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sav08:idea_of_quantifier-free_combination [2008/04/24 14:08]
vkuncak
sav08:idea_of_quantifier-free_combination [2015/04/21 17:30] (current)
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====== Idea of Quantifier-Free Combination ====== ====== Idea of Quantifier-Free Combination ======

-We wish to reason about quantifier-free ​formulas that contain different symbols, such as+We know of several classes of formulas that we can decide:
* ground formulas interpreted over arbitrary functions and relations (also called uninterpreted function symbols) - congruence closure   * ground formulas interpreted over arbitrary functions and relations (also called uninterpreted function symbols) - congruence closure
+    * $f(f(f(f(a)))) \neq a \land f(f(a))=a$
* term algebras (interpreted over Herbrand model) - unification   * term algebras (interpreted over Herbrand model) - unification
+    * $cons(a,​b)=cons(c,​d) \land a \neq c$
* real linear arithmetic - linear programming such as Simplex   * real linear arithmetic - linear programming such as Simplex
+    * $x < y \land y < x + 1$
* integer linear arithmetic - integer linear programming (branch and bound, branch and cut), reduction to SAT   * integer linear arithmetic - integer linear programming (branch and bound, branch and cut), reduction to SAT
+    * $x < y \land y < x + 1$

-We would like to separate a quantifier-free ​formula into constraints ​that talk only about individual theories, and solve each constraint separately.+We wish to reason about quantifier-free ​formulas ​that contain all these different symbols in the same formula.

-We are checking satisfiability. $F$ is satisfiable iff each disjunct in its disjunctive normal form is satisfiable. ​ We therefore consider conjunctions of literals.+The idea is to separate a quantifier-free formula into constraints that talk only about individual theories, and solve each constraint separately.

-Consider a conjunction of literals $C$.  If we can group literals into blocks, $C \leftrightarrow C_1 \land \ldots \land C_n$ and +We are checking satisfiability. $F$ is satisfiable iff each disjunct in its disjunctive normal form is satisfiable. ​
-  if one of the $C_i$ is unsatisfiable,​ then $C$ is unsatisfiable+\begin{equation*}
+    F \ \ \leftrightarrow \ \ \bigvee_{i=1}^n C_i
+\end{equation*}
+We therefore consider conjunctions of literals $C_i$.
+
+Consider a conjunction of literals $C$.  If we can group literals into blocks ​
+\begin{equation*}
+    ​C \leftrightarrow C_1 \land \ldots \land C_n
+\end{equation*
+If one of the $C_i$ is unsatisfiable,​ then $C$ is unsatisfiable.

The idea is to separate conjuncts into those specific to individual theories, and then solve each $C_i$ using a specialized decision procedure $P_i$ The idea is to separate conjuncts into those specific to individual theories, and then solve each $C_i$ using a specialized decision procedure $P_i$

-An important question is completeness:​ if each $C_i$ is satisfiable,​ is $C_1 \land \ldots \land C_n$ satisfiable? ​ We will show that, under certain conditions, this holds.+An important question is **completeness**: if each $C_i$ is satisfiable,​ is $C_1 \land \ldots \land C_n$ satisfiable? ​ We will show that, under certain conditions, this holds.