Logic: Lecture 6, 23 August 2012
--------------------------------

First order logic

Review of FOL syntax and semantics

  L = (R,F,C)
  L-structures and interpretations
  
  Assignments and free variables

  Terminology: Closed term = no variables

  Sentences and logical consequene

Expressing properties in FOL

  Groups

    Axioms G1-G3
    (Left) cancellation
    Abelian groups

    Element of order 2 ... element of order k
    All elements have a finite order ("torsion group")
       "Infinite" formula
       Cannot express even with infinite set of finite formulas

  Equivalence relations
    Ax r(x,x)
    Ax Ay r(x,y) <=> r(y,x)
    Ax Ay Az (r(x,y) and r(y,z)) => r(x,z)

  Strict linear orders
    Ax ~(x < x)
    Ax Ay Az (x < y and y < z) => x < z
    Ax Ay (x < y or x = y or y < x)

  Fields
    (F,+,.,0,1)
    (F,+,0) is an abelian group
    . is associative and commutative, identity 1, 0 =/= 1, every
      element other than 1 has right inverse wrt .
    . distributes over +

Satisfiability of FOL: Henkin's reduction

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

   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

   {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

Need enough terms!


======================================================================