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I First order logic - help with translation algorithm between

  1. Jun 5, 2016 #1
    given a dictionary [tex]\Sigma = \left \{f(),g(),R(,),c_0,c_1,c_2 \right \}[/tex] and a sentence [itex]\phi[/itex] over [itex]\Sigma[/itex], I need to find an algorithm to translate [itex]\phi[/itex] to [itex]\psi[/itex] over [itex]\Sigma'[/itex] where [itex]\Sigma' = \left \{Q(,,,), = \right \}[/itex] (Q is a 4-place relation symbol), so that [itex]\psi[/itex] is valid iff [itex]\phi[/itex] is valid.

    I understand that I am supposed to eliminate function symbols using the equality relation in [itex]\Sigma'[/itex], so that [itex] f()[/itex] in [itex]\Sigma[/itex] is translated to a relation symbol [itex]\ F(,) [/itex] , so that [itex]\ F(a,b)[/itex] holds iff [itex]\ f(a)=b [/itex] (and likewise for [itex] g() [/itex]).

    the constants can be translated to 1-ary relation symbols.

    Therefore, I have an intermediate dictionary
    [tex]\Sigma'' = \left \{F(,),G(,),R(,),C_0(),C_1(),C_2(), = \right \}[/tex]

    I need to somehow encode the six relation symbols (3 binary and 3 unary) in [itex] Q(,,,) [/itex]. Is there a particular way to do this, is this related to equivalence classes?

    thank you.
  2. jcsd
  3. Jun 10, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
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