Logic: Lecture 7, 28 August 2012
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  Cardinality
    At least two elements: Ex Ey ~(x = y)
    At least k elements:   Ex1 ... Exn And_{i =/= j} ~(xi = xj)

    At most one element:   Ax Ay (x = y)
    At most k-1 elements:  Ax1 ... Axn Or_{i =/= j} (xi = xj)

    Structure is not finite: Infinite family

Satisfiability of FOL: Henkin's reduction

   Prime formulas

     Ex r(x) or t = t'
     Ax s(x) = Ex s(x)

   Every FOL formula is a boolean combination of prime formulas

   Propositional satisfiability: Assign {tt,ff} to prime formulas

Prop: If I is an L-interpretation, exists v such that 
      I |= phi iff v |= phi

Cor: If X is FO satisfiable, X is propositionally satisfiable

Converse not true

   Add [ (c = d and d = e) => (c = e) ]

   Add     Ex phi(x) => phi(t), for some witness t
         ~ Ex phi(x) => ~phi(t) , arbitrary t

Need enough terms!

Witnessing expansion

 L = (R,F,C)

 L_0 = L, C_0 = emptyset

 C_1 = {c_phi | Ex phi in Formulas(L_0)}
 L_1 is (R,F,C U C_1)

 Assume we have L_n = (R,F,C U C_1 U ... U C_n)

 C_n+1 = {c_phi | Ex phi in Formulas(L_n) - Formulas(L_n-1)}
 L_n+1 is (R,F,C U C_1 U ... U C_n U C_n+1)
 
 C_H = C_1 U ... U C_n ...

 L_H = (R,F,C U C_H)
 

 Henkin axiom:
    Ex phi(x) => phi(c_phi(x))

 Quantifer axiom:
    phi(t) => Ex phi(x), t closed

 Equality axioms:

    reflexitivy, symmetry and transitivity for all terms
    functions: equal arguments given equal results
    relations: equal tuples have equal membership properties

 Phi_H : all instances of Henkin axioms
 Phi_Q : all instances of quantifier axioms
 Phi_Eq : all instances of equality axioms

Phi_H and Phi_Q are sentences.  
Phi_Eq are not sentences but are true under all assignments.

FO satisfiability

Theorem: Let X be a set of formulas over L. 
The following are equivalent.

1. There is an L-interpretation I = (M,sigma) that satisfies X
2. There is an L_H-interpretation I' = (M',sigma') that satisfies x
3. X U Phi_H U Phi_Q U Phi_Eq is propositonally satisfiable

2 => 1  Immediate

1 => 3  Need to extend M = (S,i) to cover L_H
        i'(c) = i(c) for c in C
        For c = c_phi in C_H, 
           if I |= Ex phi, map i'(c_phi) to o s.t. I |= phi(o)
              otherwise map i'(c_phi) arbitrarily
           (need to do this in stages for C_1, C_2, ...)
   
        Now, from I' we can derive v such that
        v |= X U Phi_H U Phi_Q U Phi_Eq

3 => 2  We have a valuation v over all prime formulas in L_H such
        that v |= X U Phi_H U Phi_Q U Phi_Eq

        Need to define an interpretation I = ((S,i),sigma)

        First S:

        Define and equivalence relation on terms, 
           t ~ t'  iff  v |= t = t'

        Elements of S are equivalence classes [t] (check that
           this is an equivalence relation)

        ... to be continued

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