Logic: Lecture 24, 23 November 2012
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Extending MSO beyond (N,0,S).

Two dimensions: underlying structure is the infinite binary tree:

  Domain is {0,1}*, finite sequence over 0/1
    Each node in the complete infinite binary tree is a word over
    {0,1} 

  Each word w has two successors, s0(w) = w0 and s1(w) = w1,
    hence this MSO theory is called S2S (Second order theory of 2
    Successors) 

  In retrospect, MSO over (N,0,S) is S1S, for 1 Successor

  S2S has two orderings:

    <   : tree order (x < y if there is a path from x to y)
    lex : lexicographic ordering over nodes (words over {0,1})

  lex is definable using < for any finitely branching tree.

Theorem (Rabin, 1969):  S2S is decidable

  Via automata on infinite trees.  Combinatorially complicated proof.

Regular tree:

  Infinite binary tree labelled by finite alphabet Sigma.

  Subtree at each node structurally isomorphic to original tree.

  Regular tree: With labels, only finitely many distinct subtrees.

  S2S corresponds to languages of regular trees.

S2S is conjectured to be a maximally decidable theory.  

  If we add any relation R to S2S that is not already
    expressible in S2S, then S2S + R is undecidable

  e.g. EqualDepth(x,y) : "x and y at same depth from root"

Within S2S we can embed many other structures.

  Constructing such an embedding is tricky.

An alternative approach is to define operations that preserve MSO
  decidability. (See slides).

  MSO interpretations
    e.g. S3S can be interpreted in S2S

  Graph unfoldings

When is MSO undecidable?

Tiling problem:

  Positive quadrant of lattice points in xy-plane: 
    {(i,j) | i >= 0, j >= 0}
  
  Finite set of tiles  : T = {t_1,t_2,...,t_N}
  
  Constraints on tiles : Right,Up, both binary relations over T x T

  A valid tiling is an assignment of tiles to lattice points
    that respects the constraints Right and Up

    t_i at (m,n) and t_j at (m+1,n) only if Right(t_i,t_j)
    t_i at (m,n) and t_j at (m,n+1) only if Up(t_i,t_j)

  Given T, Up, Right, is a tiling possible?

  Undecidable!

MSO over this grid

  Represent tiling by set variables X_1,X_2,...,X_N

  Can write an MSO formula

    EX_1 EX_2 ... EX_N ( X1,X2,...,XN is an assignment of tiles
                         to all grid points such that Right and
                         Up constraints are satisfied)

  This formula is satisfiable iff a valid tiling exists.

  Hence MSO over the grid is undecidable.

Back to S2S

  Adding predicates such as EqualDepth allows us to MSO interpret
    the grid in the infinite binary tree.

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