# Bounds on Growth and Impossibility of Collapse for Point Vortex Systems

Samuel Zbarsky

###### Abstract

We consider 2D point vortex systems and, under certain conditions on the
masses of the point vortices, prove that collapse is impossible and provide
bounds on the growth of the system. The bounds are typically of the form
$O(t^{a})$ for some $a<1/2$, but we obtain various results for various
assumptions on the masses.

##  1 Introduction

We discuss point vortex systems in $\mathbb{R}^{2}$. This is a model for
2-dimensional incompressible fluids where vorticity is concentrated around
certain points. The model is defined by having $n$ point vortices with nonzero
signed masses $m_{1},\ldots,m_{n}$ and trajectories $x_{i}(t)$ which satisfy
the ODE

$\frac{d}{dt}x_{i}(t)=\sum_{j\neq i}m_{j}K(x_{i},x_{j}).$

where

$K(x,y)=\frac{(x-y)^{\perp}}{|x-y|^{2}}.$

Here $(z_{1},z_{2})^{\perp}=(-z_{2},z_{1})$ and we assume that at $t=0$, no
two point vortices are at the same point.

We are interested in two related questions:

  1. 1.

Under what conditions on the masses can we say that the ODE has global
existence?

  2. 2.

For given masses, what bounds can we give on solutions to the ODE?

We begin with a discussion of previous work on the behavior of point vortex
systems. A more thorough introduction can be obtained by reading [10] and [2].
First, we note that there are two other ways to think of point vortex systems.
One is to identify $\mathbb{R}^{2}$ with the complex plane, and use

$K(z_{1},z_{2})=\frac{i}{\ \overline{z_{1}-z_{2}}\ }$ (1)

to rewrite the ODE. Another approach is to note that a point vortex system is
a Hamiltonian system with the position and momentum variables being the $x$
and $y$ coordinates of vortices (suitably rescaled) and the Hamiltonian being

$H=\sum_{1\leq i<j\leq n}m_{i}m_{j}\log|x_{i}-x_{j}|.$

Both of these points of view are useful for proving certain results, but we
will not adopt them in the present paper.

The ODE has the following conserved quantities, which are easy to check by
direct computation:

  1. 1.

$X=\sum m_{i}x_{i}$. This is actually two conserved quantities (the $x$ and
$y$ coordinates). When the total mass $\sum m_{i}$ is nonzero, we can divide
$X$ by it to get conservation of center of mass.

  2. 2.

$I=\sum m_{i}|x_{i}|^{2}$ is the second moment (physically, it corresponds to
the fluid’s angular momentum).

  3. 3.

$\mathcal{E}=\sum_{1\leq i<j\leq n}m_{i}m_{j}\log|x_{i}-x_{j}|$ is the energy.

  4. 4.

$\tilde{I}=\sum_{1\leq i<j\leq n}m_{i}m_{j}|x_{i}-x_{j}|^{2}$, obtained from
an appropriate linear combination of $I$ and $|X|^{2}$.

It is easy to see that one-vortex systems are stationary and that two-vortex
systems have circular or linear orbits. For three vortices the system is
completely integrable (recall the Hamiltonian formulation) and its trajectory
has been analyzed in [1]. For four vortices, the special case where the sum of
masses and $X$ are both 0 also gives an integrable system [4].

Also using the Hamiltonian formulation, [6] showed using KAM theory that for
generic collections of masses, there is a positive measure set of initial data
that leads to quasiperiodic motion, constructed hierarchically with nearby
pairs of vortices in high-frequency orbits, such pairs of vortices orbiting
each other at a much larger distance with much larger period, and so on.

There have also been a large number of results obtaining relative equilibria,
that is configurations which are stationary, rotate, or translate uniformly,
and analyzing their behavior (see [3] and [11] for surveys of results in this
direction). The only result we’ll state along these lines is that three
vortices at the vertices of an equilateral triangle, whatever their masses,
give a relative equilibrium [12]. There are also self-similarly expanding or
contracting configurations, which we discuss in Section 2.

Finally, there is the question of rigorously justifying the point vortex
model, that is obtaining solutions to the ODE as limits of smooth solutions to
the 2D Euler equation by replacing each point vortex with a localized smooth
nonpositive or nonnegative vorticity function. This justification is obtained
for different assumptions on the vorticity function in [9], [8], [14], [15]
(in order of weakening assumptions and stronger conclusions).

We now turn to a discussion of when we can tell based on masses of point
vortices that we have global existence and boundedness of solutions. For one,
two, or three vortices, this can be seen when Conditions 1 and 2 below are
satisfied by using complete integrability mentioned above. For more than 3
vortices, we note that by standard ODE theory, a point vortex system can fail
to have global existence only if

$\liminf_{s\to T^{-}}\min_{1\leq i<j\leq k}|x_{i}(s)-x_{j}(s)|=0.$

Following an argument found in [10], we note that when all the masses have the
same sign, conservation of $I$ gives us an upper bound on the solution while
it exists, and then conservation of $\mathcal{E}$ gives us a lower bound on
the distance between vortices for all time, both easily computed from the
initial data. We aim to obtain results in this vein when the masses can have
different signs.

Outside of Section 2, we typically assume that the $n$ point vortices satisfy
the following conditions:

######  Condition 1 (No Translation Condition).

No nonempty subset has total mass 0.

and

######  Condition 2 (No Spiral Condition).

Take any partition

$S=S_{0}\sqcup S_{1}\sqcup\cdots\sqcup S_{k}$

of the indices into sets where $k\geq 2$ and all the sets with the possible
exception of $S_{0}$ are nonempty. If we let

$\Omega_{i}=\sum_{j\in S_{i}}m_{j}$

then we have

$\sum_{1\leq i<j\leq k}\Omega_{i}\Omega_{j}\neq 0.$

In Section 2, we explain why we impose these conditions and give some examples
of bad behavior that can occur if they are not satisfied.

The only previous results in these directions that we are aware of for more
than three vortices of varying sign and which don’t depend on particularities
of the initial configuration come from [10] and are as follows:

  1. 1.

Assume the No Translation Condition (Condition 1) is satisfied. Then over a
fixed time interval, there is a bound (depending on the masses) for how much
the position of any vortex can change.

  2. 2.

Assume the No Translation Condition is satisfied. Then for Lebesgue-almost all
initial data, there is global existence. For a fixed time interval, they also
quantify the measure of initial data which will lead to two vortices coming
within distance $\epsilon$ of each other.

We prove the following results, stated here informally. Note that if we
reverse signs of masses and reflect the system over the $x$ axis, we will get
identical evolution of the system, so all statements below have corresponding
statements with signs reversed.

######  Theorem 3.

  1. 1.

Assume the No Translation Condition (Condition 1) is satisfied. Then the
position of vortices can change by at most $C\sqrt{t}$ (see Theorem 6 for the
precise statement).

  2. 2.

Assume the No Translation Condition is satisfied. Then the only way the ODE
can break down is by several vortices colliding at a point (see the discussion
at the end of Section 3).

  3. 3.

Assume that all the masses are positive, except one that is negative and
bigger in absolute value than all the others combined. Then the solution has
global existence and is bounded (see Theorem 7 for the precise statement).

  4. 4.

For certain other open sets of masses, we have global existence and growth
slower than $\sqrt{t}$. See Theorem 11 and Section 7 for a discussion of what
open sets we obtain this for and what bounds we obtain.

Based on the methods used in the proof, one might hope that the following more
general result holds:

######  Conjecture 4.

Suppose that the masses satisfy the No Translation and No Spiral Conditions.
Then any solution has global existence and grows as $o(\sqrt{t})$.

Probably a somewhat stronger, more narrowly tailored to avoiding specific bad
behaviors, version of this conjecture would also hold.

###  1.1 Bounds on the kernel and notation

Later on, we will need the following bounds on the kernel $K$:

$\displaystyle|DK(x,y)|$ $\displaystyle\leq\frac{C}{|x-y|^{2}}$ (2)
$\displaystyle|D^{2}K(x,y)|$ $\displaystyle\leq\frac{C}{|x-y|^{3}}$
$\displaystyle|D^{3}K(x,y)|$ $\displaystyle\leq\frac{C}{|x-y|^{4}}$

We will let $C=C(m_{1},\ldots,m_{n})$ be a big enough positive constant,
possibly denoting a different constant on different lines. When we use $O$
notation, the implicit constant will be allowed to depend on all the masses.

We will use $[n]$ to denote $\\{1,\ldots,n\\}$.

###  1.2 Acknowledgements

The author wishes to express his gratitude to Theodore Drivas for discussions
of related problems and for encouraging him to finish writing up the present
work. This work was supported by the National Science Foundation Graduate
Research Fellowship Program under Grant No. DGE-1656466.

##  2 Examples

If the No Translation Condition (Condition 1) is not satisfied, the simplest
example of bad behavior is a pair of vortices with $m_{1}=-m_{2}$ translating
at constant speed, in which case the solution grows linearly. We can also have
some other vortices in the system, as long as the pair of vortices with masses
$\pm m$ do not come too close to other vortices (for which we need a suitably
favorable starting configuration). Then they can separate out, with distance
from them to any other vortices growing linearly, and it is not hard to check
that in the limit, they will translate with speed $C+O(1/t)$. In addition, one
can expect, though we do not in general know how to prove except in the cases
of relative equilibria, similar behavior for systems with more than two
vortices for which total vorticity is 0, but $X\neq 0$. However, if the No
Translation Condition is satisfied, then conservation of the center of mass
should rule out such behaviors. This is why we need to include the No
Translation Condition.

If the No Spiral Condition is not satisfied, more specifically if we have

$\sum_{1\leq i<j\leq n}m_{i}m_{j}=0,$

then there are numerous constructed examples of solutions which self-similarly
spiral outwards, growing as $\sqrt{t}$. An analysis of such self-similarly
evolving 3-vortex system can be found in [2]. Some self-similarly evolving 4
and 5 vortex systems are constructed and analyzed in [13]. Some numerics for
self-similarly evolving systems with more vortices may be found in [7]. By
reflecting these solutions, or by reversing the signs of the vorticities, we
can construct examples that self-similarly spiral inwards and collapse in
finite time. However, if

$\sum_{1\leq i<j\leq n}m_{i}m_{j}\neq 0,$

then such self-similarly spiraling solutions are impossible, as they would
violate conservation of the energy $\mathcal{E}$.

It is also possible to add some vortices far away (and slightly perturb
initial positions) and still obtain a collapse [5]. Similarly, if we have a
system spiraling outwards with distances between vortices growing like
$\sqrt{t}$, it seems plausible that we can replace each vortex with a cluster
of vortices having the same total mass, and still get growth of the system as
$\sqrt{t}$; in fact, we will construct such solutions in Corollary 16. It is
in order to preclude these possibilities that we introduce the No Spiral
Condition. We note that to break these mechanisms, we can use slightly weaker
conditions.

##  3 Movement bound

We have the following statement, which essentially says that each vortex of a
set can move by at most $C\sqrt{T}$ over time $T$, assuming they stay far
enough away from other vortices. The proof is identical to a proof in [10],
but with dependence on the time interval quantified. It is interesting as a
result in its own right, but it is also a model for the proof of Theorem 11.

######  Lemma 5.

Suppose we have a finite set of masses $m_{1},\ldots,m_{n}$ satisfying the No
Translation Condition (Condition 1). Then for any subset $S\subseteq[n]$,
there exists a constant $C_{S}$ depending on all the masses so that for any
$T$, if we have a solution of the ODE on time interval $[t_{1},t_{1}+T)$ with

$|x_{i}(s)-x_{j}(s)|>\sqrt{T}$ (3)

for all $i\in S,j\notin S$, $s\in[t_{1},t_{1}+T]$ then for any
$s\in[t_{1},t_{1}+T]$ and $i\in S$, we have that
$|x_{i}(s)-x_{i}(t_{1})|<C_{S}\sqrt{T}$.

###### Proof.

We induct on the set $S$. If $|S|=1$, then at any point in time (3) tells us
that the velocity is at most $\frac{C}{\sqrt{T}}$, so the total distance
traveled is at most $C\sqrt{T}$.

Assume the statement holds for all proper subsets of $S$. We choose $C_{S}$
sufficiently large, depending on the masses and on $C_{A}$ for all $A\subseteq
S$, to make the following argument go through.

First, we note that the center of mass of $S$, denoted $x_{S}$, does not move
from internal interactions of $S$, so it only moves due to the external field
coming from vortices not in $S$. Due to (3), this external velocity field
applies velocity at most $\frac{\tilde{C}}{\sqrt{T}}$ to each vortex in $S$,
so it applies velocity at most $\frac{\hat{C}}{\sqrt{T}}$ to the center of
mass $x_{S}$, so $x_{S}$ moves by at most $\hat{C}\sqrt{T}$. Now, suppose for
the sake of contradiction that for some $i\in S$ and some time $s$, we have
$|x_{i}(s)-x_{i}(t_{1})|\geq C_{S}\sqrt{T}$. Then, by taking $C_{S}>3\hat{C}$,
we know that either at time $s$ or at time $t_{1}$, we have

$|x_{i}-x_{S}|\geq\frac{C_{S}}{3}\sqrt{T}.$

Without loss of generality, this occurs at time $s$ (otherwise we simply use
$s$ to denote the time $t_{1}$). Because the total mass of $S$ is not 0, this
implies that at time $s$, we have that for some $j\in S$,

$|x_{i}-x_{j}|\geq\frac{\sum_{k\in S}m_{k}}{\sum_{k\in
S}|m_{k}|}\frac{C_{S}}{3}\sqrt{T}.$

We now can split $S$ into two clusters that are sufficiently far apart. In the
worst case, the vortices of $S$ are evenly spaced with $x_{i}$ and $x_{j}$ at
the two endpoints. Whatever the arrangement, there must be some partition

$S=A\sqcup B$

with $A$, $B$ nonempty and

$|x_{k}(s)-x_{\ell}(s)|>\frac{\sum_{k\in S}m_{k}}{\sum_{k\in
S}|m_{k}|}\frac{C_{S}}{3(|S|-1)}\sqrt{T}$ (4)

for all $k\in A,\ell\in B$. We now just need that the coefficient in front of
$\sqrt{T}$ in (4) is greater than $C_{A}+C_{B}+1$ for all possible partitions
$S=A\sqcup B$, which is achievable by taking $C_{S}$ sufficiently large. We
then obtain a contradiction and complete the proof by using the inductive
hypothesis and making the bootstrap assumption that clusters $A$ and $B$
maintain a spatial separation of $\sqrt{T}$ for the entire time interval
$[t_{1},t_{1}+T]$ (all the times of which are at time interval at most $T$
forward or backward from $s$). ∎

We can now apply the lemma with $S=[n]$ to obtain

######  Theorem 6.

Suppose we have a finite set of masses $m_{1},\ldots,m_{n}$ satisfying the No
Translation Condition. Then there exists some constant
$C=C(m_{1},\ldots,m_{n})$ so that for any $T$, assuming that we have a
solution of the ODE on time interval $[t_{1},t_{1}+T)$, we have that
$|x_{i}(s)-x_{i}(t_{1})|<C\sqrt{T}$.

We make the observation that if we want to get a better constant $C$ in the
theorem statement, we can abandon the induction structure of the proof, and
instead solve an optimization problem with various constants attached to
various clusters $S\subseteq[n]$. While doing this is not particularly
worthwhile, we note it here because we will make an analogous observation in a
more interesting context after the proof of Theorem 11.

We can also use Theorem 6 to classify ways in which solutions can break down
when the No Translation Condition is satisfied.

Suppose that the ODE has a solution on time $[0,T)$ that does not continue
past $T$. It is easy to see from standard ODE existence and uniqueness that
the only way this happens is if

$\liminf_{s\to T^{-}}\min_{1\leq i<j\leq k}|x_{i}(s)-x_{j}(s)|=0.$

From this it follows that there is at least one pair $(i,j)$ satisfying $1\leq
i<j\leq k$ with

$\liminf_{s\to T^{-}}|x_{i}(s)-x_{j}(s)|=0.$

Applying Theorem 6 over intervals $[T-\epsilon,T)$, we obtain that

$\lim_{s\to T^{-}}|x_{i}(s)-x_{j}(s)|=0.$

If this holds, we say that we have a _collision_ of vortices $i$ and $j$. It
is easy to see that collision is an equivalence relation, so at time $T$, we
have one or several collisions of several vortices each, and the distance
between a given collision and other vortices has some positive lower bound on
$[T-\epsilon,T)$. Note that since we use Theorem 6, we need the No Translation
Condition for this argument. In fact, it may be possible to construct a blowup
where a pair of vortices with mass $\pm 1$ moves around faster and faster,
moving closer and thus getting faster each time it scatters off of another
vortex, and has no limiting position.

##  4 One big negative

######  Theorem 7.

Suppose that $m_{2},\ldots m_{n}>0$ and $m_{1}<0$ with
$-m_{1}>m_{2}+\cdots+m_{n}$. Then the ODE solution exists and is bounded for
all time.

###### Proof.

First, we prove existence. Suppose for the sake of contradiction that we have
a collision at time $T$, and that $S\subset[n]$ is the set of vortices
involved in that collision (since the No Translation Condition holds, this is
the only way that breakdown can occur by the discussion at the end of Section
3). We let $R>0$ and $\delta>0$ be chosen such that on the time interval
$[T-\delta,T]$, the distance between any vortex in the collision and any
vortex outside the collision is lower bounded by $R$ and the distance between
any two of the colliding vortices is upper bounded by $R/100$. Then we have
that

$\tilde{I}_{S}=\sum_{\begin{subarray}{c}i,j\in S\\\
i<j\end{subarray}}m_{i}m_{j}|x_{i}-x_{j}|^{2}$

is an almost-conserved quantity, in the sense that it doesn’t change due to
interactions of the vortices in the collision. Then we use the bound on $DK$
in (2) to get that the velocity difference between vortices $i,j\in S$ arising
from the outside sources is at most

$C\frac{|x_{i}-x_{j}|}{R^{2}},$

so

$\left|\frac{d}{dt}\tilde{I}_{S}\right|\leq\sum_{\begin{subarray}{c}i,j\in
S\\\ i<j\end{subarray}}C\frac{|x_{i}-x_{j}|^{2}}{R^{2}}$ (5)

We now have two cases.

If $1\notin S$, then we immediately get first, that $\tilde{I}_{S}$ is
positive shortly before time $T$, and second, that the right-hand side of
$\eqref{ISODE}$ is bounded by $C\tilde{I}_{S}/R^{2}$, so

$\left|\frac{d}{dt}\tilde{I}_{S}\right|\leq\frac{C}{R^{2}}\tilde{I}_{S}$

so $\tilde{I}_{S}$ cannot go to 0 as $t\to T^{-}$, which contradicts the
vortices in $S$ colliding.

Now suppose $1\in S$. We will assume wlog that $S=[k]$ and we will translate
our coordinates in a time-dependent way so that $x_{1}(t)=0$ (this does not
change any of the values $\tilde{I}_{S}$, the right-hand side of (5), or the
two quantities we are comparing below). Then we have that

$\displaystyle\sum_{2\leq i<j\leq k}m_{i}m_{j}|x_{i}-x_{j}|^{2}$
$\displaystyle=\left(\sum_{i=2}^{k}m_{i}\right)\sum_{i=2}^{k}m_{i}|x_{i}|^{2}-\sum_{i=2}^{k}\sum_{j=2}^{k}m_{i}m_{j}x_{i}\cdot
x_{j}$
$\displaystyle=\left(\sum_{i=2}^{k}m_{i}\right)\sum_{i=2}^{k}m_{i}|x_{i}|^{2}-\left|\sum_{i=2}^{k}m_{i}x_{i}\right|^{2}$
$\displaystyle\leq\left(\sum_{i=2}^{k}m_{i}\right)\sum_{i=2}^{k}m_{i}|x_{i}|^{2}$
$\displaystyle\leq\frac{\sum_{i=2}^{k}m_{i}}{-m_{1}}\left(-m_{1}\sum_{i=2}^{k}m_{i}|x_{i}-x_{1}|^{2}\right).$

Thus, if we look at both the positive and the negative terms of
$\tilde{I}_{S}$ (where the positive terms do not involve $m_{1}$ and the
negative ones do), we see that the absolute value of the sum of the negative
terms is bigger than the sum of the positive ones by a factor of at least

$1+\epsilon=\frac{-m_{1}}{\sum_{i=2}^{n}m_{i}}.$

From this we get that

$\sum_{1\leq i<j\leq n}^{n}|x_{i}-x_{1}|^{2}\leq-C\tilde{I}_{S}.$

From this we get that $\tilde{I}_{S}$ is negative shortly before time $T$.
Also, combining this with $\eqref{ISODE}$, we get that

$\left|\frac{d}{dt}\tilde{I}_{S}\right|\leq\frac{-C}{R^{2}}\tilde{I}_{S}$

so $\tilde{I}_{S}$ cannot go to 0 as $t\to T^{-}$, which contradicts the
vortices in $S$ colliding. This finishes the proof of global existence and
uniqueness of solutions to the ODE.

We now prove boundedness, using conservation of the center of mass and second
moment. Without loss of generality, we can assume that the center of mass of
the entire system is at 0. Then, for any $t>0$, we have that the center of
mass of vortices $2,\ldots,n$ is

$\frac{-m_{1}}{m_{2}+\cdots+m_{n}}x_{1}(t)$

so the second moment is

$\displaystyle I$ $\displaystyle=\sum_{i=1}^{n}m_{i}|x_{i}(t)|^{2}$
$\displaystyle\geq
m_{1}|x_{1}(t)|^{2}+\left(\sum_{i=2}^{n}m_{i}\right)\left|\frac{-m_{1}}{m_{2}+\cdots+m_{n}}x_{1}(t)\right|^{2}$
$\displaystyle\geq
m_{1}|x_{1}(t)|^{2}+\frac{m_{1}^{2}}{m_{2}+\cdots+m_{n}}|x_{1}(t)|^{2}$
$\displaystyle\geq c(m_{1},\ldots,m_{n})|x_{1}(t)|^{2}$

Thus $x_{1}$ is bounded. Since all the other masses are positive, the
conservation of the second moment then gives us that the entire system is
bounded. ∎

##  5 Clusters

For our proof of Theorem 11, we will need to define clusters and prove a lemma
about them, which we do in this section.

######  Definition 8.

Nonempty $S\subset[n]$ is an $a$-cluster at time $t$ if
$|x_{i}(t)-x_{j}(t)|\leq a$ for all $i,j\in S$. An $a$-cluster is $b$-isolated
if $|x_{i}(t)-x_{j}(t)|\geq b$ for all $i\in S,j\notin S$.

Some notes about this definition:

  * •

For any $1\leq i\leq n$, we have that $\\{i\\}$ is a 0-cluster.

  * •

$[n]$ is $\infty$-isolated.

  * •

If we talk about a $b$-isolated $a$-cluster, we are implicitly assuming that
$a\ll b$ (more precisely, there is some large constant $C$ depending on all
the masses and we will always have $Ca\leq b$). The constant $C$ is necessary
since we will need to use the centers of mass of the clusters and, depending
on the masses and their signs, we can have for instance that the center of
mass of an $a$-cluster is at distance $100a$ from any of its vortices.

When we talk about the distance between two clusters $d(S_{1},S_{2})$, we will
mean the distance between their centers of mass, but since we will only care
about this distance approximately, and since the clusters will be sufficiently
isolated, we could equally well take the closest distance of a vortex in one
to a vortex in the other, the furthest distance of a vortex in one to a vortex
in the other, or any other similar definition.

Now suppose we have a $b$-isolated $a$-cluster $S$. We can let

$m_{S}=\sum_{i\in S}m_{i}$

be the total mass of $S$ and

$x_{S}=\frac{\sum_{i\in S}m_{i}x_{i}}{m_{S}}$

be the center of mass of $S$ and consider the (negative) change in energy we
would get if we replaced the cluster with a single vortex of the mass $m_{S}$
located at $x_{S}$ (the center of mass of $S$). Specifically, we recall that

$\mathcal{E}=\sum_{1\leq i<j\leq n}m_{i}m_{j}\log|x_{i}-x_{j}|$

is the energy. We let $\overline{E}_{S}$ be the energy of the vortex
configuration where we replace the cluster $S$ with a single vortex of mass
$m_{S}$ located at $x_{S}$, that is

$\overline{E}_{S}=\sum_{\begin{subarray}{c}i,j\notin S\\\
i<j\end{subarray}}m_{i}m_{j}\log|x_{i}-x_{j}|+\sum_{i\notin
S}m_{i}m_{S}\log|x_{S}-x_{i}|.$

We now define

$\mathcal{E}_{S}=\mathcal{E}-\overline{E}_{S}=\sum_{i\notin S,j\in
S}m_{i}m_{j}\left(\log|x_{i}-x_{j}|-\log|x_{S}-x_{i}|\right)+\sum_{\begin{subarray}{c}i,j\in
S\\\ i<j\end{subarray}}m_{i}m_{j}\log|x_{i}-x_{j}|.$ (6)

This is approximately equal to just the interaction energy within the cluster
(the second sum on the right-hand-side), but we want this exact form of the
energy, since it allows us to prove the following lemma:

######  Lemma 9.

If $S$ is a $b$-isolated $a$-cluster, then

$\left|\frac{d}{dt}\mathcal{E}_{S}\right|\leq C\frac{a^{2}}{b^{4}}$

###### Proof.

The proof is by Taylor expansion of the kernel $K$ and obtaining
cancellations. We recall that $\mathcal{E}_{S}=\mathcal{E}-\overline{E}_{S}$.
Note that $\mathcal{E}$ is conserved, so we need only understand how
$\overline{E}_{S}$ changes. For simplicity of notation, we renumber the
vortices so that $S=[k]$. Then we let $v_{i},v_{S}$ be the velocities that
would arise from the vortex configuration where we replaced the cluster $S$
with a single vortex of mass $m_{S}$ located at $x_{S}$. Namely for $i>k$, we
have

$\displaystyle v_{i}$
$\displaystyle=m_{S}\frac{(x_{i}-x_{S})^{\perp}}{|x_{i}-x_{S}|^{2}}+\sum_{\begin{subarray}{c}k<j\leq
n\\\ j\neq
i\end{subarray}}m_{j}\frac{(x_{i}-x_{j})^{\perp}}{|x_{i}-x_{j}|^{2}}$
$\displaystyle v_{S}$ $\displaystyle=\sum_{k<j\leq
n}m_{j}\frac{(x_{S}-x_{j})^{\perp}}{|x_{S}-x_{j}|^{2}}$

and let $u_{i},u_{j},u_{S}$ be the adjustments to the velocity so that

$\displaystyle\frac{d}{dt}x_{i}$ $\displaystyle=v_{i}+u_{i}$
$\displaystyle\frac{d}{dt}x_{S}$ $\displaystyle=v_{S}+u_{S}$

Then

$\displaystyle-\frac{d}{dt}\mathcal{E}_{S}=\frac{d}{dt}\overline{E}_{S}$
$\displaystyle=\sum_{k<i<j\leq
n}m_{i}m_{j}\frac{\frac{d}{dt}(x_{i}-x_{j})}{|x_{i}-x_{j}|^{2}}\cdot(x_{i}-x_{j})+\sum_{k<i\leq
n}m_{i}m_{S}\frac{\frac{d}{dt}(x_{i}-x_{S})}{|x_{i}-x_{S}|^{2}}\cdot(x_{i}-x_{S})$
$\displaystyle=\sum_{k<i<j\leq
n}m_{i}m_{j}\frac{v_{i}-v_{j}}{|x_{i}-x_{j}|^{2}}\cdot(x_{i}-x_{j})+\sum_{k<i\leq
n}m_{i}m_{S}\frac{v_{i}-v_{S}}{|x_{i}-x_{S}|^{2}}\cdot(x_{i}-x_{S})$
$\displaystyle\qquad+\sum_{k<i<j\leq
n}m_{i}m_{j}\frac{u_{i}-u_{j}}{|x_{i}-x_{j}|^{2}}\cdot(x_{i}-x_{j})+\sum_{k<i\leq
n}m_{i}m_{S}\frac{u_{i}-u_{S}}{|x_{i}-x_{S}|^{2}}\cdot(x_{i}-x_{S})$
$\displaystyle=\sum_{k<i<j\leq
n}m_{i}m_{j}\frac{u_{i}-u_{j}}{|x_{i}-x_{j}|^{2}}\cdot(x_{i}-x_{j})+\sum_{k<i\leq
n}m_{i}m_{S}\frac{u_{i}-u_{S}}{|x_{i}-x_{S}|^{2}}\cdot(x_{i}-x_{S})$ (7)

where all the terms involving $v_{i}$ or $v_{S}$ cancel by the same
calculation that gives conservation of energy for a point vortex system. To
bound the right-hand side of (7), we start by bounding $u_{i}$ and $u_{S}$. We
note that by the same calculation which gives conservation of the center of
mass of a point vortex system, the interactions within the cluster have no net
effect on the derivative of $x_{S}$, so

$u_{S}=\frac{d}{dt}x_{S}-v_{S}=\sum_{j=1}^{k}\sum_{i=k+1}^{n}m_{i}\frac{m_{j}}{m_{S}}\left(\frac{(x_{j}-x_{i})^{\perp}}{|x_{j}-x_{i}|^{2}}-\frac{(x_{S}-x_{i})^{\perp}}{|x_{S}-x_{i}|^{2}}\right).$

We now Taylor expand $K(x,y)=\frac{(x-y)^{\perp}}{|x-y|^{2}}$ around
$x=x_{S}$, see that the linear terms cancel by the definition of the center of
mass, and use (2) to get that

$|u_{S}|\leq C\frac{a^{2}}{b^{3}}.$ (8)

We now bound $u_{i}$ for $i>k$ by calculating

$u_{i}=\frac{d}{dt}x_{i}-v_{i}=\sum_{j=1}^{k}m_{j}\left(\frac{(x_{i}-x_{j})^{\perp}}{|x_{i}-x_{j}|^{2}}-\frac{(x_{i}-x_{S})^{\perp}}{|x_{i}-x_{S}|^{2}}\right).$

We now Taylor expand $K(x,y)=\frac{(x-y)^{\perp}}{|x-y|^{2}}$ around
$y=x_{S}$, see that the linear terms cancel by the definition of the center of
mass, and use (2) to get that

$|u_{i}|\leq C\frac{a^{2}}{b^{3}}.$ (9)

Together, (9) and (8) give a bound of $C\frac{a^{2}}{b^{4}}$ for all terms on
the right-hand side of (7) except those where $|x_{i}-x_{j}|\leq b/10$. For
those terms, we get

$\displaystyle u_{i}-u_{j}$
$\displaystyle=\frac{d}{dt}x_{i}-v_{i}-\left(\frac{d}{dt}x_{j}-v_{j}\right)$
$\displaystyle=\sum_{\ell=1}^{k}m_{\ell}\left(\frac{(x_{i}-x_{\ell})^{\perp}}{|x_{i}-x_{\ell}|^{2}}-\frac{(x_{i}-x_{S})^{\perp}}{|x_{i}-x_{S}|^{2}}\right)-m_{\ell}\left(\frac{(x_{j}-x_{\ell})^{\perp}}{|x_{j}-x_{\ell}|^{2}}-\frac{(x_{j}-x_{S})^{\perp}}{|x_{j}-x_{S}|^{2}}\right).$

We now Taylor expand $K(x,y)=\frac{(x-y)^{\perp}}{|x-y|^{2}}$ around
$x=x_{i},y=x_{S}$, and see that the first term that doesn’t cancel comes from
$D_{x}D^{2}_{y}f$, which from (2) gives us that

$|u_{i}-u_{j}|\leq C\frac{a^{2}|x_{i}-x_{j}|}{b^{4}}.$ (10)

We now plug (8), (9), and (10) into (7) to obtain the lemma statement. ∎

##  6 No collisions and growth slower than $T^{1/2}$

Let $\kappa=\kappa(n)$ be a sufficiently large constant, and define the
following condition:

######  Condition 10.

For some $0\leq h\leq n$, we have that $m_{1},\ldots,m_{h}$ are negative,
$m_{h+1},\ldots,m_{n}$ are positive, and for $1\leq i\leq h<j\leq n$, we have
$\kappa|m_{i}|<m_{j}$.

######  Theorem 11.

If $m_{1},\ldots,m_{n}$ satisfy Condition 10 then we get no collisions and
growth of the ODE solution as $O(T^{1/2-\epsilon(n)})$.

We will be clear where in the proof we use Condition 10.

The proof of this theorem is inspired by the proof of Theorem 6, except that
we use a different notion instead of position (the shifted logarithmic
distance $\overline{d}$ between vortices defined below) and a different notion
instead of center of mass (the average distance $\hat{d}$ defined below).
Analogously to the way in which conservation of $X$ together with the No
Translation Condition cause spatial separation if any vortex travels too far,
we have that conservation of the energy $\mathcal{E}$ together with the No
Spiral Condition (Condition 2) causes separation in scales if the shifted
logarithmic distance between some two vortices changes by a large factor.

Analogously to Lemma 5, we have the following lemma from which the theorem
immediately follows:

######  Lemma 12.

There exist constants $\alpha(k)$ for $2\leq k\leq n$ satisfying the
following. Suppose we have a finite set of masses $m_{1},\ldots,m_{n}$
satisfying Condition 10. Suppose further we have a cluster $S\subseteq[n]$,
and a partition into subclusters $S=S_{1}\sqcup\cdots\sqcup S_{k}$ with $k\geq
2$, and that over a time interval $[t_{1},t_{2}]$ with $t_{2}-t_{1}\leq T$ we
have the following bootstrap assumptions:

  1. 1.

all distances are less than $T^{1/2-\epsilon(n)}$

  2. 2.

$\displaystyle\text{each $S_{i}$ is a }b_{i}\text{-isolated }$
$\displaystyle\left(\frac{b_{i}^{2}}{\sqrt{T}}\right)\text{-cluster}$
$\displaystyle\text{$S$ is an }f\text{-isolated }$
$\displaystyle\left(\frac{f^{2}}{\sqrt{T}}\right)\text{-cluster}.$

for some $b_{i},f$ independent of time. Then, as long as $T$ is sufficiently
large, if at times $s_{1},s_{2}\in[t_{1},t_{2}]$ and for some $1\leq i<j\leq
k$, we have that

$d(S_{i},S_{j})=T^{1/2-p_{1}}\qquad\text{and}\qquad
d(S_{i},S_{j})=T^{1/2-p_{2}}$

respectively, then

$\frac{p_{1}}{p_{2}}<\alpha(k)$ (11)

Figure 1: This is an illustration of the clusters in the statement of Lemma 12

In other words, if different scales stay sufficiently separated, as quantified
by bootstrap assumption 2, then the distance between two clusters cannot
change too much, as quantified by (11).

###### Proof.

We will prove this by induction on $k$, after introducing some definitions.

We will define an appropriately shifted logarithmic notion of distance:

$\overline{d}(S_{i},S_{j})=\frac{1}{2}-\frac{\log d(S_{i},S_{j})}{\log T}.$

That is, if $\overline{d}(S_{i},S_{j})=p$, then $d(S_{i},S_{j})=T^{1/2-p}$.
Note that $\overline{d}(S_{i},S_{j})\leq 0$ would correspond to a distance
between vortices of $\geq\sqrt{T}$, and we will be proving that this is never
achieved. Also, $\overline{d}$ is an inverted distance in the sense that it
decreases when actual distance increases. Finally, note that the conclusion of
the lemma we are proving is that $\overline{d}$ changes by at most a factor of
$\alpha(k)$.

Now we define the _interaction energy_ of $S_{1},S_{2},\ldots,S_{k}$ via

$E(S_{1},\ldots,S_{k})=\mathcal{E}_{S}-\sum_{i=1}^{k}\mathcal{E}_{S_{i}}.$

The reason we use this precise expression is that Lemma 9 gives bounds on how
fast $E(S_{1},\ldots,S_{k})$ can change, but the reason we care about
$E(S_{1},\ldots,S_{k})$ is that it is approximately equal to the energy of a
point vortex configuration with $k$ point vortices, one at the center of mass
of each cluster. More specifically, we can use (6) to compute

$\displaystyle E(S_{1},\ldots,S_{k})$
$\displaystyle=\sum_{i=1}^{k}\sum_{\ell\in S_{i}}\sum_{j\notin
S}m_{i}m_{j}\big{(}\left(\log|x_{\ell}-x_{j}|-\log|x_{S}-x_{j}|\right)-\left(\log|x_{\ell}-x_{j}|-\log|x_{S_{i}}-x_{j}|\right)\big{)}$
$\displaystyle\qquad+\sum_{i=1}^{k}\sum_{\begin{subarray}{c}j,\ell\in S_{i}\\\
j<\ell\end{subarray}}m_{j}m_{\ell}\log|x_{j}-x_{\ell}|-m_{j}m_{\ell}\log|x_{j}-x_{\ell}|$
$\displaystyle\qquad+\sum_{1\leq i<j\leq k}\sum_{\ell_{1}\in
S_{i}}\sum_{\ell_{2}\in
S_{j}}m_{\ell_{1}}m_{\ell_{2}}\Big{(}\log|x_{\ell_{1}}-x_{\ell_{2}}|-(\log|x_{\ell_{1}}-x_{\ell_{2}}|-\log|x_{S_{i}}-x_{\ell_{2}}|)$
$\displaystyle\qquad\qquad\qquad-(\log|x_{\ell_{2}}-x_{\ell_{1}}|-\log|x_{S_{j}}-x_{\ell_{1}}|)\Big{)}$
$\displaystyle=\sum_{i=1}^{k}\sum_{\ell\in S_{i}}\sum_{j\notin
S}O(1)+\sum_{1\leq i<j\leq k}\sum_{\ell_{1}\in S_{i}}\sum_{\ell_{2}\in
S_{j}}m_{\ell_{1}}m_{\ell_{2}}(\log d(S_{i},S_{j})+O(1))$
$\displaystyle=\sum_{1\leq i<j\leq k}m_{S_{i}}m_{S_{j}}\log
d(S_{i},S_{j})+O(1).$ (12)

Note that in the calculation above, we could have error bounds better than
$O(1)$, but it suffices for our purposes.

We now define a notion of average distance between clusters as (essentially)
the shifted logarithmic distance we would need between every pair of clusters
to get the correct energy:

$\hat{d}(S_{1},\ldots,S_{k})=\frac{1}{2}-\frac{E(S_{1},\ldots,S_{k})}{(\log
T)\sum_{1\leq i<j\leq k}m_{S_{i}}m_{S_{j}}}$

where the denominator is nonzero by the No Spiral Condition (Condition 2).
Note that by $\eqref{eq:ES1Skapprox}$, we have

$\displaystyle\hat{d}=\frac{\sum_{1\leq i<j\leq
k}m_{S_{i}}m_{S_{j}}\overline{d}(S_{i},S_{j})}{\sum_{1\leq i<j\leq
k}m_{S_{i}}m_{S_{j}}}+O(1/\log T).$ (13)

Note that while this is some sort of notion of “average” distance, there are
signed weights involved, so it is possible for it to be larger or smaller than
all the pairwise shifted logarithmic distances.

We now turn to proving the lemma by induction on $k$. For $k=2$, we note that
bootstrap assumption 2 together with Lemma 9 gives us that

$\left|\frac{d}{dt}E(S_{1},S_{2})\right|\leq\left|\frac{d}{dt}\mathcal{E}_{S}\right|+\left|\frac{d}{dt}\mathcal{E}_{S_{1}}\right|+\left|\frac{d}{dt}\mathcal{E}_{S_{2}}\right|=O\left(\frac{1}{T}\right),$

so $E(S_{1},S_{2})$ changes by $O(1)$ over the relevant time interval. This
means that $\hat{d}(S_{1},S_{2})$ changes by at most $O(1/\log T)$, so
$\overline{d}(S_{1},S_{2})$ changes by $O(1/\log T)$. Since bootstrap
assumption 1 tells us that $\overline{d}(S_{1},S_{2})\geq\epsilon$, and since
we can take $T$ sufficiently large, this finishes the proof for the case
$k=2$.

Now suppose $k\geq 3$. Assume for the sake of contradiction that there are
some $1\leq i<j\leq k$ so that $\overline{d}(S_{i},S_{j})$ varies by a factor
of at least $\alpha(k)$ and take a minimal time interval on which such
variation occurs. Bootstrap assumption 2 together with Lemma 9 gives us that

$\left|\frac{d}{dt}E(S_{1},\ldots,S_{k})\right|\leq\left|\frac{d}{dt}\mathcal{E}_{S}\right|+\sum_{i=1}^{k}\left|\frac{d}{dt}\mathcal{E}_{S_{i}}\right|=O\left(\frac{1}{T}\right),$

so $E(S_{1},\ldots,S_{k})$ changes by $O(1)$ over the relevant time interval.
This means that $\hat{d}(S_{1},\ldots,S_{k})$ changes by at most $O(1/\log
T)$. Now, by bootstrap assumption 1 we know that $\overline{d}(S_{i},S_{j})$
will always be at least $\epsilon(n)$. Since we are assuming
$\overline{d}(S_{i},S_{j})$ varies by a factor of at least $\alpha(k)$
throughout the time interval, we have there must be some time
$s\in[t_{1},t_{2}]$ with

$\frac{\hat{d}(S_{1},\ldots,S_{k})}{\overline{d}(S_{i},S_{j})}\geq\frac{\sqrt{\alpha(k)}}{2}\qquad\text{or}\qquad\frac{\hat{d}(S_{1},\ldots,S_{k})}{\overline{d}(S_{i},S_{j})}\leq\frac{2}{\sqrt{\alpha(k)}}$
(14)

where we used bootstrap assumption 1 to ensure that either $\hat{d}$ is small
enough that the second part of (14) is satisfied, or $\hat{d}$ is big enough
that the $O(1/\log T)$ change is small.

We can take $s$ to be the earliest time that (14) holds for some $i,j$. We
will assume for now, to be proven later,

######  Claim 13.

If we have the bootstrap assumptions of Lemma 12, as well as Condition 10 and
(14), then there exist some $1\leq i_{1}<j_{1}\leq k$ and $1\leq
i_{2}<j_{2}\leq k$ with

$\frac{\overline{d}(S_{i_{1}},S_{j_{1}})}{\overline{d}(S_{i_{2}},S_{j_{2}})}\geq(3\alpha(k-1)^{2})^{k^{2}}$
(15)

We are now almost done. We take all pairwise distances of the form
$\overline{d}(S_{i},S_{j})$ and list them in increasing order. From (15), we
know that two consecutive numbers in the list must differ by a factor of at
least $3\alpha(k-1)^{2}$. Let $z$ be some number in this gap. Introduce an
equivalence relation on $[k]$ via $i\sim j$ if $d(S_{i},S_{j})<z$. This
equivalence relation gives us subclusters $U_{1},\ldots,U_{\ell}$ for $\ell<k$
and with each $U_{j}$ having as subclusters various $S_{j}$, with fewer than
$k$ subclusters per $U_{j}$ (see Figure 2 for an illustration).

Figure 2: This is an illustration of the clusters $U_{i}$

We now apply the inductive hypothesis both to each $U_{j}$ and to
$S=U_{1}\sqcup\cdots\sqcup U_{\ell}$, going either forward or backward in
time, to get a contradiction. The value $3\alpha(k-1)^{2}$ in the argument
above was chosen so that the shifted logarithmic distances can change by a
factor of $\alpha(k-1)$ without violating bootstrap assumption 2.

All that remains is the proof of Claim 13.

###### Proof of Claim 13.

Suppose for the sake of contradiction that the conclusion of the claim is
false. Then for all $\mu,\nu$, we have that

$\overline{d}(S_{\mu},S_{\nu})\leq(3\alpha(k-1)^{2})^{k^{2}}\overline{d}(S_{i},S_{j})$

Then (13) and choosing $\alpha(k)$ to be sufficiently large compared to
$\alpha(k-1)$ will give us that

$\displaystyle\hat{d}(S_{1},\ldots,S_{k})$
$\displaystyle\leq\frac{\sum_{1\leq\mu<\nu\leq
k}|m_{S_{\mu}}m_{S_{\nu}}|}{\left|\sum_{1\leq\mu<\nu\leq
k}m_{S_{\mu}}m_{S_{\nu}}\right|}(3\alpha(k-1)^{2})^{k^{2}}\overline{d}(S_{i},S_{j})+O(1/\log
T)$ $\displaystyle<\frac{\sqrt{\alpha(k)}}{2}\overline{d}(S_{i},S_{j}).$ (16)

To prove an inequality in the opposite direction, we note that (13) involves a
weighted average with weights $m_{S_{\mu}}m_{S_{\nu}}$. Let $\mu,\nu$ be the
indices so that $m_{S_{\mu}}m_{S_{\nu}}$ is the largest weight in absolute
value. Now Condition 10 gives us several cases (and this is the only place
this condition shows up). The first case is that $m_{S_{\mu}},m_{S_{\nu}}$ are
both negative. Then all the weights are positive. The second case is that
$m_{S_{\mu}}>0$ and $m_{S_{\nu}}<0$. Then the largest weight by absolute value
is negative, and all positive weights are smaller than it by a factor of at
least $\kappa$. The last possibility is that $m_{S_{\mu}},m_{S_{\nu}}>0$. Then
the largest weight by absolute value is positive, and all negative weights are
smaller than it by a factor of at least $\kappa$. We will phrase the argument
below assuming that $m_{S_{\mu}}m_{S_{\nu}}>0$. This handles the first and
third cases, and the second case is handled by the same argument with a few
sign reversals. Since we are assuming that the conclusion of Claim 13 is
false, we have that

$\overline{d}(S_{i},S_{j})\leq(3\alpha(k-1)^{2})^{k^{2}}\overline{d}(S_{\mu},S_{\nu})$
(17)

so (13) will give us that

$\displaystyle\hat{d}(S_{1},\ldots,S_{k})$
$\displaystyle\geq\frac{m_{S_{\mu}}m_{S_{\nu}}-k^{2}(3\alpha(k-1)^{2})^{k^{2}}\frac{m_{S_{\mu}}m_{S_{\nu}}}{\kappa}}{k^{2}m_{S_{\mu}}m_{S_{\nu}}}\overline{d}(S_{\mu},S_{\nu})+O(1/\log
T)$ $\displaystyle\geq\frac{1}{2k^{2}}\overline{d}(S_{\mu},S_{\nu})$ (18)

where we are using that $\kappa$ is sufficiently large compares to
$\alpha(k-1)$ and that $T$ is sufficiently large. We now use (17) and the fact
that $\alpha(k)$ is sufficiently large compared to $\alpha(k-1)$ to obtain

$\hat{d}(S_{1},\ldots,S_{k})>\frac{2}{\sqrt{\alpha(k)}}\overline{d}(S_{i},S_{j}).$
(19)

Together, (16) and (19) combine to contradict the hypotheses of the claim,
thus finishing its proof and the proof of Lemma 12. ∎

∎

To obtain the proof of Theorem 11, we now apply the Lemma 12 to $S=[n]$ with
$S_{i}=\\{i\\}$ for each $i$. Since all distances are initially of order 1,
this gives that at sufficiently late time $T$, all distances are in the range

$\left[cT^{-\frac{\alpha(n)}{2}},CT^{\frac{1}{2}-\frac{1}{2\alpha(n)}}\right].$

This precludes the possibility of collisions. Also, now that we have an upper
bound on distances between vortices, conservation of center of mass gives the
same bound (up to a constant factor) for their positions.

##  7 Improvements, corollaries, and comments

In this section, we discuss potential improvements to Lemma 9, Lemma 12, and
Theorem 11, as well as some corollaries. We start with improvements to the
proofs:

  1. 1.

Condition 10 is not the most general condition under which the argument in the
proof of Lemma 12 works. For instance, it will work provided the following
condition:

######  Condition 14.

If we list the masses in increasing order by absolute value, then whenever
$m_{i}$ and $m_{i+1}$ have opposite signs, we have
$\kappa(n)|m_{i}|<|m_{i+1}|$.

We can also have pairs of similar size and opposite signs, under some
conditions that are annoying to phrase correctly. The important thing is that
the No Translation and No Spiral Conditions hold and that the appropriate step
in the proof of Claim 13 goes through.

  2. 2.

Given a specific list of masses, if we want to try to push the argument
through in some more sensitive cases, we may need to forget about the
convenience of doing an induction proof and using Lemma 12, and instead do a
lot of casework based on what clusters we have and the relative scales, along
with optimization of various parameters. We also need to do such an
optimization if we want to get better upper bounds on growth of the system.
The remaining comments in this list give some ideas we can exploit when we do
such an optimization.

  3. 3.

The proof as written never used the triangle inequality; using it can give
improvements in constants, since it constrains possibilities for the distances
$\overline{d}$.

  4. 4.

The conservation of $\tilde{I}$ gives information which we can use about the
behavior of the top-level cluster if all the masses of its subclusters have
the same sign (or if one has a different sign from the rest and a bigger mass
than the sum of the rest, see the arguments in Section 4).

  5. 5.

Lemma 9 is sufficient for the results as stated in this paper, but it’s
potentially possible to improve this lemma in some time-averaged sense when a
cluster consists of only two subclusters. This is because the centers of mass
of those two subclusters approximately follow circular orbits over appropriate
timescales. This makes the behavior of the system isotropic in an appropriate
time-averaged sense, leading to higher-order cancellation for most of the
terms of (7). This doesn’t deal with the $u_{i}-u_{j}$ terms in (7) when
$|x_{i}-x_{j}|$ is small, but those may be possible to deal with by grouping
the other vortices into clusters and saying either they are far enough apart
(say $\sqrt{ab}$) to get improvement, or we can group them into a cluster.
When a cluster consists of three subclusters, it is a completely integrable
system with some introduced forcing, and it’s going to be isotropic on average
unless we are exceptionally unlucky, so maybe some improvements along similar
lines can be had there too.

Next, we will give some direct corollaries of Lemma 12. We formulate them in
terms of Condition 10, but all the earlier considerations about what
conditions we actually need apply.

######  Corollary 15.

Suppose we have a point vortex system with masses $m_{i}$ and trajectories
$x_{i}(t)$ evolving over a fixed time interval $[0,T]$, with some lower bound
on the distance between them. Suppose that for each of the three vortices, we
have a list of masses $m_{i}^{1},\ldots,m_{i}^{n_{i}}$ summing to $m_{i}$ and
satisfying Condition 10. Then at time 0, we can replace each of the point
vortices with a cluster of point vortices having masses
$m_{i}^{1},\ldots,m_{i}^{n_{i}}$, and all being within distance $\delta$ of
$x_{i}(0)$. Then for the entire time interval, the vortices of cluster $i$ are
within distance $f(\delta)$ of $x_{i}(t)$ where $f(\delta)\to 0$ as $\delta\to
0$. Here, the function $f$ is allowed to depend on all the masses of all the
vortices in each swarm and on the trajectories $x_{i}(t)$.

Note that a weaker version of this result, where all point vortices in each
cluster have the same sign, is proven in [10].

We now mention the result [16] which allows one to take a self-similarly
expanding system of 3 vortices and replace each point vortex with a vortex
patch and still get a system where the distance between the patches grows as
$\sqrt{t}$. From the point of view of stability of the 3-vortex system, the
relevant fact about the evolution of the vortex patches was that the diameter
of each vortex patch grows at most as $t^{a}$ for some $a<1/2$, which we can
still obtain if instead of a vortex patch, we have a cluster of point
vortices. To prove this corollary, one needs to combine the stability analysis
for the 3-vortex system from [16] with the bootstrap assumption that the
diameter of each cluster is less than $\epsilon t^{a}$, which we get from
Lemma 12.

######  Corollary 16.

Suppose we have a self-similarly expanding system of three vortices with
masses $m_{1},m_{2},m_{3}$ whose distance grows like $\sqrt{t}$. Suppose that
for each of the three vortices, we have a list of masses
$m_{i}^{1},\ldots,m_{i}^{n_{i}}$ summing to $m_{i}$ and satisfying Condition
10. Then for $\delta$ sufficiently small, we can replace each of the three
point vortex with a cluster of vortices having diameter at most $\delta$ and
masses $m_{i}^{1},\ldots,m_{i}^{n_{i}}$ and then at all times $t$, we will
have three clusters, each of diameter at most $t^{a}$ for some $a<1/2$ and
with the distance between clusters being at least $c\sqrt{t}$.

Note that this includes the case where all vortices in the cluster have the
same sign (since Condition 10 is then satisfied). Note also that if one
obtained stability for some self-similarly expanding system of more than three
point vortices, one would then obtain a result corresponding to Corollary 16
for that system.

Finally, we note a limitation of the method of proof used for Lemma 12. One
might hope that this proof technique could allow us to prove Conjecture 4.
This runs into the following difficulty:

Imagine we have $n=10$ and $S=\\{6,7,8,9,10\\}$. As $t$ approaches $T$, we set
all distances inside the cluster $S$ to be $\sim(T-t)^{3/2}$ and all other
distances to be $\sim T-t$. By appropriately choosing the masses, we can get
that the total energy is conserved. Then Lemma 9 tells us that the internal
energy of the cluster can change at rate at most

$\frac{C\left((T-t)^{3/2}\right)^{2}}{(T-t)^{4}}=\frac{C}{T-t},$

but this is compatible with the behavior described above and blowup at time
$T$. To forbid such examples, we need some variant of Condition 10 (or new
ideas).

## References

  * [1] H. Aref.  Motion of three vortices.  The Physics of Fluids, 22(3), 1979.
  * [2] H. Aref.  Point vortex dynamics: a classical mathematics playground.  J. Math. Phys., 48(6):065401, 23, 2007.
  * [3] H. Aref, P. Newton, M. Stremler, T. Tokieda, and D. Vainchtein.  Vortex crystals.  Advances in Applied Mechanics, 39:1–79, 12 2003.
  * [4] B. Eckhardt.  Integrable four vortex motion.  Phys. Fluids, 31(10):2796–2801, 1988.
  * [5] F. Grotto and U. Pappalettera.  Burst of point vortices and non-uniqueness of 2D Euler equations.  Archive for Rational Mechanics and Analysis, 245:89–125, 2022\.
  * [6] K. M. Khanin.  Quasi-periodic motions of vortex systems.  Physica D: Nonlinear Phenomena, 4(2):261–269, 1982.
  * [7] H. Kudela.  Collapse of $n$-point vortices in self-similar motion.  Fluid Dyn. Res., 46(3):031414, 16, 2014.
  * [8] C. Marchioro.  On the localization of the vortices.  Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1(3):571–584, 1998.
  * [9] C. Marchioro and M. Pulvirenti.  Vortices and localization in Euler flows.  Comm. Math. Phys., 154(1):49–61, 1993.
  * [10] C. Marchioro and M. Pulvirenti.  Mathematical theory of incompressible nonviscous fluids, volume 96 of Applied Mathematical Sciences, chapter 4, pages 134–177.  Springer-Verlag, New York, 1994.
  * [11] P. K. Newton.  Point vortex dynamics in the post-Aref era.  Fluid Dyn. Res., 46(3):031401, 11, 2014.
  * [12] E. Novikov.  Dynamics and statistics of a system of vortices.  Zh. Eksper. Teoret. Fiz., 68, 1974.
  * [13] E. Novikov and Y. Sedov.  Vortex collapse.  J. Exp. Theor. Phys., (50):297–301, 1979.
  * [14] P. Serfati.  Borne en temps des caractéristiques de l’équation d’Euler 2d à tourbillon positif et localisation pour le modèle point-vortex, 1998.  Manuscript.
  * [15] P. Serfati.  Tourbillons-presque-mesures spatialement bornés et équation d’Euler 2D, 1998.  Manuscript.
  * [16] S. Zbarsky.  From point vortices to vortex patches in self-similar expanding configurations.  2020\.


# BeliefBank: Adding Memory to a Pre-Trained Language Model for a Systematic
Notion of Belief

Nora Kassner1,2, Oyvind Tafjord1, Hinrich Schütze2, Peter Clark1
1Allen Institute for AI, Seattle, WA
2Center for Information and Language Processing, LMU Munich, Germany
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Although pretrained language models (PTLMs) contain significant amounts of
world knowledge, they can still produce inconsistent answers to questions when
probed, even after specialized training. As a result, it can be hard to
identify what the model actually “believes” about the world, making it
susceptible to inconsistent behavior and simple errors. Our goal is to reduce
these problems. Our approach is to embed a PTLM in a broader system that also
includes an evolving, symbolic memory of beliefs – a BeliefBank – that records
but then may modify the raw PTLM answers. We describe two mechanisms to
improve belief consistency in the overall system. First, a reasoning component
– a weighted MaxSAT solver – revises beliefs that significantly clash with
others. Second, a feedback component issues future queries to the PTLM using
known beliefs as context. We show that, in a controlled experimental setting,
these two mechanisms result in more consistent beliefs in the overall system,
improving both the accuracy and consistency of its answers over time. This is
significant as it is a first step towards PTLM-based architectures with a
systematic notion of belief, enabling them to construct a more coherent
picture of the world, and improve over time without model retraining.

##  1 Introduction

Figure 1:  The proposed architecture. The model’s raw answers are stored in a
persistent, symbolic memory (BeliefBank), and two mechanisms attempt to
improve them: (a) A constraint solver flips beliefs (e.g., the belief that “a
swallow is a fish”) that clash significantly with others. (b) A feedback
mechanism poses new questions using existing, relevant beliefs (e.g., “a
swallow is not a fish”) as the query context. We find that both consistency
and accuracy of the overall system improve. Example: The model $M$ shown in
the figure incorrectly answers “yes”, when asked “a swallow has gills?”. But
(as shown above) if reminded of its previous answer “a swallow is not a fish”,
$M$ correctly answers "no".

Intelligent agents are typically considered to have beliefs about the world –
propositions that they take as true Genin and Huber (2021). In general, a
system can be said to (appear to) believe a proposition p, e.g., “eagles are
birds”, if it produces answers consistent with p (and its other beliefs).
Pragmatically, we expect the system to (a) give a consistent answer to
different paraphrases of the question "p?" ("Are eagles birds?", "Is an eagle
a type of bird?", …), and (b) give correct answers about implications of p
("Eagles lay eggs", "Eagles have feathers", …), conditional on its other
knowledge and reasoning abilities.

Maintaining a consistent set of beliefs (a “belief system”) is a key facet of
intelligence, as it can help debug errors and encourage rational behavior.
However, although PTLMs contain substantial world knowledge Petroni et al.
(2019), their answers to probing questions can be inconsistent Elazar et al.
(2021); Kassner and Schütze (2020), even after specialized training to reduce
inconsistency Ribeiro et al. (2019); Li et al. (2019). As a result, it is
sometimes hard to pin down what a PTLM actually “believes”, making them
susceptible to inconsistent and/or irrational behavior. Our goal is a first
step to avoid these problems by embedding a PTLM in a broader system with a
clearer notion of belief (see Figure 1).

Prior work in AI, including in formal logic Genesereth and Nilsson (1987),
belief maintenance De Kleer (1986); Dechter and Dechter (1988), and
uncertainty Pearl (1986), offers models for how beliefs can be managed. Most
importantly, it posits that creating a coherent set of beliefs – a kind of
“mental model” of the world Johnson-Laird (1983) \- is a constructive process
requiring explicit representation of beliefs, and inference about their
dependencies. Based on this, our approach is to embed a PTLM in a broader
system with an evolving, symbolic memory of beliefs - a BeliefBank - along
with two mechanisms to improve belief consistency of the overall system. First
a reasoning component – a weighted MaxSAT (satisfiability) solver – reasons
about belief dependencies and revises beliefs that significantly clash with
others. Second, a feedback component poses future queries to the model using
known beliefs as context, aiming for more accurate and consistent answers from
the PTLM itself. The BeliefBank represents the overall system’s beliefs (a
“mental model”) about the world, constructed by deliberating over the noisy
output of a raw PTLM.

We explore this in a controlled experimental setting where both candidate
beliefs and constraints between them are provided. Candidate facts are simple
sentences that may be true or false, e.g., "An eagle is a bird" (T), “An eagle
is a mammal” (F). Constraints are between (variabilized) facts, e.g., “X is a
bird $\rightarrow$ X has wings”. These allow us to both probe and measure
improvement in the consistency and accuracy of a system’s beliefs, compared
with an original PTLM.

In contrast to prior work, this system does not rely on fine-tuning the PTLM.
Fine-tuning requires expensive training data, and risks destabilizing the
model’s performance on other tasks outside the scope of training. Instead, our
system functions without training data, explicitly reasoning about beliefs
using an external mechanism, thus allowing both controllability and
interpretability. Most significantly, we find that improving consistency in
this way improves accuracy, while earlier fine-tuning-based approaches report
either no accuracy gains Ribeiro et al. (2019); Minervini and Riedel (2018);
Li et al. (2019) or only slight gains Asai and Hajishirzi (2020).

We make the following contributions:

  * 1.

We show that a PTLM-based system can be given a consistent notion of belief by
augmenting the PTLM with a global memory – the BeliefBank – that can be
deliberated over. Specifically, we show that two mechanisms – constraint
reasoning and feedback – improve the overall system’s accuracy and consistency
over time.

  * 2.

We contribute a targeted dataset to measure a system’s consistency against
given constraints.

  * 3.

We provide an analysis of the failure modes and directions for future work.

This work is significant as it is a first step towards PTLM-based
architectures that have a systematic notion of belief, allowing them to
construct a more coherent picture of the world, and improve over time without
model retraining.111Dataset is available at
https://allenai.org/data/beliefbank

##  2 Related work

The idea that agents should have a belief system dates back to the earliest
years of AI, e.g., McCarthy (1959) envisioned representing a system’s beliefs
as formal propositions along with a reasoning process to identify what
follows. Multiple subfields of AI have explored ways of representing and
updating beliefs, including in formal logic Genesereth and Nilsson (1987);
Moore (1983), belief revision De Kleer (1986); Dechter and Dechter (1988), and
uncertainty Pearl (1986). Similarly, work in cognitive science has promoted
mental models – coherent, constructed representations of the way the world is
believed to be – as central to understanding and communication Johnson-Laird
(1983); Gentner and Stevens (1983); Hilton (1996). We draw on these ideas,
proposing how they can be layered on top of PTLMs, here representing beliefs
as NL statements rather than formal structures.

Although PTLMs contain extensive world knowledge Petroni et al. (2019);
Roberts et al. (2020), they can be inconsistent in their answers to probing
questions Ettinger (2020); Davison et al. (2019); Ravichander et al. (2020);
Elazar et al. (2021); Subramanian et al. (2020), making their “world model”
unclear. Although various approaches have improved answer consistency, mainly
through modified model training, e.g., Ribeiro et al. (2019); Minervini and
Riedel (2018); Li et al. (2019); Asai and Hajishirzi (2020), they have not
solved the problem. Current PTLMs still behave as a source of noisy knowledge,
rather than projecting a coherent picture of the world Kassner and Schütze
(2020).

A close analogy to our task is in knowledge graph (KG) construction. Pujara et
al. (2013) define “knowledge graph identification” as the task of building a
maximally consistent KG given noisy candidate facts and their extraction
confidences, and constraints between them. They develop a solution using
probabilistic soft logic (PSL) Broecheler et al. (2010) as their constraint
reasoner. Our reasoning component follows similar ideas, but applied to the
noisy predictions of a PTLM. On the face of it, it is not clear how to plug a
constraint solver into a PTLM, given their very different natures. Our
solution introduces a global persistent memory, making this novel combination
of technologies possible. To our knowledge this has not been done before.

Our work presents a broader system architecture in which a PTLM is embedded,
along with a dynamic, persistent memory. While there are prior neural
architectures that include an associated memory, e.g., Henaff et al. (2016);
Sukhbaatar et al. (2015); Graves et al. (2016), these components typically
play the role of a short-term working memory to help computation. In contrast,
our BeliefBank is a persistent, long-term memory, and we treat the PTLM as a
component within a larger system. Our work also differs from retrieval-
augmented architectures, e.g., RAG Lewis et al. (2020), REALM Guu et al.
(2020), that augment model input with external retrievals. Rather, our memory
is reflective, built from model outputs and reasoned over.

Our feedback mechanism uses old answers to help answer new questions. This
builds on prior work such as Self-Talk Shwartz et al. (2020), where a model
asks itself related questions to help with new answers, and can be seen as a
form of dynamic prompt engineering Liu et al. (2021). In our case, feedback is
selected from a global BeliefBank, rather than generated with templated
subqueries, potentially allowing more control over feedback selection.

Finally, our system performs a kind of continual learning Parisi et al.
(2019); Carlson et al. (2010). While recent work in this area has focused on
dynamic update of model parameters, e.g., Ebrahimi et al. (2021), our work
leaves the model fixed, and seeks improvement in the broader system in which
the model is embedded, exploring an alternative and potentially more
interpretable architecture towards this goal.

##  3 Task

Our goal is to ascribe a clearer notion of “belief” to a system that includes
a model $M$, by improving, over time, the consistency and accuracy of its
answers (compared with $M$) to a stream of questions. We measure this with
true/false probing, where we are also given a set of constraints between
answers: [vskip=0mm,leftmargin=2mm] Given:

  * •

a stream of sentences $Q$, interpreted as questions to the model

  * •

a set of constraints $C(S)$ between (the truth values of) sentences in $Q$,
each annotated with a weight $w_{i}$ (A penalty $w_{i}$ is applied if
$c_{i}\in C(S)$ is violated)

  * •

a model $M$ that takes as input a question $q\in Q$ and optionally an (NL)
context $X$ (consisting of answers to previously posed questions), and
predicts a True/False answer $A$ with confidence score $F$

Accumulate:

  * •

the True/False labels for $Q$ predicted by $M$, optionally corrected by the
constraint solver, so as to maximally improve accuracy (with respect to gold
labels) and consistency (minimize total penalties of constraint violations)

##  4 Approach

Our approach adds a memory layer, called the BeliefBank, on top of the model
to globally track beliefs. Two mechanisms are then used to manage the
BeliefBank beliefs, namely constraint reasoning and feedback, as we now
describe.

###  4.1 Definitions

Let

  * •

a belief $b_{i}$ be a triple ($s_{i}$,$l_{i}$,$w_{i}$), where

    * –

$s_{i}$ is a sentence $\in S$

    * –

label $l_{i}\in$ {T,F} denotes the system’s True/False belief about the truth
of $s_{i}$222 Strictly, the label F denotes the belief that the negation of
$s_{i}$ is true, e.g., (“a poodle is a bird”,F,…) denotes the belief “a poodle
is not a bird”.

    * –

weight $w_{i}$ is a number $\in[0,1]$ representing the system’s strength of
that belief

For example:

> ("a poodle is a dog", T, 0.9)

denotes the belief (strength 0.9) that "a poodle is a dog" is a true statement
(T).

  * •

a BeliefBank $B(S)$ = a set of beliefs over sentences $S$ = $s_{1},...,s_{n}$

  * •

a constraint $c_{i}$ = a 5-tuple of the form ($s_{i}.l_{i}\rightarrow
s_{j}.l_{j},w_{i}$) where

    * –

$s_{i},s_{j}$ are sentences $\in S$,

    * –

$l_{i},l_{j}\in$ {T,F}. If $s_{i}$ has truth value $l_{i}$, denoted
$s_{i}.l_{i}$, then $s_{j}$ is expected to have truth value $l_{j}$, denoted
$s_{j}.l_{j}$.

    * –

$w_{i}$ denotes the strength of that expectation (a penalty $w_{i}$ is applied
if violated).

For convenience, a shared variable X can be used in $s_{i},s_{j}$, allowing a
set of grounded constraints to be expressed in one statement, e.g.,
[vskip=0mm,leftmargin=2mm] (“X is a dog”.T $\rightarrow$ “X has a tail”.T,
0.8) expresses that if something is a dog, then it should (T) have a tail,
with a penalty of 0.8 applied if it does not. Mutual exclusivity is expressed
using two rules, e.g., that fish and birds are mutually exclusive is
expressed: [vskip=0mm,leftmargin=2mm] (“X is a bird”.T $\rightarrow$ “X is a
fish”.F, 1.0)
(“X is a fish”.T $\rightarrow$ “X is a bird”.F, 1.0) where “F” indicates the
conclusion should be false if the condition here is true (T).

  * •

a constraint graph $C(S)$ = a set of constraints $c_{i}$ over sentences $S$

Given a set of beliefs $B(S)$ about $S$ and a set of constraints $C(S)$, we
measure consistency using (the complement of) Li et al. (2019)’s conditional
constraint violation ($\tau$) metric, namely the fraction of constraints whose
condition $s_{i}.l_{i}$ is believed, but whose conclusion (that $s_{j}$ has
truth value $l_{j}$) is not. In other words, over all constraints $c_{i}\in
C(S)$, inconsistency $\tau$ is

$\tau=|\\{~{}c_{i}~{}|~{}\neg(s_{i}.l_{i}\rightarrow
s_{j}.l_{j})~{}\\}|~{}~{}/~{}~{}|\\{~{}c_{i}~{}|~{}s_{i}.l_{i}~{}\\}|$

i.e., the size of the set of violated constraints ($s_{i}.l_{i}\rightarrow
s_{j}.s_{j}$ is false) divided by the size of the set of applicable
constraints. We then define: [vskip=0mm,leftmargin=4mm]

consistency = 1 - $\tau$

###  4.2 Methods

We consider our system in a dynamic setting, where it receives a stream of
questions and gradually builds up a BeliefBank of answers (including revising
earlier answers). We evaluate two methods for improving the BeliefBank’s
accuracy and consistency over time:

  * Constraint solving:

Given a model $M$’s raw answers (with confidences), a constraint solver seeks
to reduce constraint violations by potentially flipping answers that maximally
clash with other answers.

  * Feedback:

Given a new question $q$, selected beliefs in the BeliefBank are provided as
context to $M$ to help it answer $q$ correctly.

Figure 1 shows these components.

####  4.2.1 Constraint Solving

Given a set of beliefs and constraints, the constraint solver has two
competing objectives: (a) flip beliefs so as to minimize constraint violations
(b) don’t flip beliefs, so as to preserve the model’s raw answers, i.e.,
minimize conflict between the model and BeliefBank. To implement this
tradeoff, the model’s answers are themselves treated as just another
constraint, e.g., the answer that "a poodle is a dog" is true (confidence 0.9)
is treated as a constraint "a poodle is a dog", with penalty 0.9 if it is
violated. To balance the two objectives (a) and (b), the model confidences are
scaled by a learned hyper-parameter $\lambda$, trained on a calibration part
of our dataset, disjoint from the data then used in experiments (Section 5).

To implement constraint solving, we translate the task into a weighted MaxSAT
(satisfiability) problem $P$, for which efficient algorithms with guarantees
exist. Each belief becomes a weighted assertion in $P$, e.g., the belief ("a
poodle is a dog", T, 0.9) is expressed in SAT syntax:
[vskip=0mm,leftmargin=4mm] 0.9  "a poodle is a dog"333In practice, strings are
replaced with numeric identifiers in SAT syntax, but for clarity we leave them
as strings here. while the constraint ("a poodle is a dog".T $\rightarrow$ "a
poodle has a tail".T, 0.8) is expressed: [vskip=0mm,leftmargin=4mm] 0.8 "a
poodle has a tail" -"a poodle is a dog" (literally: "a poodle has a tail" OR
NOT ("-") "a poodle is a dog"). We then apply the solver Z3 De Moura and
Bjørner (2008) to $P$, which outputs a set of truth assignments for all
individual sentences in $P$ so as to minimize the weighted sum of violations.
If the truth of any sentence has changed, the BeliefBank is correspondingly
updated.

####  4.2.2 Feedback

Feedback involves asking the model a question, but with the benefit of knowing
answers to prior, related questions. To use these answers in the query,
selected beliefs are added to the query context before asking the model. (Note
that the selected beliefs are not guaranteed to be correct, of course). Our
conjecture is that if the model is explicitly reminded of relevant beliefs
when answering a new question, it will answer the question more accurately and
consistently. For example, in Figure 1, when asked "Do swallows have gills?",
our model $M$ incorrectly answers "yes". But if reminded that swallows are not
fish, by asking: "CONTEXT Swallows are not fish. QUERY Do swallows have
gills?" the model now correctly answers "no".

We evaluate two policies for choosing which beliefs to feed back to $M$ when
asking question $q$ about entity $e$:

  * 1.

on topic beliefs, namely current beliefs about entity $e$, randomly selected
from the BeliefBank

  * 2.

most relevant on topic beliefs (i.e., again about $e$), using the constraint
graph to identify relevance. As the constraint graph captures potential
clashes that the answer to $q$ could cause, we use the graph to identify
beliefs that would be most affected by that answer. For example, if the
current query is: "Is a poodle an animal?", the constraint graph identifies
potential clashes that would occur if the model answered "yes", and also
clashes if it answered "no". Here, if the model answered "no", the resulting
belief ("a poodle is not an animal") would strongly clash with other beliefs
"A poodle is a dog." and "A poodle is a mammal.", so these two are strong
candidates for the context. We select the three strongest clashing beliefs
found in this way, considering both "yes" and "no" answers to $q$. If no
relevant fact is present, we use a randomly selected topic belief instead.

In both cases, three beliefs are selected, this number was empirically found
to be most effective.

##  5 Dataset

We create a dataset444Dataset is available at
https://allenai.org/data/beliefbank to test our approach in a controlled way,
allowing us to perform systematic experiments to evaluate behavior. The
dataset contains two parts, constraints and facts, defined over simple
sentences such as “a swallow is a bird.”

###  5.1 Constraints

The dataset contains two kinds of constraints:

  * positive implications:

conclusion truth value $\textrm{l}_{j}$ = T (true), e.g.,
[vskip=0mm,leftmargin=2mm] “X is a dog.T $\rightarrow$ “X has a tail.”.T

  * mutual exclusivities:

expressed as a pair of constraints with $\textrm{l}_{j}$ = F (false), e.g.,
[vskip=0mm,leftmargin=2mm] “X is a dog".T $\rightarrow$ "X is a bird.”.F
“X is a bird".T $\rightarrow$ "X is a dog.”.F expresses that an entity cannot
be both a dog and a bird at the same time.

Positive implications were manually gathered from ConceptNet Speer et al.
(2017). First, we identified 121 general concepts of interest, e.g., “mammal”,
then converted selected triples about them to constraints (Details of the
selection process are in Appendix A). For example, the ConceptNet triple
(dog,HasA,tail) becomes the constraint "X is a dog" $\rightarrow$ "X has a
tail". We also add weaker, disjunctive constraints in the backward direction,
e.g., "X has a tail" $\rightarrow$ "X is a dog" OR "X is a cat" OR …. for all
entities with tails. Mutual exclusivities were gathered from the “isa”
taxonomies in ConceptNet and WordNet Fellbaum (2005), using the approximation
that siblings in the noun hierarchy are mutually exclusive. Thus, for any pair
of siblings, we add a mutual exclusivity constraint (using two constraint
rules).

We collected 2612 constraints in this fashion (1836 forward implications,
2*388 bidirectional mutual exclusivities).

###  5.2 Constraint Weights

Constraint weights need to be set appropriately to mix well with the model’s
confidences inside the weighted SAT solver. We use a development set of 1072
facts about seven entities to set one constraint weight for the forward
direction of the implications and the mutual exclusivity rules and a second
one for the backward direction of the implications. To do this we perform a
grid search over these parameters, finding the values that result in the
highest F1 (accuracy) after running the constraint solver over the raw model’s
beliefs about these facts.

###  5.3 Facts

We also collect a set of truth-labeled facts about different entities,
relevant to the constraints. To do this, we select a new entity, e.g.,
"poodle", that is a member of one of our general concepts, e.g., "dog", then
instantiate the constraint graph with that entity (i.e., set X = "poodle"). We
then identify the leaf (source) nodes of that graph, just considering forward
implication rules, i.e., finding facts not implied by other facts in the
graph, and manually annotate their True/False labels. We then use the
implications and mutual exclusivities to infer other True/False labels for
other sentences, i.e., we propagate the annotated labels through the graph.
This provides “silver” labels for sentences reachable in this way (a subset of
all the sentences in the graph) – silver because the implications are soft,
hence not guaranteed to hold for all entities.

We repeat this for 85 entities (animals and plants), resulting in a final
dataset containing 12,525 “silver” facts (sentences + True/False labels). Note
that this data is purely for evaluation. There is no training phase or
training data. The system does not have access to any labeled data besides the
constraint rules.

##  6 Model

The fixed model $M$ that we use for our experiments is Macaw Tafjord and Clark
(2021), a state-of-the-art T5 QA model fine-tuned on $\approx$400k QA pairs.
To query the model, we pose the query (optionally with a textual context), and
let the model choose between the two answer options "yes" and "no". The model
also outputs an answer confidence, used as the belief weight.

We use the T5-large version of this model. Note that we do not retrain the
model for this work; rather, it is used as a black-box QA module in the
broader system (Figure 1). Other models could equally have been used.

##  7 Experiments

We evaluate our system in a dynamic setting in which it receives a stream of
questions, building up and revising a BeliefBank. To simplify the evaluation,
we consider questions to arrive in batches, and evaluate the BeliefBank after
each batch, measuring accuracy (F1)555 We measure accuracy with F1 (on the
True class) rather than % correct because the True/False distribution in our
dataset is unbalanced, with significantly fewer True than False answers. F1
avoids scores being dominated by negative answers. and consistency (1-$\tau$,
Section 4.1) of the BeliefBank so far, comparing with the gold labels. We
evaluate four configurations:

  * Raw model:

The BeliefBank simply records the raw model’s answers 666To the best of our
knowledge there are no other baseline models to compare to as consistency
based Q&A does not go beyond paraphrases and relies on finetuning Elazar et
al. (2021).

  * Constraint-Solving:

After each batch, the constraint solver is run over all the (raw) model
answers so far, and the BeliefBank updated accordingly.

  * Feedback:

Questions in batch $n$ are posed to the model using a context selected from
the beliefs already in the BeliefBank (batches 1 to $n-1$). We evaluate two
selection strategies:

    * Feedback (on-topic):

Random beliefs about the entity $e$ being queried about

    * Feedback (relevant):

On-topic beliefs (i.e., again about $e$) that are most relevant to the query,
as defined in Section 4.2.2

  * Feedback + Constraint-Solving:

A combination of the two.

These configurations are illustrated in Figure 2.

Figure 2: The four configurations we evaluate. In (B), the contraint-solver
(SAT solver) is run over all model M answers so far. In (C), current beliefs
are fed back as context for new questions. (D) combines the two.

OnTopic-FB = using (randomly selected) on-topic feedback from old answers for
new queries.
Relevant-FB = using most relevant on-topic feedback for new queries.
Constraints = running the constraint-solver after each batch.

Figure 3: Accuracy (left) and consistency (right) of the growing BeliefBank,
as the system answers incrementally more questions (each batch = 10% of the
queries). Relevant feedback, constraint-solving, and both, all help improve
both F1 and Consistency.

###  7.1 Results

The results are shown in Figure 3, showing the changing accuracy and
consistency of the growing BeliefBank with time, for different configurations.
Each time-step (batch) represents another 10% of the test questions being
posed to the system. (The same data is presented in tabular form in Appendix
B). Several conclusions can be drawn:

$\bullet$ Use of feedback, constraint-checking, or both, all result in a
continually improving accuracy over time. This is a significant result,
showing a larger system can continually improve even if its internal PTLM
component is fixed. (The raw accuracy of the PTLM itself is necessarily
constant).

$\bullet$ Use of the constraint-solver results in very high (~95%)
consistency, indicating that it is doing its job well, and also improving
accuracy substantially (+17% over the raw model). The constraint-solver has a
global view of the BeliefBank, and thus can balance all beliefs seen so far
with the provided constraints to make a decision.

$\bullet$ Relevant feedback results in significant consistency gains compared
with just on-topic feedback. As relevant beliefs are exactly those that may
clash with the answer to the current question (Section 4.2.2), this encourages
the model to answer consistently with those beliefs, promoting consistency. Of
course, this could hurt accuracy if those relevant beliefs were wrong,
amplifying the errors. In fact, the overall accuracy remains about the same as
with on-topic feedback, and significantly better than the model’s raw answers.

$\bullet$ The greatest gains are for feedback and constraint-solving combined,
resulting in +18% F1 (absolute) over the raw model accuracy. This suggests
that feedback and constraints can work together in a positive way.

$\bullet$ As a sanity check we also tried using random beliefs about other
entities (off-topic) as feedback, but (as expected) the results did not
significantly differ from no feedback (raw model), ending at $\approx$72% F1
and $\approx$74% consistency after batch 10.

We also evaluated a non-incremental, “omniscient” version of the BeliefBank:
Given the raw model answers to all questions, re-ask every question using
feedback selected (using relevance) from all the other answers. The resulting
accuracy was 74.5%, substantially lower than for the on-topic incremental
approaches. Interestingly, this approach’s built-in advantage (that every
question has access to answers for all other questions) does not outweigh the
built-in disadvantage (that those are the raw, rather than incrementally
corrected, answers). This is a significant result demonstrating that the
positive feedback loop of the incremental approaches can be advantageous,
where feedback feeds more accurate beliefs into the BeliefBank, improving
future feedback, etc.

###  7.2 Failure Analysis

We now provide some examples of good and bad flips to better understand the
behavior of the model.

First, as an illustration of desired behavior, the raw model incorrectly
believes that a pine is both a plant (correct) and a vertebrate (incorrect),
when queried. However, this violates a mutual exclusivity rule, so the
constraint-solver considers flipping one of these. Flipping “pine is a plant”
from T to F would result in numerous other violations, e.g., “pine is a tree”
(which the model also believes) would be violated. As a result, it prefers to
(correctly) disbelieve “pine is a vertebrate”, improving both accuracy and
consistency.

From an analysis of the data, we see that the majority of the raw model errors
are false positives – the raw model generally answers (almost) all the
positive facts correctly (recall is $\approx$98%), but mistakenly thinks many
negative facts are also true (precision is $\approx$54%). These false
positives can be rather unusual facts, e.g., “A poodle is a bathroom.”
(model’s answer: True). It is unsurprising that the model knows most of the
positive facts, as they are simple statements about common entities (“eagles
can fly”), likely seen in pre-training. However, the fact that the model makes
(what a person would view as) catastrophic errors when asked more unusual
questions, e.g., believing that “a poodle is plant”, reveals that the PTLM’s
grasp of the world is still incomplete and problematic. The constraint
mechanism proposed here essentially asks the model to think about its answers
and their consequences, so that it can spot problems that the PTLM alone does
not see, and repair them.

The constraint reasoner can also make mistakes, flipping things the wrong way
so as to improve consistency, at the expense of accuracy. For example, the raw
model correctly believes that “a rat is not a cat”. However, the constraint
solver then (incorrectly) flips this to “a rat is a cat”, because multiple
constraints weakly suggest rats are cats given other beliefs (”rats catch
mice”, “rats have tails”,…), which together add up, causing the (incorrect)
flip, including overwhelming the strong (but not infinitely strong) constraint
that “a rat is not a feline.” This illustrates that the constraint mechanism
is sensitive to the number of and weights on constraints, even with automatic
hyperparameter tuning (Section 5.2).

Similarly, the feedback mechanism is sensitive to question order, especially
if the model’s early answers are wrong, as the feedback mechanism causes the
model to pay extra (sometime disproportionate) attention to earlier context
Kassner and Schütze (2020). For example, the bad context “A poodle is not a
mammal” (from an earlier bad answer) undesirably causes the model to change
its answer for "A poodle is a dog" from true (raw model) to false.

Finally, we can only speculate why feedback improves results, in particular,
since the feedback consists of facts that came from the model itself (i.e.,
that it already knows). One explanation is that feedback may help the model
focus attention on important facts, e.g., reminding the model that “a swallow
is not a fish” should help it realize that “a swallow has gills” is False
(Figure 1). In addition, feedback may possibly help resolve some ambiguities,
e.g., the feedback “a swallow has wings” helps identify the bird sense of
“swallow”. Similar advantageous use of feedback was observed in the SelfTalk
experiments Shwartz et al. (2020).

In future work, the feedback mechanism can be improved further by training it
to respond more systematically to feedback (similar to Clark et al. (2020))
and to better balance implicit and explicit knowledge Talmor et al. (2020),
ideally incorporating different levels of confidence.

##  8 Future Work

###  8.1 Human in the Loop

Although our system is autonomous, its incremental setting combined with the
explicit representation of beliefs makes it amenable to a human in the loop.
In this setting, a human might spot an egregious bad belief in the BeliefBank,
and forcibly correct it. Then, ideally, this strong positive datapoint would
also improve the model’s accuracy on other beliefs, both in the BeliefBank and
for future questions. As a brief test of this, we allowed a human to correct
all bad beliefs (average 6) in the BeliefBank after just the first batch (10%)
of questions, and then continued as before to completion, using the
constraint-solving approach. We find that these limited interventions
increased both the final F1 and Consistency each by 2% (absolute) on top of
the gains produced by the corrected beliefs themselves. Although preliminary,
this suggests that our architecture may have value in an interactive “machine
teaching” setting, where the user is supervising and correcting the system,
and it continually improves as a result Zhu (2015).

###  8.2 Towards Deployment

Although our work has been in a constrained setting (targeted set of
relations, entities and constraints), there is a clear development path to
deployment in real QA systems to reduce the kind of irrational behavior we
have described, such as in this (real) transcript:
[vskip=0mm,leftmargin=2mm](1) Is oxygen colorless? yes
(2) What color is oxygen? blue
(3) What gas do plants produce? oxygen
(4) What color is the gas plants produce? green The basic components of our
architecture provide a framework to help avoid such irrationality. First,
(declarative versions of) questions and model answers would be persistently
stored in a BeliefBank. Second, on-topic feedback could be selected to help
answer new questions using information retrieval over the BeliefBank. Third,
given a source of constraints, e.g., a general rule of taxonomic
inheritance,777I.e., that the properties of an entity type usually apply to
all its subtypes also. constraint solving could be applied to spot and reduce
clashes. This would require a mechanism to identify when a belief satisfies a
constraint’s condition or conclusion, e.g., a state-of-the-art textual
entailment engine such as CA-MTL Pilault et al. (2021). A variant of our
system could also work without the distinction of model beliefs and
constraints: Instead of providing constraints externally we could treat them
as beliefs, e.g., query the model for mutual exclusivities "Can an entity be
an animal and a plant?" or implications: "Do dogs have tails?" directly. This
would run the risk of adding extra noise, but would eliminate the manual
effort involved in generating the constraint set, and therefore improve
scalability. Together, such developments would pave the way to real-world QA
systems that are more consistent and improve over time, rather than remain
static.

###  8.3 The Broader Research Agenda

This work only touches on a broader research agenda, namely how to expand work
on PTLMs to encompass the cognitive skills of world modeling and deliberative
reasoning (“thinking, fast and slow” Kahneman (2011)). In this broader agenda,
intelligence is not just about opaque question-answering, but also about
constructing mental models that describe how (some aspect of) the world works
Gentner and Stevens (1983). Although mental models are abstractions (hence are
approximate), they add a powerful, systematic component to understanding that
should expand its capabilities.

The BeliefBank can be seen as a simple illustration of this broader agenda. A
wider pursuit would include a richer notion of a model, perhaps with more
structure to model elements than just sentences; more sophisticated means of
model construction than just accumulating and resolving answers; and the
generation of explanations to convey the deliberative component’s behavior,
and ultimately interact with a user. Such mechanisms may be symbolic or neural
in nature, e.g., Talmor et al. (2020). Although these issues are beyond the
scope of this paper, our work points to this interesting, larger goal for PTLM
research, as well as offering a specific mechanism for belief consistency.

##  9 Conclusion

PTLMs can be inconsistent in their answers to probing questions, and can still
give (what to a person appear as) naively wrong answers. This work is a first
step towards alleviating these problems. By embedding a PTLM within a larger
system with a persistent, global memory – the BeliefBank –, a constraint-
solver and feedback mechanism, we have shown that the overall system’s
behavior is more coherent, both in terms of consistency and accuracy. The
additional memory layer can loosely be seen as the system’s “mental model”, a
representation constructed from the PTLM’s raw answers.

Our experiments were conducted in a restricted (small set of relations,
entities and constraints), controlled setting, and further development is
needed to scale to larger and more complex tasks. Nevertheless, the work here
is significant as it is a first step towards PTLM-based architectures with a
globally consistent notion of belief, allowing them to construct a more
coherent picture of the world, and continually improve with time.

## Acknowledgements

This work has been funded by the Allen Institute for AI and the European
Research Council (#740516) and by the German Federal Ministry of Education and
Research (BMBF) under Grant No. 01IS18036A. The authors of this work take full
responsibility for its content.

## References

  * Asai and Hajishirzi (2020) Akari Asai and Hannaneh Hajishirzi. 2020.  Logic-guided data augmentation and regularization for consistent question answering.  _ArXiv_ , abs/2004.10157.
  * Broecheler et al. (2010) Matthias Broecheler, Lilyana Mihalkova, and L. Getoor. 2010.  Probabilistic similarity logic.  In _UAI_.
  * Carlson et al. (2010) Andrew Carlson, J. Betteridge, Bryan Kisiel, Burr Settles, Estevam R. Hruschka, and Tom Michael Mitchell. 2010.  Toward an architecture for never-ending language learning.  In _AAAI_.
  * Clark et al. (2020) P. Clark, Oyvind Tafjord, and Kyle Richardson. 2020.  Transformers as soft reasoners over language.  In _IJCAI_.
  * Davison et al. (2019) Joe Davison, Joshua Feldman, and Alexander Rush. 2019.  Commonsense knowledge mining from pretrained models.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 1173–1178, Hong Kong, China. Association for Computational Linguistics.
  * De Kleer (1986) Johan De Kleer. 1986.  An assumption-based tms.  _Artificial intelligence_ , 28(2):127–162.
  * De Moura and Bjørner (2008) Leonardo De Moura and Nikolaj Bjørner. 2008.  Z3: An efficient SMT solver.  In _International conference on Tools and Algorithms for the Construction and Analysis of Systems_ , pages 337–340. Springer.
  * Dechter and Dechter (1988) Rina Dechter and Avi Dechter. 1988.  _Belief maintenance in dynamic constraint networks_.  University of California, Computer Science Department.
  * Ebrahimi et al. (2021) Sayna Ebrahimi, Suzanne Petryk, Akash Gokul, William Gan, J. Gonzalez, Marcus Rohrbach, and Trevor Darrell. 2021.  Remembering for the right reasons: Explanations reduce catastrophic forgetting.  _ICLR_.
  * Elazar et al. (2021) Yanai Elazar, Nora Kassner, Shauli Ravfogel, Abhilasha Ravichander, E. Hovy, Hinrich Schütze, and Yoav Goldberg. 2021.  Measuring and improving consistency in pretrained language models.  _ArXiv_ , abs/2102.01017.
  * Ettinger (2020) Allyson Ettinger. 2020.  What BERT is not: Lessons from a new suite of psycholinguistic diagnostics for language models.  _Transactions of the Association for Computational Linguistics_ , 8:34–48.
  * Fellbaum (2005) Christiane Fellbaum. 2005.  Wordnet and wordnets.  In Keith et al. Brown, editor, _Encyclopedia of Language and Linguistics, Second Edition_. Elsevier, Oxford.
  * Genesereth and Nilsson (1987) M. Genesereth and N. Nilsson. 1987.  _Logical foundations of artificial intelligence_.  Kaufmann.
  * Genin and Huber (2021) Konstantin Genin and Franz Huber. 2021.  Formal Representations of Belief.  In Edward N. Zalta, editor, _The Stanford Encyclopedia of Philosophy_ , spring 2021 edition. Metaphysics Research Lab, Stanford University.
  * Gentner and Stevens (1983) Dedre Gentner and Albert L. Stevens. 1983.  _Mental Models_.  Lawrence Erlbaum Associates.
  * Graves et al. (2016) A. Graves, Greg Wayne, M. Reynolds, Tim Harley, Ivo Danihelka, Agnieszka Grabska-Barwinska, Sergio Gomez Colmenarejo, Edward Grefenstette, Tiago Ramalho, J. Agapiou, Adrià Puigdomènech Badia, K. Hermann, Yori Zwols, Georg Ostrovski, Adam Cain, Helen King, C. Summerfield, P. Blunsom, K. Kavukcuoglu, and D. Hassabis. 2016.  Hybrid computing using a neural network with dynamic external memory.  _Nature_ , 538:471–476.
  * Guu et al. (2020) Kelvin Guu, Kenton Lee, Z. Tung, Panupong Pasupat, and Ming-Wei Chang. 2020.  Realm: Retrieval-augmented language model pre-training.  _ArXiv_ , abs/2002.08909.
  * Henaff et al. (2016) Mikael Henaff, J. Weston, Arthur D. Szlam, Antoine Bordes, and Y. LeCun. 2016.  Tracking the world state with recurrent entity networks.  In _ICLR_.
  * Hilton (1996) D. Hilton. 1996.  Mental models and causal explanation: Judgements of probable cause and explanatory relevance.  _Thinking & Reasoning_, 2:273–308.
  * Johnson-Laird (1983) P. Johnson-Laird. 1983.  _Mental Models : Towards a Cognitive Science of Language_.  Harvard University Press.
  * Kahneman (2011) D. Kahneman. 2011.  Thinking, fast and slow.  Farrar, Straus, and Giroux.
  * Kassner and Schütze (2020) Nora Kassner and H. Schütze. 2020.  Negated and misprimed probes for pretrained language models: Birds can talk, but cannot fly.  In _ACL_.
  * Lewis et al. (2020) Patrick Lewis, Ethan Perez, Aleksandara Piktus, Fabio Petroni, V. Karpukhin, Naman Goyal, Heinrich Kuttler, M. Lewis, Wen tau Yih, Tim Rocktäschel, Sebastian Riedel, and Douwe Kiela. 2020.  Retrieval-augmented generation for knowledge-intensive nlp tasks.  In _NeurIPS_.
  * Li et al. (2019) Tao Li, Vivek Gupta, Maitrey Mehta, and Vivek Srikumar. 2019.  A logic-driven framework for consistency of neural models.  In _EMNLP_.
  * Liu et al. (2021) Jiachang Liu, Dinghan Shen, Yizhe Zhang, Bill Dolan, L. Carin, and Weizhu Chen. 2021\.  What makes good in-context examples for GPT-3?  _ArXiv_ , abs/2101.06804.
  * McCarthy (1959) John W. McCarthy. 1959.  Programs with common sense.  In _Proc. Tedding Conf. on the Mechanization of Thought Processes_ , pages 75–91.
  * Minervini and Riedel (2018) Pasquale Minervini and S. Riedel. 2018.  Adversarially regularising neural NLI models to integrate logical background knowledge.  In _CoNLL_.
  * Moore (1983) R. Moore. 1983.  Semantical considerations on nonmonotonic logic.  In _IJCAI_.
  * Parisi et al. (2019) G. I. Parisi, Ronald Kemker, Jose L. Part, Christopher Kanan, and S. Wermter. 2019\.  Continual lifelong learning with neural networks: A review.  _Neural networks : the official journal of the International Neural Network Society_ , 113:54–71.
  * Pearl (1986) J. Pearl. 1986.  Fusion, propagation, and structuring in belief networks.  _Artif. Intell._ , 29:241–288.
  * Petroni et al. (2019) F. Petroni, Tim Rocktäschel, Patrick Lewis, A. Bakhtin, Yuxiang Wu, Alexander H. Miller, and S. Riedel. 2019.  Language models as knowledge bases?  In _EMNLP_.
  * Pilault et al. (2021) Jonathan Pilault, Amine Elhattami, and C. Pal. 2021.  Conditionally adaptive multi-task learning: Improving transfer learning in nlp using fewer parameters & less data.  In _ICLR_.
  * Pujara et al. (2013) J. Pujara, H. Miao, L. Getoor, and William W. Cohen. 2013.  Knowledge graph identification.  In _International Semantic Web Conference_.
  * Ravichander et al. (2020) Abhilasha Ravichander, Eduard Hovy, Kaheer Suleman, Adam Trischler, and Jackie Chi Kit Cheung. 2020.  On the systematicity of probing contextualized word representations: The case of hypernymy in BERT.  In _Proceedings of the Ninth Joint Conference on Lexical and Computational Semantics_ , pages 88–102, Barcelona, Spain (Online). Association for Computational Linguistics.
  * Ribeiro et al. (2019) Marco Tulio Ribeiro, Carlos Guestrin, and Sameer Singh. 2019.  Are red roses red? Evaluating consistency of question-answering models.  In _ACL_.
  * Roberts et al. (2020) Adam Roberts, Colin Raffel, and Noam Shazeer. 2020.  How much knowledge can you pack into the parameters of a language model? In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 5418–5426, Online. Association for Computational Linguistics.
  * Shwartz et al. (2020) Vered Shwartz, Peter West, Ronan Le Bras, Chandra Bhagavatula, and Yejin Choi. 2020\.  Unsupervised commonsense question answering with self-talk.  In _EMNLP_.
  * Speer et al. (2017) Robyn Speer, Joshua Chin, and Catherine Havasi. 2017.  Conceptnet 5.5: An open multilingual graph of general knowledge.  In _AAAI_.
  * Subramanian et al. (2020) Sanjay Subramanian, Ben Bogin, Nitish Gupta, Tomer Wolfson, Sameer Singh, Jonathan Berant, and Matt Gardner. 2020.  Obtaining faithful interpretations from compositional neural networks.  In _ACL_.
  * Sukhbaatar et al. (2015) Sainbayar Sukhbaatar, Arthur D. Szlam, J. Weston, and R. Fergus. 2015.  End-to-end memory networks.  In _NeurIPS_.
  * Tafjord and Clark (2021) Oyvind Tafjord and Peter Clark. 2021.  General-purpose question-answering with Macaw.  _arXiv_ , abs/2109.02593.
  * Talmor et al. (2020) Alon Talmor, Oyvind Tafjord, Peter Clark, Yoav Goldberg, and Jonathan Berant. 2020\.  Leap-of-thought: Teaching pre-trained models to systematically reason over implicit knowledge.  In _NeurIPS_.
  * Zhu (2015) Xiaojin Zhu. 2015.  Machine teaching: An inverse problem to machine learning and an approach toward optimal education.  In _AAAI_.

Appendix: BeliefBank: Adding Memory to a Pre-Trained Language
Model for a Systematic Notion of Belief

Accuracy (F1) after batch $\rightarrow$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---
Raw model | 69.2 | 70.0 | 70.0 | 69.3 | 69.3 | 69.5 | 69.2 | 69.0 | 69.1 | 69.3
OnTopic-FB | 69.2 | 74.3 | 76.0 | 76.4 | 77.0 | 77.3 | 77.5 | 77.5 | 77.8 | 77.8
Relevant-FB | 69.2 | 73.8 | 75.1 | 75.8 | 76.8 | 77.2 | 77.7 | 78.4 | 78.9 | 79.5
Constraints | 72.3 | 75.2 | 76.8 | 78.3 | 80.1 | 81.6 | 82.4 | 83.3 | 84.9 | 85.8
Relevant-FB + Constraints | 72.4 | 78.5 | 80.5 | 82.1 | 83.6 | 84.2 | 85.1 | 86.1 | 86.7 | 86.6
Table 1: Experimental results for accuracy (F1), as plotted in Figure 3, here shown in tabular form. Consistency ($1-\tau$) after batch $\rightarrow$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---
Raw model | 73.2 | 73.7 | 74.0 | 72.8 | 72.4 | 72.3 | 72.3 | 72.2 | 72.4 | 72.5
OnTopic-FB | 73.2 | 72.9 | 73.6 | 73.3 | 73.5 | 72.6 | 72.5 | 72.6 | 72.5 | 72.3
Relevant-FB | 73.2 | 77.8 | 79.3 | 80.1 | 80.6 | 79.7 | 80.1 | 80.5 | 80.8 | 81.0
Constraints | 98.2 | 97.1 | 96.5 | 96.2 | 96.3 | 96.4 | 96.1 | 96.1 | 96.0 | 96.0
Relevant-FB + Constraints | 98.8 | 98.2 | 97.6 | 97.8 | 97.9 | 97.7 | 97.3 | 97.2 | 97.1 | 97.0
Table 2: Experimental results for consistency ($1-\tau$), as plotted in Figure
3, here shown in tabular form.

##  Appendix A Selecting Constraint Rules from ConceptNet

As described in Section 5.1, positive implication (constraint) rules were
manually gathered from ConceptNet Speer et al. (2017). First, we identified
121 general concepts of interest, e.g., “mammal”, choosing concepts with high
occurrence ($>$ 100 times) in ConceptNet, avoiding significantly ambiguous
terms (e.g., “bat”), and filtering out plurals and obscure concepts. For these
entities, we then collected all ConceptNet facts involving 6 relations: IsA,
HasA, MadeOf, PartOf, HasProperty, and CapableOf, and re-expressed them as
constraints. For example, the ConceptNet triple (dog, HasA, tail) gives rise
to the constraint "X is a dog" $\rightarrow$ "X has a tail." (Triples are
converted into English sentences using simple templates). We then manually
filter theses constraints for factual correctness. We also add weaker,
disjunctive constraints in the backwards direction, e.g., "X has a tail"
$\rightarrow$ "X is a dog" OR "X is a cat" OR …. for all entities with tails.
(These backwards rules discourage the trivial solution that everything is
false.) Finally, two hyperparameters for weights on forward and backwards
rules are set by automatic calibration (Section 5.2).

##  Appendix B Experimental results in table form

Tables 1 and 2 contain the numerical data for the experimental results plotted
in Figure 3.