I encountered a problem while trying to define some rules over bit vectors in z3. If I try to solve the following file with z3
(declare-rel FunnyFun ((_ BitVec 64)))
(declare-var A (_ BitVec 64))
(declare-var B (_ BitVec 64))
(rule (=> (= B (bvadd A #x0000000000000001))
(FunnyFun B)))
(declare-rel q1 ())
(rule (=> (FunnyFun #x0000000000000001) q1))
(query q1)
I get the error
(error "query failed: Rule contains infinite sorts in rule <null>:
FunnyFun(#0) :-
(= (:var 0) (bvadd (:var 1) #x0000000000000001)).
")
Funnily enough z3 instantly gives the expected result (sat) when using smaller bit widths:
(declare-rel FunnyFun ((_ BitVec 60)))
(declare-var A (_ BitVec 60))
(declare-var B (_ BitVec 60))
(rule (=> (= B (bvadd A #x000000000000001))
(FunnyFun B)))
(declare-rel q1 ())
(rule (=> (FunnyFun #x000000000000001) q1))
(query q1)
Is this a bug or am I missing some constraints (I assumed BitVec can have an arbitrary bit width)?
I tried different z3 version (4.6.0, 4.8.3 and 4.8.5) and all of them showed this behavior.
This limitation is by design. See here: https://github.com/Z3Prover/z3/issues/1698
Related
I have a set of optimization problems in QF_BV. How can I efficiently solve them? For example, how can I use (check-sat-using (then simplify bit-blast) ...?
What follows is a fragment of a problem.
(declare-const p1_1 (_ BitVec 36))
(declare-const ps1_1 (_ BitVec 36))
(declare-const pe1_1 (_ BitVec 36))
(assert (= (_ bv0 36) (bvand ps1_1 (bvadd ps1_1 (_ bv1 36)))))
(assert (= (_ bv0 36) (bvand pe1_1 (bvadd pe1_1 (_ bv1 36)))))
(assert (and
(= p1_1 (bvxor ps1_1 pe1_1))
(not (= (_ bv0 36) (bvlshr ps1_1 (_ bv4 36))))
(=> (not (= p1_1 (_ bv0 36))) (= (concat ps1_1 #b1) (bvlshr (concat pe1_1 #b1) (_ bv4 37))))
))
(assert-soft (not (= p1_1 (_ bv0 36))) :weight 1)
(check-sat)
(get-model)
(get-objectives)
The optimizing engine (i.e., the engine that's used when you have assert-soft) doesn't work with tactics as far as I'm aware. It has its own solver and thus doesn't benefit from the general machinery of tactics.
You might want to ask and double check at https://github.com/Z3Prover/z3/discussions to see if there might be some other trick to orchestrate the behavior of the optimizing solver. (Though I doubt such a facility exists as of today.) Please report back here what you find!
I am using the inbuilt interpolation feature on the latest (unstable branch) version of Z3. It works fine with SMT2 formulas containing integers. It does however, throw a iz3proof_itp::proof_error and a subsequent iz3translate::unsupported error (See Below) for the following SMT2 program -
(set-option :produce-models true)
(set-logic QF_AUFBV)
(declare-fun a () (_ BitVec 32))
(declare-fun b () (_ BitVec 32))
(declare-fun x1 () (_ BitVec 32))
(declare-fun x2 () (_ BitVec 32))
(declare-fun x3 () (_ BitVec 32))
(declare-fun y1 () (_ BitVec 32))
(declare-fun y2 () (_ BitVec 32))
(compute-interpolant
(= a (_ bv0 32))
(= b (bvneg (_ bv2 32)))
(= x1 (_ bv1 32))
(= y1 (_ bv0 32))
(= x2 (bvadd x1 a))
(= x3 (bvadd x2 b))
(= y2 (bvadd y1 a))
(bvsge x3 (_ bv0 32))
)
I tried it on the online version on rise4fun, and it worked fine. So after a bit of debugging, I found that the error is thrown from inside the function find_congruence_position in file iz3proof_itp.cpp.
So I made the following simple (maybe dangerous) change to the function to take care of the proof_error atleast for now -
Changing if(x == arg(arg(con,0),i) && (y == arg(arg(con,1),i)) at line 2431
to if((x == arg(arg(con,0),i) && (y == arg(arg(con,1),i))) || (y == arg(arg(con,0),i) && (x == arg(arg(con,1),i))))
I simply or'd the condition with its copy where x and y are interchanged - I had found that x and y sometimes have their values interchanged, maybe due to some proof techniques.
This did take care of the problem, I found that using equality and non-equality, along with bvadd or bvneg with BitVecs while computing interpolants worked. For example the following file worked -
(set-option :produce-models true)
(set-logic QF_AUFBV)
(declare-fun a () (_ BitVec 32))
(declare-fun b () (_ BitVec 32))
(compute-interpolant
(= a (_ bv0 32))
(= b (bvadd a (_ bv1 32)))
(= b (_ bv0 32))
)
But then I tried using relational operators, like bvsgt or bvsge, and it threw a new error -
terminate called after throwing an instance of 'iz3translation::unsupported'
I looked more into it, found out that the expression causing the problem was -
(not ((_ bit2bool 2) x2)) - it was assigned a PR_TH_LEMMA type and the UNKNOWN_THEORY kind. It seems that there is no support for such operations.
Since the online version is working, I'd like to know if it's possible to obtain that version. I read the previous questions and answers on StackOverflow, and I'm a bit confused. Some say BitVec theory is not supported (Although these posts are old), but then how is the online version working? Or am I doing something wrong? Any help is highly appreciated.
Actually, interpolation for bit vector arithmetic is not supported. It's interesting that it works on the web version. That version is quite old, however, and I think it predates the source-available version of Z3. I can think of two possible reasons why it works:
1) That version of interpolating Z3 used the "foci" prover as a backup. Whenever it encountered a part of the proof it didn't understand, it would package that part up as a lemma and reprove it using foci. The current version does not use foci (which is a third-party tool not available in source) and relies entirely on the proof generated by Z3.
2) The older version might have been doing bit-blasting in a different way. If all the non-local proof steps are purely propositional (using just the resolution rule) then it is easy to compute an interpolant.
As far as I understand it, however, computing interpolants from an efficient bit-vector solver (using all the known pre-processing tricks) is an open problem.
How does one use the 'repeat' and 'rotate_left' bitvector operations?
More generally, where can I find detailed documentation of bitvector operations in the SMT2 scripting format used by Z3?
Everything I find seems to just go to tutorials, or broken links:
https://github.com/Z3Prover/z3/wiki/Documentation
http://research.microsoft.com/en-us/um/redmond/projects/z3/old/documentation.html
Trying to understand "repeat", "rotate_left", and "rotate_right" by guessing has been frustating. I cannot figure out how to use them. For example
(display (repeat #b01))
(display (repeat #b01 3))
(display (repeat 3))
(display (rotate_left #b0001 2))
gives
"repeat expects one non-zero integer parameter"
"repeat expects one argument"
"operator is applied to arguments of the wrong sort"
"rotate left expects one argument"
Where is the documentation? Hoping they didn't explain because all of this is standard, I also looked at smt-lib.org but that doesn't list these details either. So frustrating.
In addition to dejvuth's answer:
The SMT language is well documented (see smt-lib.org), for this particular issue the FixedSizeBitVectors theory and the QF_BV logic definition are relevant. The latter contains the definition for repeat:
((_ repeat i) (_ BitVec m) (_ BitVec i*m))
- ((_ repeat i) x) means concatenate i copies of x
Apart from those, David Cok wrote an excellent SMT2 tutorial.
The names of functions in the Z3 API is the same as in SMT2 where syntax permits, in this case prefixed with Z3_mk_ to indicate that they are functions that construct Z3 expressions.
For your example, you should write something like this
(declare-const a (_ BitVec 2))
(declare-const b (_ BitVec 6))
(assert (= a #b01))
(assert (= b ((_ repeat 3) a)))
(declare-const c (_ BitVec 4))
(declare-const d (_ BitVec 4))
(assert (= c #b0001))
(assert (= d ((_ rotate_left 2) c)))
(check-sat)
(get-model)
You will get
sat
(model
(define-fun d () (_ BitVec 4)
#x4)
(define-fun c () (_ BitVec 4)
#x1)
(define-fun b () (_ BitVec 6)
#b010101)
(define-fun a () (_ BitVec 2)
#b01)
)
A good document that I usually use is its API.
(declare-const a Int)
(declare-const b Int)
(declare-const c (_ BitVec 32))
(declare-const d (_ BitVec 32))
(assert (= b (bv2int c)))
(assert (= c (int2bv a)))
(check-sat)
I am confused about the exception "int2bv expects one parameter" caused by the code above, how to use function int2bv correctly?
This is because int2bv is a parametric function and the SMT2 syntax for these is (_ f p1 p2 ...), so in this case the correct syntax is
((_ int2bv 32) a)
Note that int2bv is essentially treated as uninterpreted; the API documentation says:
"NB. This function is essentially treated as uninterpreted. So you cannot expect Z3 to precisely reflect the semantics of this function when solving constraints with this function." (from here)
I'm playing around with Z3's QBVF solver, and wondering if it's possible to extract values from an existential assertion. To wit, let's say I have the following:
(assert (exists ((x (_ BitVec 16))) (forall ((y (_ BitVec 16))) (bvuge y x))))
This basically says that there is a "least" 16-bit unsigned value. Then, I can say:
(check-sat)
(get-model)
And Z3-3.0 happily responds:
sat
(model (define-fun x!0 () (_ BitVec 16)
#x0000)
)
Which is really cool. But what I want to do is to be able to extract pieces of that model via get-value. Unsurprisingly, none of the following seem to work
(get-value (x))
(get-value (x!0))
In each case Z3 rightly complains there's no such constant. Clearly Z3 has that information as evidenced by the response to the (check-sat) call. Is there any way to access the existential value automatically via get-value, or some other mechanism?
Thanks..
In Z3, get-value only allows the user to reference “global” declarations.
The existential variable x is a local declaration. Thus, it can’t be accessed using get-value.
By default, Z3 eliminates existential variables using a process called “skolemization”.
The idea is to replace existential variables with fresh constants and function symbols.
For example, the formula
exists x. forall y. exists z. P(x, y, z)
is converted into
forall y. P(x!1, y, z!1(y))
Note that z becomes a function because the choice of z may depend on y.
Wikipedia has an entry on skolem normal form
That being said, I never found a satisfactory solution for the problem you described.
For example, a formula may have many different existential variables with the same name.
So, it is not clear how to reference each instance in the get-value command in a non-ambiguous way.
A possible workaround for this limitation is to apply the skolemization step “by hand”, or at least for the variables you want to know the value.
For example,
(assert (exists ((x (_ BitVec 16))) (forall ((y (_ BitVec 16))) (bvuge y x))))
is written as:
(declare-const x (_ BitVec 16))
(assert (forall ((y (_ BitVec 16))) (bvuge y x)))
(check-sat)
(get-value x)
If the existential variable is nested in a universal quantifier such as:
(assert (forall ((y (_ BitVec 16))) (exists ((x (_ BitVec 16))) (bvuge y x))))
(check-sat)
(get-model)
A fresh skolem function can be used to obtain the value of x for each y.
The example above becomes:
(declare-fun sx ((_ BitVec 16)) (_ BitVec 16))
(assert (forall ((y (_ BitVec 16))) (bvuge y (sx y))))
(check-sat)
(get-model)
In this example, sx is the fresh function. The model, produced by Z3, will assign an interpretation for sx. In version 3.0, the interpretation is the identity function. This function can be used to obtain the value of x for each y.