Unification Modulo Builtins

Abstract

Combining rewriting modulo an equational theory and SMT solving introduces new challenges in the area of term rewriting. One such challenge is unification of terms in the presence of equations and of uninterpreted and interpreted function symbols. The interpreted function symbols are part of a builtin model which can be reasoned about using an SMT solver. In this article, we formalize this problem, that we call unification modulo builtins. We show that under reasonable assumptions, complete sets of unifiers for unification modulo builtins problems can be effectively computed by reduction to usual E-unification problems and by relying on an oracle for SMT solving.

A Maude Prototype

Since the algorithm needs to manipulate terms (for instance, to compute the abstractions) we use the metalevel capabilities of Maude to implement the following functionalities:

• For a given meta-representation of a term, getAbstraction returns the abstraction of a term w.r.t. the set of builtins sorts, which need to be provided explicitly. When computing abstractions, fresh variables are generated;

• The E-unifiers of the abstractions are computed by unifyAbstractions;

• The unsatisfiable formulas are filtered out by filterUnsatUMBs;

• The E-unification modulo builtins algorithm that we propose is implemented by completeSetOfUMBs, which gets as input the meta-representations of two terms and returns the unifiers modulo builtins in two steps: it first generates the substitution-formula pairs with completeSetOfUMBsUnfiltered, and then it eliminates the unsatisfiable solutions using the aforementioned filterUnsatUMBs.

We show how to use our prototype to find the E-unifiers modulo builtins of the E-unification modulo builtins problem $\textsf {n} \mapsto 1, \textsf {cnt} \mapsto 42 =Z, \textsf {n} \mapsto 1$, introduced in Example 12. First, Maude finds a complete set of ACI-unifiers for the abstractions of the two terms. Then, from these unifiers we generate the pairs $\{(\tau _1', \phi _1'), (\tau _2', \phi _2') \}$, as shown in Example 12):

The variables abs0, abs1, ..., are generated during the abstraction process, while %1, %2, ...are generated by the Maude’s variant unifier.

Finally, completeSetOfUMBs – the main function in our prototype – filters out the first unifier, which has an unsatisfiable constraint. Because the interaction between Maude and the SMT solver only supports integers and booleans, we have encoded identifiers (of sort Id) into integers before sending the formula to the SMT solver. The solution is:

The result is essentially that the variable Z must be $\textsf {cnt} \mapsto 42$, since %1 = abs2:Id = count and %2 = abs3:Id = 42.

We now show how our prototype solves the four E-unification modulo builtins problems discussed in the Introduction:

• 1. $\textsf {n} \mapsto N =\textsf {cnt} \mapsto C' + N', \textsf {n} \mapsto N' + 3$

• 2. $\textsf {n} \mapsto 2 \times N + 1, \textsf {cnt} \mapsto C =\textsf {cnt} \mapsto C' + N', \textsf {n} \mapsto N' + 3$

• 3. $\textsf {n} \mapsto 2 \times N, \textsf {cnt} \mapsto C =\textsf {cnt} \mapsto C' + N', \textsf {n} \mapsto N' + 3$

• 4. $\textsf {n} \mapsto 1, \textsf {cnt} \mapsto C =\textsf {cnt} \mapsto C' + N', \textsf {n} \mapsto N' + 3$

The set of unifiers modulo builtins is computed by our Maude prototype for each case as shown below:

For the first E-umb problem, the tool returns noUMBResults, which means that it does not have any solution. For the other examples, the prototype finds the unifiers, as expected.