How can I access the last element in Z3 list?
(declare-const lst (List Int))
(assert (not (= lst nil)))
(assert (= (head lst) 1))
(assert (= (last_element lst) 2))
(check-sat)
AFAIK, Z3 has no built-in function to access the last element of a list. Since SMT-Lib2 does not support recursive functions (see this answer), you have to declare and axiomatise an uninterpreted last_element function yourself.
Declaration:
(declare-fun last_element ((List Int) (Int)) Bool)
Axiom "nil has no last element":
(assert (forall ((x Int)) (!
(not (last_element nil x))
:pattern ((last_element nil x))
)))
Axiom "If xs is the list x:nil then x is the last element of xs":
(assert (forall ((xs (List Int)) (x Int)) (!
(implies
(= xs (insert x nil))
(last_element xs x))
:pattern ((last_element xs x))
)))
Axiom "If x is the last element of the tail of xs, then it is also the last element of xs itself":
(assert (forall ((xs (List Int)) (x Int)) (!
(implies
(last_element (tail xs) x)
(last_element xs x))
:pattern ((last_element xs x))
)))
See this rise4fun-link for an example.
Please note: The linked example switches of model based quantifier instantiation (MBQI, see the Z3 guide) and thus relies on E-matching. This is also the reason why explicit patterns are provided, see this question. If you want to give MBQI a try you might have to change the axiomatisation, but I hardly know anything about MBQI.
Related
I defined my own version of list concat based on Haskell as below:
(declare-datatypes ((MyList 1))
((par (T) ((cons (head T) (tail (MyList T))) (nil)))))
(declare-fun my-concat ( (MyList T1) (MyList T1) ) (MyList T1))
(assert (forall ((xs (MyList T1)) (ys (MyList T1)) (x T1))
(ite (= (as nil (MyList T1)) xs)
(= (my-concat xs ys) ys)
(= (my-concat (cons x xs) ys) (cons x (my-concat xs ys))))))
I am wondering why z3 is not able to reason about the following?
(assert (not (= (my-concat (cons 4 (as nil (MyList Int))) (as nil (MyList Int)))
(cons 4 (as nil (MyList Int))))))
(check-sat) ; runs forever
Loading your file
If I load your program to z3 as you've given it, it says:
(error "line 3 column 33: Parsing function declaration. Expecting sort list '(': unknown sort 'T1'")
(error "line 4 column 29: invalid sorted variables: unknown sort 'T1'")
(error "line 8 column 79: unknown function/constant my-concat")
That's because you cannot define "polymorphic" functions in SMTLib. At the user-level, only fully monomorphic functions are allowed. (Though internally SMTLib does provide polymorphic constants, there's no way for the user to actually create any polymorphic constants.)
So, I'm not sure how you got to load that file.
Monomorphisation
Looks like you only care about integer-lists anyhow, so let's just modify our program to work on lists of integers. This process is called monomorphisation, and is usually automatically done by some front-end tool before you even get to the SMT-solver, depending on what framework you're working in. That's just a fancy way of saying create instances of all your polymorphic constants at the mono-types they are used. (Since you mentioned Haskell, I'll just throw in that while monomorphisation is usually possible, it's not always feasible: There might be way too many variants to generate making it impractical. Also, if you have polymoprhic-recursion then monomorphisation doesn't work. But that's a digression for the time being.)
If I monomorphise your program to Int's only, I get:
(declare-datatypes ((MyList 1))
((par (T) ((cons (head T) (tail (MyList T))) (nil)))))
(declare-fun my-concat ( (MyList Int) (MyList Int) ) (MyList Int))
(assert (forall ((xs (MyList Int)) (ys (MyList Int)) (x Int))
(ite (= (as nil (MyList Int)) xs)
(= (my-concat xs ys) ys)
(= (my-concat (cons x xs) ys) (cons x (my-concat xs ys))))))
(assert (not (= (my-concat (cons 4 (as nil (MyList Int))) (as nil (MyList Int)))
(cons 4 (as nil (MyList Int))))))
(check-sat)
(get-info :reason-unknown)
When I run z3 on this, I get:
unknown
(:reason-unknown "smt tactic failed to show goal to be sat/unsat (incomplete quantifiers)")
So, it doesn't loop at all like you mentioned; but your file didn't really load in the first place. So maybe you were working with some other contents in the file as well, but that's a digression for the current question.
But it still doesn't prove it!
Of course, you wanted unsat for this trivial formula! But z3 said it's too hard for it to deal with. The reason-unknown is "incomplete quantifiers." What does that mean?
In short, SMTLib is essentially logic of many-sorted first order formulas. Solvers are "complete" for the quantifier-free fragment of this logic. But adding quantifiers makes the logic semi-decidable. What that means is that if you give enough resources, and if smart heuristics are in play, the solver will eventually say sat for a satisifiable formula, but it may loop forever if given an unsat one. (It might get lucky and say unsat as well, but most likely it'll loop.)
There are very good reasons why above is the case, but keep in mind that this has nothing to do with z3: First-order logic with quantifiers is semi-decidable. (Here's a good place to start reading: https://en.wikipedia.org/wiki/Decidability_(logic)#Semidecidability)
What usually happens in practice, however, is that the solver will not even answer sat, and will simply give up and say unknown as z3 did above. Quantifiers are simply beyond the means of SMT-solvers in general. You can try using patterns (search stack-overflow for quantifiers and pattern triggers), but that usually is futile as patterns and triggers are tricky to work with and they can be quite brittle.
What course of action do you have:
Honestly, this is one of those cases where "give up" is good advice. An SMT solver is just not a good fit for this sort of a problem. Use a theorem-prover like Isabelle, ACL2, Coq, HOL, HOL-Light, Lean, ... where you can express quantifiers and recursive functions and reason with them. They are built for this sort of thing. Don't expect your SMT solver to handle these sorts of queries. It's just not the right match.
Still, is there anything I can do?
You can try SMTLib's recursive-function definition facilities. You'd write:
(declare-datatypes ((MyList 1))
((par (T) ((cons (head T) (tail (MyList T))) (nil)))))
(define-fun-rec my-concat ((xs (MyList Int)) (ys (MyList Int))) (MyList Int)
(ite (= (as nil (MyList Int)) xs)
ys
(cons (head xs) (my-concat (tail xs) ys))))
(assert (not (= (my-concat (cons 4 (as nil (MyList Int))) (as nil (MyList Int)))
(cons 4 (as nil (MyList Int))))))
(check-sat)
Note the define-fun-rec construct, which allows for recursive definitions.
And voila, we get:
unsat
But this does not mean z3 will be able to prove arbitrary theorems regarding this concat function. If you try anything that requires induction, it'll either give up (saying unknown) or loop-forever. Of course, as the capabilities improve some induction-like proofs might be possible in z3 or other SMT-solvers, but that's really beyond what they're designed for. So, use with caution.
You can read more about recursive definitions in Section 4.2.3 of http://smtlib.cs.uiowa.edu/papers/smt-lib-reference-v2.6-r2017-07-18.pdf
To sum up
Do not use an SMT-solver for reasoning with quantified or recursive definitions. They just don't have the necessary power to deal with such problems, and it's unlikely they'll ever get there. Use a proper theorem prover for these tasks. Note that most theorem-provers use SMT-solvers as underlying tactics, so you get the best of both worlds: You can do some manual work to guide the inductive proof, and have the prover use an SMT-solver to handle most of the goals automatically for you. Here's a good paper to read to get you started on the details: https://people.mpi-inf.mpg.de/~jblanche/jar-smt.pdf
alias makes valid points, but I don't completely agree with the final conclusion to " not use an SMT-solver for reasoning with quantified or recursive definitions". In the end, it depends on which kind of properties you need to reason about, what kind of answers you need (is unsat/unknown OK, or do you need unsat/sat and a model?), and how much work you're willing to invest :-)
For example, SMT-based program verifiers such as Dafny and Viper quickly verify the following assertions about lists:
assert [4] + [] == [4]; // holds
assert [4,1] + [1,4] == [4,1,1,4]; // holds
assert [4] + [1] == [1,4]; // fails
Both tools can be used online, but the websites are rather slow and not terribly reliable. You can find the Dafny example here and the Viper example here.
Here is the relevant SMT code that Viper generates:
(set-option :auto_config false) ; Usually a good idea
(set-option :smt.mbqi false)
;; The following definitions are an excerpt of Viper's sequence axiomatisation,
;; which is based on Dafny's sequence axiomatisation.
;; See also:
;; https://github.com/dafny-lang/dafny
;; http://viper.ethz.ch
(declare-sort Seq<Int>) ;; Monomorphised sort of integer sequences
(declare-const Seq_empty Seq<Int>)
(declare-fun Seq_length (Seq<Int>) Int)
(declare-fun Seq_singleton (Int) Seq<Int>)
(declare-fun Seq_index (Seq<Int> Int) Int)
(declare-fun Seq_append (Seq<Int> Seq<Int>) Seq<Int>)
(declare-fun Seq_equal (Seq<Int> Seq<Int>) Bool)
(assert (forall ((s Seq<Int>)) (!
(<= 0 (Seq_length s))
:pattern ((Seq_length s))
)))
(assert (= (Seq_length (as Seq_empty Seq<Int>)) 0))
(assert (forall ((s1 Seq<Int>) (s2 Seq<Int>)) (!
(implies
(and
(not (= s1 (as Seq_empty Seq<Int>)))
(not (= s2 (as Seq_empty Seq<Int>))))
(= (Seq_length (Seq_append s1 s2)) (+ (Seq_length s1) (Seq_length s2))))
:pattern ((Seq_length (Seq_append s1 s2)))
)))
(assert (forall ((s Seq<Int>)) (!
(= (Seq_append (as Seq_empty Seq<Int>) s) s)
:pattern ((Seq_append (as Seq_empty Seq<Int>) s))
)))
(assert (forall ((s Seq<Int>)) (!
(= (Seq_append s (as Seq_empty Seq<Int>)) s)
:pattern ((Seq_append s (as Seq_empty Seq<Int>)))
)))
(assert (forall ((s1 Seq<Int>) (s2 Seq<Int>) (i Int)) (!
(implies
(and
(not (= s1 (as Seq_empty Seq<Int>)))
(not (= s2 (as Seq_empty Seq<Int>))))
(ite
(< i (Seq_length s1))
(= (Seq_index (Seq_append s1 s2) i) (Seq_index s1 i))
(= (Seq_index (Seq_append s1 s2) i) (Seq_index s2 (- i (Seq_length s1))))))
:pattern ((Seq_index (Seq_append s1 s2) i))
:pattern ((Seq_index s1 i) (Seq_append s1 s2))
)))
(assert (forall ((s1 Seq<Int>) (s2 Seq<Int>)) (!
(=
(Seq_equal s1 s2)
(and
(= (Seq_length s1) (Seq_length s2))
(forall ((i Int)) (!
(implies
(and (<= 0 i) (< i (Seq_length s1)))
(= (Seq_index s1 i) (Seq_index s2 i)))
:pattern ((Seq_index s1 i))
:pattern ((Seq_index s2 i))
))))
:pattern ((Seq_equal s1 s2))
)))
(assert (forall ((s1 Seq<Int>) (s2 Seq<Int>)) (!
(implies (Seq_equal s1 s2) (= s1 s2))
:pattern ((Seq_equal s1 s2))
)))
; ------------------------------------------------------------
; assert Seq(4) ++ Seq[Int]() == Seq(4)
(push)
(assert (not
(Seq_equal
(Seq_append (Seq_singleton 4) Seq_empty)
(Seq_singleton 4))))
(check-sat) ; unsat -- good!
(pop)
; assert Seq(4, 1) ++ Seq(1, 4) == Seq(4, 1, 1, 4)
(push)
(assert (not
(Seq_equal
(Seq_append
(Seq_append (Seq_singleton 4) (Seq_singleton 1))
(Seq_append (Seq_singleton 1) (Seq_singleton 4)))
(Seq_append
(Seq_append
(Seq_append (Seq_singleton 4) (Seq_singleton 1))
(Seq_singleton 1))
(Seq_singleton 4)))))
(check-sat) ; unsat -- good!
(pop)
; assert Seq(4) ++ Seq(1) == Seq(1, 4)
(push)
(assert (not (Seq_equal
(Seq_append (Seq_singleton 4) (Seq_singleton 1))
(Seq_append (Seq_singleton 1) (Seq_singleton 4)))))
(check-sat) ; unknown -- OK, since property doesn't hold
(pop)
Given a typical list datatype:
(declare-datatypes (X)(
(Lst
(nil)
(insert
(head X)
(tail Lst)))))
I'm trying to create a function that returns whether a given element is in a given list. It looks like I have to use an uninterpreted function. I've tried different approaches without success. For example:
(declare-fun is-in-lst ((Int)(Lst Int)) Bool)
(assert (forall((elem Int)(lst (Lst Int)))
(=
(is-in-lst elem lst)
(and
(is-insert lst)
(or
(= elem (head lst))
(is-in-lst elem (tail lst)))))))
(declare-const l1 (Lst Int))
(assert (is-in-lst 6 l1))
(assert (is-in-lst 5 l1))
Is this doable in Z3? If so, what would be the best way to address the problem?
Thank you
The latest SMTLib standard allows for recursive definitions. Also List is predefined in Z3, so you don't need to define it yourself. You can code your function as:
(define-fun-rec elem ((e Int) (l (List Int))) Bool
(ite (= l nil) false (or (= e (head l)) (elem e (tail l)))))
(declare-const l1 (List Int))
(assert (elem 6 l1))
(assert (elem 5 l1))
(check-sat)
(get-value (l1))
For this, z3 responds:
sat
((l1 (let ((a!1 (insert 6 (insert 4 (insert 7 (insert 5 nil))))))
(insert 0 (insert 1 (insert 3 (insert 2 a!1)))))))
You might have to get a recent version of Z3 from their nightly builds, as support for recursive-functions is rather new and thus the latest official release (4.6.0) may or may not work with this. (You can get nightly builds from https://github.com/Z3Prover/bin/tree/master/nightly.)
For formula with uninterpreted sort B Z3 prints unsat but when I replace sort B to Int it prints timeout(second script). I would like to understand the reason for it.
first:
(declare-sort A)
(declare-sort B)
(declare-fun f (B) A)
(declare-fun f-inv (A) B)
(declare-const b0 B)
(declare-const b1 B)
(assert (forall ((x B)) (= (f-inv (f x)) x)))
(assert (not (= (f b0) (f b1))))
(check-sat)
second:
(declare-sort A)
(declare-fun f (Int) A)
(declare-fun f-inv (A) Int)
(assert (forall ((x Int)) (= (f-inv (f x)) x)))
(assert (not (= (f 0) (f 1))))
(check-sat)
The reason is that Z3 is not able to decide neither "first" and "second".
For "first" I am obtaining : "sat".
For "first" when the line
(assert (not (= (f b0) (f b1))))
is replaced by
(assert (= (f b0) (f b1)))
the result is "sat". In other words Z3 is not able to decide "first". The effective result from Z3 is "timeout" for "first" and "second".
Z3 has problems with combinations of uninterpreted functions.
Do you agree?
Both versions are now sat when tried online on Compsys tools.
Is there a way to concat two lists in z3? Similar to the # operator in ML? I was thinking of defining it myself but I don't think z3 supports recursive function definitions, i.e.,
define-fun concat ( (List l1) (List l2) List
(ite (isNil l1) (l2) (concat (tail l1) (insert (head l1) l2)) )
)
2021 Update
Below answer was written in 2012; 9 years ago. It largely remains still correct; except SMTLib now explicitly allows for recursive-function definitions, via define-fun-rec construct. However, solver support is still very weak, and most properties of interest regarding such functions can still not be proven out-of-the-box. Bottom line remains that such recursive definitions lead to inductive proofs, and SMT-solvers are simply not equipped to do induction. Perhaps in another 9 years they will be able to do so, presumably allowing users to specify their own invariants. For the time being, theorem-provers such as Isabelle, Coq, ACL2, HOL, Lean, etc., remain the best tools to handle these sorts of problems.
Answer from 2012
You are correct that SMT-Lib2 does not allow recursive function definitions. (In SMT-Lib2, function definitions are more like macros, they are good for abbreviations.)
The usual trick is to declare such symbols as uninterpreted functions, and then assert the defining equations as quantified axioms. Of course, as soon as quantifiers come into play the solver can start returning unknown or timeout for "difficult" queries. However, Z3 is pretty good at many goals arising from typical software verification tasks, so it should be able to prove many properties of interest.
Here's an example illustrating how you can define len and append over lists, and then prove some theorems about them. Note that if a proof requires induction, then Z3 is likely to time-out (as in the second example below), but future versions of Z3 might be able to handle inductive proofs as well.
Here's the permalink for this example on the Z3 web-site if you'd like to play around: http://rise4fun.com/Z3/RYmx
; declare len as an uninterpreted function
(declare-fun len ((List Int)) Int)
; assert defining equations for len as an axiom
(assert (forall ((xs (List Int)))
(ite (= nil xs)
(= 0 (len xs))
(= (+ 1 (len (tail xs))) (len xs)))))
; declare append as an uninterpreted function
(declare-fun append ((List Int) (List Int)) (List Int))
; assert defining equations for append as an axiom
(assert (forall ((xs (List Int)) (ys (List Int)))
(ite (= nil xs)
(= (append xs ys) ys)
(= (append xs ys) (insert (head xs) (append (tail xs) ys))))))
; declare some existential constants
(declare-fun x () Int)
(declare-fun xs () (List Int))
(declare-fun ys () (List Int))
; prove len (insert x xs) = 1 + len xs
; note that we assert the negation, so unsat means the theorem is valid
(push)
(assert (not (= (+ 1 (len xs)) (len (insert x xs)))))
(check-sat)
(pop)
; prove (len (append xs ys)) = len xs + len ys
; note that Z3 will time out since this proof requires induction
; future versions might very well be able to deal with it..
(push)
(assert (not (= (len (append xs ys)) (+ (len xs) (len ys)))))
(check-sat)
(pop)
While Levent's code works, if you're willing to set a bound on the recursion depth, Z3 normally has much less trouble with your assertions. You don't even need to rely on MBQI, which often takes far too much time to be practical. Conceptually, you'll want to do:
; the macro finder can figure out when universal declarations are macros
(set-option :macro-finder true)
(declare-fun len0 ((List Int)) Int)
(assert (forall ((xs (List Int))) (= (len0 xs) 0)))
(declare-fun len1 ((List Int)) Int)
(assert (forall ((xs (List Int))) (ite (= xs nil)
0
(+ 1 (len0 (tail xs))))))
(declare-fun len2 ((List Int)) Int)
(assert (forall ((xs (List Int))) (ite (= xs nil)
0
(+ 1 (len1 (tail xs))))))
... and so on. Writing all of this down manually will probably be a pain, so I'd recommend using a programmatic API. (Shameless plug: I've been working on Racket bindings and here's how you'd do it there.)
how can I make a datatype that contains a set of another objects. Basically, I am doing the following code:
(define-sort Set(T) (Array Int T))
(declare-datatypes () ((A f1 (cons (value Int) (b (Set B))))
(B f2 (cons (id Int) (a (Set A))))
))
But Z3 tells me unknown sort for A and B. If I remove "Set" it works just as the guide states.
I was trying to use List instead but it does not work. Anyone knows how to make it work?
You are addressing a question that comes up on a regular basis:
how can I mix data-types and arrays (as sets, multi-sets or
data-types in the range)?
As stated above Z3 does not support mixing data-types
and arrays in a single declaration.
A solution is to develop a custom solver for the
mixed datatype + array theory. Z3 contains programmatic
APIs for developing custom solvers.
It is still useful to develop this example
to illustrate the capabilities and limitations
of encoding theories with quantifiers and triggers.
Let me simplify your example by just using A.
As a work-around you can define an auxiliary sort.
The workaround is not ideal, though. It illustrates some
axiom 'hacking'. It relies on the operational semantics
of how quantifiers are instantiated during search.
(set-option :model true) ; We are going to display models.
(set-option :auto-config false)
(set-option :mbqi false) ; Model-based quantifier instantiation is too powerful here
(declare-sort SetA) ; Declare a custom fresh sort SetA
(declare-datatypes () ((A f1 (cons (value Int) (a SetA)))))
(define-sort Set (T) (Array T Bool))
Then define bijections between (Set A), SetA.
(declare-fun injSA ((Set A)) SetA)
(declare-fun projSA (SetA) (Set A))
(assert (forall ((x SetA)) (= (injSA (projSA x)) x)))
(assert (forall ((x (Set A))) (= (projSA (injSA x)) x)))
This is almost what the data-type declaration states.
To enforce well-foundedness you can associate an ordinal with members of A
and enforce that members of SetA are smaller in the well-founded ordering:
(declare-const v Int)
(declare-const s1 SetA)
(declare-const a1 A)
(declare-const sa1 (Set A))
(declare-const s2 SetA)
(declare-const a2 A)
(declare-const sa2 (Set A))
With the axioms so far, a1 can be a member of itself.
(push)
(assert (select sa1 a1))
(assert (= s1 (injSA sa1)))
(assert (= a1 (cons v s1)))
(check-sat)
(get-model)
(pop)
We now associate an ordinal number with the members of A.
(declare-fun ord (A) Int)
(assert (forall ((x SetA) (v Int) (a A))
(=> (select (projSA x) a)
(> (ord (cons v x)) (ord a)))))
(assert (forall ((x A)) (> (ord x) 0)))
By default quantifier instantiation in Z3 is pattern-based.
The first quantified assert above will not be instantiated on all
relevant instances. One can instead assert:
(assert (forall ((x1 SetA) (x2 (Set A)) (v Int) (a A))
(! (=> (and (= (projSA x1) x2) (select x2 a))
(> (ord (cons v x1)) (ord a)))
:pattern ((select x2 a) (cons v x1)))))
Axioms like these, that use two patterns (called a multi-pattern)
are quite expensive. They produce instantiations for every pair
of (select x2 a) and (cons v x1)
The membership constraint from before is now unsatisfiable.
(push)
(assert (select sa1 a1))
(assert (= s1 (injSA sa1)))
(assert (= a1 (cons v s1)))
(check-sat)
(pop)
but models are not necessarily well formed yet.
the default value of the set is 'true', which
would mean that the model implies there is a membership cycle
when there isn't one.
(push)
(assert (not (= (cons v s1) a1)))
(assert (= (projSA s1) sa1))
(assert (select sa1 a1))
(check-sat)
(get-model)
(pop)
We can approximate more faithful models by using
the following approach to enforce that sets that are
used in data-types are finite.
For example, whenever there is a membership check on a set x2,
we enforce that the 'default' value of the set is 'false'.
(assert (forall ((x2 (Set A)) (a A))
(! (not (default x2))
:pattern ((select x2 a)))))
Alternatively, whenever a set occurs in a data-type constructor
it is finite
(assert (forall ((v Int) (x1 SetA))
(! (not (default (projSA x1)))
:pattern ((cons v x1)))))
(push)
(assert (not (= (cons v s1) a1)))
(assert (= (projSA s1) sa1))
(assert (select sa1 a1))
(check-sat)
(get-model)
(pop)
Throughout the inclusion of additional axioms,
Z3 produces the answer 'unknown' and furthermore
the model that is produced indicates that the domain SetA
is finite (a singleton). So while we could patch the defaults
this model still does not satisfy the axioms. It satisfies
the axioms modulo instantiation only.
This is not supported in Z3. You can use arrays in datatype declarations, but they can't contain "references" to the datatypes you are declaring. For example, it is ok to use (Set Int).