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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 12:59] vkuncak |
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Note the following equivalent statements of our target condition: | 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 $w$) | ||
- | So, we work with formula $G$ denoting $\lnot F$, interpreted over Herbrand model, and we express the condition that all Turing machine computation histories are non-accepting. | + | So, we work with formula $G$ denoting $\lnot F$, interpreted over Herbrand model, and we express the condition that all Turing machine computation histories are non-accepting. In other words, $F$, when interpreted over Herbrand interpretations, says that there exists an accepting Turing computation history. |
The fact that we work over Herbrand models helps us talk about computation histories, because we can encode strings and reachability. | The fact that we work over Herbrand models helps us talk about computation histories, because we can encode strings and reachability. | ||