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sav08:atomic_diagram_normal_form [2009/05/12 22:43] vkuncak |
sav08:atomic_diagram_normal_form [2009/05/12 23:24] vkuncak |
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| We next consider a syntactic normal form that helps us understand the decidability of the combination problem. | We next consider a syntactic normal form that helps us understand the decidability of the combination problem. | ||
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| ===== Flat Form ===== | ===== Flat Form ===== | ||
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| \[ | \[ | ||
| C[t] \ \ \leadsto \ \ (x=t) \land C[x] | C[t] \ \ \leadsto \ \ (x=t) \land C[x] | ||
| + | \] | ||
| + | |||
| + | **Example:** Represent $f(x)+y < z$ as | ||
| + | \[ | ||
| + | x_1 = f(x) \land s_1 = x_1 + y \land s_1 < z | ||
| \] | \] | ||
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| Assume $K$ variables. ($K = |FV(C)|$) | Assume $K$ variables. ($K = |FV(C)|$) | ||
| - | Number of $R(y_1,\ldots,y_n) = K^n$ | + | Number of $R(y_1,\ldots,y_n)$ is: $K^n$ |
| - | Number of $x=y$ atomic formulas: $K^2$ | + | Number of $x=y$ atomic formulas is: $K^2$ |
| - | Number of $x=f(y_1,\ldots,y_n)$ literals: $K^(n+1)$ | + | Number of $x=f(y_1,\ldots,y_n)$ literals is: $K^{n+1}$ |
| Note: if we did not have only flat literals, we could have infinitely many atomic formulas, because of arbitrarily large terms. | Note: if we did not have only flat literals, we could have infinitely many atomic formulas, because of arbitrarily large terms. | ||