Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision | |||
|
sav08:deciding_quantifier-free_fol_over_ground_terms [2008/05/19 14:08] vkuncak |
sav08:deciding_quantifier-free_fol_over_ground_terms [2015/04/21 17:30] (current) |
||
|---|---|---|---|
| Line 10: | Line 10: | ||
| Is the following formula satisfiable ? | Is the following formula satisfiable ? | ||
| - | \[ | + | \begin{equation*} |
| a=f(a) \land a \neq b | a=f(a) \land a \neq b | ||
| - | \] ++|No. When interpreting variables as constants, two syntactically different terms are different ($a \neq f(a)$).++ | + | \end{equation*} ++|No. When interpreting variables as constants, two syntactically different terms are different ($a \neq f(a)$).++ |
| And the following ? | And the following ? | ||
| - | \[ | + | \begin{equation*} |
| (f(a)=b \lor (f(f(f(a))) = a \land f(a) \neq f(b))) \land f(f(f(f(a)) \neq b \land b=f(b) | (f(a)=b \lor (f(f(f(a))) = a \land f(a) \neq f(b))) \land f(f(f(f(a)) \neq b \land b=f(b) | ||
| - | \] | + | \end{equation*} |
| ===== Axioms for Term Algebras ===== | ===== Axioms for Term Algebras ===== | ||
| Term algebra axioms for function symbols (recall [[Unification]]): | Term algebra axioms for function symbols (recall [[Unification]]): | ||
| - | \[ | + | \begin{equation*} |
| \forall x_1,\ldots,x_n,y_1,\ldots,y_n.\ f(x_1,\ldots,x_n)=f(y_1,\ldots,y_n) \rightarrow \bigwedge_{i=1}^n x_i=y_i | \forall x_1,\ldots,x_n,y_1,\ldots,y_n.\ f(x_1,\ldots,x_n)=f(y_1,\ldots,y_n) \rightarrow \bigwedge_{i=1}^n x_i=y_i | ||
| - | \] | + | \end{equation*} |
| and | and | ||
| - | \[ | + | \begin{equation*} |
| \forall x_1,\ldots,x_n,y_1,\ldots,y_m.\ f(x_1,\ldots,x_n) \neq g(y_1,\ldots,y_m) | \forall x_1,\ldots,x_n,y_1,\ldots,y_m.\ f(x_1,\ldots,x_n) \neq g(y_1,\ldots,y_m) | ||
| - | \] | + | \end{equation*} |
| for $g$ and $f$ distinct symbols. | for $g$ and $f$ distinct symbols. | ||
| Unless the language is infinite, we have that each element is either a constant or is result of application of some function symbols: | Unless the language is infinite, we have that each element is either a constant or is result of application of some function symbols: | ||
| - | \[ | + | \begin{equation*} |
| \forall x. \bigvee_{f \in {\cal L}} \exists y_1,\ldots,y_{ar(f)}.\ x=f(y_1,\ldots,y_{ar(f)}) | \forall x. \bigvee_{f \in {\cal L}} \exists y_1,\ldots,y_{ar(f)}.\ x=f(y_1,\ldots,y_{ar(f)}) | ||
| - | \] | + | \end{equation*} |
| Additional axiom schema for finite terms: | Additional axiom schema for finite terms: | ||
| - | \[ | + | \begin{equation*} |
| \forall x. t(x) \neq x | \forall x. t(x) \neq x | ||
| - | \] | + | \end{equation*} |
| if $t(x)$ is a term containing $x$ but not identical to $x$. | if $t(x)$ is a term containing $x$ but not identical to $x$. | ||
| Line 46: | Line 46: | ||
| Represent each disequation $t_1 \neq t_2$ as | Represent each disequation $t_1 \neq t_2$ as | ||
| - | \[ | + | \begin{equation*} |
| x_1 = t_1 \land x_2 = t_2 \land x_1 \neq x_2 | x_1 = t_1 \land x_2 = t_2 \land x_1 \neq x_2 | ||
| - | \] | + | \end{equation*} |
| where $x_1,x_2$ are fresh variables. | where $x_1,x_2$ are fresh variables. | ||
| Line 76: | Line 76: | ||
| Represent $cons(x,y)=z$ by | Represent $cons(x,y)=z$ by | ||
| - | \[ | + | \begin{equation*} |
| x=car(x) \land y = cdr(z) \land \lnot atom(z) | x=car(x) \land y = cdr(z) \land \lnot atom(z) | ||
| - | \] | + | \end{equation*} |
| here | here | ||
| * $atom(x)$ is negation of $Is_{cons}$ | * $atom(x)$ is negation of $Is_{cons}$ | ||