Finite Model Theory Lecture 11 Second Order Logic

Outline Chapter 7 in the textbook: Games for SO, application to reachability Buchi’s theorem Star Free Languages Tree automata

Buchi’s Theorem Proof Part 1: regular languages µ MSO Every regular language is in 9MSO [in class] Every regular language is in 8 MSO [in class]

Buchi’s Theorem MSO µ regular languages This is harder: let f 2 MSO. Need to show that the set {M | M ² f} is regular Proof 1 (not in the book): induction on f [in class] Proof 2 (from the book): using types [in class]

Applications Corollary Over strings, 9 MSO = 8 MSO = MSO Does this hold in general ?

Applications Consider s = {<} A structure M = a string in {1}* = a number What sets of natural numbers are definable in MSO ?

Applications Proposition Hamiltonean path on undirected graphs is not definable in MSO Proof in class

Star-free regular languages Let S = {a,b,c} (can be larger in general) A regular language is defined by the grammar: E ::= ; E ::= e | a | b | c E ::= E [ E E ::= E.E E ::= E* A star-free regular language: drop E*, add E (need overline, not underline)

Star-free Regular Languages Which of the following are star-free ? Theorem L is star-free iff it is in FO. L is regular iff it is in MSO {ab, accb, ca} S* a* {a,b}* ((a [ b).c)*

Ranked Trees A ranked alphabet is a set S and r : S ! N For a 2 S, r = r(a) ¸ 0 is it’s rank; also write ar Rank(S)=max(r(a) | a 2 S) 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

Tree Automtata Notations: Trees(S) A tree language is L µ Tree(S) This generalizes languages over strings (why ?)

Example Symbolic expressions: S = {a,b,c,f1,g2,h2,k3} h f h k b b a c

Not needed Tree Automata A = (S, Q, q0, d, F) where: q0 2 Q, F µ Q d is a set of tuples of the form: (w, a, q) w 2 Q*, a 2 S, q 2 Q, s.t. |w| = r(a) Execution: given a tree T, A executes bottom up or top down [show in class] A language L µ Tree[S] is a regular tree language iff there exists A s.t. L = {T | T accepted by A}

Tree Automata Example: S = {0,1,:, Æ, Ç} A = (S, Q, q0, d, F) Q = {false, true} q0 = (not needed) F = { true } 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)

Ranked Trees as Structures Let’s restrict to Rank(S) = 2 easy to generalize Let s = {<, (Pa)a 2 S, succ1, succ2} 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

MSO and Regular Tree Languages Theorem A set of trees is definable in MSO iff it is regular Proof: almost identical to Buchi’s theorem What are star-free regular tree languages ?