Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision | ||
sav08:idea_of_quantifier-free_combination [2008/04/24 13:50] vkuncak |
sav08:idea_of_quantifier-free_combination [2009/05/13 10:37] vkuncak |
||
---|---|---|---|
Line 1: | Line 1: | ||
====== 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) | + | * ground formulas interpreted over arbitrary functions and relations (also called uninterpreted function symbols) - congruence closure |
- | * term algebras (interpreted over Herbrand model) | + | * $f(f(f(f(a)))) \neq a \land f(f(a))=a$ |
- | * integer linear arithmetic | + | * term algebras (interpreted over Herbrand model) - unification |
- | * real linear arithmetic | + | * $cons(a,b)=cons(c,d) \land a \neq c$ |
+ | * 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 | ||
+ | * $x < y \land y < x + 1$ | ||
- | 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 wish to reason about quantifier-free formulas that contain all these different symbols in the same formula. |
- | 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. | + | 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 literlas $C$. If we can represent it as separate conjunction $C_1$,...,$C_n$ then | + | 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 | + | \[ |
+ | F \ \ \leftrightarrow \ \ \bigvee_{i=1}^n C_i | ||
+ | \] | ||
+ | We therefore consider conjunctions of literals $C_i$. | ||
+ | |||
+ | Consider a conjunction of literals $C$. If we can group literals into blocks | ||
+ | \[ | ||
+ | C \leftrightarrow C_1 \land \ldots \land C_n | ||
+ | \] | ||
+ | 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. |