YouTube Video https://youtu.be/6q15ytIOE3U Speaker Dr. Leemon Baird | Co-Founder & Chief Scientist | Hedera Hashgraph Abstract Dr. Leemon Baird will present an introduction to how mathematical proofs can be checked by a computer, and how this can be useful in projects where security and correctness is important. These formal methods can increase our confidence that a consensus algorithm is truly Asynchronous Byzantine Fault Tolerant (aBFT), or that a given program correctly implements that algorithm. Examples will be shown from the proofs that have been developed for the Hashgraph consensus algorithm, using the Coq Proof Assistant.

1. Dr.Leemon Baird
Co-founder + Chief Scientist

2. Formal Methods

3. FORMAL METHODS
Math proofs, checked by computer
Prove an algorithm is “good”
Prove source code is “correct”

4. FORMAL METHODS
Math proofs, checked by computer
•Coq proof assistant
Prove an algorithm is “good”
Prove source code is “correct”

5. FORMAL METHODS
Math proofs, checked by computer
•Coq proof assistant
Prove an algorithm is “good”
•Hashgraph consensus is ABFT
Prove source code is “correct”

6. FORMAL METHODS
Math proofs, checked by computer
•Coq proof assistant
Prove an algorithm is “good”
•Hashgraph consensus is ABFT
Prove source code is “correct”
•Java implements Hashgraph consensus
•Source code open review

7. Hashgraph
ABFT proof
Order
Median
Received
Famous
Consensus
Decision
Progress
Vote
Round
Sees
HashgraphFacts
Hashgraph
Majority
Calculate
Cardinality
Decide
Relation
Tact
18 Libraries

8. Library Sees
SEES.HTML

9. SEES.HTML

10. x
y
DEFINITION OF SEES IN SEES.V

11. z z’
x
y
DEFINITION OF SEES IN SEES.V

12. x is an ancestor of y
AND there are no two events z and z’ for which:
z is an ancestor of y
AND z’ is an ancestor of y
AND z and z’ are “forks”
AND z and x have the same creator
x @= y
/ ~ exists z z’,
z @= y
/ z’ @= y
/ fork z z’
/ creator z = creator x.
Say “sees x y” which means event y can see event x, meaning:
Definition sees ( x y : event) : Prop :=
z z’
x
y
DEFINITION OF SEES IN SEES.V

13. Informal lemma
Informal proof
Math journal lemma
Math journal proof
Coq lemma
Coq proof
1
2
3
4
5
6
MATH PROOFS

14. v
x
Strongly Seeing Lemma:
If any event strongly sees x, then no event strongly sees a fork of x.
INFORMAL LEMMA

15. Informal
Proof
Strongly Seeing Lemma:
If any event strongly sees x, then no event strongly
sees a fork of x.
v
x
> 2/3

16. Strongly Seeing Lemma:
If any event strongly sees x, then no event strongly
sees a fork of x.
v
x
w
y
Informal
Proof > 2/3 > 2/3

17. Strongly Seeing Lemma:
If any event strongly sees x, then no event strongly
sees a fork of x.
v
x
> 2/3
w
y
> 2/3> 1/3Informal
Proof

18. Strongly Seeing Lemma:
If any event strongly sees x, then no event strongly
sees a fork of x.
v
x
w
y
< 1/3
Informal
Proof > 2/3 > 2/3> 1/3

19. Strongly Seeing Lemma:
If any event strongly sees x, then no event strongly
sees a fork of x.
v
x
w
y
a
b
Informal
Proof
< 1/3
> 2/3 > 2/3> 1/3

20. Strongly Seeing Lemma:
If any event strongly sees x, then no event strongly
sees a fork of x.
v
x
w
y
a
b
Informal
Proof
< 1/3
> 2/3 > 2/3> 1/3

21. MATH JOURNAL LEMMA

22. MATH JOURNAL PROOF

23. Coq Lemma
Statement

24. Coq Lemma
Proof
Proof.
1 subgoal
______________________________________(1/1)
forall (W : world) (x y v w : event),
member W v ->
member W w -> fork x y -> stsees x v -> stsees y w -> False

25. Proof.
intros W x y v w Wv Ww Hfork Hssx Hssy.
1 subgoal
W : world
x, y, v, w : event
Wv : member W v
Ww : member W w
Hfork : fork x y
Hssx : stsees x v
Hssy : stsees y w
______________________________________(1/1)
False
Coq Lemma
Proof

26. Proof.
intros W x y v w Wv Ww Hfork Hssx Hssy.
Coq Lemma
Proof
1 subgoal
W : world
x, y, v, w : event
Wv : member W v
Ww : member W w
Hfork : fork x y
Hssx : stsees x v
Hssy : stsees y w
______________________________________(1/1)
False

27. Proof.
intros W x y v w Wv Ww Hfork Hssx Hssy.
destruct Hssx as (v' & Hv & Hmajx).
Coq Lemma
Proof
1 subgoal
W : world
x, y, v, w : event
Wv : member W v
Ww : member W w
Hfork : fork x y
v' : event
Hv : v' @= v
Hmajx : supermajority
(fun a : peer =>
exists z : event,
creator z = a / sees x z / sees z v') every
Hssy : stsees y w
______________________________________(1/1)
False

28. Proof.
intros W x y v w Wv Ww Hfork Hssx Hssy.
destruct Hssx as (v' & Hv & Hmajx).
destruct Hssy as (w' & Hw & Hmajy).
so (supermajority_intersect_3 _#5 eq_peer_decide
supermajority_honest Hmajx Hmajy) as
(a & Hhonest & Hseesx & Hseesy).
destruct Hseesx as (q & Hcrq & Hxq & Hqv').
destruct Hseesy as (r & Hcrr & Hyr & Hqw').
assert (q @= r / r @= q) as Hordered.
{
so (ancestor_decide q r) as [? | Hnqr]; auto.
so (ancestor_decide r q) as [? | Hnrq]; auto.
exfalso.
so (world_forks W _ Hhonest) as H.
destruct H.
exists q, r.
do2 3 split; auto.
{
apply (world_closed _ _ v); auto.
eapply star_trans; eauto.
apply sees_impl_ancestor; auto.
}
{
apply (world_closed _ _ w); auto.
eapply star_trans; eauto.
apply sees_impl_ancestor; auto.
}
{
do2 2 split.
{
subst a; auto.
}
{
contradict Hnqr.
apply self_ancestor_impl_ancestor; auto.
}
{
contradict Hnrq.
apply self_ancestor_impl_ancestor; auto.
}
}
}
cut (forall x y q r,
fork x y
-> sees x q
-> sees y r
-> q @= r
-> False).
{
intros H.
destruct Hordered; [eapply (H x) | eapply (H y)];
eauto using fork_symm.
}
repeat (match goal with H : _ |- _ => clear H end).
intros x y q r Hfork Hxq Hyr Hqr.
destruct Hyr as (Hyr & Hnofork).
destruct Hnofork.
exists x, y.
do2 3 split; auto using fork_creator.
destruct Hxq as (Hxq & _).
eapply star_trans; eauto.
Qed.
Coq Lemma
Proof

29. Prove code correctness
• Krakatoa: Prove Java code implements the
algorithm
• Coq: automatically generate code
(OCaml, Haskell or Scheme)
Extend the algorithms
Prove the compiler, JVM, OS, microprocessor
NEXT STEPS

30. THE TRUST LAYER OF THE INTERNET