A Simple Sound Combination Method

The following method need not be complete, but applies to any formulas, not just quantifier-free formulas.

Suppose we wish to unsatisfiability of a class of formulas ${\cal F}$ and we have provers $P_i$ which can prove formulas in language ${\cal L}_i$ for $1 \le i \le n$.

Suppose each prover has approximation function $\alpha_i : {\cal F} \to {\cal F}_i$ such that

  • $\alpha_i(F) \in {\cal L}_i$
  • $F \models \alpha_i(F)$

In other words, $\alpha_i$ approximates an arbitrary formula with a weaker formula in language understood by $P_i$.

To prove formula $F$, for each $i$, apply $P_i$ to check satisfiability of $\alpha_i(F)$.

  • If any of the provers concludes that formula is unsatisfiable, then $F$ is unsatisfiable (soundness)
  • If no prover finds unsatisfiability, we cannot say much in general, depending on the approximation functions

Improvement of precision: if $F$ is equivalent to disjunction $\vee_{j=1}^m D_j$ then to prove $F$ unsatisfiable, for each $1 \le j \le m$ check that $D_j$ is unsatisfiable by applying each of the provers.

Defining Approximations

We can define approximation for arbitrary first-order formulas.

Start with $\alpha^1$, flip on negation.

$\alpha^p(F_1 \land F_2) = \alpha^p(F_1) \land \alpha^p(F_2)$

$\alpha^p(F_1 \lor F_2) = \alpha^p(F_1) \lor \alpha^p(F_2)$

$\alpha^p(\lnot F) = \lnot \alpha^{1-p}(F)$

$\alpha^p(\forall x.F) = \forall x. \alpha^p(F)$

$\alpha^p(\exists x.F) = \exists x. \alpha^p(F)$

$\alpha^p(A) = A$ for any supported atomic formula

$\alpha^1(A) = true$ for unsupported atomic formula

$\alpha^0(A) = false$ for unsupported atomic formula

Lemma: $\alpha^0(F) \models F \models \alpha^1(F)$

What does $\alpha^1$ do on a conjunction of flat literals?


Consider decision procedure $P_U$ for uninterpreted symbols and $P_A$ for Presburger arithmetic. Let approximation $\alpha_U$ convert formula to negation normal form, and then drop all literals that contain arithmetic elements. Similarly, $\alpha_A$ converts to NNF, then drops all literlas that contain symbols not presented in Presburger arithmetic (e.g. uninterpreted function symbols).

Example (from Calculus of Computation Textbook, page 210, Example 10.1): Let $F$ be

    1 \le x \land x \le 2 \land f(x) \neq f(1) \land f(x) \neq f(2)

This formula is unsatisfiable. However,

   \alpha_U(F) = true

   \alpha_A(F) = 1 \le x \land x \le 2

and both approximations are satisfiable in resulting theories.

Instead of doing approximation directly, let us transform the original formula into formula $F'$ where function application and arithmetic are separated:

   1 \le x \land x \le 2 \land y_1=1 \land f(x) \neq f(y_1) \land y_2=2 \land f(x) \neq f(y_2)

Variables $x$ and $y_1$ both appear in the formula, so let us convert $F'$ into disjunction of $(F' \land x=y_1) \lor (F' \land x \neq y_1)$. We then check satisfiability for each of the disjuncts. Consider, for example, $F' \land x = y_1$, which is formula $F''$

   1 \le x \land x \le 2 \land y_1=1 \land f(x) \neq f(y_1) \land y_2=2 \land f(x) \neq f(y_2) \land x = y_1

We have

   \alpha_U(F'') = f(x) \neq f(y_1) \land f(x) \neq f(y_2) \land x=y_1

which is unsatisfiable. Then, considering formula $F' \land x=y_1$, we can do further case analysis on equality of two variables, say $x=y_2$. For $F' \land x \neq y_1 \land x = y_2$ we similarly obtain unsatisfiability of $\alpha_U$-approximation. The remaining case is $F' \land x \neq y_1 \land x \neq y_2$. For this formula, denoted $F'''$, we have

   \alpha_A(F''') = 1 \le x \land x \le 2 \land y_1=1 \land y_2=2 \land x \neq y_1 \land x \neq y_2

which is unsatisfiable. Therefore, by transforming formula into disjunction of formulas, we were able to prove unsatisfiability. Two things helped precision

  • separating literals into literals understood by individual theories (unlike mixed literals $f(x) \neq f(1)$)
  • doing case analysis on equality of variables

These are the key techniques that we use in methods that are complete for quantifier-free combinations of procedures that reason about disjoint function and predicate symbols.