Thursday, April 12, 2012

quantifier elimination over bit vectors produces overly complicated results

I am using the ELIM_QUANTIFIERS feature of Z3 to eliminate quantifiers from formulas over bit vectors. I ran into the following situation where Z3 produces a correct, yet overly complicated result, and I was wondering whether there exists a way to re-write my problem or perhaps a configuration option that would lead to the simple result I expect.

First, here is an example that works as expected. It states that for a bit vector of length 4 there exists a bit vector that is equal to it.

(set-option ELIM_QUANTIFIERS true)
(declare-fun a () BitVec[4])
(simplify (exists ((x BitVec[4]))
(= a x)))

Z3 generates the following output for this example.


However, if I add a negation:

(set-option ELIM_QUANTIFIERS true)
(declare-fun a () BitVec[4])
(simplify (exists ((x BitVec[4]))
(not (= a x))))

then Z3 produces the following output, which lists all possible values of the vector instead of just returning "true".

(let (($x54 (= (_ bv0 4) a)))
(let (($x55 (not $x54)))
(let (($x61 (= (_ bv2 4) a)))
(let (($x62 (not $x61)))
(let (($x68 (= (_ bv6 4) a)))
(let (($x69 (not $x68)))
(let (($x75 (= (_ bv4 4) a)))
(let (($x76 (not $x75)))
(let (($x82 (= (_ bv12 4) a)))
(let (($x83 (not $x82)))
(let (($x89 (= (_ bv8 4) a)))
(let (($x90 (not $x89)))
(let (($x95 (= (_ bv1 4) a)))
(let (($x96 (not $x95)))
(let (($x102 (= (_ bv5 4) a)))
(let (($x103 (not $x102)))
(let (($x109 (= (_ bv13 4) a)))
(let (($x110 (not $x109)))
(let (($x116 (= (_ bv9 4) a)))
(let (($x117 (not $x116)))
(let (($x123 (= (_ bv3 4) a)))
(let (($x124 (not $x123)))
(let (($x130 (= (_ bv7 4) a)))
(let (($x131 (not $x130)))
(let (($x137 (= (_ bv14 4) a)))
(let (($x138 (not $x137)))
(let (($x144 (= (_ bv10 4) a)))
(let (($x145 (not $x144)))
(let (($x151 (= (_ bv11 4) a)))
(let (($x152 (not $x151)))
(let (($x158 (= (_ bv15 4) a)))
(let (($x159 (not $x158)))
(or $x159 $x152 $x145 $x138 $x131 $x124 $x117 $x110 $x103 $x96 $x90 $x83 $x76 $x69 $x62 $x55)))))))))))))))))))))))))))))))))

For longer bit vectors, e.g., of size 32 or more, Z3 does not produce a result in a reasonable time, as it is presumably enumerating all possible values of the 32-bit variable.

Note that in this particular case I could just use check-sat to check the validity of the formula; however in the general case I am interested in obtaining a quantifier-free expression equivalent to the original formula, rather than just checking its validity.

I am using Z3 v3.2 for Linux.

1 comment:

  1. Thanks this is a good simple example showing a limitation with the approach used for bit-vectors. It is really only appropriate for formulas where the number of solutions is relatively small. There is no way to toggle bit-vectors on/off during elimination this way. It is a valid point to improve on in Z3. There are a couple of potential work-arounds: First of all, the Boolean quantifier elimination, while it is also rudimentary (it just enumerates satisfiable instances and eliminates the Boolean variable) it does produce the simpler result 'true'. when you bit-blast your bit-vector from this example. A second avenue is to consider whether your problems can be reformulated in the UFBV fragment where satisfiability does not require quantifier elimination.