# On the geometry of free nilpotent groups

Artem Semidetnov  and  Ruslan Magdiev  Laboratory of Continuous Mathematical
Education (School 564 of St. Petersburg), nab. Obvodnogo kanala 143, Saint
Petersburg, Russia<EMAIL_ADDRESS>St. Petersburg State University,
14th Line, 29b, Saint Petersburg, 199178 Russia<EMAIL_ADDRESS>
###### Abstract.

In this article, we study geometric properties of nilpotent groups. We find a
geometric criterion for the word problem for the finitely generated free
nilpotent groups. By geometric criterion we mean a way to determine whether
two words represent the same element in a free nilpotent group of rank $r$ and
class $k$ by analyzing their behaviour on the Cayley graph of the free
nilpotent group of rank $r$ and class $k-1$.

###### Key words and phrases:

Geometry, Nilpotent groups, free, Cayley graph, word problem, criterion

##  1\. Introduction and background

Nilpotent groups are a long-standing topic in the group theory. In relatively
recent years they have started to attract attention from a geometric point of
view. As Gromov proved in [7], a group as a metric space has polynomial growth
if and only if it is virtually nilpotent. It has become a world-recognized
example of the geometric uniqueness of nilpotent groups.

As in most varieties of groups, the key to understanding the whole structure
is studying the free objects of the variety. In the case in question, these
are the free nilpotent groups. Apparently, any finitely-generated free
nilpotent group can be determined by only two positive integers which are the
number of generators and the class of nilpotency. The complexity of the
geometric structure of these groups increases with these numbers.

The most studied non-abelian free nilpotent group has the smallest determining
values among all of the non-abelian free nilpotent groups. These values are 2
and 2. This group is also known as the discrete Heisenberg group
$H(\mathbb{Z})$. By its original definition, it coincides with the group of
upper triangular matrices over the ring of integers. However, we are
interested in the following presentation:

$H(\mathbb{Z})=\langle a,b\;|\;[a,[a,b]]=[b,[a,b]]=1\rangle.$

With respect to this presentation the following fact is true: two words drawn
in its Cayley graph end in the same point if and only if special-defined
closures of projections on the horizontal plane have the same area. The main
result of the present article is to generalize this statement to all finitely
generated free nilpotent groups.

Figure 1. Cayley graph of $H(\mathbb{Z})$ with respect to the generating set
$\\{a,b,[a,b]\\}$.

###  1.1. Notions

Let $G$ be a group. For $a,b\in G$ we put $[a,b]=aba^{-1}b^{-1}$ and
$[x_{1},x_{2},\ldots,x_{n}]$ for $[[[x_{1},x_{2}],x_{3}]\ldots,x_{n}]$. For
the sake of clarity we define commutator in the degenerate case of $n=1$ as
$[x_{1}]=x_{1}$. Within a group we will write $[H,K]$ for the subgroup
generated by all commutators $[h,k]$ with $h$ ranging over $H\leq G$ and $k$
ranging over $K\leq G$, so that in particular $[G,G]$ is the usual commutator
subgroup of $G$. Define a sequence of subgroups inductively as
$\gamma_{1}(G)=G,\;\gamma_{s}(G)=[\gamma_{s-1}(G),G]$. The group is said to be
nilpotent of class $k$ if this sequence converges to trivial subgroup in
exactly $k+1$ steps:

$\\{1\\}=\gamma_{k+1}(G)\vartriangleleft\gamma_{k}(G)\vartriangleleft\ldots\vartriangleleft\gamma_{3}(G)\vartriangleleft\gamma_{2}(G)\vartriangleleft\gamma_{1}(G)=G.$

For $r,k\in\mathbb{N}$ define the free nilpotent group of rank $r$ and class
$k$ as the quotient group $F_{r}/\gamma_{k+1}(F_{r})$, where $F_{r}$ is a
unique free non-abelian group of rank $r$. For a basis of $F_{r}$, say,
$\\{s_{1},\ldots,s_{r}\\}$ put
$\mathcal{J}_{r,k}=\\{[s_{i_{1}},\ldots,s_{i_{k}}]\;|\;\forall i_{j}\\}$.
Using this notion we define the standard presentation for a free nilpotent
group of rank $r$ and class $k$:

$N_{r,k}=\langle s_{1},s_{2},\ldots,s_{r}\;|\;\mathcal{J}_{r,k+1}\rangle$

All of the definitions of geometric-group-theory terms may be found in [5].
However, here we want to put emphasis on the following construction. Let $G$
be a group and $S$ be a fixed set of generators for this group. For a given
word $w$ over an alphabet $S\cup S^{-1}$ there’s a way to ’draw’ this word on
$\text{\bf Cay}(G,S)$. It consists of elements represented by prefixes of $w$
which are connected by edges in a consequential order.

Obviously, two words represent the same element in the group, if and only if
their curves end in the same vertex on the Cayley graph. You may see the
curves that correspond to the words $aba^{-1}b^{-1},a^{2}b^{2}a^{-2}b^{-2}$
and $a^{4}b^{4}a^{-4}b^{-4}$ in the discrete Heisenberg group on the figure 1.

In this article, the idea of a word in $F(S)$ being able to represent elements
in all of the groups with generating sets equinumerous to $S$ is used several
times. That is why for $w\in F_{r}$ denote $[w]_{k}$ as the element of
$N_{r,k}$ represented by $w$.

###  1.2. Known results on $N_{r,k}$

Going back to free nilpotent groups, throughout this paper we will consider
them only with the standard generating sets, so instead of $\text{\bf
Cay}(N_{r,k},\\{s_{1},\ldots,s_{r}\\})$ we will write just $\text{\bf
Cay}(N_{r,k})$. Now let’s pick $r,k>1$. We avoid considering cases $r=1$ or
$k=1$ because they are abelian. There is a very helpful normal form for
elements of $N_{r,k}$ that uses so-called Mal’tsev bases. A precise definition
is given in [10] but in our special case, we will just call this name sets
$\mathcal{J}_{r,k}$. Put $\Delta_{r,k}=|\mathcal{J}_{r,k}|$ and denote
$c(r,k):=|\Delta_{r,1}|+\ldots+|\Delta_{r,k}|$.

Due to the fact that a Mal’tsev basis $\mathcal{J}_{r,j}$ is equinumerous to
the set $\\{1,\ldots,\Delta_{r,j}\\}$ we can order these sets linearly. Choose
such an order $\preccurlyeq$. Define the canonical form of an element $g$ as

(1.1)
$u_{g}=\prod_{w_{i}\in\mathcal{J}_{r,k}}w_{i}^{\alpha^{k}_{i}}\cdot\ldots\cdot\prod_{w_{i}\in\mathcal{J}_{r,2}}w_{i}^{\alpha^{2}_{i}}\cdot\prod_{s_{i}\in
S}s_{i}^{\alpha^{1}_{i}}$

where the products are ordered in correspondence to $\preccurlyeq$. Though it
may seem confusing, it is just a product of elements of
$\cup\mathcal{J}_{r,i}$ for $i=\overline{1,k}$ in some powers
$\alpha_{i}^{j}$, uniquely defined by the element $g$.

###  1.3. Geometric characteristic for words and the main result.

Here we define a geometric characteristic for words in $F_{r}$ presenting
elements of $N_{r,k}$. The main result, that is Theorem A draws a connection
between this characteristic and the word problem for $N_{r,k}$.

The (1.1) product gives rise to a coordinate form of elements of $N_{r,k}$:
each of them can be associated with an array of integers, given by powers of
the elements of $\cup\mathcal{J}_{r,i}$111in the example above these are
$\alpha_{i}^{j}$, with $i=\overline{1,k}$. As $c(r,k)=\Delta_{r,k}+c(r,k-1)$
for an element $g$ we will simply write

$g=(T_{1},T_{2},\ldots,T_{\Delta_{r,k}},t_{c(r,k-1)},\ldots,t_{1})$

and call $T_{i}$ and $t_{j}$ for $i=\overline{1,\Delta_{r,k}}$ and
$j=\overline{1,c(r,k-1)}$ coordinates. We distinguish them by capitalization
because of the folowing fact. Let $\pi_{k}$ be the canonical epimorphism
$N_{r,k}\to N_{r,k-1}$ which kernel is $\gamma_{k}(N_{r,k})$. It is easy to
see that under that mapping

$(T_{1},T_{2},\ldots,T_{\Delta_{r,k}},t_{c(r,k-1)},\ldots,t_{1})\mapsto(t_{c(r,k-1)},\ldots,t_{1})$

The coordinate system defines a way to immerse $\text{\bf Cay}(N_{r,k})$ into
$\mathbb{R}^{c(r,k)}$ via the map identifying coordinates in $N_{r,k}$ and
$c(r,k)$-tuples in $\mathbb{R}^{c(r,k)}$.

As there is a chosen order $\preccurlyeq$ on $\mathcal{J}_{r,k}$ denote
$K_{i}$ the $i$-th element in $\mathcal{J}_{r,k}$. It is easy to see that
$K_{i}$ represents the element

$(0,\ldots,0,1,0,\ldots,0)$

where the only non-zero coordinate is the $i$-th.

######  Definition 1.1.

Let $w$ be a word in $F_{r}$ and $k\in\mathbb{N}$. Let
$i\in\\{1,\ldots,\Delta_{r,k}\\}$. Then $K_{i}=[k,s]$ where
$s\in\\{s_{1},\ldots,s_{r}\\}$ and $k\in\mathcal{J}_{r,k-1}$. Denote by
$\overline{s},\overline{k}\in\mathbb{R}^{c(r,k-1)}$ the images of $s,k$ under
the immersion defined above. Let ${\bf W}\subset\mathbb{R}^{c(r,k)}$ be the
curve for the word $w$ on the $\text{\bf Cay}(N_{r,k})$. We say that the
orthogonal projection of ${\bf W}$ on the plane
$\text{span}(\overline{s},\overline{k})$ is the projection of the word $w$ on
the $i$-th coordinate.

Sometimes we will abuse notation and say that the projection is being
considered not on the $i$-th coordinate but on the $K_{i}$-th coordinate.

Now we define area for a projection of a word $w$ on the $i$-th coordinate for
some $i$. Obviously, area enclosed be some curve can be defined only in the
case when the said curve is closed. It is explained below that the curve for a
word $w$ on $\text{\bf Cay}(N_{r,k})$ projected on $\text{\bf Cay}(N_{r,k-1})$
is closed if and only if it represents an element of $\gamma_{k}(N_{r,k})$.

Define closure of a word $w$ representing $g\in N_{r,k}$ as the
$u_{\pi_{k}(g^{-1})}$. As it is explained below the curve’s projection for
$\omega\cdot u_{\pi_{k}(g^{-1})}$ projection on $\text{\bf Cay}(N_{r,k-1})$ is
closed. Call the oriented area of the polygon defined as the projection of
$\omega\cdot u_{\pi_{k}(g^{-1})}$ onto the $i$-th coordinate the area of $w$’s
projection onto the $i$-th coordinate.

Note that closure of $K_{i}$ is the empty word. For each
$K_{i}\in\mathcal{J}_{r,k}$ denote $C_{i}$ for its projection’s area on the
$i$-th coordinate.

Our main result is the following fact:

Theorem A. A word $w\in F_{r}$ represents an element $g\in N_{r,k}$ with
coordinates $(T_{1},T_{2},\ldots,T_{\Delta_{r,k}},t_{c(r,k-1)},\ldots,t_{1})$
if and only if $w$ represents an element $(t_{c(r,k-1)},\ldots,t_{1})$ in
$N_{r,k-1}$ and areas of its standard closure’s projections onto
$T_{i}$-coordinates equal to $C_{i}T_{i}$ for $i=\overline{1,\Delta_{r,k}}.$

##  2\. Some properties of $N_{r,k}$

In this section, we elaborate on some properties of free nilpotent groups.
Again, pick constants $r,k>1$.

######  Proposition 2.1.

Center of the $N_{r,k}$ coincides with $\gamma_{k}(N_{r,k})$.

######  Remark 2.2.

A word $w$ represents a closed curve on $\text{\bf Cay}(N_{r,k-1})$ if and
only if $[w]_{k}\in\gamma_{k}(N_{r,k})$.

######  Lemma 2.3.

The following facts are true

  1. (1)

Area of a word $u\in\mathcal{J}_{r,k+1}$ or its inverse projected on any
$i$-th coordinate equals zero.

  2. (2)

Let $w,v\in F(S)$. If $[w]_{k}=[v]_{k}$ then on any $i$-th coordinate areas of
their projections coincide.

  3. (3)

Let $w\in\mathcal{J}_{r,k}$ and $v\in F(S)$ and let their $w$-projection areas
be equal to $\Sigma_{w}$ and $\Sigma_{v}$ respectively. Then the area of $w$
projection of $vw$ equals to $\Sigma_{v}+\Sigma_{w}$.

######  Lemma 2.4.

If $g\in N_{r,k}$ has the following coordinates

$g=(T_{1},\ldots,T_{\Delta_{r,k}},0\ldots,0),$

and each of all its areas on $i$-th projections are zeros, then $g=1$.

###  2.1. Values of $c(r,k)$ and $\Delta_{r,k}$.

######  Proposition 2.5 (see [12] 7, 11.2.2).

Numbers $\Delta_{r,k}$ are given by necklace polynomials:

$\Delta_{r,k}=\frac{1}{k}\sum_{d\,|\,k}\mu(d)r^{k/d}$

Where $\mu$ is the m$\ddot{o}$bius function.

Also we prove the following recursion property for $c(r,k)$:

######  Lemma 2.6.

Value $c(r,k)$ satisfies following relation

$c(r,k)=(r+1)\cdot c(r,k-1)-r\cdot c(r,k-2)$

with $c(r,1)=r$, $c(r,2)=r(r+1)/2$ for $r>2$ and $c(2,2)=1$. As a corollary,
$c(2,k)=2^{k-1}+1$.

Also, we should include the notion of null sequences. For any group $G=\langle
S\;|\;R\rangle$ consider $w\in F(S)$. Word $w$ can be broken down into a
product of two words (possibly empty), $w=uv$. Also, for any $r\in R$ and
$s\in S$ words $uss^{-1}v$ and $usrs^{-1}v$ represent the same element in $G$
as $w$. Actually, any two words representing the same element in $G$ can be
transformed into each other using these alterations. They are called null-
sequences (see [5]).

###  2.2. Examples on $N_{2,2}$ and $N_{2,3}$

The Discrete Heisenberg group is the only group for which the geometric
criterion has already been known. It was introduced in [2] and widely used in
[1, 2, 3, 9]. Its $\pi_{2}$ epimorphism is actually the orthogonal projection
of $\text{\bf Cay}(N_{2,2},\\{a,b\\})$ onto $\text{\bf
Cay}(\mathbb{Z}\times\mathbb{Z},\\{a,b\\})$ given by the abelenization map.
Moreover, the corresponding short exact sequence is split

$1\longrightarrow\mathbb{Z}\longrightarrow
N_{2,2}\longrightarrow\mathbb{Z}\times\mathbb{Z}\longrightarrow 1$

Figure 2. Projections of words in $N_{2,2}$ on $\text{\bf Cay}(N_{2,1})$.

The criterion in $N_{2,2}$ is quite easy to comprehend: two words $w,v\in
F(a,b)$ represent the same element in the discrete Heisenberg group if and
only if they, being pictured on $\text{\bf Cay}(\mathbb{Z}\times\mathbb{Z})$,
end in the same point $(m,n)$ on this graph and the oriented areas of their
closures $wa^{-n}b^{-m},va^{-n}b^{-m}$ coincide.

Now let’s consider $N_{2,3}$. The theorem states that we should look at
$\text{\bf Cay}(N_{2,2})$. The set $\mathcal{J}_{2,3}$ consists of two
elements: $\mathcal{J}_{2,3}=\\{[a,a,b],[b,a,b]\\}$. Exact computations show
that areas of their projections are 2 for $[a,a,b]$ and 1 for $[b,a,b]$. It is
easy to see that the closure for a word $w$ ending in
$(k,m,n)\in\mathbb{Z}^{3}$ on $\text{\bf Cay}(N_{2,2})$ is
$wa^{-n}b^{-m}[a,b]^{-k}$.

Figure 3. Word $w=aabba^{-1}ba^{-1}b^{-1}abb$ presented on $\text{\bf
Cay}(N_{2,2})$ and projected to planes $x=0$ and $y=0$ in $(x,y,z)$ coordinate
system.

For example, picture 3 shows the word $w=aabba^{-1}ba^{-1}b^{-1}abb$. It is
drawn in grey, and red lines represent its closure. It is easy to see that
areas of figures are 4 and 10 for projections on first and second coordinates
respectively, which is drawn in red has projections with areas 4 and 10.
Consequently, $[w]_{3}=(8,10,5,4,1)$.

##  3\. Proofs

######  Remark 3.1.

For any element

$g=(T_{1},T_{2},\ldots,T_{\Delta_{r,k}},t_{c(r,k-1)},\ldots,t_{1})$

An area of the word $u_{g}$ projected on $i$-th coordinate equals to
$C_{i}T_{i}$ for all $i=\overline{1,\Delta_{r,k}}$

###### Proof of Lemma 2.6.

This proof consists of two computations:

We define graph of commutators $\Gamma_{r,k}$. Its set of vertices is
$V(\Gamma_{r,k})=\cup\mathcal{J}_{r,j}$ for $j=\overline{1,k}$ and the
commutator $[s_{1},\ldots,s_{j}]$ for $j\leq k$ is connected with

  1. (1)

$s_{1}$

  2. (2)

$[s_{2},\ldots,s_{j}]$

  3. (3)

$[s,s_{1},s_{2},\ldots,s_{j}]$ for all $s\in S$ if $j<k$.

You can see $\Gamma_{2,4}$ on picture 4. Obviously, $c(r,k)=|\Gamma_{r,k}|$.
Let’s compute number of elements that adds to $\Gamma_{r,k}$ in
$\Gamma_{r,k+1}$. (Obviously, there’s an embedding
$i:\Gamma_{r,k}\rightarrow\Gamma_{r,k+1}$). It is easy to see that any element
of $\Gamma_{r,k+1}\backslash i(\Gamma_{r,k})$ is connected with exactly two
elements: one in $i(\Gamma_{r,k})\backslash i(\Gamma_{r,k-1})$ and one in $S$.
So, number of elements in $\Gamma_{r,k+1}\backslash i(\Gamma_{r,k})$ equals to
number of edges in complete bipartite graph $K_{m,n}$, where
$m=|i(\Gamma_{r,k})\backslash i(\Gamma_{r,k-1})|$ and $n=|S|=r$. Thus,

$c(r,k)=r\cdot|\Gamma_{r,k}\backslash\Gamma_{r,k-1}|+c(r,k-1)=\\\
=r(c(r,k-1)-c(r,k-2))+c(r,k-1)=(r+1)\cdot c(r,k-1)-r\cdot c(r,k-2)$

For $k=2$ put $c_{k}:=c(2,k)$. Then it admits linear recurrent relation of a
form $c_{k}=3c_{k-1}-2c_{k-2}$.It has corresponding characteristic polynomial
$q^{2}-3q+2$. It has two roots: $2,1$. Then any sequence $d_{n}=A\cdot
2^{k}+B$ has the same kind of linear recurrent relation. So, by solving system
of equations $d_{1}=c_{1},\,d_{2}=c_{2}$ for $A,B$ we will find that form for
$c_{k}$. Recall that $c_{1}=2,c_{2}=1$.

$\begin{cases}2=2A+B\\\ 1=4A+B\end{cases}$

It is easy to see that $A=1/2,B=1$. So we have $c_{k}=c(2,k)=2^{k-1}+1$.

Figure 4. Graph of commutators $\Gamma_{2,4}$.

∎

###### Proof of Lemma 2.2.

Obviously, the word $w$ represents a closed curve on $\text{\bf
Cay}(N_{r,k-1})$ if and only if $[w]_{k-1}$ is the identity element in
$N_{r,k-1}$. It is equivalent to the fact that $[w]_{k}\in\ker(\pi_{k})$. As
it was noticed above, $\ker(\pi_{k})=\text{Cent}(N_{r,k})$. ∎

###### Proof of Lemma 2.3.

  1. (1)

Note that any $u\in\mathcal{J}_{r,k+1}$ can be seen as $[s,v]$ for some $s\in
S$ and $v\in\mathcal{J}_{r,k}$. So, $u=svs^{-1}v^{-1}$. By multiplication in
$N_{r,k}$ it is easy to see that curve of $svs^{-1}$ ends in 0 and segments
corresponding to $s$ and $s^{-1}$ on projection coincide. Moreover,
projections of $v$ and $v^{-1}$ are isometric in $\mathbb{R}^{2}$ and oriented
in different ways. Thus, area of the projection is $0$.

  2. (2)

Let $[w]_{k}=[v]_{k}$. Then $[wv^{-1}]_{k}=1$, which means (see [5]) that
there’s a null-sequence of $wv^{-1}$ leading to the empty word. Now we prove
that any null-transformation does not change area of a word’s projection. Let
$u=u_{1}u_{2}$ and $s\in S$. Obviously, $u_{1}ss^{-1}u_{2}$ as a curve
represents curve $u_{1}u_{2}$ with glued to $[u_{1}]_{k}$ segment that ends in
$[u_{1}s]_{k}$, so its area did not change. The second transformation is
$u\mapsto u_{1}srs^{-1}u_{2}$, where $r\in\mathcal{J}_{r,k+1}$ is a relation
word. It is easy to see that its curve consists of $u_{1}u_{2}$ curve with
curve of $srs^{-1}$ glued to $[u_{1}]_{k}$. Note, that it has zero area of
projection because of the previous point in this lemma. Note, that inverse
null-transformations need not to be analyzed separately, because we have shown
that images’ area always coincide with preimages’.

  3. (3)

As $[w]$ lies in $\text{Cent}(N_{r,k})$, words $vw$ and $wv$ represent the
same element in $N_{r,k}$. Obviously, area of $wv$ equals to
$\Sigma_{w}+\Sigma_{v}$. So using previous point of this lemma we obtain what
we desired so hardly.

∎

###### Proof of the lemma 2.4.

Suppose $w$ represents a non-identity element $g\in N_{r,k}$ and it has above-
mentioned coordinates. Then consider $u_{g}$. It has areas that are equal to
$C_{i}T_{i}$ on the corresponding planes. Then by lemma 2.3 we get that areas
of $w$ coincide with these numbers. Hence, they are no zeros. ∎

###### Proof of the main theorem.

Start with the necessity. As

$[w]_{k}=g=(T_{1},T_{2},\ldots,T_{\Delta_{r,k}},t_{c(r,k-1)},\ldots,t_{1})$

its image under $\pi_{k}$ represents $(t_{c(r,k-1)},\ldots,t_{1})\in
N_{r,k-1}$. That’s the first part. Next, consider $u_{g}$. As
$[u_{g}]_{k}=[w]_{k}$, there exists a null-sequence that transforms $u_{g}$ to
$w$. As it was noted above, areas of projections on $T_{i}$ coordinates of
$u_{g}$ equal to $C_{i}T_{i}$ with $i=\overline{1,\Delta_{r,k}}$. Also, null-
sequences do not change these areas. Consequently, an area of projection of
$w$ on $T_{i}$ coordinate equals to $C_{i}T_{i}$ for every
$i=\overline{1,\Delta_{r,k}}$

Sufficiency. Let

$[w]_{k}=(T_{1}^{\prime},\ldots,T_{\Delta_{r,k}}^{\prime},t_{c(r,k-1)},\ldots,t_{1}).$

Let us prove that $w$ represents the same element as $u_{g}$. Consider the
word $v=wu_{g}^{-1}$. The following holds for it:

$[v]_{k}=(T_{1}^{\prime},\ldots,T_{\Delta_{r,k}}^{\prime},t_{c(r,k-1)},\ldots,t_{1})(T_{1},T_{2},\ldots,T_{\Delta_{r,k}},t_{c(r,k-1)},\ldots,t_{1})^{-1}=\\\
=(T_{1}^{\prime},\ldots,T_{\Delta_{r,k}}^{\prime},0,\ldots,0)\cdot(0,\ldots,0,t_{c(r,k-1)},\ldots,t_{1})(T_{1},\ldots,T_{\Delta_{r,k}},0,\ldots,0)^{-1}\cdot(0,\ldots,0,t_{c(r,k-1)},\ldots,t_{1})^{-1}=\\\
=(T_{1}^{\prime},\ldots,T_{\Delta_{r,k}}^{\prime},0,\ldots,0)(T_{1},\ldots,T_{\Delta_{r,k}},0,\ldots,0)^{-1}$

Consequently, it suffices to check that the last product equals to the
identity.

By the lemma 2.3 and the condition on areas of $w$ stated in the theorem we
can state that all the areas of projections of the closed curve $wu_{g}^{-1}$
are equal to zero. Then, by the 2.4 we get $[wu_{g}^{-1}]_{k}=1$. Then
$[w]_{k}=g$.

∎

## References

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# Taking the Intentional Stance Seriously, or
“Intending” to Improve Cognitive Systems

Will Bridewell

###### Abstract

Finding claims that researchers have made considerable progress in artificial
intelligence over the last several decades is easy. However, our everyday
interactions with cognitive systems (e.g., Siri, Alexa, DALL-E) quickly move
from intriguing to frustrating. One cause of those frustrations rests in a
mismatch between the expectations we have due to our inherent, folk-
psychological theories and the real limitations we experience with existing
computer programs. The software does not understand that people have goals,
beliefs about how to achieve those goals, and intentions to act accordingly.
One way to align cognitive systems with our expectations is to imbue them with
mental states that mirror those we use to predict and explain human behavior.
This paper discusses these concerns and illustrates the challenge of following
this route by analyzing the mental state ‘intention.’ That analysis is joined
with high-level methodological suggestions that support progress in this
endeavor.

<EMAIL_ADDRESS>
U.S. Naval Research Laboratory, Washington, DC 20375 USA

##  1 Introduction

Largely due to commercial interest and support there has been a dramatic shift
from projects in artificial intelligence (AI) that are mind inspired and
theory driven toward ones that are brain inspired and data driven. In the span
of a decade, machine learning approaches that draw inspiration from
neuroscience and mathematics eclipsed research informed by psychology,
philosophy, and economic theory. Even before deep learning dominated the
field, researchers in AI replaced structured representations of intentional
states such as belief, desire, intention, and choice with mathematical
representations that provide formal guarantees on inference and learning. In
the process, beliefs became probability distributions, actions became the
outputs of learned policies, and rationality was equated with an optimality
criterion. Simultaneously, the internals of models and systems in AI became
increasingly opaque to the point that predicting and explaining their behavior
became impractical, if not impracticable.

In response, there has been a call for explainable AI (XAI) from the Defense
Advanced Research Projects Agency (DARPA) to “enable human users to
understand, appropriately trust, and effectively manage the emerging
generation of artificially intelligent partners” [33]. In practice,
researchers take this statement to mean that a causal explanation of an AI
system’s output is necessary for people to assign credence and to diagnose
failures [14, 11]. However, the imagined human-machine partnerships inspiring
this call extend beyond deciding whether to accept machine responses and
determining how to improve machine accuracy. Advice on this point is given by
Clancey and Hoffman [*]Clancey2021 and Hoffman and colleagues [*]ihmc2021.

Looking at how we work with other people can provide a sense of what might be
required for successful collaborations with AI partners. When we jointly act
with people, we rely heavily on the application of folk-psychological
theories: our understanding of intentional states like knowledge, beliefs,
desires, intentions, hopes, and concerns. Taking what Dennett [*]dennett87
calls the intentional stance, involves using these theories to guide
prediction, explanation, and planning so that we can navigate our social
relationships.111The word ‘intentional’ is used in two ways in this paper.
When referring to the intentional stance, the reference is to intentionality,
which is philosophical jargon for mental states being about some form of
content. Beliefs, desires, hopes, wishes, and so on are about some other
subject (e.g., I believe that X, I hope that X). In contrast, intentional
action most often refers to planning to do something (e.g., I intend to go to
the store later today, I intend to learn calculus) or the act of doing
something on purpose (e.g., You intentionally spilled my drink!). Although
there is a possibility of confusion, the intended meaning should be clear in
context. Contrast that with how we view our interactions with ordinary
machines like toasters or televisions. In those cases, Dennett points out that
we usually take a design stance, which is informed by our knowledge of the
functions those machines were built to carry out. We describe a toaster by
saying that you put your bread in the slots, press down on the lever, and
after some time your toast is ready. This description does not refer to mental
states any more than it refers to physical laws. All that matters for
prediction, explanation, and planning is a knowledge of the toaster’s
function. However, as intelligent machines become general purpose surrogates
for people, the potential behaviors and interactions will increase in
complexity to the point that the design stance becomes unmanageable for
everyday interactions. The functionality of the machines and the rules guiding
their behavior will become too complex to anticipate or learn. We see this
already with intelligent systems like Amazon’s Alexa and Apple’s Siri, which
are meant to simulate limited forms of human interaction and are constantly
growing more capable. Furthermore, people not only naturally apply their folk-
psychological concepts when explaining robotic behavior [9] but will also
forfeit their careers to hold on to that interpretation [30, 31].

I assume that easing human-machine collaboration requires closing the gap
between the behavior of our artificially intelligent partners and the
expectations we form from our folk-psychological theories. Likewise, de Graaf
and Malle [*]DeGraaf2017a argue that explanations given by an AI system need
to be couched in the terms of folk psychology for people to accurately
calibrate their trust in that system [28]. If this is the case, then the AI
community needs to determine or specify the extent to which its systems align
with those theories. After a brief digression on how the intentional stance is
viewed within AI, I examine intention and its relationship to intentional
action in detail to illustrate the challenges of accepting this assumption.
Finally, I discuss how to evaluate AI systems that include artificial
intention and how scientific progress toward a full understanding of intention
can be made.

##  2 Searching for Intentionality in Modern AI

Although AI has been through a radical shift in focus, the move to deep
learning and neural network approaches has not eliminated taking the
intentional stance. We still describe AlphaGo as wanting to win a game and
intentionally placing stones in locations that it believes are most strategic.
That there are no structural analogs to the intentions, beliefs, or desires
within the system is irrelevant. When Lee Se-dol was playing AlphaGo, his move
predictions could not be based on the actual implementation and could only be
driven by his assessment of what a rational opponent, trying to win, would do.
And, when explaining a move by AlphaGo that surprised Lee (because it appeared
irrational to him), there is a natural inclination to use a variety of
intentionality-laden words to describe the process, including ‘discover’ and
‘understand’ [21]. Such language tends to be used loosely and metaphorically,
which serves to mask the inadequacy of the explanation. Readers are left to
infer how the words apply, and whether they are experts or not,
misunderstandings are unavoidable.

Some AI researchers might react to the loose use of the language of
intentional states by resorting to formalization. This is the strategy behind
the development of Belief-Desire-Intention (BDI) frameworks, which provide
definitions of belief, desire, and intention and the sort of guarantees on
rationality that accompany normative theories [26]. These frameworks are used
to guide system design and to let people both predict behavior and explain how
an agent’s desires drive its behavior, at least under the aegis of a normative
rationality. Building a system that adheres to such a theory of how
intelligent systems should, in principle, behave does not imply that such
systems characterize or describe how humans do behave. Moreover, capturing
some of the characteristics of beliefs, desires, and intentions in a logic
does not imply capturing all relevant characteristics that people bring to
bear in their folk-psychological theories. For instance, knowing someone’s
plans facilitates the prediction of what kinds of objects, actions, and events
will be important to them in pursuit of that plan, but there may be no such
relationship in a system built using a BDI framework. When describing BDI
agents, we may use the language of intentionality, but the terms have taken a
meaning related to but not nearly as rich as their use in folk psychology.

How then should we use the language of intentionality to communicate the way
in which an intelligent system operates? The answer is rooted in empirical
methodology. The next section delineates some of the functions and properties
of intentional action. This list serves two roles. First, showing that a
single intentional state is multifaceted acts as an argument that the
referents of terms like ‘intention’ and ‘belief’ are more complex than is
traditionally assumed in computational research. Second, the list provides, at
a high level, target phenomena or capabilities that can be used when testing
and evaluating a system. Importantly, the list is not a comprehensive
accounting of functions and properties. As with attempts to achieve behavior
as complex as autonomous driving, establishing a guide for system design and
testing requires a serious and sustained interdisciplinary effort [29, 35].
The result would be the selection of desirable features, the development of
corresponding test scenarios, and agreed upon levels of functionality. The
last of these would enable people to align their expectations with the
capabilities of an intelligent system.

##  3 A Closer Look at Intentional Action

When researchers talk about intentional states or other psychological
concepts, they typically carve up their characteristics so that they are only
talking about them from a limited perspective. Sometimes researchers take
these limited perspectives to be the core character of the concept within a
specific, circumscribed scope. Consider belief. We might say that someone
would be surprised if they (a) believed that a toaster was plugged in, (b)
followed the normal directions for toasting bread, and (c) observed that their
bread had not been toasted. The expectation based on the toaster’s design has
been violated. This relationship between belief and surprise expresses an
important characteristic of belief but says nothing about other aspects such
as strength of conviction, effects on mood, changes in expectations, or how
the experience warrants other beliefs. Therefore, it would be premature to
declare that a system that connects these thin representations of beliefs,
expectations, and surprise to model this case has beliefs in a folk-
psychological sense. To examine the complexity of intentional states and to
motivate a testing and evaluation methodology, let us take a closer look at
intention.

Within AI, intention and resultant intentional action were initially analyzed
in the context of speech acts [1] and were eventually associated with planning
[4]. This emphasis was due largely to the influence of Bratman [*]Bratman1984
who worked to shift attention from what he called present-directed intention,
which supports intentional action in the moment, toward future-directed
intention, which is a commitment to a later action. To illustrate the
difference, the law is particularly concerned with intentional action and
distinguishes among crimes planned in advance (i.e., premeditated), crimes of
passion whose outcomes were intended only in the moment, and crimes of
negligence where the negative effects were unintended. Although the
relationship between plans and intentions had been discussed since the early
days of cognitive science [22], the connection remained unexploited in AI
until the 1980s and eventually led to the development of BDI frameworks and
agent-oriented programming [27].

The relationship to planning is only one aspect of intention, and I will
examine intention from three different perspectives to appreciate its
richness. In addition to intention’s connection with plans, I will consider
its connection to both action and control.

###  3.1 Intention Through the Lens of Planning

I have already mentioned the high-level intersection between intentions and
plans, and here I highlight three characteristics that Bratman discusses in
detail. The first is that intentions enable planning by storing the results of
deliberation about which actions to take in the future. Without this ability,
there would be no connection between plan creation at time $t_{0}$ and actions
at later time $t_{n}$. This is not to say that intentions always exist as a
fully specified set of actions. Steps in plans can be more or less abstract in
the sense of, “I intend to go to the bakery later,” or they may include
concrete times, locations, and subplans. Intentions can also be incomplete or
open ended such as, “I intend to buy a loaf of rye bread and maybe something
else.” The important characteristic is that they serve as the results of
deliberative activity, indicating that options were considered and a decision
was made.

Remembering the results of deliberation is of little use unless you are
committed to that decision. This is the second characteristic taken from
Bratman and expanded on by Cohen and Levesque [*]Cohen1990a. By committing
oneself to a future course of action, a person stabilizes their activity. As a
consequence, they are better able to make predictions about their future
states. More importantly, when a person tells others about their plans, those
people can make predictions under the supposition that in most cases people
see their intentions through. In this manner, predictability provides
information about reliability and enables others to build trust in someone’s
performance. So, there is social pressure for a person to keep true to their
publicly announced intentions. In addition, one person can use another
person’s plans to inform their own, which can enable coordination and
cooperation across a seemingly infinite diversity of jointly pursued
activities.

Because intentions encode commitments to act according to earlier decisions,
they facilitate a third characteristic, which is the ability to coordinate
with one’s future self. In practice, this means that intentions provide anchor
points for reasoning about the future. If you decide to walk to the bakery,
some plans (e.g., stopping at a neighboring store on the way) are made more
feasible, whereas other plans are ruled out or constrained (e.g., your
purchases are limited to what you can comfortably carry). Moreover, unrelated
plans also take these commitments into account. For instance, intending to buy
bread means that even before going to the bakery, you can plan to have a
sandwich for lunch the next day. On a larger scale, when we consider long-term
goals for our lives, we make plans like attending college, pursuing a major,
and applying at specific companies all in service of that goal and all in
relationship to each other even though those plans may not come to fruition
for years. So, whether at the small scale or the large scale, intentions
provide a means to organize our lives over time. A primary value of intending
is that it biases people toward preserving the results of their earlier
deliberation.

###  3.2 Intention Through the Lens of Action

Whereas Bratman’s philosophical analysis emphasizes the role of intention in
planning, Malle and Knobe [*]Knobe1997 set out to determine how intentions and
intentional actions are related in our folk-psychological intuitions. As a
starting point, Bratman, Malle, and Knobe would all agree that a causal
history connecting an intention with a specific action does not necessitate
that the action was intentional. For instance, suppose that someone playing
their first game of darts intends to hit the bullseye on every throw of a
round. The first dart hits, and the second and third fly wide of their target.
Did the person intentionally hit the bullseye? Speaking strictly in terms of
causal history, they did, but because we know that the person lacks the skill
required to hit their mark regularly, we may attribute the bullseye to
beginner’s luck or call it a fortunate accident. Confirming this intuition,
Malle and Knobe’s experiments found that people see skill, the ability to
reliably follow through on an intention, as an important component of
intentional action.

Additionally, Malle and Knobe’s experiments identified awareness as an
expected characteristic of intentional action. In their model, awareness
refers to an agent knowing that it is fulfilling the intention through its
action. To illustrate, consider the following two scenarios. Suppose that
Amelia decides to give away her old winter coat, puts it in a box, and takes
that box to a donation center. Clearly, she intentionally donated her coat.
Compare that to a case where Amelia’s mom finds the coat, puts it in the box,
and heads to work while Amelia is collecting other items. After searching for
the coat unsuccessfully, Amelia gives up and takes the box to the donation
center. When she returns home, she asks her mom about the coat and discovers
that she had already donated it. Although Amelia intended to donate the coat,
she was not aware that she was doing so at the time. Note that the limited
scope of this definition of awareness means that we do not have to fully
determine what awareness is. Instead, we might only stipulate that an agent
can report which intentions it is carrying out at a particular time.

The awareness condition of intentional action is easy to confuse with other
phenomena, such as when our actions have unintentional consequences. We may
not have known that an outcome was possible or we might have been careless in
our execution. For example, someone might desire to read a book, intend to
reach for it, and knock over a cup of coffee in the process. Reaching for the
book was intentional, but spilling the coffee was not. How can the same action
be both intentional and unintentional? Anscombe [*]anscombe79 accounted for
these situations by saying that actions are intentional under a description.
For instance, the person’s action may have been intentional under the
description reaching-for-the-book but unintentional under the descriptions
knocking-over-the-cup and extending-an-arm. The lack of awareness is not what
makes knocking-over-the-cup unintentional, the action under that description
was never intended in the first place. Of course, if every time a person
reached for a book, they knocked over a cup, we might suspect that they were
using the book as excuse for being a nuisance. Therefore, an agent should be
clear and accurate in describing their intentions so that others can continue
to trust them.

###  3.3 Intention Through the Lens of Control

Although folk psychology usually refers to the relationships among intentional
and other cognitive states, people also have expectations about how these
states are manifested. Pacherie [*]Pacherie2009 and Mylopoulos and Pacherie
[*]Mylopoulos19 present one way of describing this link between cognition and
action that centers on three categories of intention: distal, proximal, and
motor. That model arranges these forms of intention in a cascade with distal
intentions sitting at the top and playing a role in planning (cf. Bratman’s
future-directed intentions). The distal intentions are representationally
abstract and influence the selection of proximal intentions, which are
situated and concrete. This process involves grounding conceptual information
from the distal intention with perceptual information. Proximal intentions, in
turn, parameterize motor intentions, which guide the selection of an
appropriate motor program. As an example, you might form a distal intention to
grab a pen from your desk. The corresponding proximal intention would pick out
a specific pen, and the motor intention would select a motor program for
grabbing it. In addition to providing a connection between cognition and
action, Pacherie’s model also allows for an exchange of information across
representational levels. For instance, a failure to grab a pen could lead to
executing a different motor program, selecting a different pen, or replanning
using a new performance constraint.

Along these lines, if someone is driving and finds that their regular route is
blocked, we expect them to find a new path. If a person is writing software
and realizes that their programming skills are inadequate, we expect them to
learn new material or seek assistance. Likewise, if a robot finds a door it
cannot open, we would expect it to find another way of reaching its goal
whether that involves acquiring a new skill, constructing a tool, seeking
assistance from nearby people, or creating a new plan to move forward. These
abilities imply that intentions contain processes for guiding and monitoring
activities with each category of intention playing a distinctive role [25].
Moreover, we expect that people will avoid plans that are unachievable (i.e.,
with no possibility of guidance) and will either avoid or discount plans where
they cannot effectively monitor progress. In short, intending to do something
without also believing that you can control yourself or others in the
necessary way through guidance and monitoring is irrational.

Guidance and monitoring processes work, in part, by drawing our attention to
information relevant to our current intentions. Watzl identifies attention as
the reshaping of priority structures that arrange occurrent, subject-level,
mental states, and he writes that intentions “constrain how [a subject’s]
priority structures are going to evolve in the future” [*][][p. 143]watzl2017.
Similarly, Wu, who views attention as “selection for action,” writes that “the
subject is attuned during [intentional] action to relevant information such
that it is deployed to inform the subject’s response” [*][][p. 98]Wu2016. In
other words, intentions configure attentional priorities which then control
behavior. From a folk-psychological perspective, we seem to know this when we
talk about distractions that pull us off task and failures that occur while
multitasking. Consider someone driving down the highway who decides to adjust
their radio to a new station. Suppose the driver misses their exit even though
it was clearly marked and their GPS was providing directions. In this case,
visually attending to the radio guided the pressing of buttons or turning of
knobs, aurally attending to the music enabled monitoring for content and
volume, and those attentional priorities led to missing the sign for the exit
and the instructions from the GPS. The driver’s intention to change the radio
station overrode the attentional priorities that would have kept them on
route.

The driver in this example might have avoided missing the exit if they had
used what Gollwitzer [*]Gollwitzer1999 calls implementation intentions as
protection. Implementation intentions are small plans that look like simple
rules. For instance, someone on a diet might say, “When I order dinner at the
restaurant, I will skip dessert.” Even if that person forgets their intention
throughout the day, opening the menu may trigger the memory of the earlier
commitment. Similarly, the driver could have said, “When I am near my exit, I
will make sure to listen to the GPS.” As with plans, these simpler intentions
are commitments we make to ourselves, but instead of giving step-by-step
guidance, they tune our behavior as we engage in some broader activity.
Thinking back to folk psychology, this ability means that we expect that
intentions can have triggering conditions and can be combined in ways that
tune our actions in the moment.

##  4 Progress Toward a Computational Folk-Psychology

Recall that the purpose of examining intention is twofold. First, gaining
familiarity with the depths to which others have analyzed intentional states
reveals the limitations in existing representational approaches. Second,
elaborating on the characteristics of intention found in folk psychology and
summarized in Table 1 provides a starting point for a discussion about
evaluation and progress. To the first point, consider how the elements in
Table 1 map onto a program like AlphaGo. Viewed through the lens of action,
the program appears to have some aspects of intention in that it clearly has
the necessary skills, it can report what action it is taking (here, the action
and the report are identical), and it is clearly only acting under the
description of making moves appropriate to the game of Go. However, through
the lenses of planning and control, there is no obvious mapping. AlphaGo
commits to its current action not a future one, does not store deliberations
that affect future play, and does not coordinate with its future actions.
There appear to be no representations of intentions that separately enables
reasoning and feedback across abstract, situationally grounded, and motor-
program levels. Each move is made in isolation, so there is no time-extended
strategy that requires guidance and monitoring, and there are no triggering
conditions that result in pre-planned or stereotypical behaviors. This is a
brief and informal analysis of one system, but to the second point, how in
general should we determine whether a system aligns with our folk-
psychological intuitions and how do we define progress?

Table 1: Functions and properties of intentions from a folk-psychological perspective Planning |  Store the results of deliberation
---|---
Commit to future action
Coordinate with future self
Action |  Assume skill to carry out intention
Assume current intention can be reported
Require clarity and truth in report
Control |  Incorporate multi-level representations
Include guidance and monitoring processes
Include triggering conditions

Taking the extensive work on autonomous driving as an example, we can begin to
think about how evaluation of intentional states might be standardized. In a
report by the National Highway Traffic Safety Administration [[,
NHTSA;]]ndot2018, there is a list of categories of behavioral competencies for
driving that are subdivided into specific ones. For instance, tactical
maneuvers like parking are broken down into specific behaviors including
navigating a parking lot, locating open spaces, and making appropriate parking
maneuvers. Similarly, for object and event detection and response capabilities
the category of traffic control devices and infrastructure includes following
local driving laws and reacting to speed limit changes. These competencies are
broad in scope because they are meant to guide the development of test
scenarios. (Examples of these scenarios are too detailed to include here but
are available in Appendix B of NHTSA’s report.) In the case of intentions,
Table 1 and future extensions to it are meant to serve the same purpose. One
evaluation scenario could combine committing to future actions with
assumptions that the tested system has the skill to execute those actions and
can monitor their performance for success. The test scenario will need to
state how these capabilities connect with the exact tasks or behaviors that
the system is meant to carry out.

On a pessimistic side, autonomous driving research has also shown that the
more we know about how to implement driving, the more complicated and detailed
the activity appears to be. There are seemingly endless corner cases, whether
they involve challenges to computer vision such as treating the moon or
restaurant signs as traffic lights [18] or control failures such as attempting
to turn into oncoming vehicles or light-rail lines [15]. Many of the problems
that arise are not about driving per se but are instead tied to the broader
set of challenges related to acting in an open-ended environment that is
suffused with dynamicity. Driving in the world, it turns out, is not like
playing shogi, go, or chess. In a different sense, implementing intentional
states is not like building an autonomous car. Parking is not believing. For
instance, we know how to determine if a car is properly parked within a space,
but how do we determine whether a belief is properly held by an agent? If the
functional roles of intentional states differ situationally, then the
challenges are even greater. We know little about what “implementing
intention” might require compared to addressing task-centered domains like
driving. That is not to say that intention, belief, desire, and hope are as
nebulous as intelligence. But, to the extent that intentional states are
moving targets forever on the horizon, it may be necessary to rethink what a
progressive research program looks like.

My suggestion would be to adopt the apophatic methodology originally developed
for a science of consciousness [5]. As a topic of scientific research,
consciousness is resistant to objective measurement. Consequently, scientists
develop proxy measures to target in their evaluations. For instance, one might
say that a robot is conscious because it passes a mirror test [32] similarly
to how one might pronounce that a program is intelligent because it wins chess
games [12]. These kinds of claims beg the question of consciousness or
intelligence (i.e., intelligence just is the ability to play a board game),
and people are right to respond with incredulity and doubt. Apophatic science
offers an alternative route for progress.

The core of this methodology should be familiar as it draws from Newell’s
[*]newell73 prescriptions for progress in cognitive science. The central idea
applied to AI is to implement a series of computational models such that each
one accounts for more intelligence-relevant phenomena than its predecessor.
Additionally, the fidelity to mechanisms that enable human intelligence,
according to the researchers developing the models, should also increase.
Comparatively, the validation step of the apophatic methodology is radical.
Once a model meets its original design criteria, it is recognized as
inadequate—the criteria are insufficient for capturing intelligence
computationally. The next step is to ask, “What’s missing?” Given an answer,
the set of criteria are extended, and the expanding list of properties and
functions collects the not-quite-it of intelligence as the potentially
unending process, summarized in Table 2, continues.

Table 2: The apophatic methodology in a nutshell.

Given a phenomenon $P$ and an initial set of functional roles $X$
loop $(X)$
$P=X$
Implement $P$ in model $M$
Verify that $M$ is a correct implementation of $P$
Reject $M$ as insufficient
Identify functional role $a$
recur $(X\cup a)$

Even if intentional states turn out to be easier to objectively define than
intelligence or consciousness, the discussion in Section 3 suggests that they
are not circumscribed as what we find in current implementations (e.g., as
plans or policies). The good news is that, from a practical standpoint, only a
subset of the known functional roles and properties may be required by any
individual cognitive system. Nevertheless, by methodically exploring
intentions, beliefs, and other intentional states we increase our ability to
identify which properties and functional roles will satisfy the needs of any
distinct task. To illustrate this point, the success of AlphaGo against human
challengers suggests that including other functional roles of intention is
unnecessary to meet the system’s purpose. If the program were a
multifunctional game player [17] that was required to justify its moves by
appeal to deliberation (“Why did you place that piece?”), then more work would
be required.

For the apophatic method to operate effectively, it is important to embrace
stakeholder involvement. By this, I mean not only that we should collaborate
with researchers in other disciplines who have important contributions to make
regarding intentionality but also that we should engage with the communities
that would benefit from the systems we build. This level of interaction has
taken place in the realm of autonomous vehicles, with people providing
information on successful drives and identifying challenging corner cases. We
have also seen this advice being given by groups working on DARPA’s XAI
program [16]. People approach software with different needs and goals in mind
and a one-size-fits-all explanation is not going to be sufficient in all
scenarios. As Hoffman and colleagues note, “By hearing first-hand from the
different stakeholders about what they need in terms of explanations,
developers will be better able to help stakeholders develop good mental models
of a system” [*][][p. 3]ihmc2021. When it comes to intentions, a planning
system that is not connected with execution may not need to commit to a future
action, but a user would certainly expect it to store the results of its
deliberation. Compare that case with a joint planning and execution system
where users would expect it not only to store its plan but also to treat those
results as commitments to the planned actions.

As may be obvious by now, the questions “What is intention and how do we model
it?” are too large to answer in this paper. That task falls to the research
community in collaboration with other stakeholders. Undoubtedly multiple
working groups will be needed, each focusing on a facet of folk psychology
(e.g., intention, knowledge, belief, and desire) and meeting to ensure
agreement and coherency across high-level concepts. Overall, the goal should
be to reduce ambiguity when describing intentionality. By this, I mean that
whenever researchers claim that their systems have beliefs or intentions, they
should be able to point to specific properties and functional roles, tie those
to the tasks that they are addressing with their systems, and explain why
these are sufficient.

Figure 1: A recent post on Twitter by Gary Marcus [*]marcus22, a New York
University Professor Emeritus and founder of two AI companies. The responses
he received were varied and although anyone was free to reply, the contents
reflected those in the AI community. (Note: ’immerses’ is a typo for
’immense’.)

There is often considerable handwringing about how progress in AI is measured
and why the field fails to meet some long-established goals. Figure 1 provides
only one of the more recent and public examples of this conversation. To
address that question for an entire discipline covering almost 70 years of
effort is too much to ask. As an alternative, I have attempted to provide a
response for one aspect of AI that is especially relevant to cognitive
systems. My hope is that instead of contributing to some debate regarding who
or what is at fault, I have illustrated the scale of the challenges we face,
pointed to recent advances in evaluative approaches, and indicated a method
for measuring scientific progress in AI.

Acknowledgements This paper emerged from conversations with Paul Bello and has
benefitted from the suggestions of Andrew Lovett. The author’s effort was
supported by the Office of Naval Research through the Science of Artificial
Intelligence program award number N0001422WX00033. Distribution Statement A.
Approved for public release; distribution is unlimited.

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