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sav08:first-order_logic_is_undecidable [2008/04/03 12:56] vkuncak |
sav08:first-order_logic_is_undecidable [2008/04/03 12:57] vkuncak |
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| Technical difficulty: no simple way to express sequences, reachability, or finiteness in first order logic. | Technical difficulty: no simple way to express sequences, reachability, or finiteness in first order logic. | ||
| - | Note the following equivalences: | + | Note the following equivalent statements of our target condition: |
| - | $F$ is valid iff $M$ accepts $w$ | + | ($F$ is valid) iff ($M$ accepts $w$) |
| - | ($\lnot F$ is unsatisfiable) iff $M$ accepts $w$ | + | ($\lnot F$ is unsatisfiable) iff ($M$ accepts $w$) |
| - | ($\lnot F$ is satisfiable) iff $M$ does not accept $w$ | + | ($\lnot F$ is satisfiable) iff ($M$ does not accept $w$) |
| (($\lnot F$) is true in some Herbrand model) iff ($M$ does not accept $w$) | (($\lnot F$) is true in some Herbrand model) iff ($M$ does not accept $w$) | ||