Nested Words. A model of linear and hierarchical structure in data. Examples of such data

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2 Nested Words A model of linear and hierarchical structure in data. s o m e w o r d Examples of such data Executions of procedural programs Marked-up languages such as XML Annotated linguistic data

3 Example: Execution of a Structured Program as a Nested Word 3 procedures

4 Example: Execution of a Structured Program as a Nested Word 3 procedures

5 Example: Execution of a Structured Program as a Nested Word 3 procedures

6 Example: Execution of a Structured Program as a Nested Word 3 procedures

7 Example: Execution of a Structured Program as a Nested Word 3 procedures

8 Example: Execution of a Structured Program as a Nested Word 3 procedures

9 Example: Execution of a Structured Program as a Nested Word 3 procedures

10 Example: Execution of a Structured Program as a Nested Word 3 procedures

11 Example: Execution of a Structured Program as a Nested Word 3 procedures

12 Example: Execution of a Structured Program as a Nested Word 3 procedures

13 Example: Execution of a Structured Program as a Nested Word 3 procedures

14 Example: Execution of a Structured Program as a Nested Word 3 procedures

15 Example: Execution of a Structured Program as a Nested Word 3 procedures

16 Example: Execution of a Structured Program as a Nested Word 3 procedures

17 Example: Execution of a Structured Program as a Nested Word 3 procedures

Nested Words Automata (NWA) Special kinds of pushdown automata whose push/pop actions are directed by the hierarchical structure in the input nested word.

18 Nested Words Automata (NWA) Special kinds of pushdown automata whose push/pop actions are directed by the hierarchical structure in the input nested word. On every open-tag, the automaton pushes On every close-tag, the automaton pops On data with no hierarchical connections, no stack operation is made. A language of nested words is said to be regular if it can be recognized by a NWA

19 Properties of Regular Languages of Nested Words Automata Deterministic nested word automata are as expressive as their non-deterministic counterparts The class is closed under: union intersection complementation concatenation Kleene-* prefixes reversal homomorphism The following problems are decidable: emptiness membership inclusion equivalence

20 Weakness of Nested Words But what if an exceptions is thrown?

21 Weakness of Nested Words But what if an exceptions is thrown? r 3

22 Weakness of Nested Words But what if an exceptions is thrown? p 5 p 6 r 3

23 Weakness of Nested Words But what if an exceptions is thrown? p 5 r 3

24 Weakness of Nested Words But what if an exceptions is thrown? p 5 r 3 r 3 p 5 p 6 p 5

25 Other Examples Python program

Colored Nested Words - Encoding Explicitly, using a graph with several types of edges (linear, mono-chromatic and bi-chromatic) and defining the constraints they should

26 Colored Nested Words - Encoding Explicitly, using a graph with several types of edges (linear, mono-chromatic and bi-chromatic) and defining the constraints they should satisfy. Implicitly, using colored/types parenthesis. (p 0 (p 1 p 1 p 3 (q 0 q 1 q 2 (r 0 r 1 r 2 r 3 ) p 5 p 6 ) [p 0 (p 1 p 1 p 3 [q 0 q 1 q 2 [r 0 r 1 r 2 r 3 ) p 5 p 6 ]

27 Regularity How can we define regularity of a language of colored nested words? Intuitively, we d like the language obtained by inserting the missing closing tags to be a well-colored regular language of nested words. (p 0 (p 1 p 1 p 3 (q 0 q 1 q 2 (r 0 r 1 r 2 r 3 ) p 5 p 6 ) (p 0 (p 1 p 1 p 3 (q 0 q 1 q 2 (r 0 r 1 r 2 -) -) r 3 ) p 5 p 6 )

28 Regularity Suppose we have such a mapping from CNW to NW f(w) = w such that w has all the missing closing tags Definition: Let L be a language of CNW. Then L is regular if { f(w) w L } is a regular lang. of NW

29 How can we process regular CNW? w Transducer Implementing f f(w) NWA But how can we implement this transducer? It is not regular, and not regular NW, because it needs to pop more than one letter when reading a rescuing closing tag!

30 Colored Nested Words Automata The pushdown automaton marks the pushed letters with the respective color A read of an x-colored close-tag pops from the stack all symbols of y-colored symbols up to the first x-colored symbol (as well as this letter itself) R(n) Q(n) try P(n) r 3

31 Colored Nested Words Automata The pushdown automaton marks the pushed letters with the respective color A read of an x-colored close-tag pops from the stack all symbols of y-colored symbols up to the first x-colored symbol (as well as this letter itself) R(n) Q(n) try P(n) p 5 r 3

32 Blind vs. Sighted CNA Can the automaton view all the letters in the stack above the matching x-colored letter? Or can it just see that x-colored letter? We call the first version Sighted CNA R(n) Q(n) try P(n)

33 Blind vs. Sighted CNA Can the automaton view all the letters in the stack above the matching x-colored letter? Or can it just see that x-colored letter? We call the first version Sighted CNA And the second version Blind CNA R(n) Q(n) try P(n) Theorem 1 Sighted CNA and Blind CNA have the same expressive power

34 Blind vs. Sighted CNA Theorem 2 Sighted CNA and Blind CNA accept all the regular languages of colored nested words

35 Other Examples broken html

36 CNA html example Pop trans Push trans Internal ε-trans

37 CNA - Examples Accepts word where #a s between () is even

38 CNA - Examples Accepts word where #a s between () is even and #a s between [] is odd

39 CNA - Examples Accepts word where #a s between () is even and #a s between [] is odd a ( may be unmatched if encapsulated by []

40 Properties of Regular Languages of Colored Nested Words Automata Deterministic nested word automata are as expressive as their non-deterministic counterparts The class is closed under: union intersection complementation concatenation Kleene-* prefixes reversal homomorphism The following problems are decidable: emptiness membership inclusion equivalence

41 Grammar Characterization There exists a grammar characterization for NWs It classifies variables to two sets One that disallows pending calls One that disallows pending returns There exists a grammar characterization for CNWs It classifies variables C +2 sets One per each color ))... ) (..) ((.)()()() ) ( ().( One that disallows pending calls One that disallows pending returns )) ) (..) ((.) ( () ( ) ( ().(

Colored Nested Words

Colored Nested Words Rajeev Alur and Dana Fisman Department of Computer and Information Sciences, University of Pennsylvania, Philadelphia PA, USA talur,fismanu@cis.upenn.edu Abstract. Nested words allow