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Chapter 7 in the textbook: |
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Games for SO, application to reachability |
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Buchi’s theorem |
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Star Free Languages |
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Tree automata |
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Proof |
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Part 1: regular languages µ MSO |
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Every regular language is in 9MSO
[in class] |
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Every regular language is in 8 MSO
[in class] |
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MSO µ regular languages |
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This is harder: let f 2 MSO. Need to show that the set {M | M ² f} is
regular |
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Proof 1 (not in the book): induction on f [in
class] |
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Proof 2 (from the book): using types [in class] |
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Corollary
Over strings, 9 MSO = 8 MSO = MSO |
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Does this hold in general ? |
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Consider s = {<} |
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A structure M = a string in {1}* = a
number |
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Proposition Hamiltonean path on undirected
graphs is not definable in MSO |
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Proof in class |
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Let S = {a,b,c}
(can be larger in general) |
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A regular language is defined by the
grammar:
E ::= ;
E ::= e | a
| b |
c
E ::= E [ E
E ::= E.E
E ::= E* |
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A star-free regular language:
drop E*,
add E (need overline, not
underline) |
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Which of the following are star-free ? |
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Theorem
L is star-free iff it is in FO.
L is regular iff it is in MSO |
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A ranked alphabet is a set S and r : S ! N |
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For a 2 S, r = r(a) ¸ 0 is it’s rank; also write
ar |
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Rank(S)=max(r(a) | a 2 S) |
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A ranked tree T is a tree labeled with symbols
in S s.t. each node x labeled with a has exactly r(a) children |
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Notations: |
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Trees(S) |
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A tree language is L µ Tree(S) |
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This generalizes languages over strings (why ?) |
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Symbolic expressions:
S =
{a,b,c,f1,g2,h2,k3} |
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A = (S, Q, q0, d, F) where:
q0
2 Q, F µ Q
d is a set of
tuples of the form: |
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(w, a, q) w 2 Q*,
a 2 S, q 2 Q, s.t. |w| = r(a) |
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Execution: given a tree T, A executes
bottom up or top down [show in
class] |
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A language L µ Tree[S] is a regular tree
language iff there exists A s.t. L = {T | T accepted by A} |
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Example: S = {0,1,:, Æ, Ç} |
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A = (S, Q, q0, d, F) |
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Q = {false, true} |
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q0 = (not needed) |
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F = { true } |
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d = (e,
0, false), (e, 1, true)
(false, :,
true), (true, :, false)
(false.false, Æ,
false), …, (true.true, Æ, true)
(false.false, Ç,
false), …, (true.true, Ç, true) |
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Let’s restrict to Rank(S) = 2
easy to generalize |
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Let s = {<, (Pa)a 2
S, succ1, succ2} |
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Consider only structures in STRUCT[s] that are
trees, where < is the ancestor-descendant relation, and succ1,
succ2 are the left and right child respectively |
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Theorem A set of trees is definable in MSO iff
it is regular |
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Proof: almost identical to Buchi’s theorem |
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What are star-free regular tree languages ? |
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Notes
