Logic: Lecture 8, 27 August 2015
--------------------------------

Substitution:

  Recall that phi(x1,...,xn) means FV(phi) subset of {x1,..,xn}

  phi(t1,...,tn) is the formula where each free xi is uniformly
  replaced by ti

  phi(t1,...,tn) = phi(x1,...,xn)[x1 |-> t1, ..., xn |-> tn]

  ti need not be closed --- be careful about free variables being
  "captured" by quantifiers:

  phi(x) = Ey ~(x = y)
  phi(y) = Ey ~(x = y)[x |-> y] = Ex ~(y = y)

  The meaning has changed: phi(y) is not satisfiable in any
  model; a free y is "captured" by Ey

  Instead, rename bound variable before substitution

  phi(x) = Ez ~(x = z)  --->  phi(y) = Ez ~(y = z)

  Formalizing this idea of renaming and substitution correctly is
  tedious and error-prone.  Exercise!

Satisfiability of FOL: Henkin's reduction

  Aim is to show completeness for FOL.  Can we reuse ideas from
  propositional logic?

  In propositional logic, we followed this sequence:

  1. Completeness:  |= phi implies |- phi

  2. Compactness: X |= phi implies Y |= phi for some finite
     subset Y of X

  3. Strong Completeness: X |= phi implies X |= phi

  We established that 2 and 3 were equally strong: from one we
  can prove the other.

  Also, clearly 3 implies 1: Take X = empty set to get
  completeness from strong completeness.

  For FOL, we start with compactness.  Recall that compactness
  was proved using finite satisfiability, so we begin by looking
  at satifiability in FOL.
  
  Can we "translate" FOL to propositional logic?

  First attempt: assign each atomic formula of FOL a new
  proposition
  
  This exposes some basic facts, but not all

    r(t,t') and ~r(t,t')      : clearly not satisfiable
    Ax r(x,t) and ~Ex r(x,t)  : more tricky

  FOL syntax has quantifiers.  Need to account for these in our
  translation to propositional logic.

  Prime formulas: atomic formulas + formulas of the form Ex phi

  For instance:
    t = t'
    r(x,y)
    Ex f(x) = t

  Assign each prime formula a new proposition

  Fact: Every FOL formula is a boolean combination of prime formulas

  Notation:
    For prime alpha, let p_alpha denote corresponding proposition
    For arbitrary formula phi, let prime(phi) denote translation
      of phi into propositional logic

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

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

        For each atomic formula alpha, set v(p_alpha) = tt iff
	I |= alpha

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

  Converse not true

    {c = d, d = e, ~(c = e)}
    {Ax (r(x) => s(x)), Ax r(x), Ex ~s(x)}

    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

    I |= Ex phi(x)  iff
    there is an a in S such that I[x |-> a] |= phi(x)

    What if we can't find a witness term t such that t maps to a?
    Then phi(t) does not hold for any t --- the element a has no
    "name" in our language.

    Need more 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)
    C_2 = {c_phi | Ex phi in Formulas(L_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 propositionally satisfiable