Logic: Lecture 22, 08 November 2012
----------------------------------
 
Recall formal MSO syntax, word w defines a model ^w

For free variables, fix an assignment to the variables:

  ^w |= phi(i1,...,im,I1....,In)

MSO definable languages
 
  Sentence phi  L = { w | ^w |= phi }

FO fragment, FO definable languages

Theorem [Buechi, Elgot, Trakhtenbrot 1960]
  A language (of nonempty words) is MSO definable iff it is regular.

Eliminating min and max

  psi(min) ---> Ey (psi(y) and ~Ez z < y)
  psi(max) ---> Ey (psi(y) and ~Ez y < z)

Eliminate <

  x < y ---> Ex [ ~X(x) and Az Az'(X(z) and S(z,z') -> X(z')) and X(y) ]

    
Encoding a model for phi(x1,..,xl,X1,..,Xm) 

Word a_0 .. a_{n-1} plus an encoding of the assignment to the variables

- Encode the assignment by the characteristic function
- x_i |-> k_i is a 0/1 sequence of length n with exactly one 1 at k_i 
- X_i |-> K_i is a 0/1 sequence of length n with ones at each k in K_i

Alphabet is now Sigma x {0,1}^{l+m}

Proof of BET Theorem:

A. Encoding automata in MSO
----------------------------

- A = (Q = {q_1,...,q_m}, Delta, q_1, F)
    Assume initial state is q_1

- Use set variables X_1,...,X_m to describe runs
    X_j is those positions in the run where the state is q_j

  EX_1 EX_2 ... EX_m [
         X_j's form a partition 
     and X_1(min) 
     and Ax Ax S(x,y) -> a valid transition is applied
     and the move from max can lead to a final state
  ]

  EX_1 EX_2 ... EX_m [
         Ax Or_j X_j(x) and ~Ex And_i,j X_i(x) and X_j(x)
     and X_1(min) 
     and Ax Ax S(x,y) -> 
         Or_{q_j,a,q_k in Delta} X_j(x) and Q_a(x) and X_k(y) 
     and Or_{q_j,a,f in Delta, f in F} X_j(max) and Q_a(max)
  ]

Note:

  - Only existential set quantifiers needed to encode automata: EMSO
  - Can make do with log n quantifiers using binary encoding for n states

B. Encoding MSO in automata
---------------------------

Reduce to MSO_0 -- no individual variables

- X subseteq Y, X subseteq P_a
- Sing(X)  [ "X is a singleton"]
- S(X,Y)   [ "X = {x}, Y = {y} and S(x,y)" ]
- X < Y    [ "X = {x}, Y = {y} and x < y" ]

Can transform MSO to MSO_0 by induction

e.g phi(X_1) = Ax (P_a(x) -> Ey (S(x,y) and X_1(y))

    phi(X_1) = AX (Sing(X) and X subseteq P_a ->
                   EY (Sing(Y) and S(X,Y) and Y subseteq X_1)

Now, inductively build NFAs for MSO_0 formulas:

1. X_1 subseteq X_2

2. X_1 subseteq P_a

3. Sing(X_1)

4. S(X_1,X_2)

5. X_1 < X_2

For existential quantification, use projection:

Lemma: Let f: Sigma -> Gamma. Extend f to words in the obvious way. L is
       regular implies f(L) is regular.

Proof: Given automaton A for L, replace labels on transitions:
       (q,a,q') becomes (q,f(a),q').  New automaton accepts f(L).


Given automaton for phi(X_1,X_2,...,X_n), automaton for
EX_n phi(X_1,X_2,...,X_n) is obtained using the canonical
projection
 
  (a,x_1,...,x_{n-1},x_n) |->  (a,x_1,...,x_{n-1})

Take the automaton for phi(X_1,...,X_n) and omit the last
component of each transition label.

Complexity
----------

Individual operations are polynomial, except complementation,
which requires subset construction, so exponential.

Overall worst-case complexity is a tower of exponentials of
unbounded height [Meyer, Stockmeyer 1971].

More precisely, 

  There is no translation of MSO-formulas phi to automata in which
  the number of states of the resulting automaton can be bounded
  by a function of the form  2^{2^{2^ ...{2^n}...}} where the
  height of the exponential is some constant k.

======================================================================