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sav08:idea_of_quantifier-free_combination [2008/04/23 10:13] vkuncak created |
sav08:idea_of_quantifier-free_combination [2008/04/24 14:08] vkuncak |
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| We wish to reason about quantifier-free formulas that contain different symbols, such as | We wish to reason about quantifier-free formulas that contain different symbols, such as | ||
| - | * ground formulas interpreted over arbitrary functions and relations (also called uninterpreted function symbols) | + | * ground formulas interpreted over arbitrary functions and relations (also called uninterpreted function symbols) - congruence closure |
| - | * term algebras (interpreted over Herbrand model) | + | * term algebras (interpreted over Herbrand model) - unification |
| - | * integer linear arithmetic | + | * real linear arithmetic - linear programming such as Simplex |
| - | * real linear arithmetic | + | * integer linear arithmetic - integer linear programming (branch and bound, branch and cut), reduction to SAT |
| - | For this, we would like to separate a quantifier-free formula into constraints that talk only about individual theories, and solve each constraint separately. | + | We would like to separate a quantifier-free formula into constraints that talk only about individual theories, and solve each constraint separately. |
| - | Note that we are checking satisfiability, and $F$ is satisfiable iff each disjunct in its disjunctive normal form is satisfiable. We therefore consider conjunctions of literals. | + | We are checking satisfiability. $F$ is satisfiable iff each disjunct in its disjunctive normal form is satisfiable. We therefore consider conjunctions of literals. |
| - | Consider a conjunction of literlas $C$. If we can represent it as separate conjunction $C_1$,...,$C_n$ then | + | Consider a conjunction of literals $C$. If we can group literals into blocks, $C \leftrightarrow C_1 \land \ldots \land C_n$ and |
| * if one of the $C_i$ is unsatisfiable, then $C$ is unsatisfiable | * 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 them using specialized decision procedures. | + | 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 $C_1 \land \ldots C_n$ 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. |