I have tried to solve 2^x=4 with Z3, by putting the following on the Z3 website: https://rise4fun.com/z3/tutorial.
(declare-const x Real)
(declare-const y Real)
(declare-const z Real)
(assert (=(^ 2 x) 4))
(check-sat)
(get-model)
Z3 produced
unknown
(model
)
Am I misusing Z3?
Problems involving exponentials are typically beyond the reach of z3, or general-purpose SMT solvers. This doesn't mean that they cannot solve them: Theory of reals is decidable. But they may not have the right "heuristics" to kick in to answer every query involving exponentials as sat/unsat. You can search stack-overflow for keywords like nnf, non-linear, etc., to see a plethora of questions regarding queries that involve such difficult terms.
Having said that, there's a separate line of research called delta-satisfiability that can help with these sorts of problems to a great extent. Note that delta-satisfiability is different than regular satisfiability. It means either the formula is satisfiable, or a delta-perturbation of it is. The most prominent such solver is dReal, and you can read all about it here: http://dreal.github.io/
For your query, dReal says:
[urfa]~/qq>dreal a.smt2
delta-sat with delta = 0.001
(model
(define-fun x () Real [2, 2])
(define-fun y () Real [-0.0005, 0.0005])
(define-fun z () Real [-0.0005, 0.0005])
)
(You didn't actually use y and z in your query, so you can ignore those outputs.)
As you can see dReal determines x must be in the range [2, 2], i.e., it must be 2. But it also says delta-sat with delta = 0.001: This means it has ensured the correctness within that factor. (You can tweak the factor yourself, making it arbitrarily small, but not zero.) When you have problems that arise from physical systems, delta-sat is the right choice in modeling them in the SMT-world.
Related
I would like to use Z3 to prove the accuracy of certain floating point expressions. My plan was to:
Implement the expression as Reals.
Implement the same expression as Float32().
Use fpRealToFP to create correspondences between the FP and Real consts.
Prove that the results are within some margin of error.
However, despite fpRealToFP existing, it seems to fail on even the most trivial tests.
>>> import z3
>>> r = z3.Real("r")
>>> f = z3.Const("f", z3.Float32())
>>> z3.solve(f > z3.fpRealToFP(z3.RNE(), r, z3.Float32()))
failed to solve
>>> z3.solve(z3.fpToReal(f) > r)
failed to solve
Is there a reasonable way to approach this type of problem in Z3, or is this simply not something that it can do?
One approach that I've considered is to replace the Real part with a higher precision float, i.e. prove that the F32 result is within some MoE of the F128 result. This may be good enough to convince myself of probable correctness, but it wouldn't technically be a proper proof on its own.
Floating-point from/to real conversions are extremely expensive. But I'm surprised z3 can't even handle the example you posted, as there are literally no constraints at all. Here's the SMTLib equivalent:
(set-logic ALL)
(set-option :produce-models true)
(declare-fun r () Real)
(declare-fun f () (_ FloatingPoint 8 24))
(assert (fp.gt f ((_ to_fp 8 24) roundNearestTiesToEven r)))
(check-sat)
(get-model)
z3 responds unknown, but cvc5 does much better on it:
$ cvc5 a.smt2
sat
(
(define-fun r () Real (- 1.0))
(define-fun f () (_ FloatingPoint 8 24) (fp #b0 #b11111110 #b11111111111111111111111))
)
So, you might get better mileage by trying CVC5 on your examples. But keep in mind that once you have complicated constraints, all solvers will have hard time coping with these sorts of problems. Real/FP conversions create highly non-linear terms, and so far as I know no solver can deal with such problems all that well, at least not for the time being.
Given this formula,
(p & (x < 0)) | (~p & (x > 0)).
How could I get these 2 "parametric" models in Z3:
{p=true, x<0}
{p=false, x>0}
When I submit this SMTLIB program to Z3,
(declare-const p Bool)
(declare-const x Int)
(assert (or (and p (< x 0)) (and (not p) (> x 0))))
(check-sat)
(get-model)
(assert (or (not p) (not (= x -1))))
(check-sat)
(get-model)
(exit)
it gives me concrete models instead (e.g. {p=true, x=-1}, {p=true, x=-2}, ...).
You can't.
SMT solvers do not produce non-concrete models; it's just not how they work. What you want is essentially some form of "simplification" in steroids, and while you can use an SMT solver to help you in simplifying expressions, you'll have to build a tool on top that understands the kind of simplifications you'd like to see. Bottom line: What you'd consider "simple" as a person, and what an automated SMT-solver sees as "simple" are usually quite different from each other; and given lack of normal forms over arbitrary theories, you cannot expect them to do a good job.
If these sorts of simplifications is what you're after, you might want to look at symbolic math packages, such as sympy, Mathematica, etc.
There are several posts regarding the conversion of bit vectors
to integers (and vice versa) in z3. See for example here,
here and here.
The documentation says that Z3_mk_bv2int is uninterpreted:
"...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..."
However, I could not find a simple example where it does fail
to reflect the expected semantics.
For instance, whenever I use queries like this:
(declare-const s String)
(declare-const someBitVec10 (_ BitVec 10))
(assert (= s "74g\x00!!#2#$$"))
(assert (str.in.re (str.at s (bv2int someBitVec10)) (re.range "1" "3")))
(check-sat)
(get-value (s someBitVec10))
I get a correct answer (index should 7, and it is)
sat
((s "74g\x00!!#2#$$")
(someBitVec10 #b0000000111))
Could anyone please provide a simple example where z3's
bv2int and/or int2bv fail?? thanks!
This issue is now resolved, as it turns out that both int2bv and bv2int are indeed interpreted. The documentation hasn't been updated, and this might have caused the confusion (at least in my case it did). All the details are in this GitHub/z3/issues post.
Using smtlib I would like to make something like modulo using QF_UFNRA. This disables me from using mod, to_int, to_real an such things.
In the end I want to get the fractional part of z in the following code:
(set-logic QF_UFNRA)
(declare-fun z () Real)
(declare-fun z1 () Real)
(define-fun zval_1 ((x Real)) Real
x
)
(declare-fun zval (Real) Real)
(assert (= z 1.5));
(assert (=> (and (<= 0.0 z) (< z 1.0)) (= (zval z) (zval_1 z))))
(assert (=> (>= z 1.0) (= (zval z) (zval (- z 1.0)))))
(assert (= z1 (zval z)))
Of course, as I am asking this question here, implies, that it didn't work out.
Has anybody got an idea how to get the fractional part of z into z1 using logic QF_UFNRA?
This is a great question. Unfortunately, what you want to do is not possible in general if you restrict yourself to QF_UFNRA.
If you could encode such functionality, then you can decide arbitrary Diophantine equations. You would simply cast a given Diophantine equation over reals, compute the "fraction" of the real solution with this alleged method, and assert that the fraction is 0. Since reals are decidable, this would give you a decision procedure for Diophantine equations, accomplishing the impossible. (This is known as Hilbert's 10th problem.)
So, as innocent as the task looks, it is actually not doable. But that doesn't mean you cannot encode this with some extensions, and possibly have the solver successfully decide instances of it.
If you allow quantifiers and recursive functions
If you allow yourself quantifiers and recursive-functions, then you can write:
(set-logic UFNRA)
(define-fun-rec frac ((x Real)) Real (ite (< x 1) x (frac (- x 1))))
(declare-fun res () Real)
(assert (= (frac 1.5) res))
(check-sat)
(get-value (res))
To which z3 responds:
sat
((res (/ 1.0 2.0)))
Note that we used the UFNRA logic allowing quantification, which is required here implicitly due to the use of the define-fun-rec construct. (See the SMTLib manual for details.) This is essentially what you tried to encode in your question, but instead using the recursive-function-definition facilities instead of implicit encoding. There are several caveats in using recursive functions in SMTLib however: In particular, you can write functions that render your system inconsistent rather easily. See Section 4.2.3 of http://smtlib.cs.uiowa.edu/papers/smt-lib-reference-v2.5-draft.pdf for details.
If you can use QF_UFNIRA
If you move to QF_UFNIRA, i.e., allow mixing reals and integers, the encoding is easy:
(set-logic QF_UFNIRA)
(declare-fun z () Real)
(declare-fun zF () Real)
(declare-fun zI () Int)
(assert (= z (+ zF zI)))
(assert (<= 0 zF))
(assert (< zF 1))
(assert (= z 1.5))
(check-sat)
(get-value (zF zI))
z3 responds:
sat
((zF (/ 1.0 2.0))
(zI 1))
(You might have to be careful about the computation of zI when z < 0, but the idea is the same.)
Note that just because the encoding is easy doesn't mean z3 will always be able to answer the query successfully. Due to mixing of Real's and Integer's, the problem remains undecidable as discussed before. If you have other constraints on z, z3 might very well respond unknown to this encoding. In this particular case, it happens to be simple enough so z3 is able to find a model.
If you have sin and pi:
This is more of a thought experiment than a real alternative. If SMTLib allowed for sin and pi, then you can check whether sin (zI * pi) is 0, for a suitably constrained zI. Any satisfying model to this query would ensure that zI is integer. You can then use this value to extract the fractional part by subtracting zI from z.
But this is futile as SMTLib neither allows for sin nor pi. And for good reason: Decidability would be lost. Having said that, maybe some brave soul can design a logic that supported sin, pi, etc., and successfully answered your query correctly, while returning unknown when the problem becomes too hard for the solver. This is already the case for nonlinear arithmetic and the QF_UFNIRA fragment: The solver may give up in general, but the heuristics it employs might solve problems of practical interest.
Restriction to Rationals
As a theoretical aside, it turns out that if you restrict yourself to rationals only (instead of actual reals) then you can indeed write a first-order formula to recognize integers. The encoding is not for the faint of heart, however: http://math.mit.edu/~poonen/papers/ae.pdf. Furthermore, since the encoding involves quantifiers, it's probably quite unlikely that SMT solvers will do well with a formulation based on this idea.
[Incidentally, I should extend thanks to my work colleagues; this question made for a great lunch-time conversation!]
I would like to construct an SMT formula having a number of assertions over integer linear arithmetic and Boolean variables as well as some assertions over real non-linear arithmetic and again Boolean variables. The assertions over integers and reals share only the Boolean variables. As an example, consider the following formula:
(declare-fun b () Bool)
(assert (= b true))
(declare-fun x () Int)
(declare-fun y () Int)
(declare-fun z () Int)
(assert (or (not b) (>= (+ x y) (- x (+ (* 2 z) 1)))))
(declare-fun r () Real)
(assert (or (not b) (= (+ (* r r) (* 3 r) (- 4)) 0)))
If I feed z3 with this formula, it immediately reports "unknown". But if I remove the integer part of it, I get the solution right away, which satisfies the constraint with variable "r". I presume this means that the non-linear constraint on its own is not hard for the solver. The issue should be in mixing (linear) constraints over integers and (non-linear) constraints over reals.
So my question is the following. What is the correct way to handle this kind of mixed formulas using z3 (if there is any)? My understanding of DPLL(T) is that it should be able to deal with such formulas using different theory solvers for different constraints. Please, correct me if I am wrong.
As George said in his comment, the non-linear solver in Z3 is rather fragile and the out-of-the-box performance isn't great. That said, there are a number of questions and answers about this problem here on stackoverflow, e.g., see these:
Z3 Performance with Non-Linear Arithmetic
How does Z3 handle non-linear integer arithmetic?
Z3 : strange behavior with non linear arithmetic
Non-linear arithmetic and uninterpreted functions
Z3 Theorem Prover: Pythagorean Theorem (Non-Linear Artithmetic)
Which techniques are used to handle Non-linear Integer Real problems in z3?
Satisfiablity checking in non-linear integer arithmetic by approximation