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sav08:finite_model_and_infinite_herbrand_model_example [2008/04/01 23:12] vkuncak |
sav08:finite_model_and_infinite_herbrand_model_example [2009/05/14 12:40] vkuncak |
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| - | ====== Formulas with Finite Model but Infinite Herbrand Model ====== | + | ====== Formulas with only Finite Models but Infinite Set of Ground Terms ====== |
| Example: | Example: | ||
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| What are possible cardinalities for models? ++| they all have exactly two elements.++ | What are possible cardinalities for models? ++| they all have exactly two elements.++ | ||
| - | What is Herbrand model for this formula? ++++| | + | What are ground terms for this formula? ++++| |
| \[ | \[ | ||
| GT = \{ a, b, f(a), f(b), f(f(a)), f(f(b)), \ldots \} | GT = \{ a, b, f(a), f(b), f(f(a)), f(f(b)), \ldots \} | ||
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| Cardinality of Herbrand model is ++|infinite!++ | Cardinality of Herbrand model is ++|infinite!++ | ||
| - | By Herbrand theorem, does every satisfiable first-order logic formula have Herbrand model? ++| Only if formula has no equality. ++ | + | 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. ++ |