Differences
This shows you the differences between two versions of the page.
sav08:herbrand_universe_for_equality [2008/04/02 22:58] vkuncak |
sav08:herbrand_universe_for_equality [2015/04/21 17:30] |
||
---|---|---|---|
Line 1: | Line 1: | ||
- | ====== Herbrand Universe for Equality ====== | ||
- | |||
- | Recall example of formula $F$: | ||
- | \[ | ||
- | P(a) \land (\forall x.\ P(x) \rightarrow \lnot P(f(x))) \land (\forall x. x=a \lor x=b) | ||
- | \] | ||
- | Replace '=' with 'eq': | ||
- | \[ | ||
- | P(a) \land (\forall x.\ P(x) \rightarrow \lnot P(f(x))) \land (\forall x. eq(x,a) \lor eq(x,b)) | ||
- | \] | ||
- | call the resulting formula $F'$. | ||
- | |||
- | Consider Herbrand model $I_H = (GT,\alpha_H)$ for the set $\{F'\} \cup AxEq$. The relation $\alpha_H(eq)$ splits $GT$ into two sets: the set of terms eq with $a$, and the set of terms eq with $b$. The idea is to consider these two partitions as domain of new interpretation, denoted $([GT], \alpha_Q)$. | ||
- | |||
- | ===== Constructing Model for Formulas with Equality ===== | ||
- | |||
- | Let $S$ be a set of formulas in first-order logic with equality and $S'$ result of replacing '=' with 'eq' in $S$. Suppose $S' \cup AxEq$ is satisfiable. | ||
- | |||
- | Let $I_H = (GT,\alpha_H)$ be Herbrand model for $S \cup AxEq$. We construct a new model as [[Interpretation Quotient Under Congruence]] of $(GT,\alpha)$ under congruence 'eq'. Denote quoient structure by $I_Q = ([GT],\alpha_Q)$. By theorem on quotient structures, $S$ is true in $I_Q$. Therefore, $S$ is satisfiable. | ||
- | |||
- | ===== Herbrand-Like Theorem for Equality ===== | ||
- | |||
- | **Theorem:** For every set of formulas with equality $S$ the following are equivalent | ||
- | - $S$ has a model; | ||
- | - $S' \cup AxEq$ has a model (where $AxEq$ are [[Axioms for Equality]] and $S'$ is result of replacing '=' with 'eq' in $S$); | ||
- | - $S$ has a model whose domain is the quotient $[GT]$ of ground terms under some congruence. | ||