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sav08:finite_models_imply_decidability [2008/04/02 17:06]
vkuncak created
sav08:finite_models_imply_decidability [2008/04/03 13:04] (current)
vkuncak
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====== Finite Models Imply Decidability ====== ====== Finite Models Imply Decidability ======

-Prover and counterexample finder running in parallel.+Now for some positive results.

+**Theorem:​** Suppose that $S$ is a subset of all first-order logic sentences such that for all $F \in S$,
+  * $\lnot F \in S$, and
+  * if $F$ has a model, then $F$ has a finite model
+Then there exists an algorithm that, given a formula from $F \in S$, determinies whether $F$ is valid.
+
+Note: such algorithm is called a decision procedure for the class of formulas $S$.
+
+Note: we have just shown that such algorithm does not exists for the set of all sentences in first-order logic. ​ There is only an algorithm (e.g. resolution) that will terminate when formula is valid, but need not terminate on invalid formulas.
+
+**Proof:**
+
+Prover and counterexample finder running in parallel.