Summary:
For some Exp forms, Exp.solve is not complete, and this is necessary
since the result of solve is a substitution that needs to encode the
input equation as a conjunction of equations each between a variable
and an exp. This is tantamount to, stronger even than, the theory
being convex. So Exp.solve is not complete for some exps, and some of
those have constructors that perform some simplification. For example,
`(1 ≠ 0)` simplifies to `-1` (i.e. true). To enable deductions such as
`((x ≠ 0) = b) && x = 1 |- b = -1` needs the equality solver to
substitute through subexps of simplifiable exps like = and ≠, as it
does for interpreted exps like + and ×. At the same time, since
Exp.solve for non-interpreted exps cannot be complete, to enable
deductions such as `((x ≠ 0) = (y ≠ 0)) && x = 1 |- y ≠ 0` needs the
equality solver to congruence-close over subexps of simplifiable exps
such as = and ≠, as it does for uninterpreted exps.
To strengthen the equality solver in these sorts of cases, this diff
adds a new class of exps for = and ≠, and revises the equality solver
to handle them in a hybrid fashion between interpreted and
uninterpreted.
I am not currently sure whether or not this breaks the termination
proof, but I have also not managed to adapt usual divergent cases to
break this. One notable point is that simplifying = and ≠ exps always
produces genuinely simpler and smaller exps, in contrast to
e.g. polynomial simplification and gaussian elimination.
Note that the real solution to this problem is likely to be to
eliminate the i1 type in favor or a genuine boolean type, and
translate all integer operations on i1 to boolean/logical ones. Then
the disjunction implicit in e.g. equations between disequations would
appear as actual disjunction, and could be dealt with as such.
Reviewed By: jvillard
Differential Revision: D15424823
fbshipit-source-id: 67d62df1f
Summary:
Interpreted subexps were not handled correctly: they must not be in
the carrier, but their non-interpreted subexps should.
Reviewed By: jvillard
Differential Revision: D14344291
fbshipit-source-id: 995896640
Summary:
- Change representation of Concat expressions from curried binary
operator to an nary one. This enables normalizing Concat expressions
with respect to associativity.
- Generalize Exp.solve to return a map rather than a pair of exps, to
enable expressing cases where solving an equation yields multiple
equations.
- Strengthen solver with implied equalities between sums of sizes of
concatenations of byte arrays.
Reviewed By: ngorogiannis
Differential Revision: D14297865
fbshipit-source-id: b40871559
Summary:
This patch adds an embarrassingly inefficient implementation of a
decision procedure for equality in the theories of uninterpreted
functions and linear arithmetic. A Shostak-style approach, where a
single congruence closure structure is shared by all theories, is
used. This is mostly based on the corrected variant of Shostak's
algorithm from:
Harald Rueβ and Natarajan Shankar. 2001. Deconstructing Shostak. In
Proceedings of the 16th Annual IEEE Symposium on Logic in Computer
Science (LICS '01).
Reviewed By: jvillard
Differential Revision: D14251655
fbshipit-source-id: 8a080145f
Josh Berdine