Logic: Lecture 14, 20 September 2012
------------------------------------

Recall:

Definition: m-equivalence

  G = (V,E), H = (W,F), \bar{a} a tuple over V, \bar{b} a tuple over W

  (G,\bar{a}) =m= (H,\bar{b}) iff for every phi(\bar{x}) of
  quantifier rank <= m, 
  (G,\bar{a}) |= \phi(\bar{x}) iff (H,\bar{b}) |= \phi(\bar{x})
  

Definition: m-isomorphism
  (G,\bar{a}) ~0~ (H,\bar{b}) if there is a partial isomorphism

  (G,\bar{a}) ~m~ (H,\bar{b}) if

  - For each c in V, exists d in W, (G,\bar{a}c) ~(m-1)~
    (H,\bar{b}d)
  - Vice versa

Definition:
  Finitely isomorphic:  (G,\bar{a}) ~f~ (H,\bar{b}) iff
  (G,\bar{a}) ~m~ (H,\bar{b}) for all m >= 0

Theorem (Fraisse)
  G ~f~ H iff G == H
  
Ehrenfeucht games:

Two players, Spoiler and Duplicator and two structures M = (S,i) and 
M' = (s',i').

Spoiler first chooses r, number of rounds.

Each round j, 1 <= j <= r is as follows:

1. Spoiler chooses a_j in S and Duplicator responds with b_j in S'

    OR

2. Spoiler chooses b_j in S' and Duplicator responds with a_j in S

Duplicator wins if (a_1,...,a_r) and (b_1,...,b_r) are partially isomorphic.  Otherwise Spoiler wins.

Theorem:
  M ~f~ M' iff Duplicator has a winning strategy.

Proof:
  If M ~f~ M' then M ~i~ M' for every i.  Starting from any r,
  Duplicator is guaranteed to be able to match every choice of
  Spoiler.

  If not(M ~f~ M') then for some r, not (M ~r~ M').  Spoiler
  chooses that r and picks an invalidating witness from M or M'
  at each stage.

Examples:

Successor axioms:

  Ax (~(x = 0) <=> Ey s(y) = x)
  AxAy (s(x) = s(y) => x = y)
  { Ax ~(s(s(...(s(x)...))) = x | m >= 1 }
         ---------
          m-times

Let X be the set of successor axioms.

Thm: All models of X are elementary equivalent.  
     That is, M |= X and M' |= X implies M == M'
 
Assuming the theorem:

  For any sentence phi, we have X |= phi or X |= ~phi.
  X is said to be a "complete theory".

  For a complete theory, using the completeness theorem we have a
  decision procedure (effective algorithm) for the question
  "X |= phi"

  Since X |= phi iff X |- phi, we can systematically enumerate
  all proofs of the form X |- beta. At some finite point, we must
  find a proof of X |- phi or X |- ~phi.

Proof of theorem:

   By Fraisse's theorem, it suffices to show that M ~f~ M'.

   See pdf lecture notes.

2. Evenness is not expressible.

   -- empty signature: Proposition 3.3, page 25 of Libkin

   -- with <, pages 28-30 of Libkin

4. Use inexpressibleness of evenness to show connectivity is not
   expressible even for finite graphs

   -- page 37 of Libkin
 
======================================================================