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sav07_lecture_3 [2007/03/22 13:25] yuanjianwz |
sav07_lecture_3 [2007/03/22 13:43] 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// |
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| + | We try to figure out the boundary M. | ||
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| + | Another example: | ||
| + | //¬t1 < t2// | ||
| + | reduce to | ||
| + | //t2+1≤t1// | ||
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| + | How about the not equal ? | ||
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| + | //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. | ||