# Constrained Horn Clauses - Invariants

# BTOR2

Btor is a model checking format supported by bitwuzla.

Hmm. Interesting.
So btor is intended for hardware model checking. There is a single transition relation. If you want more, you should model a program counter.
btormc is a boolector based model checker. It iteratively unwraps the state relation
K induction mode –kind
Refers to variables and calculations by number. Many operations are predicated on sort. You can repeat sort definitions
`20 sort bitvec 1`

defines at 20 a bitvector sort of length 1

Ok doesn’t allow duplicate symbol declaration
`21 state 20 fred`

defines a symbol of sort “20” called fred

`22 add 1 4 2`

add takes a sort. It isn’t inferred. `eq`

also takes a sort? Can it not be bitvec 1?

`b`

bad states and `j`

justice states?
bad, constraint, fair, output.
`constraint`

lets you assume something (an invariant)
`input`

for un constraied stuff

Yeah, it all makes sense. Especially in a hardware context.

How do you put suggested hand written invariants in?

```
; int i = 1, factorial = 1;
; assert (i <= 2 || !(factorial & 1));
; for (;;) {
; factorial *= i;
; i++;
; assert (i <= 2 || !(factorial & 1));
; }
1 sort bitvec 4
2 one 1
3 state 1 factorial ; defining 2 state variables
4 state 1 i
5 init 1 3 2 ; initialization of factorial= 1
6 init 1 4 2 ; initialization of i = 1
7 add 1 4 2 ; expression i + 1
8 mul 1 3 4 ; expression factorial * i
9 next 1 4 7 ; next state i = i + 1
10 next 1 3 8 ; this is setting factorial = factorial * i
11 ones 1 ; b1111
12 sort bitvec 1
13 eq 12 4 11 ; predicate i == b1111 overflow condition?
14 bad 13
15 slice 12 3 0 0
16 constd 1 3
17 ugt 12 4 16 ; comparing i > 3
18 and 12 17 15
19 bad 18
```

# Resources

eldarica https://github.com/uuverifiers/eldarica web Impressive seeming examples.

```
• CEGAR and predicate abstraction, such as HSF [20],
Duality [30], and ELDARICA;
• IC3/PDR, such as the PDR engine in Z3 [21]. The algorithm implemented in SPACER [28] extends IC3/PDR by
maintaining both under- and over-approximations during
analysis;
• Transformation of Horn clauses, such as VeriMAP [13]
and Rahft [26];
• Machine learning, such as SynthHorn [33], FreqHorn
[17] and HoIce [11], which progressively drive concrete
invariant samples and use machine learning classification
techniques to find the inductive invariant.
```

liquid fixpoint Not entirely sure what this is, but it is a solverified version of liquid haskell ish? Using liquid types as a horn clause problem specification language?

cartesian predciate abstraction

Analysis and Transformation of Constrained Horn Clauses for Program Verification∗ Good review article. Bottom up CHC = DMT?

higher order chc https://research-information.bris.ac.uk/ws/portalfiles/portal/142259264/popl18_p253.pdf

magic set for chc tools for chc solving constrained horn cluases using interpolation mcmillan Rybalchenko 2013 Solving Constrained Horn Clauses Modulo Algebraic Data Types and Recursive Functions Forward vs backwards chaining. One could also consider a datalog like execution model.

Linear logic and CHC. Perhaps a more convenient language for frames.

https://github.com/Z3Prover/z3/discussions/5093 Interpolants with CHCs and List. Using List to represent a stack. Seems reasonable.. Lists of questionable support.

Consider using sexp macro expansion to chc.

higher order model checking? Kobayashi? There’s some stuff here https://ericpony.github.io/z3py-tutorial/fixpoint-examples.htm that didn’t register See the bottom

https://insights.sei.cmu.edu/blog/ghihorn-path-analysis-in-ghidra-using-smt-solvers/ ghihorn - using horn solver on ghidra pcode

(https://www.youtube.com/watch?v=kbtnye_q3PA&t=677s&ab_channel=VSS-IARCS)

Constrained Horn Clauses: Z3 has kind of a prolog in it

Maurizio Proietti: Removing Algebraic Data Types from Constrained Horn Clauses… (IJCAR A) adtrem - interestintg Program trasnformations from prolog can be used on CHC to remove or deforest annoying structures. mentions a CVC4+induction mode

rusthorn +holce? Towards Automatic Verification of Unsafe Rust with Constrained Horn Solvers

Horn clauses are a logical view on the form of programs allowed in prolog programs. They can explained in a couple different ways. https://www.youtube.com/watch?v=hgw59_HBU2A
One way is to describe them as logical statements of the form `a /\ b /\ c /\ d -> e`

, a conjunction of literals implying a single conclusion predicate. The reason this form is nice is because it lends itself to back-chaining. If we need to prove `e`

we can look for all the rules that have `e`

in their head and try them one byu one, recursively seeing if we can also backchain the rules requirements.
Alternatively we can easily drive the forward. We can look at each clause in turn, see if we satisfy the requirements, and if so add the conclusion to the database of things we know. This is roughly how a datalog program is typically run.
We can also throw variables into the mix and lift the `a/b/c/d/e`

into being predicates.

There is a sense when you are constructing a normal z3 query that you need to be talking about a kind of fixed arena. You need to define enough variables to describe all the possibilities you want z3 to explore. If you have a query where a system could go down road A and take 20 steps and road B and take 3 steps, you need to spell out enough variables ahead of time to encode all these steps.

Quantifiers kind of let you get around this. There are different mechanisms in Z3 for working with quantifiers. One is the ematching engine that looks for patterns in the stuff you have lying around and instantiates forall quantifiers with those patterns. Another is the horn clause engine.

Constrained logic programming and constrained horn clauses are the same thing. The first name comes from the logic programming/prolog community and the second from the verification/smt solver community. https://www.metalevel.at/prolog/clpz Basically, it seems you call it one or the other depending on whether you’re tacking constraints onto a prolog implementation or a “prolog” onto some kind of constraint engine like an smt solver.

Z3 currently has a bounded model checker, datalog, and a mode called spacer as distinct engines for solving horn clauses. It has had different options such as the “duality” engine in the past that are now defunct.

I do say and feel that you need a rough picture of what a tool/library is doing to be able to use it.

The most obvious way to me to check a system is to unroll it out in time. This is bounded model checking (BMC). If it finds a counterexample, great! That’s a real counterexample If not, well, you haven’t proven anything that useful without more analysis. You probably have gained some confidence in the system though.

IC3/PDR (Property Directed Reachability) is a kind of model checking algorithm that doesn’t unroll executions out into a giant query. Instead it maintains an approximate representation of reachable sets N step out in time. This representation is basically as a logical formula, which you can query and refine by using SMT queries. The spacer algorithm is some kind of twist on these algorithms.

### What is a query?

Z3 has a separate interface you can use to define prolog like rules, or you can phrase them in the ordinary smtlib interface and specify to use the horn solver. It is somewhat confusing that the return codes of `sat/unsat`

mean opposite things depending on the mode you’re using. Using the fixedpoint interface, `sat`

means the query succeeded, like how a prolog query might succeed. This means there was a way to successfully chain together the horn clauses
https://stackoverflow.com/questions/39403644/%E2%88%83-queries-and-%E2%88%80-queries-with-z3-fixedpoint-engine

I rather like the perspective from Miller and Nadathur where they describe a prolog query as intuitinistic proof search. A query `p(X)`

creates an executions that corresponds to a proof of `exists x. p(x)`

However prolog is usually considered in a background of classical logic, and Z3 certainly is a classical logic engine. A successful query is a proof of unsat by adding `(not p)`

or equivalently `(=> p false)`

from the perspective of the sat solver. It is trying to backchain a proof of `false`

or forward chain finding `p`

to be true and then immediately finding false. The resolution proof of false is the analog of https://www.javatpoint.com/ai-resolution-in-first-order-logic
The production of learned clauses is a form of resolution proof. The DRAT format records a trace of the SAT execution. It records the clauses you need to resolve together to make lemma clauses eventually leading to a contradiction.
Classically, if you want to prove `p`

, a uniform way of doing so is to add `not p`

as an assumption and try to prove false.

### Program Verification

Reachable

Just as you can write a functional program to emulate the execution of some imperative code or assembly, you can write a prolog program to do the same. In these pure languages, this is achieved by explicitly passing state as a parameter.

To actuallyl get the output state. start(State, EndState) :- body_start(State, State1), block2(State1, EndState). block2(State1, )

You could query a program to give back those states for which there is an error.

main_err(State) :- body_start(State, State1), block2_err(State1). main_err(State) :- err_inside_main(State). block2(State1)

How does one

```
for(int i = 0; i < 10; i++){ y += 5; }
```

```
main(Y) :- loop_entry(0, Y).
loop_entry(I, Y) :- I >= 10, loop_exit(Y).
loop_entry(I, Y) :- I < 10, I1 = I + 1, Y1 = Y + 5, loop_entry(I1, Y1).
```

```
(define-fun main (Int Int) Bool)
(define-fun loop-entry (Int Int) Bool)
(define-fun loop-exit (Int Int) Bool)
```

## Tricks

- Tracking inputs.
- Product Program
- WP using let
- Program counter exists.
- Memory modelling -

“partial evaluations”
You can ask for the program counter to stay in a known set. This is an invariant
You can ask that only certain PC -> PC edges exist. This is more difficult. It requires remembering prev_PC? No, but not tracked through the predciate. Just put a ghost `prev_PC := PC`

before every

Once you know that you can fuse the firs program counter variables with the predicate itself. Then you havethe CFG flavored version.
You can also pre-fuse linear program flow.

Getting a counterexample - the counterexample is in the UNSAT proof. But you can instead try negating the initial condition clause and remove it’s forall binder to expose the initial variables? But then the forall of each statement are a disaster. Hmm. Maybe flip Bad -> Good and make horn clauses equalities? Yeah. This doesn’t work. nvm

https://www.cs.stevens.edu/~ejk/papers/cav21-preprint.pdf - relational verficiation using some enhancement on CHC

I guess the most basic version of invaraint generation is to use an abstract interpetation. If the invaraint falls within the epxressive power of the abstract domain and it terminates (not too much erroenous widening) Then you can discover invaraints in the abstract domain.

Z3, eldraica, hsf?

http://theory.stanford.edu/~nikolaj/nus.html - Bjorner talk

http://seahorn.github.io/blog/ https://github.com/seahorn/seahorn-tutorial

https://www.youtube.com/watch?v=yJQZ7sG8xSM&ab_channel=Rust - horn clasues generation from rust - eldarica

https://www.cs.utexas.edu/~tdillig/cs395/esc-houdini.ps houdini leino and flanagan. https://www.cs.utexas.edu/~isil/fmcad-tutorial.pdf abduction https://theory.stanford.edu/~aiken/publications/papers/pldi19.pdf

# Ideas

How to help along the solver. I think, maybe, that if you can guess parts of the invariant that helps. (declare-const pvar ) (define-fun p () (and known_invariants pvar )

This goes into both heads an bodies. Is that okay? Or have a reach-exit, reach-entry split. Fill this in with info from user annotation and abstract analysis.

isil dillig - mistral an abduction generating smt solver? On LIA, get minimal satisyinfyng assignemtn (pins least variables) and quantifier eliminate the rest

Interesting z3 tidbit - use unsat core to get minimal satisfying assignment unsat core tricks in general are something to think about. Partial models to avoid the else case https://stackoverflow.com/questions/41425514/partial-assignments-in-z3

http://www.cs.cmu.edu/~aldrich/courses/17-355-19sp/notes/slides27-daikon.pdf - daikon - dynamic checking of guessed invaraints. Fuzzing basically.

C2I LoopInvGen ICE-DT CODE2Inv

CHC clauses allows us to get predictaes out. These predicate solutions are over approximations of possible state sufficient to satisfy the problem.

https://arxiv.org/abs/2002.09002 rusthorn. Trick involving old new values to model borrowing from creusot talk

https://arieg.bitbucket.io/pdf/gurfinkel_ssft15.pdf IC3/PDR

deadlock empire using CHC? Well deadlock empire has counterexamples.

Recursion and Loops are where it comes handy. We can reduce the straight line behavior (and branching) with ordinary wp. But the loops are tougher. Function calls may not be recursive of course. If we previously establidshed

Keeping overapproximations of Reachable sets. Generalize counterexamnple and push them backwards

Z3 implementation. https://github.com/pddenhar/Z3-IC3-PDR

really nice description http://www.cs.tau.ac.il/~msagiv/courses/asv/IC3.pdf

ic3/pdr vs interpolation based use unsat cores

https://github.com/Z3Prover/z3/blob/master/examples/python/mini_ic3.py

There is some interpolation happening in SPACER

The SAT version Maintain cnf fomrulas (sets of clauses)for each time step A cube = an assignment = a conjunction of literals. We manitain a queue of counterexamples If the counterexample is in F0, then we found a bad trace NewLemma - take negation of counterexample.. This is candidate clauses (possibly we may delete some literals). We want to exclude an errant counterexample.

Push - pick a clauses in Fi. split it into two pieces. if a subpiece is not in Fi+1 unfold - we initialize a new Fi to have no clauses (no constraints). Candidate will pick any bad state now. Predecessor- m0 and m1 is a combined state picked. m1 -> m means we may add literals to m. We need to find a state that ends up in m. We pushed the counterexample backwards. That makes sense. And we might find multiple. Requeue - if this counterexample can’t be reached from previous guy, push it forward.

```
init = And(p, q, r, yada)
sets = [init]
def check_unreachable():
for F, Fn in zip(sets, sets[1:]):
if check(Implies(Fn,F)):
retrun True
return False
queue = []
while true:
check_reachable()
check_unreachable
```

So if I used spacer naively with a predicate per programs point, each would be solved in the PDR manner.

Hmm. So the horn clauses don’t give you anything in regards to wp itself. They are useful for interprcoedural and invaraints?

Interesting stuff in here. Locking examples. Concurrency In raw smtlib but also using tool https://github.com/seahorn/seahorn-tutorial

Hmm. The tutorial shows the block-like CHC form But then takes macro steps using wp to fuse out?

Hmm. So the horn clauses aren’t because they are a natural modelling framework, but because they have a natural solution method.

CHC and CLP are the same thing, different communities

init -> inv inv /\

https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/nbjorner-yurifest.pdf

Maybe there is some connection to the datalog stuff? One predicate per line of program? Or you could have program point as varaible?

Lifting blocks to clauses. How would one model a control flow diagram in prolog? block1in(x’) :- blockout(x) /\ blocktransition(x,x’)

blockin(x) :- (blockout( wp() ))

block1in :- block7out

which direction should be in/out? depends on wp vs sp?

Alternatively an explciit variable for control point block(n, )

Or should entire block be a predicate? block(invars, outvars) :- blocka(outvars, invars’)

https://www.youtube.com/watch?v=bTPSCVzp1m8&ab_channel=WorkshoponSoftwareCorrectnessandReliability2013

The expnasion / unrolling operator.

https://arxiv.org/abs/2104.04224 - A theory of heap[s for cxonstrained horn clauses

Houdini dillig daikon - random execution traces infer stuf “invasraint inference” Aiken - https://ti.arc.nasa.gov/m/events/nfm2013/pubs/AikenNFM13.pdf “sounds like daikon” Notes on z3 codebase muz = muZ. fixedpoint solvers. There are a vairety

PDR / IC3 always come up. I should try to know what those are

Meta foregin code base techniques

- checkout build files - Gives a hierarchy of the structure of the project. Dependency sorted.
- checkout tests

muz has: datalog, bmc, and spacer mode, and xform? :print-certificatie fives “inductive invataint” even in datalog?

data driven horn clause solver. They use decision trees? That’s interesting. What about a scikit-learn loop, Or polynomial fitting? Good invariants candidates over bit vectors? That cbat thing sounds kind of good. One of those domains. There was also that thing. That paper for comparative whatever that was guessing useful conditions inside.

Constrained horn clauses

https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/MSR-TR-2013-6.pdf

https://chc-comp.github.io/

https://spacer.bitbucket.io/

https://seahorn.github.io/

https://theory.stanford.edu/~arbrad/papers/Understanding_IC3.pdf

Programming Z3 tutorial Spacer jupyter tutorial https://github.com/agurfinkel/spacer-on-jupyter/blob/master/Dagstuhl2019.ipynb https://arieg.bitbucket.io/pdf/synasc2019.pdf

IC3 / PDR unbounded model checking

This is somehow more than a prolog. It’s inferring *predicates*

https://people.eecs.berkeley.edu/~sseshia/219c/#books modelchecking course.

Subgoal induction. ??? Seems tailor fitted to CHC. From Mitchel Wand paper refernces to Morris paper “inductionless induction / narrowing” are the things cody seemed to find this reminsent of

Wait, so SAT is solving the problem, but unsat is a counterexample trace?

```
from z3 import *
x = BitVec("x",16)
x1 = BitVec("x'",16)
s = SolverFor("HORN")
i16 = BitVecSort(16)
Inv = Function('mc', i16, BoolSort())
'''
x = 0
while(x < 10)
x++
assert(x < 11)
'''
init = ForAll(x, Implies(x == 0, Inv(x)))
loop = ForAll([x,x1], Implies( And(x < 10, Inv(x), x1 == x + 1), Inv(x1) ) )
post = ForAll(x, Implies(Inv(x), x < 11)) #ForAll(x, Implies(And( x < 11 , Inv(x)), False))
s.add(init, loop, post)
s.check()
s.model().eval( Inv(x) )
```

You can use horn clauses to perform craig interpolation. A => I I /\ B => False Hmmmmmm. But what do you use craig interpolation for? IC3 I think. You can build over approximations of intermiedate sets.

Running a prolog query:

forall(x, plus(s(x), ) ) plus(x,y,z) => False Or True => plus(x,y,z) ? with no quantification?