Suppose we have a knowledge base $K$ : $$K=\{(\phi_i, \alpha_i) : i=1..n\}$$ Such as:
- $\phi_i$ : are propositional formulas (CNF format for example).
- $\alpha_i$ : are integers ($\alpha_i \in \mathcal{N})$.
And we have two functions $S$ and $M$, such as for each logical interpretation $\omega$ :
- $M(\omega)= max\{\alpha_i : (\phi_i, \alpha_i) \in K, \omega \not\vDash \phi\}$
- $S(\omega)= \sum\{\alpha_i : (\phi_i, \alpha_i) \in K, \omega \not\vDash \phi\}$
Can we define a function $F$ such that, for each logical interpretation $\omega$: $$M(\omega)=F(S(\omega))$$ or $$S(\omega)=F(M(\omega))$$
Can a such function exists? else, can we proof its nonexistence?
Example: let $K=\{(a \lor b, 6), (\lnot{a} \lor c, 4), (c, 3), (a, 1), (\lnot{b},1))\}$ a knowledge base, consider the interpretation $ab\lnot{c}$, we have :
- $M(ab\lnot{c})=4$
- $S(ab\lnot{c})=4+3+1=8$
$S(\omega)=F(M(\omega))$ is not possible, consider $\{(a, 1), (b, 1), (b \vee c, 1)\}$. Then $M(\lnot a \lnot bc) = M(\lnot abc) = 1$, but $S(\lnot a \lnot bc) \neq S(\lnot abc)$, and $F(1)$ can't possibly equal both values.
Can you provide a similar counterexample for $M(\omega)=F(S(\omega))$?