LARA

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Idea of Quantifier-Free Combination

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) - congruence closure
  • term algebras (interpreted over Herbrand model) - unification
  • real linear arithmetic - linear programming such as Simplex
  • integer linear arithmetic - integer linear programming (branch and bound, branch and cut), reduction to SAT

We would like to separate a quantifier-free formula into constraints that talk only about individual theories, and solve each constraint separately.

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 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

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.