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sav08:idea_of_quantifier-free_combination [2009/05/13 10:37] vkuncak |
sav08:idea_of_quantifier-free_combination [2015/04/21 17:30] |
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| - | ====== Idea of Quantifier-Free Combination ====== | ||
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| - | 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 | ||
| - | * $f(f(f(f(a)))) \neq a \land f(f(a))=a$ | ||
| - | * 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 | ||
| - | * $x < y \land y < x + 1$ | ||
| - | * integer linear arithmetic - integer linear programming (branch and bound, branch and cut), reduction to SAT | ||
| - | * $x < y \land y < x + 1$ | ||
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| - | We wish to reason about quantifier-free formulas that contain all these different symbols in the same formula. | ||
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| - | The idea is to separate a quantifier-free formula into constraints that talk only about individual theories, and solve each constraint separately. | ||
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| - | We are checking satisfiability. $F$ is satisfiable iff each disjunct in its disjunctive normal form is satisfiable. | ||
| - | \[ | ||
| - | F \ \ \leftrightarrow \ \ \bigvee_{i=1}^n C_i | ||
| - | \] | ||
| - | We therefore consider conjunctions of literals $C_i$. | ||
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| - | 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. | ||
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| - | The idea is to separate conjuncts into those specific to individual theories, and then solve each $C_i$ using a specialized decision procedure $P_i$ | ||
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| - | 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. | ||