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Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
sav07_lecture_3_skeleton [2007/03/21 11:01] vkuncak |
sav07_lecture_3_skeleton [2007/03/21 12:28] vkuncak |
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R( c ) -> error=false | R( c ) -> error=false | ||
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R(havoc x) = frame(x) | R(havoc x) = frame(x) | ||
- | R(assume F) = F[x:=x_0, y:=y_0, error:=error_0] | + | R(assume F) = F[x:=x_0, y:=y_0, error:=error_0] & frame() |
R(assert F) = (F -> frame) | R(assert F) = (F -> frame) | ||
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T ::= var | T + T | K * T (terms) | T ::= var | T + T | K * T (terms) | ||
A ::= T=T | T <= T (atomic formulas) | A ::= T=T | T <= T (atomic formulas) | ||
- | F ::= F & F | F|F | ~F (formulas) | + | F ::= A | F & F | F|F | ~F (formulas) |
To get full Presburger arithmetic, allow existential and universal quantifiers in formula as well. | To get full Presburger arithmetic, allow existential and universal quantifiers in formula as well. | ||
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Proof: small model theorem. | Proof: small model theorem. | ||
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* solution of Ax=b (A regular) has as components rationals of form p/q with bounded p,q | * solution of Ax=b (A regular) has as components rationals of form p/q with bounded p,q | ||
* duality of linear programming | * duality of linear programming | ||
- | * obtains bound $M = n(ma)^{2m+1}$, which needs $(2m+1)(\log n + \log m + \log a)$ bits | + | * obtains bound $M = n(ma)^{2m+1}$, which needs $\log n + (2m+1)\log(ma)$ bits |
* we could encode the problem into SAT: use circuits for addition, comparison etc. | * we could encode the problem into SAT: use circuits for addition, comparison etc. | ||
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* Presburger Arithmetic (PA) bounds: {{papadimitriou81complexityintegerprogramming.pdf}} | * Presburger Arithmetic (PA) bounds: {{papadimitriou81complexityintegerprogramming.pdf}} | ||
* Specializing PA bounds: http://www.lmcs-online.org/ojs/viewarticle.php?id=43&layout=abstract | * Specializing PA bounds: http://www.lmcs-online.org/ojs/viewarticle.php?id=43&layout=abstract | ||
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