First order logic - help with translation algorithm between

[ENgez]

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

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

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

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

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

thank you.

 

[mmwave]

Yes, you can encode the six relation symbols in Q(,,,) by using equivalence classes. This means that you can use Q(a,b,c,d) to represent the relation symbols F(a,b), G(a,b), R(a,b), C_0(a), C_1(b), and C_2(c). For example, if Q(a,b,c,d) represents F(a,b), then Q(a,b,c,d) holds if and only if F(a,b) holds. Similarly, if Q(a,b,c,d) represents G(a,b), then Q(a,b,c,d) holds if and only if G(a,b) holds. This way, you can use Q(,,,) to represent all six relation symbols in \Sigma''.

To ensure that \psi is valid if and only if \phi is valid, you can use the properties of equivalence classes. For example, if Q(a,b,c,d) represents F(a,b), then Q(a,b,c,d) holds if and only if F(a,b) holds. This means that if \psi is valid, then Q(a,b,c,d) holds for some values of a,b,c,d, which means that F(a,b) holds for those values as well. Similarly, if \phi is valid, then F(a,b) holds for some values of a,b, which means that Q(a,b,c,d) holds for those values as well, making \psi valid.

In summary, you can use equivalence classes to encode the six relation symbols in Q(,,,) and ensure that \psi is valid if and only if \phi is valid.