, d = (Σ, r, P), P = {r xy, x c, y bc, b c, c ε}, { α Σ if α {x, y}, : µ(α) = α otherwise. So L(d ) = {rxccxybccbccyr} nd L(d) = {rccbccbccr}.

14 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Definition We cll (specilized) DTD d strongly recognizble iff L(d) is recognizble.

15 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Exmple (non-recursive DTD) Agin consider the DTD d = (Σ, r, P) where Σ = {r,, b, c} nd P = {r, bc + c, b c, c ε} The DTD d is not recursive nd the lnguge L(d) cn be represented by the regulr expression r (cc + bccbcc) r. Hence, this DTD is strongly recognizble.

16 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Exmple (recursive DTD) Let d = (Σ, r, P) where Σ = {r, } nd P = {r, + ε}. The DTD d is obviously recursive. Moreover L(d) = {r n n r n 1}. Hence, d is not strongly recognizble.

17 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Theorem Theorem A specilized DTD is strongly recognizble iff it is non-recursive. [Segoufin & Vinu, 2002]

18 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Proof Step 1 Let d = (Σ, Σ, d, µ) be specilized DTD. Step 1: d is strongly recognizble d is non-recursive: Let d be strongly recognizble. Then there exists n FSA A which ccepts L(d).

19 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Proof Step 1 Suppose Σ is recursive with respect to d.

20 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Proof Step 1 Suppose Σ is recursive with respect to d. Then d nd d re recursive nd there exists tree t D d such tht repets on pth of t. So [t] hs the form [t] = ru 1 v 1 wv 2 u 2 r where u 1 u 2 nd v 1 v 2 re well-blnced words.

21 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Proof Step 1 Suppose Σ is recursive with respect to d. Then d nd d re recursive nd there exists tree t D d such tht repets on pth of t. So [t] hs the form [t] = ru 1 v 1 wv 2 u 2 r where u 1 u 2 nd v 1 v 2 re well-blnced words. Since is recursive we cn repet the prts v 1 nd v 2 nd the trees n > 0: [t n ] = ru 1 (v 1 ) n w(v 2 ) n u 2 r re lso in L(d ) nd A ccepts µ([t n ]).

22 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Proof Step 1 However, with the Myhill-Nerode theorem we cn show tht L(d) is not regulr: There is n infinite number of equivlence clsses of strings over Σ Σ becuse i, j 1: i j µ(ru 1 (v 1 ) i ) µ(ru 1 (v 1 ) j ).

23 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Proof Step 1 However, with the Myhill-Nerode theorem we cn show tht L(d) is not regulr: There is n infinite number of equivlence clsses of strings over Σ Σ becuse i, j 1: i j µ(ru 1 (v 1 ) i ) µ(ru 1 (v 1 ) j ). This is contrdiction to the ssumption tht L(d) is regulr nd tht A recognizes L(d), hence, d nd d cn not be recursive.

24 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Proof Step 2: FSA Construction Let d = (Σ, Σ, d, µ) be specilized DTD where Σ = {r,, b}, Σ = {ρ, α, β}, d = (Σ, ρ, P ), P = {ρ α, α β + ε, β ε} nd µ(ρ) = r, µ(α) =, µ(β) = b Since d is not recursive there exists strongly vlidting FSA. Our utomt A b for every b Σ re

25 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Proof Step 2: FSA Construction Let d = (Σ, Σ, d, µ) be specilized DTD where Σ = {r,, b}, Σ = {ρ, α, β}, d = (Σ, ρ, P ), nd P = {ρ α, α β + ε, β ε} µ(ρ) = r, µ(α) =, µ(β) = b Since d is not recursive there exists strongly vlidting FSA. Our utomt A b for every b Σ re α A ρ : q 0,ρ r r

26 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Proof Step 2: FSA Construction Let d = (Σ, Σ, d, µ) be specilized DTD where Σ = {r,, b}, Σ = {ρ, α, β}, d = (Σ, ρ, P ), nd P = {ρ α, α β + ε, β ε} µ(ρ) = r, µ(α) =, µ(β) = b Since d is not recursive there exists strongly vlidting FSA. Our utomt A b for every b Σ re A α : q 0,α β

27 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Proof Step 2: FSA Construction Let d = (Σ, Σ, d, µ) be specilized DTD where Σ = {r,, b}, Σ = {ρ, α, β}, d = (Σ, ρ, P ), P = {ρ α, α β + ε, β ε} nd µ(ρ) = r, µ(α) =, µ(β) = b Since d is not recursive there exists strongly vlidting FSA. Our utomt A b for every b Σ re A β : q 0,β b b

28 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Proof Step 2: FSA Construction α A ρ : q 0,ρ r r Now we build the trget utomton step by step. Our A 0 is equl to A ρ. The following re:

29 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Proof Step 2: FSA Construction A α : q 0,α β A 1 : r ε ε r β

30 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Proof Step 2: FSA Construction A β : q 0,β b b A 2 : r ε ε r ε b b ε

31 Preliminries DTD Document Type Definition References Strong Vlidtion Strong Vlidtion Proof Step 2: FSA Construction A 2 : r ε ε r ε b b ε Since A 2 contins no symbols from Σ nymore the following utomt A 3, A 4,... will be the sme. So A 2 is the desired utomton.

32 Preliminries DTD Document Type Definition References Vlidting well-formed XML Documents Exmple (recognizble DTD) Consider the DTD d = (Σ, r, P) where Σ = {r, } nd P = {r, + ε} gin.

33 Preliminries DTD Document Type Definition References Vlidting well-formed XML Documents Exmple (recognizble DTD) Consider the DTD d = (Σ, r, P) where Σ = {r, } nd P = {r, + ε} gin. There is regulr lnguge L R such tht L(d) = [T Σ ] L R.

34 Preliminries DTD Document Type Definition References Vlidting well-formed XML Documents Exmple (recognizble DTD) Consider the DTD d = (Σ, r, P) where Σ = {r, } nd gin. P = {r, + ε} There is regulr lnguge L R such tht L(d) = [T Σ ] L R. Let L R for exmple be L(r r). Then L R = {r m n r m 1, n 0} nd [T Σ ] L R = {r n n r n 1} = L(d).

35 Preliminries DTD Document Type Definition References Vlidting well-formed XML Documents Exmple (recognizble DTD) Consider the DTD d = (Σ, r, P) where Σ = {r, } nd gin. P = {r, + ε} There is regulr lnguge L R such tht L(d) = [T Σ ] L R. Let L R for exmple be L(r r). Then L R = {r m n r m 1, n 0} nd [T Σ ] L R = {r n n r n 1} = L(d). Or let L R for exmple be L(r r). So L R is mbiguous.

36 Preliminries DTD Document Type Definition References Vlidting well-formed XML Documents Exmple (not recognizble DTD) Let d = (Σ, b, P) be DTD where Σ = {, b, c} nd P = {b b + bc + ε, ε, c ε}. b b b b b c c Figure: Grphicl representtion for tree in D d.

37 Preliminries DTD Document Type Definition References References Bder, Prof. Dr.-Ing. Frnz (mrch). Skript zur Lehrvernstltung Grundlgen der Theoretischen Informtik. Segoufin, Luc, & Vinu, Victor. (2002). Vlidting streming XML documents. Pges of: Symposium on principles of dtbse systems. Assocition for Computing Mchinery.