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msol_over_strings [2007/05/10 18:15] vkuncak |
msol_over_strings [2007/05/11 16:58] ghid.maatouk |
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| **Using transitive closure and successors:** | **Using transitive closure and successors:** | ||
| * Constant zero: $(x=0) = One(x) \land \lnot (\exists y. One(y) \land s(y,x))$ | * Constant zero: $(x=0) = One(x) \land \lnot (\exists y. One(y) \land s(y,x))$ | ||
| - | * Addition by constant: $(x = y + c) = (\exists y_1,\ldots,y_{c-1}. s(y,y_1) \land s(y_1,y_2) \land \ldots \land s(y_{k-1},x))$ | + | * Addition by constant: $(x = y + c) = (\exists y_1,\ldots,y_{c-1}. s(y,y_1) \land s(y_1,y_2) \land \ldots \land s(y_{c-1},x))$ |
| * Ordering on positions in the string: $(u \leq v) = ((u,v) \in \{(x,y)|s(x,y))\}^*$ | * Ordering on positions in the string: $(u \leq v) = ((u,v) \in \{(x,y)|s(x,y))\}^*$ | ||
| * Reachability in $k$-increments, that is, $\exists k \geq 0. y=x + c\cdot k$: $\mbox{Reach}_c(u,v) = ((u,v) \in \{(x,y)\mid y=x+c\})$ | * Reachability in $k$-increments, that is, $\exists k \geq 0. y=x + c\cdot k$: $\mbox{Reach}_c(u,v) = ((u,v) \in \{(x,y)\mid y=x+c\})$ | ||