First order logic - help with translation algorithm between

  • #1
75
0

Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
18,086
7,510
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?
 

Related Threads on First order logic - help with translation algorithm between

  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
6
Views
4K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
3
Views
689
  • Last Post
2
Replies
35
Views
7K
  • Last Post
Replies
1
Views
1K
Replies
13
Views
8K
  • Last Post
2
Replies
35
Views
7K
Replies
2
Views
1K
Replies
7
Views
2K
Top