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sav08:finite_model_and_infinite_herbrand_model_example [2009/05/14 12:40] vkuncak |
sav08:finite_model_and_infinite_herbrand_model_example [2015/04/21 17:30] |
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| - | ====== Formulas with only Finite Models but Infinite Set of Ground Terms ====== | ||
| - | |||
| - | Example: | ||
| - | \[ | ||
| - | P(a) \land (\forall x.\ P(x) \rightarrow \lnot P(f(x))) \land (\forall x. x=a \lor x=b) | ||
| - | \] | ||
| - | |||
| - | Is this formula satisfiable? ++| yes++ | ||
| - | |||
| - | What are possible cardinalities for models? ++| they all have exactly two elements.++ | ||
| - | |||
| - | What are ground terms for this formula? ++++| | ||
| - | \[ | ||
| - | GT = \{ a, b, f(a), f(b), f(f(a)), f(f(b)), \ldots \} | ||
| - | \] | ||
| - | ++++ | ||
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| - | Cardinality of Herbrand model is ++|infinite!++ | ||
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| - | By Herbrand theorem, does every satisfiable first-order logic formula have Herbrand model? ++| Only if formula has no equality! We made this assumption when proving Herbrand theorem. ++ | ||
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