LARA

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sav08:atomic_diagram_normal_form [2009/05/12 22:59]
vkuncak 		sav08:atomic_diagram_normal_form [2009/05/12 23:30]
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.
 +
  
 ===== 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
 +\[
 +    x_1 = f(x) \land s_1 = x_1 + y \land s_1 < z
 +\]
 +
 +
 +
  
 ===== Finiteness of Flat Literals with Fixed Variables ===== ===== Finiteness of Flat Literals with Fixed Variables =====
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 Number of $x=f(y_1,​\ldots,​y_n)$ literals is: $K^{n+1}$ 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. 
 + 
 +**Example:​** When we have relation symbols $<$ and no function symbols, and consider 3 variables $x$,​$y$,​$z$,​ possible atomic formulas are: 
 +  * $x=x$, $y=y$, $z=z$, $x=y$, $y=x$, $x=z$, $z=x$, $y=z$, $z=y$ 
 +  * $x<x$, $y<y$, $z<z$, $x<y$, $y<x$, $x<z$, $z<x$, $y<z$, $z<y$ 
 +and there are 3^2+3^2=18 atomic formulas.
  
 ===== Atomic Diagram Normal Form ===== ===== Atomic Diagram Normal Form =====