Logic: Lecture 12, 10 September 2015
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(For details, see the chaper by Barwise, pp 37-40)

A "sequent" G |- D consists of two list of formulas
G = {g1,g2,...,gm} and D = {d1,d2,...,dn}.

The intention is that if assume all of the formulas in G, we can
prove at least one of the formulas in D.

    And_i g_i |= Or_j d_j

A formal system in the Sequent Calculus has very few axioms and
many rules, unlike Hilbert style systems that have many axioms
and few rules.

Sequent Calculus for propositional logic:

  Axiom:

   G, phi |- D, phi   


  (And |-)                           (|- And)              
                                                           
   G, phi, psi |- D                   G |- D, phi    G |- D, psi 
   -------------------                -------------------------- 
   G, phi and psi |- D                G |- D, phi and psi        


  (Or |-)                            (|- Or)               
                                                           
   G, phi |- D     G, psi |- D        G |- D, phi, psi
   ---------------------------        ------------------
   G, phi or psi |- D                 G |- D, phi or psi        


  (Not |-)                           (|- Not)              
                                                           
   G |- D, phi                        G, phi |- D
   -------------                      ------------
   G, ~phi |- D                       G |- D, ~phi


  (Implies |-)                       (|- Implies)

   G |- D, phi     G, psi |- D       G, phi |- D, psi
   ---------------------------       ------------------
   G, phi -> psi |- D                G |- D, phi -> psi


Derivations:

An inverted tree, where the formula to prove is the root (at the
bottom), each leaf (top) is an axiom, and every branching node
corresponds to a valid rule.

Example:

  phi, ~psi |- phi, psi      (Axiom)
  ------------------------
  phi and ~psi |- phi, psi   (And !-)    phi |- phi, theta    (Axiom)
  --------------------------             -------------------
  phi and ~psi |- phi or psi (|- Or)     phi |- phi or theta  (|- Or)
  ----------------------------------------------------------
      ((phi and ~psi) or theta |- (phi or theta)              (Or !-)


Soundess:

For each rule, verify that if And_i g_i |= Or_j d_j holds for the
sequent above, it also holds for the sequent below

Completeness:

Suppose G = {g1,g2,...,gm} and  D = {d1,d2,...,dn} such that
And_i g_i |= Or_j d_j.   Construct a derivation for G |- D as
follows.

Apply rule for top level connective in G or D.  (Choose any
formula in G or D to simplify.  Top level connective is unique.
Exactly one rule applies).  This process stops when each leaf
node looks like {p1,p2,..,pk} |- {q1,q2,...,ql} where each pi, qj
is an atomic proposition.

If each leaf node has some intersection between left and right
list, we have a valid proof.

If not, define a valuation v such that v(pi) = tt for all i and
v(qj) = ff for all j. Verify that each rule preserves this
property: at every level in the derivation, if the v(g'i) = tt
and v(d'j) = ff for the sequent G' |- D' above, then v(g"i) = tt
and v(d"j) = ff for the sequent G" |- D" below. This means that
for the root node, v(gi) = tt for all gi in G and v(dj) = ff for
all dj in G, which contradicts the assumption that And_i g_i |=
Or_j d_j.


Sequent Calculus for first order logic:

Add the following:

  (Equality axioms)

   G |- D, (t = t)         where t is any term

  (Equality rule)

   G, phi(t1) |- D, phi(t1)       where E is (t1 = t2) or (t2 = t1)
   ---------------------------
   G, E, phi(t2) |- D, phi(t2)


  (A |-)                            (|- A)              
                                                           
   G, phi(t) |- D                    G |- D, phi(v)       v not
   -------------------               ---------------      free in
   G, Av phi(v) |- D                 G |- D, Av phi(v)    G, D


  (E |-)                            (|- E)              
                                                           
   G, phi(v) |- D       v not        G |- D, phi(t)       
   -------------------  free in      ---------------    
   G, Ev phi(v) |- D    G, D         G |- D, Ev phi(v)

  (Cut)

   G, phi |- D    G |- D, phi
   --------------------------
   G |- D

Cut is the equivalent of Modus Ponens.  Note that we did not need
Cut in our sequent calculus rules for propositional logic.

Completeness for first order logic:

Proof structure is similar to the Hilbert style completeness
proof.

- Suppose X |= phi.  Then, propositionally,
  X U phi_E U phi_Q U phi_H |= phi

- Hence Y U {~phi} is not satisfiable for some finite subset Y of
  lhs

- Split Y as {a_i} coming from X U phi_E U phi_Q and {b_j}
  coming from phi_H

- Replace all Henkin constants by fresh variables to get
  {a'_i} and {b'_j}.  Note that phi' = phi, and a'_i =
  a_i for a_i in X.

- By assumption,

  |= a'_1 -> (a'_1 -> ... -> (a'_k -> (b'_1 -> ... -> (b'_l -> phi)...)

- Need to derive

  |- a'_1 -> (a'_1 -> ... -> (a'_k -> (b'_1 -> ... -> (b'_l -> phi)...)

- Formalas a'_i are directly derivable.  Apply Cut repeatedly
  to eliminate them.

- Derive the following using the proof system above

  If G, (psi -> theta) |- D then G |- D, psi and G, theta |- D

  Use Cut repeately to eliminate the b'_j


Cut elimination:

Note that the completness proof for propositional logic does
not use Cut.  However, the completeness proof for first order
logic, via a reduction to propositional logic, uses Cut heavily.

Is Cut really needed?

Theorem (Gentzen):  Any proof using Cut can be effectively
  transformed to one that does not use Cut

(Proof is beyond the scope of this course.)

The "cut elimination" theorem is probably the earliest
non-trivial example of a result in "proof theory" --- a direct
statement about the structure and transformation of proofs.

Cut elimination is analogous to substituting a lemma every time
it is used by the proof of the lemma directly, so proofs blow up
in size.

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