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Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
sav07_lecture_3 [2007/03/22 13:25] yuanjianwz |
sav07_lecture_3 [2007/03/22 13:36] yuanjianwz |
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Proof: small model theorem. | Proof: small model theorem. | ||
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==== Small model theorem for Quantifier-Free Presburger Arithmetic (QFPA) ==== | ==== Small model theorem for Quantifier-Free Presburger Arithmetic (QFPA) ==== | ||
- | he idea is to reduce the case, for example: | + | The idea is to reduce the case, for example: |
∃x,y,z.F | ∃x,y,z.F | ||
reduce to | reduce to | ||
- | ∃x,y,z ≤ M(F).F | + | ∃x≤ M,y≤ M,z ≤ M.F |
+ | We try to figure out M. | ||
+ | |||
+ | Another example: | ||
+ | ¬t1 < t2 | ||
+ | reduce to | ||
+ | t2+1≤t1 | ||
+ | |||
+ | How about the not equal ? | ||
+ | t1≠t2 | ||
+ | can be reduced to | ||
+ | (t1 < t2 ) ∨ (t2 < t1) => (t1 ≤ t2-1) ∨ (t2 ≤ t1-1) | ||
First step: transform to disjunctive normal form. | First step: transform to disjunctive normal form. |