assertionAssertion conjectureConjecture definitionDefinition
hypothesisHypothesis remarkRemark noteNote observationObservation
problemProblem questionQuestion algorithmAlgorithm exampleExample
notationNotation In memory of Dick Askey, 4th June 1933 – 9th October 2019
42C05 (primary), 33C45 (secondary) This research was conducted during the
second author’s PhD, funded by the EPSRC Centre for Doctoral Training in the
Mathematics of Planet Earth [EP/L016613/1].

# Integral Representations of Ultraspherical Polynomials II

N. H. Bingham and Tasmin L. Symons<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In the first part, by the first author’s work of 1972, an integral
representation for an ultraspherical polynomial of higher index in terms of
one of lower index and an infinite series was obtained. While this
representation works well from a theoretical point of view, it is not
numerically satisfactory as it involves polynomials of high degree, which are
numerically unstable. Here we sum this series to obtain an integral, which is
numerically tractable.

##  1 Introduction

As in [7], we write $W_{n}^{\lambda}(x)$ for the ultraspherical (or
Gegenbauer) polynomial of degree $n$ and index $\lambda>0$ (see e.g. [1, p.
302], [21, §4.4]), normalised so that $W_{n}^{\lambda}(1)=1$. This choice of
normalisation, due to Bochner [15], is convenient probabilistically; the first
part [7] was probabilistically motivated [8]; so too is this sequel (see §5).
The $W_{n}^{\nu}$ are orthogonal polynomials on the interval $[-1,1]$ with
respect to the probability measure $G_{\nu}$, where

$G_{\nu}(dy)=\frac{\Gamma(\nu+1)}{\sqrt{\pi}\Gamma(\nu+1/2)}(1-y^{2})^{\nu-1/2}dy:$
(1)
$\int_{-1}^{1}W_{m}^{\nu}(x)W_{n}^{\nu}(x)G_{\nu}(dx)={\delta}_{mn}{\omega}^{\nu}_{n},$

where

$\omega^{\nu}_{m}=\frac{n+\nu}{\nu}\frac{\Gamma(n+2\nu)}{n!\Gamma(2\nu)}.$ (2)

######  Theorem 1.1 ([7])

If $0<\nu<\lambda$, $x\in[-1,1]$, there exists a probability measure
$M_{\nu}^{\lambda}(x;dy)$ on $[-1,1]$ such that

$W^{\lambda}_{n}(x)=\int_{-1}^{1}W^{\nu}_{n}(y)\,M^{\lambda}_{\nu}(x;dy).$ (3)

Moreover, when $\lambda\neq\nu$ the measure $M^{\lambda}_{\nu}$ is absolutely
continuous with density

$M^{\lambda}_{\nu}(x;dy)=G_{\nu}(dy)\sum_{m=0}^{\infty}\omega^{\nu}_{m}W^{\lambda}_{m}(x)W^{\nu}_{m}(y).$
(4)

The series in 4 is transparent from a theoretical point of view (it is derived
in [7] from the earlier work of Askey and Fitch [3] by an Abel-limit
operation), but unsuitable for numerical use as it involves polynomials of
high degree, which oscillate wildly. Our purpose here is to circumvent this by
giving an explicit formula for the sum of the infinite series as a double
integral, which is numerically tractable. Our result, Theorem 3.1 below, is
interesting in its own right, completing the integral representations in [7]
by showing the dependence on the higher index, $\lambda$, in a more convenient
and structurally revealing way.

##  2 Preliminaries

The Poisson kernel for the Jacobi polynomials reduces in the ultraspherical
case to the generating function

$\sum_{n=0}^{\infty}{\omega}_{n}^{\nu}r^{n}W_{n}^{\nu}(x)=(1-r^{2})/(1-2rx+x^{2})^{\nu+1},\quad
r\in(-1,1),$ (5)

cf. [7, (2.1)]. Note that this is not the usual generating function for the
ultraspherical polynomials [21, §4.7.23].

Askey and Fitch [3] showed that for $x,y\in[-1,1]$, $r\in(-1,1)$,
$0\leq\nu<\lambda\leq\infty$, the series

$\sum_{n=0}^{\infty}{\omega}_{n}^{\nu}r^{n}W_{n}^{\lambda}(x)W_{n}^{\nu}(y)$
(6)

converges to a non-negative sum-function, which leads to a corresponding
probability measure $M^{\lambda}_{\nu}(x)$ satisfying

$W_{n}^{\lambda}(x)=\int_{-1}^{1}W_{n}^{\nu}(y)M^{\lambda}_{\nu}(x;dy),\quad
n=0,1,2,\ldots.$ (7)

Here (see [7]) we may take $0\leq\nu\leq\lambda\leq\infty$, $x\in[-1,1]$. Some
cases give Dirac laws: if $x=\pm 1$, $M^{\lambda}_{\nu}(\pm 1)=\delta_{\pm 1}$
(as $W_{n}^{\lambda}(\pm 1)=(\pm 1)^{n}$). If $\lambda=\nu$, then
$M^{\lambda}_{\lambda}(x)={\delta}_{x}$ (as there is no projection to be
done); so we may restrict to $\nu<\lambda$ as before. Now [7, Lemma 1] gives
the Abel-limit operation explicitly: for $x,y\in(-1,1)$, we may take $r=1$
here to get

$m^{\lambda}_{\nu}(x;y):=\sum_{n=0}^{\infty}{\omega}_{n}^{\nu}W_{n}^{\lambda}(x)W_{n}^{\nu}(y)\geq
0,$ (8)

a non-negative function in $L_{1}(G_{\nu})$, finite-valued unless $x=y$ and
$\nu<\lambda\leq\nu+1$. It is in fact the Radon-Nikodym derivative
$dM^{\lambda}_{\nu}(x;dy)/dG_{\nu}(dy)$:

$M^{\lambda}_{\nu}(x;dy)=G_{\nu}(dy)\cdot
m^{\lambda}_{\nu}(x;y)=G_{\nu}(dx)\cdot\sum_{n=0}^{\infty}{\omega}_{n}^{\nu}W_{n}^{\lambda}(x)W_{n}^{\nu}(y).$
(9)

Following [7], for $\lambda>\nu$ write $H_{\nu}^{\lambda}$ for the probability
measure of Beta type on $[0,1]$ given by the Sonine law

$H_{\nu}^{\lambda}(dx):=\frac{2\Gamma(\lambda+\frac{1}{2})}{\Gamma(\nu+\frac{1}{2})\Gamma(\lambda-\nu)}\cdot
x^{2\nu}(1-x^{2})^{\lambda-\nu-\frac{1}{2}}dx.$ (10)

This occurs in Sonine’s first finite integral for the Bessel function [22, p.
373]: for

$\displaystyle\Lambda_{\mu}(t)$
$\displaystyle:=\Gamma(\nu+1)J_{\nu}(t)(t/2)^{-\mu},$ (11)
$\displaystyle\Lambda_{\lambda-\frac{1}{2}}(t)$
$\displaystyle=\int_{0}^{1}\Lambda_{\nu-\frac{1}{2}}(ut)H_{\nu}^{\lambda}(du)$
(12)

(the drop by a half-integer in parameter here reflects the drop in dimension
in ${\mathbb{S}}^{d}\subset{\mathbb{R}}^{d+1}$; see §4 below).

For the product of $W_{n}$ terms in (8), we need Gegenbauer’s multiplication
theorem for the ultraspherical polynomials [22, p. 369],

$W_{n}^{\nu}(x)W_{n}^{\nu}(y)=\int_{-1}^{1}W_{n}^{\nu}(xy+\sigma\sqrt{1-x^{2}}\sqrt{1-y^{2}})G_{\nu-\frac{1}{2}}(d\sigma).$
(13)

To cope with the drop in index (dimension) in (8), we need the Feldheim-
Vilenkin integral [7, (2.11)], [1, p.315], [3],

$\displaystyle W_{n}^{\lambda}(x)=$
$\displaystyle\left[\frac{2\Gamma(\lambda+\frac{1}{2})}{\Gamma(\nu+\frac{1}{2})\Gamma(\lambda-\nu)}\right]\int_{0}^{1}u^{2\nu}(1-u^{2})^{\lambda-\nu-1}$
$\displaystyle\cdot[x^{2}-x^{2}u^{2}+u^{2}]^{\frac{1}{2}n}W_{n}^{\nu}\left(\frac{x}{\sqrt{x^{2}-x^{2}u^{2}+u^{2}}}\right)du.$
(14)

##  3 The result

We can now formulate our result.

###### Theorem 3.1

For $r\in(-1,1)$, the sum of the Askey-Fitch series (8) above is given by the
integral (15) below:

$\int_{0}^{1}H_{\nu}^{\lambda}(du)\int_{-1}^{1}G_{\nu-\frac{1}{2}}(dv)\,\frac{\left[1-r^{2}(x^{2}-x^{2}u^{2}+u^{2})\right]}{I^{\nu+1}},$
(15)

where $I$ is given by

$I:=1-2r\cdot\frac{xy+uv\sqrt{1-x^{2}}\sqrt{1-y^{2}}}{\sqrt{x^{2}-x^{2}u^{2}+u^{2}}}+\frac{(xy+uv\sqrt{1-x^{2}}\sqrt{1-y^{2}})^{2}}{(x^{2}-x^{2}u^{2}+u^{2})}.$
(16)

Moreover, this holds also for $r=1$ unless $\nu<\lambda\leq\nu+1$.

######  Proof 3.2.

We sum the series by reducing it to the generating function (5). There are two
steps: reduction of $\lambda$ to $\nu$ by the Feldheim-Vilenkin integral (14)
and reduction of two $W_{n}$ terms to one by Gegenbauer’s multiplication
theorem (13).

We follow [7]. As there, we may substitute for $W_{n}^{\lambda}(x)$ from (14)
into the series (8) and integrate termwise, rewriting (8) as

$\int_{0}^{1}H_{\nu}^{\lambda}(du)\sum_{n=0}^{\infty}{\omega}_{n}^{\nu}(r[x^{2}-x^{2}u^{2}+u^{2}]^{\frac{1}{2}})^{n}\cdot
W_{n}^{\nu}(y)W_{n}^{\nu}\left(\frac{x}{\sqrt{x^{2}-x^{2}u^{2}+u^{2}}}\right).$
(17)

We use Gegenbauer’s multiplication theorem (13) with

$r\mapsto r\sqrt{x^{2}-x^{2}u^{2}+u^{2}},$

and replace the product of $W_{n}^{\nu}$ factors in the above, at the cost of
another integration over $G_{\nu-\frac{1}{2}}(dv)$, by a single $W_{n}^{\nu}$
term, with argument

$\frac{xy}{\sqrt{x^{2}-x^{2}u^{2}+y^{2}}}+v\sqrt{1-y^{2}}.\sqrt{1-\frac{x^{2}}{x^{2}-x^{2}u^{2}+u^{2}}}=\frac{xy+uv\sqrt{1-x^{2}}\sqrt{1-y^{2}}}{\sqrt{x^{2}-x^{2}u^{2}+u^{2}}}.$
(18)

The integrand is now of the form
$\sum\omega^{\nu}_{n}r^{n}W^{\nu}_{n}(\cdot)$, and the result now follows from
the generating function (5).

This result completes and complements the work in [3] and [7] by displaying
the dependence on the higher index $\lambda$ in a structurally revealing way:
for simplicity, let $r=1$ so that

$I=\left(1-\frac{xy+uv\sqrt{1-x^{2}}\sqrt{1-y^{2}}}{\sqrt{x^{2}-x^{2}u^{2}+u^{2}}}\right)^{2},$
(19)

and (15) is given by

$\int_{0}^{1}H_{\nu}^{\lambda}(du)\int_{-1}^{1}G_{\nu-\frac{1}{2}}(dv)\frac{1-(x^{2}-x^{2}u^{2}+u^{2})}{I^{\nu+1}}.$
(20)

Using the definition of $H^{\lambda}_{\nu}$ and the probability measure
$G_{\nu+\frac{1}{2}}$ and simplifying, (15) becomes

$\displaystyle\frac{2}{\sqrt{\pi}}\int_{0}^{1}\frac{\Gamma(\lambda+\frac{1}{2})}{\Gamma(\lambda-\nu)}u^{2\nu}(1-u^{2})^{\lambda-\nu-\frac{1}{2}}\int_{-1}^{1}(1-v^{2})^{\nu-\frac{1}{2}}\left[\frac{1-(x^{2}-x^{2}u^{2}+u^{2})}{I^{2(\nu+1)}}\right]\,dvdu.$
(21)

Note that the higher index $\lambda$ occurs only in the outer integral.
Moreover, the interactions between the indexes in the outer integral occurs
only in the Gamma function $\Gamma(\lambda-\nu)$ and the power
$\lambda-\nu-1/2$ of $(1-u^{2})$.

Figure 1: Numerical evaluation of series (15) for $\lambda=3.0$. $\nu=0.5$.

##  4 Dimension walks

Write ${\cal P}_{\nu}$ for the class of functions $f$ on $[-1,1]$ which are
mixtures of $W_{n}^{\nu}$, i.e., of the form

$f(x)=\sum_{n=0}^{\infty}a_{n}W_{n}^{\nu}(x),\quad\sum a_{n}=1,a_{n}\geq 0$
(22)

for some probability law $a=\\{a_{n}\\}_{0}^{\infty}$ (the ultraspherical
series converges uniformly as $|W_{n}^{\nu}(x)|\leq 1$). The classes ${\cal
P}_{\nu}$ are decreasing in $\nu\in[0,\infty]$, and are continuous in $\nu$,
in that

$\bigcap\\{{\cal P}_{\mu}:0\leq\mu<\nu\\}={\cal P}_{\nu},\qquad\bigcup\\{{\cal
P}_{\mu}:\nu<\mu\leq\infty\\}={\cal P}_{\nu}$

([9, Th. 1]). While the parameters $\lambda$, $\nu>0$ are continuous here, the
half-integer values are particularly important. With ${\mathbb{S}}^{d}$ the
$d$-sphere – the unit sphere in Euclidean $(d+1)$-space ${\mathbb{R}}^{d+1}$,
a $d$-dimensional Riemannian manifold – the relevant index for the
ultraspherical polynomial is $\nu$, where

$\nu=\frac{1}{2}(d-1).$

With $\nu<\lambda$ as above, the higher dimension corresponding to $\lambda$
will be written $d^{\prime}$ (so $\lambda=\frac{1}{2}(d^{\prime}-1)$). Then,
as in [8], [7], the passage from $\lambda$ to $\nu<\lambda$ corresponds to
projection from the $d^{\prime}$-sphere to the $d$-sphere. The limiting case
$\nu=\infty$ gives $W_{n}^{\infty}(x)=x^{n}$, and ${\cal P}_{\infty}$ is the
class of probability generating functions, or the class of positive definite
functions on the unit sphere in Hilbert space ([9, Lemma 2], [20]).

Covariance functions on spheres are very valuable in applications to Planet
Earth (see §5). Operations which preserve positive-definiteness are useful in
the construction of new families of such covariance functions. Two such
operations, coined ‘walks on dimensions’, one changing the dimension by 1, and
the other by 2, were proposed for positive-definite functions on spheres by
Beatson and zu Castell [5], [6]. The one-step walks in [5] are based on the
Riemann-Liouville operators, but lack the highly desirable semi-group
property, in which passage from $\lambda$ to $\nu$ and then $\nu$ to $\mu$ is
the same as passage from $\lambda$ to $\mu$ directly.

######  Theorem 4.1.

For $f\in{\cal P}_{\nu}$ as in (22),

$f(x)=\sum_{n=0}^{\infty}a_{n}W_{n}^{\lambda}(x)\in{\cal P}_{\lambda}.$ (23)

######  Proof 4.2.

$\displaystyle\int_{-1}^{1}f(y)M^{\lambda}_{\nu}(x;dy)$
$\displaystyle=\int_{-1}^{1}\sum_{n=0}^{\infty}a_{n}W_{n}^{\nu}(y)M^{\lambda}_{\nu}(x;dy)$
(24)
$\displaystyle=\sum_{n=0}^{\infty}a_{n}\int_{-1}^{1}W_{n}^{\nu}(y)M^{\lambda}_{\mu}(x;dy)$
(25)
$\displaystyle=\sum_{n=0}^{\infty}a_{n}W^{\lambda}_{n}(x)\in\mathcal{P}(\mathbb{S}^{d^{\prime}}),$
(26)

interchange of the sum and integral in 25 being justified by the uniform
convergence of the Schoenberg expansion.

######  Corollary 4.3.

The operation of passing from $f(x)\in{\cal P}_{\nu}$ to
$\int_{-1}^{1}f(y)M_{\nu}^{\lambda}(x,dy)\in{\cal P}_{\lambda}$ in the theorem
has the semigroup property.

######  Proof 4.4.

The mixture coefficients $a_{n}$ are unchanged by this operation, and so
remain unchanged under further operations of the same type.

##  5 Complements

###  5.1 Hypergroups and symmetric spaces.

Hypergroups are ‘locally compact spaces with a group-like structure on which
the bounded measures convolve in a similar way to that on a locally compact
group’, to quote from the standard work on this important subject, [14, p.1].
The probabilistic setting of random walks on spheres [8, p.196-197] that
inspired both [7] and this paper, its sequel, is in hypergroup language that
of the Bingham (or Bingham-Gegenbauer) hypergroup. This in turn was inspired
by Kingman’s work on random walks with spherical symmetry [18], which gives
the Kingman (or Kingman-Bessel) hypergroup. The theory for spheres and for
spherical symmetry give the prototypical examples of symmetric spaces of rank
one of compact type (constant positive curvature) and of Euclidean type (zero
curvature); these are complemented by the case of constant negative curvature,
the hyperbolic or Zeuner hypergroups [24]. For background on symmetric spaces
we refer to Helgason [17], for spaces of constant curvature to Wolf [23], and
for compact symmetric spaces to Askey and Bingham [2].

We note that the Kingman situation (Euclidean space with spherical symmetry)
may be recovered from the spherical one here by letting the radius of the
sphere tend to infinity. The Bessel functions in the Kingman theory arise from
radialisation of the Fourier transform in Euclidean space under spherical
symmetry [16, II.7].

###  5.2 Gaussian processes, path properties, Tauberian theorems.

The positive definite functions in the classes ${\cal P_{\nu}}$ of §4 serve as
covariances of Gaussian processes parametrised by spheres. Their distributions
are determined by the sequence $a=\\{a_{n}\\}$ (the angular power spectrum) of
the Schoenberg expansion coefficients above. In particular, the rate of decay
of the $a_{n}$ governs the path properties: the faster the decay, the smoother
the paths. For details, see [13]. Crucial here is Malyarenko’s theorem [19,
Ch. 4]. This rests on a Tauberian theorem of the first author [10], which in
turn derives from work of Askey and Wainger [4]. Here it is necessary to move
from the one-parameter family of ultraspherical polynomials $W_{n}^{\nu}$ to
the two-parameter family of Jacobi polynomials $J_{n}^{\alpha,\beta}$
containing it ([1, Ch. 6], [21, Ch. IV]).

###  5.3 Sphere cross line.

The motivation for much of the interest in positive definite functions on
spheres derives from its applications in geostatistics. Here one has both
spatial dependence and temporal evolution, and so one is dealing with
geotemporal processes. For background here, see e.g. [11], [12].

## Postscript

To close, the first author takes pleasure in noting the half-century between
Part I [7] (which derives from his own PhD of 1969) and the present Part II
(which derives from the second author’s PhD of 2020). We both take pleasure in
dedicating the paper to the memory of Dick Askey, whose influence pervades it.
Dick was a famous expert on special functions, but was interested in their
applications, including those to probability. When [2] was written, he used to
dine out by saying, with tongue in cheek, “I’ve just written a paper with
Bingham on Gaussian processes – whatever they are.”

## References

  * [1] G. E. Andrews, R. Askey, and R. Roy.  Special Functions.  Cambridge University Press, 1999.
  * [2] R. Askey and N. H. Bingham.  Gaussian processes on compact symmetric spaces.  Z. Wahrschein., 37:127 – 143, 1976.
  * [3] R. Askey and J. Fitch.  Integral representations for Jacobi polynomials, and some applications.  J. Math. Anal. Appl., 25:411–437, 1969.
  * [4] R. Askey and S. Wainger.  On the behaviour of special classes of ultraspherical polynomials i & ii.  J. Anal. Math., 15:193 – 244 and 245 – 262, 1965.
  * [5] R. K. Beatson and W. Zu Castell.  One-step recurrences for stationary random fields on the sphere.  SIGMA, 12:19p, 2016.
  * [6] R. K. Beatson and W. Zu Castell.  Dimension hopping and families of strictly positive definite zonal basis functions on spheres.  J. Approximation Theory, 221:22 – 37, 2017.
  * [7] N. H. Bingham.  Integral representations for ultraspherical polynomials.  J. London Math. Soc., 6:1 – 11, 1972.
  * [8] N. H. Bingham.  Random walks on spheres.  Z. Wahrscheinlichkeitstheorie verw. Geb., 22:169 – 192, 1972.
  * [9] N. H. Bingham.  Positive definite functions on spheres.  Proc. Cambridge Phil. Soc., 73:145–156, 1973.
  * [10] N. H. Bingham.  Tauberian theorems for Jacobi series.  Proc. London Math. Soc., 36:285 – 309, 1978.
  * [11] N. H. Bingham, Aleksandar Mijatović, and Tasmin L. Symons.  Brownian manifolds, negative type and geo-temporal covariances.  Communications on Stochastic Analysis, 10:421 – 432, 2016.
  * [12] N. H. Bingham and Tasmin L. Symons.  Dimension walks on the sphere cross line.  Statist. Probab. Lett., 147:12 – 17, 2019.
  * [13] N. H. Bingham and Tasmin L. Symons.  Gaussian random fields on the sphere and sphere cross line.  Stoc. Proc. and Appl., 2019.  https://doi.org/10.1016/j.spa.2019.08.007. (To appear in Larry Shepp memorial issue.)
  * [14] W. R. Bloom and H. Heyer.  Harmonic analysis of probability measures on hypergroups, volume 20 of De Gruyter Studies in Math.  Walter de Gruyter, 1994.
  * [15] S. Bochner.  Positive zonal functions on spheres.  Proc. nat. Acad. Sci. (USA), 40:1141–1147, 1954.
  * [16] S. Bochner and K. Chandrasekharan.  Fourier transforms.  Annals of Mathematics Studies 19. Princeton University Press, 1949.
  * [17] S. Helgason.  Differential geometry, Lie groups and symmetric spaces.  Academic Press, 1978.
  * [18] J. F. C. Kingman.  Random walks with spherical symmetry.  Acta. Math., 109:11 – 53, 1963.
  * [19] A. A. Malyarenko.  Invariant random fields on spaces with a group action.  Springer, 2013.
  * [20] I. J. Schoenberg.  Positive definite functions on spheres.  Duke Math. J., 9:96 – 108, 1942.
  * [21] G. Szegő.  Orthogonal polynomials.  American Mathematical Society, Colloquium Publiations Volume XXIII, 1939\.
  * [22] G. N. Watson.  A treatise on the theory of Bessel functions.  Cambridge University Press, 2nd edition, 1962.
  * [23] J. A. Wolf.  Spaces of constant curvature.  Amer. Math. Soc., 6th edition, 2011.
  * [24] H. Zeuner.  On hyperbolic hypergroups.  In Probability measures on groups, VIII (Oberwolfach, 1985), Lecture Notes in Math., pages 216–224. Springer, 1986.

N. H. Bingham
Department of Mathematics
Imperial College London
South Kensington Campus
London, SW7 1AZ
UK Tasmin L. Symons
Telethon Kids Institute
15 Hospital Avenue
Perth, WA 6009
Australia


# Recognition of Mental Adjectives in An Efficient and Automatic Style

Fei Yang
MathAI Lab
<EMAIL_ADDRESS>
###### Abstract

In recent years, commonsense reasoning has received more and more attention
from academic community. We propose a new lexical inference task, _Mental_ and
_Physical_ Classification (MPC), to handle commonsense reasoning in a
reasoning graph. _Mental_ words relate to mental activities, which fall into
six categories: Emotion, Need, Perceiving, Reasoning, Planning and
Personality. _Physical_ words describe physical attributes of an object, like
color, hardness, speed and malleability. A BERT model is fine-tuned for this
task and active learning algorithm is adopted in the training framework to
reduce the required annotation resources. The model using ENTROPY strategy
achieves satisfactory accuracy and requires only about 300 labeled words. We
also compare our result with SentiWordNet to check the difference between MPC
and subjectivity classification task in sentiment analysis.

##  1 Introduction

In the field of artificial intelligence, commonsense reasoning refers to the
capacity that a machine understands the nature of scenes commonly encountered
by humans every day, and makes reasonable and appropriate reactions, mimicking
human cognitive abilities. Through commonsense reasoning, humans are capable
of intricate reasoning relating to fundamental domains including time, space,
naive physics, and naive psychology [6]. Therefore, a good starting point is
understanding how time, space, naive physics affect human’s mind, exploring
possible causal relationships. For example, let’s consider a review "This
saltwater taffy had great flavors and was very soft and chewy. I loved it and
I would highly recommend this candy!". The concepts "great flavors", "soft",
"chewy" describe physical attributes of the saltwater taffy and the concepts
"love", "recommend" describe mental activities of the reviewer. Here concept
refers to word or phrase in natural language. If a seven years old child reads
this review, the child would understand that the mental activities are caused
by the taffy’s physical attributes. Figure 1 shows a possible reasoning graph
existed in the child’s mind. The words "great flavors", "soft", "chewy"
indicate that this taffy is edible with a positive effect. This effect greatly
satisfies the reviewer’s need of food and then this strong satisfaction
invokes the reviewer’s emotion of love with an reaction "I love it". That
strong satisfaction also invokes the reviewer’s need of friendship positively,
with an reaction that the reviewer would like to share this taffy with
friends.

Figure 1: A reasoning graph between a physical event and mental reactions.
_Edible::Positive_ means positive effect over a physical attribute _Edible_.
In mentality part, _Need$Food::Positive_ , _Need$Friendship::Positive_ mean
positive effect over _Food_ and _Friendship_ respectively, which belong to
_Need_ category [17]. _Need$Food::Positive_ invokes _Love_ belonging to
_Emotion_ category [20]. Other tags not invoked are omitted.

To let a machine figure out a similar reasoning graph, the first step is
recognizing which concept is _Physical_ and which one is _Mental_. Then all
concepts are mapped to numerous more granular tags, like _Edible::Positive_ or
_Need$Friendship::Positive_ shown in Figure 1. Last, all tags are linked
together to form a powerful reasoning graph. The first step cannot be skipped
because the coarsest reasoning path, _Physical_ -> _Mental_ , provides causal
concept pairs, facilitating design of more fine-grained tags. Under this
research plan, we propose a task of _Mental_ and _Physical_ Classification
(MPC) at lexical level in this work. Each adjective extracted from Amazon Fine
Food Reviews dataset [18] is inferred with a binary tag, _Mental_ or
_Physical_ , by a fine-tuned BERT model. A _Mental_ adjective describes mental
activities, like emotion, need, reasoning, while a _Physical_ one shows
physical attributes of an object, like color, hardness, speed and
malleability. Although our inference methods have been for adjectives, they
can be directly applied to other word classes. The inferred tags of MPC only
reveals that an adjective is more likely to express a mental view or a
physical view, as a word might have different senses.

Besides MPC, dozens of binary or multi-value tags, like _Emotion_ or _Need_
category will be developed in the follow-up research work. Moreover, in order
to improve the reasoning performance, these tags might need to be updated or
new tags join the reasoning graph. This continuous and rapid iterative process
makes it impossible to annotate all the words at once. In fact, what this
project really needs is the ability to tag all the words automatically relying
on zero or very low annotation resources. Therefore, we consider active
learning methods to train a BERT [7] model for MPC in this work. ENTROPY [13],
CORESET [25], CAL [16] and Random strategies are implemented and evaluated.
The experiment results indicate that ENTROPY outperforms others and achieves
_Mental_ F1 0.72 and _Physical_ F1 0.87 on testset, with only around 300 words
are annotated for training.

The definition of MPC task bears some similarity to subjectivity
classification which is one task of sentiment analysis [14], and classify
whether a piece of text is objective or subjective. To investigate the
difference between these tasks, our result by ENTROPY is compared with
SentiWordNet [24; 2]. We find that 41.5% of the _Mental_ adjectives bear
objective meanings, which indicates the notion of MPC is quite different from
subjectivity classification. Adjective examples are listed out to illustrate
this difference in Table 4.

The main contributions of this paper include the following three points: (1) a
new task MPC is proposed to handle commonsense reasoning, (2) active learning
is introduced to solve MPC efficiently, relying on only a small size of
annotated words, (3) a dataset with the inferred MPC tags is released publicly
for future research.

##  2 Related Work

Commonsense Reasoning.  Reasoning between mentality and physics has been
studied by the research community in recent years. The mental reason of
affective events is explained based on seven common human needs [8].
Event2Mind studies two kind of mental state, intent and emotion, which are
inferred by deep learning models given physical events described by short
text-free phrases [22]. ATOMIC considers two more kind of mental state,
planning and personality, under the same task setting of Event2Mind [23].
Reasoning between physical events are studied by [36] and [33]. Previous works
provide no clear explanation about "how" and "why" in commonsense reasoning,
which is the core question that our research works try to address.

Sentiment Analysis.  Subjectivity classification and sentiment classification
are two sub-topics of sentiment analysis [14]. Subjectivity classification is
to determine whether a content is objective or subjective. On the other hand,
sentiment classification is utilized for subjective content to identify the
sentiment polarity, that is, whether the author expresses a positive or
negative opinion. One approach to sentiment analysis is using lexicons where
each word is assigned with scores showing it is neutral, positive or negative
[32; 10; 11; 31]. These scores are known as prior polarity, that is,
irrespective of the context, whether the word convey a positive or negative or
neutral connotation [35]. One popular lexical resource is SentiWordNet [24; 2]
which associates polarity scores to each synset of WordNet [19]. Early
researches in this domain focus on adjectives, as adjectives express the
majority of subjective meaning in a piece of writing [11; 31]. Under the same
consideration, we also focus on adjectives for MPC first in this work.

Active Learning.  When machine learning or deep learning algorithms are
considered to solve NLP tasks, one of most common challenges is lack of
labeled data and limited annotation resources due to project budget. To
efficiently make use of annotation resources, only the most valuable samples
are hoped to be selected out for human labeling. Active learning provides a
set of algorithms to fulfill this goal [26]. ENTROPY is an uncertainty-based
method, choosing the sample with the highest predicted entropy [13]. However,
the problem with this approach is that there is a risk of picking outliers or
similar samples [26]. To increase diversity of the selected samples, CORESET
[25] chooses the furthest sample in the embedding space from the samples
already selected in previous iterations. CAL [16] finds the most contrastive
sample to its nearest neighbors by calculating KL divergence, leveraging both
uncertainty and diversity.

BERT.  In recent years BERT [7] has become one of the most famous pre-training
language models and has shown effectiveness in many natural language
processing tasks. These include sentiment analysis [28], semantical similarity
[9], question answering [21] and entailment inference [34]. BERT is pre-
trained on the BooksCorpus (800M words) [38] and English Wikipedia (2,500M
words). By pre-training on such large text data, BERT grasps rich semantic
information. The most common usage of BERT is fine-tuning it over downstream
tasks, trained with data from downstream tasks to update all its pre-trained
parameters. By this way, both the rich semantic information from pre-training
and the features from downstream tasks are taken advantage of to achieve an
excellent performance.

##  3 Data and Annotation

Task Definition.  In this work, we define a binary classification task,
inferring a word is _Mental_ or _Physical_. The notion of _Mental_ relates to
mental activities, which fall into six categories: Emotion, Need, Perceiving,
Reasoning, Planning and Personality. Personality are regarded as the external
manifestation of persistent mental activities. Detailed definition of each
category and word examples are shown in Table 4. Other words are defined as
_Physical_ describing physical attributes of an object, like color, hardness,
speed and malleability. _Mental_ words usually have abstract meanings, but
_Physical_ words have more concrete meanings that can be observed in the
world. This difference can be used as a simple reference to determine which
class a word belongs to. The inferred class only reveals that a word is more
likely to express a mental view or a physical view, as a word might have
different senses. The main reason we choose lexical level rather than sense
level for MPC, is to facilitate subsequent research and reduce development
complexity.

Data Process.  Amazon Fine Food Reviews dataset [18] 111This dataset is
distributed under CC0: Public Domain License. Download url:
https://www.kaggle.com/datasets/snap/amazon-fine-food-reviews is used as
corpus for MPC task, as this dataset contains reasoning between physics (food
description) and mentality (people’s opinion) in our daily life. It has more
than 0.5 million reviews of Amazon fine foods from Oct 1999 to Oct 2012. We
use only text column and remove all other columns like ProductId, UserId,
ProfileName for anonymization considerations. Data process contains three
steps. An processing example is given in Table 1. First, each piece of review
is splitted into words and each word is classified into part of speech (POS-
tagging) 222Penn Treebank POS tags are used. See details in
https://www.ling.upenn.edu/courses/Fall_2003/ling001/penn_treebank_pos.html.
Then (adjective, noun) pairs are recognized and extracted out, where the noun
appeared immediately after the adjective in review sentences. The goal of this
step is to make sure the extracted adjectives are used in daily life to
describe objects or states, like "roasted beans", "angry complaint", and
counteract possible errors from POS-tagging. Finally, adjectives from these
pairs are validated by checking if they have definition text from WordNet.
After deduplication, 7292 adjectives are obtained. We use version 3.6.7 of
NLTK package for POS-tagging and WordNet calling.

|  Review: I have found them all to be
---
of good quality.
|  Step 1: Pos-tagging.
---
Result: ("I", "PRP"),
("have", "VBP"), ("found", "VBN"),
("them", "PRP"), ("all", "DT"), ("to", "TO"),
("be", "VB"), ("of", "IN"), ("good", "JJ"),
("quality", "NN"), (".", ".")
|  Step 2: Detect (adjective, noun) pairs.
---
Result: ("good", "quality")
|  Step 3: Validate adjectives.
---
Result: "good"
Table 1:  Review process pipeline. Input is "I have found them all to be of
good quality." and after processing the word "good" is outputted.

Annotation.  Each adjective is annotated by two annotators checking word
definition from WordNet, and disagreements are adjudicated by another expert.
All participants are experienced volunteers and they are notified how their
annotations are used in this work. Examples of words and their definitions are
presented in Appendix A. For words with different senses, annotation results
are mainly based on the frequency of daily use. For instance, although the
word "cold" has a _Mental_ sense, "feeling or showing no enthusiasm", it’s
labeled as _Physical_ since it is used more frequently with the gloss "having
a low or inadequate temperature or feeling a sensation of coldness or having
been made cold by e.g. ice or refrigeration".

A testset consisting of 100 words is annotated for measuring model
performance. It contains 26% _Mental_ words and 74% _Physical_ words. Among
the _Mental_ words, 12% of them have annotation disagreements while this
number drops to 5% for _Physical_ words. This difference indicates that
_Mental_ words are more likely to be misclassified. Total disagreement over
this dataset between two annotators is 7% . Statistics of the testset is
summarized in Table 2. For each active learning strategy, a dataset for
training and validation is annotated, which has no overlap with the testset.

Class | Total | Disagreement | Rate
---|---|---|---
_Mental_ | 26 | 3 | 12%
_Physical_ | 74 | 4 | 5%
Table 2:  Total word numbers, disagreement numbers and rate of disagreement of
the two classes in the testset. Difference of disagreement rates indicates
that _Mental_ words are more likely to be misclassified.

##  4 Methods

We use active learning framework to train a binary classifier for MPC task,
which is shown in Algorithm 1. An unlabeled word pool $\mathcal{U}$ is set up
consisting of the extracted adjectives. The random strategy is used to select
a word for annotation in the first iteration, while in other iterations
different active learning strategies are used. We aim to annotate $K_{1}$
positives and $K_{2}$ negatives in each iteration, which are put into a
labeled word pool $\mathcal{D}_{labeled}$. A threshold $M$ is set to control
the total number of annotation of each iteration, in case that the active
learning strategy fails to find another positive or negative sample. At the
end of each iteration, a BERT model is fine-tuned over
$\mathcal{D}_{labeled}$. When iterations end, the BERT model with best
performance over testset is employed in pipeline for inference.

Algorithm 1 Active Learning Framework

0: Unlabeled word pool $\mathcal{U}$, number of positive samples $K_{1}$ and
negative samples $K_{2}$ and maximum annotated samples $M$ per iteration,
number of iterations $T$

1: $\mathcal{D}_{labeled}=\\{\\}$

2: $t=0$

3: while $t<T$ do

4: $\mathcal{D}_{pos},\mathcal{D}_{neg}=\\{\\},\\{\\}$

5: $m=0$

6: while True do

7: if $t=1$ then

8: $w_{new}\leftarrow$ Randomly select a word from $\mathcal{U}$

9: else

10: $w_{new}\leftarrow$ Select a word from $\mathcal{U}$ by a specific
strategy

11: end if

12: Annotate $w_{new}$ with a class label $C$

13: $\mathcal{U}=\mathcal{U}\setminus\\{w_{new}\\}$

14: $m=m+1$

15: if $C$ is positive and $|\mathcal{D}_{pos}|<K_{1}$ then

16: $\mathcal{D}_{pos}=\mathcal{D}_{pos}\cup\\{(w_{new},C)\\}$

17: end if

18: if $C$ is negative and $|\mathcal{D}_{neg}|<K_{2}$ then

19: $\mathcal{D}_{neg}=\mathcal{D}_{neg}\cup\\{(w_{new},C)\\}$

20: end if

21: if ($|\mathcal{D}_{pos}|=K_{1}$ and $|\mathcal{D}_{neg}|=K_{2}$) or $m=M$
then

22: break

23: end if

24: end while

25: $\mathcal{D}_{labeled}=\mathcal{D}_{labeled}\cup\mathcal{D}_{pos}\cup\
\mathcal{D}_{neg}$

26: Fine-tune a BERT over $\mathcal{D}_{labeled}$

27: $t=t+1$

28: end while

BERT fine-tuning and inference procedure is shown in Figure 2. As WordNet maps
words into sets of cognitive synonyms, each expressing a distinct concept,
therefore more than one piece of definition text are provided by WordNet for a
given word. For example, "shining" belongs to three clusters as an adjective
with three different definitions: (1) marked by exceptional merit, (2) made
smooth and bright by or as if by rubbing; reflecting a sheen or glow, and (3)
reflecting light. All of them are aggregated as one piece of text, serving as
input of BERT with a special token _[CLS]_ at head. We use the final hidden
state of _[CLS]_ as BERT output, which is then connected with a dropout layer
[29] and a linear layer. Sigmoid node is added after the linear layer to
transform logits into the probability of positive class. For fine-tuning, a
standard cross entropy loss is computed to update all parameters of the BERT
model and the subsequent linear layer.

Figure 2: Definitions of words are concatenated with _[CLS]_ at head as input
of BERT. A dropout and a linear layer are connected to BERT sequentially. At
last, a sigmoid node outputs the probability of being positive. The output
probability is consumed for inference, or as input of cross entropy loss for
fine-tuning. We use a word "shining" with its definition text as an input
example.

##  5 Experiments and Results

We compare four active learning strategies, considering their classification
performance and annotation resource consumption, to find which strategy is
most suitable for MPC task given a limited project budget. _Mental_ class
serves as positive and _Physical_ serves as negative in training. F1 scores of
_Mental_ and _Physical_ classes over testset are computed respectively. The
average of the number of labeled samples in each iteration is recorded.

ENTROPY.  Samples with the highest predicted entropy are selected [13]. For
binary classification, the closer the prediction probability of a sample is to
0.5, the higher its entropy is. Therefore, at the starting of each iteration,
the BERT model from last iteration outputs the probability of all the words in
$\mathcal{U}$ and the word whose probability is closest to 0.5 is selected.

CORESET.  Samples that are furthest away from the samples selected in previous
iterations are chosen to enlarge the semantic diversity [25]. FastText [3] is
used to represent a word by embedding vector, as fastText works well in word-
level semantic textual similarity (STS) tasks [37]. In each iteration, a word
is selected as follows:

$w_{new}=\mathop{\arg\max_{w\in\mathcal{U}}}{\mathop{\min_{v\in\mathcal{D}_{labeled}}}{L_{2}(\phi(w),\phi(v))}},$
(1)

where $L_{2}(\cdot,\cdot)$ computes $L_{2}$ distance between two vectors and
$\phi(\cdot)$ returns the embedding vector of a word.

The most contrastive sample to its nearest neighbors by calculating KL
divergence is chosen.

CAL.  The most contrastive sample to its nearest neighbors by calculating KL
divergence is chosen [16]. Given a word $w$ in unlabeled word pool
$\mathcal{U}$, nearest 10 words are selected as neighbors in labeled word pool
$\mathcal{D}_{labeled}$ by $L_{2}$ distance. The average KL divergence between
$w$ and its neighbors is computed as a measure of contrastive degree. The word
with the largest value of this measure is selected.

Random.  Select a word $w$ in unlabeled word pool $\mathcal{U}$ randomly.

(a) _Mental_ F1

Iteration | ENTROPY | CORESET | CAL | Random
---|---|---|---|---
3 | 0.61 | 0.69 | 0.62 | 0.61
4 | 0.72 | 0.71 | 0.64 | 0.64
5 | 0.70 | 0.69 | 0.68 | 0.64

(b) _Physical_ F1

Iteration | ENTROPY | CORESET | CAL | Random
---|---|---|---|---
3 | 0.82 | 0.66 | 0.79 | 0.71
4 | 0.87 | 0.83 | 0.80 | 0.66
5 | 0.85 | 0.81 | 0.76 | 0.69

Table 3: Averaged F1 scores after 3,4,5 iterations. ENTROPY outperforms the
other three, achieving the highest _Mental_ F1 0.72 and _Physical_ F1 0.87 at
iteration 4.

All strategies share the same experimental settings as following: total
iterations $T$ = 5, number of positive samples $K_{1}$ = 20, number of
negative samples $K_{2}$ = 20, maximum annotation number $M$ = 120. In each
iteration, BERT fine-tuning takes totally 20 epochs with learning rate 2e-5
and batch size 32. Learning rate drops to 1/10 of the original level after 10
epochs. We split $\mathcal{D}_{labeled}$ by 80% - 20% as trainset and devset.
If BERT outputs a value greater than 0.5, the word is considered to belong to
_Mental_ , otherwise _Physical_. Winner model $\mathcal{M}_{t}$ is the one
with maximum accuracy over devset. We hypertune BERT with different values of
learning rate {1e-5, 2e-5, 1e-4} and batch size {32, 64, 128} for ENTROPY
strategy. The best result over devset is achieved at 2e-5 learning rate and 32
batch size.

Our BERT implementation is provided by Hugging Face and we choose "bert-base-
uncased" version which contains 110M parameters and does not make a difference
between lowercase and uppercase words.333https://huggingface.co/bert-base-
uncased We use the Adam optimizer with 0.001 weight decay [15]. The size of
the linear layer is 768 which is the same size of BERT final hidden state.
Dropout with a probability of 0.3 is applied in the network. Training
framework is based on Pytorch Lightning (version 1.5.8) which could greatly
boosts training efficiency. All experiments use this network architecture.

Category | Definition | Example
---|---|---
Emotion |  | Plutchik’s wheel of emotions [20].
---
| favored, scorned, frisky,
---
trustworthy, gripping, stilted
Need |  | Maslow’s hierarchy of needs [17].
---
| devout, deserving, protective,
---
hired, wealthy, rewarding
Perceiving |  | individuals use information to form perceptions
---
of themselves and others based on
social categories. [1].
| sensuous, ubiquitous, instinctive,
---
detected, recognized, perceivable
Reasoning |  | Deduction based on classical
---
logic [27] or mental
models [12].
| suitable, predominate, critical,
---
substandard, relevant, causal
Planning |  | Mental time travel
---
[30].
| committed, aimless, exploited,
---
unplanned, purposeful, executed
Personality |  | Stable patterns of behavior, cognition,
---
and emotion [4].
| whimsical, squeamish, shy,
---
punctual, entrepreneurial, intrepid
Table 4:  _Mental_ adjective examples which are classified as objective in
subjectivity classification. The definition of each mental category is
provided.

Each strategy is run three times with different random seeds and the averaged
F1 scores over testset after three, four, five iterations are reported in
Table 3. ENTROPY outperforms the other three, achieving the highest _Mental_
F1 0.72 and _Physical_ F1 0.87 at iteration 4. The reason that CAL fails might
be we don’t find semantically similar neighbors as the size of
$\mathcal{D}_{labeled}$ is too small. Table 5 shows annotation resource
consumption. ENTROPY requires 60~70 labeled words per iteration, which means
totally only 300 labeled words are needed to deliver an applicable classifier.
CORESET and Random need more annotations than ENTROPY. CAL could not provide
enough positive and negative samples after 120 words are annotated for some
iterations. Precision and recall scores are presented in Appendix B.

Strategy | Words/Iter | EnoughSamples
---|---|---
ENTROPY | 60 ~ 70 | Yes
CORESET | 80 ~ 120 | Yes
CAL | 50 ~ 120 | No
Random | 80 ~ 100 | Yes
Table 5:  Range of annotation number per iteration. ENTROPY requires the
lowest annotation resource. CAL could not provide enough positive and negative
samples after 120 words are annotated for some iterations.

##  6 Comparison with SentiWordNet

As the notion of _Mental_ and _Physical_ is to some extent similar to
"subjective" and "objective" in the subjectivity classification task [14] of
sentiment analysis, we’d like to investigate the difference between them. We
choose to compare our result with SentiWordNet [24; 2] which is the most used
lexicon in social opinion mining studies [5]. SentiWordNet is a lexical
resource which labels each synset from WordNet [19] as "positive", "negative"
or "neutral". The used version of SentiWordNet is 3.0, which is based on
WordNet 3.0.

SentiWordNet 3.0 associates each synset with three numerical scores _PosScore_
, _NegScore_ and _ObjScore_ which show how positive, negative, and neutral the
words contained in the synset are [2]. All three scores range from 0 to 1 and
their sum is 1. We focus on adjective synsets and classify each of them into
two classes: _SubSyn_ , if the maximum of the three scores is _PosScore_ or
_NegScore_ ; otherwise, _ObjSyn_. For an adjective that belongs to more than
one synset, it owns different senses, perhaps having both subjective and
objective meanings. Therefore, at lexical level, an adjective is labeled by
this rule: _Subjective_ , if it only belongs to _SubSyn_ synsets; _Objective_
, if it only belongs to _ObjSyn_ synsets; _Dual_ , if if belongs to both
_SubSyn_ and _ObjSyn_ synsets. Table 6 shows the distribution of _Subjective_
, _Objective_ and _Dual_ adjectives in _Mental_ and _Physical_ classes. We
find that 43% of the _Mental_ adjectives are labeled as _Objective_. This
indicates the notions of _Mental/Physical_ are different from
_Subjective/Objective_. In fact, many _Objective_ adjectives bear mental
functionalities. Some adjective examples are listed to illustrate this point
in Table 4 in six categories: Emotion, Need, Perceiving, Reasoning, Planning
and Personality.

Class | Subjective | Objective | Dual | Total
---|---|---|---|---
Mental | 28% | 43% | 29% | 100%
Physical | 9% | 74% | 17% | 100%

Table 6:  Distribution of _Subjective_ , _Objective_ and _Dual_ adjectives in
_Mental_ and _Physical_ classes. Only 28% of those words in _Mental_ fall into
_Subjective_ , while 43% belong to _Objective_. This indicates many
_Objective_ adjectives bear mental functionalities under MPC definition.

##  7 Conclusion

Aiming to explicitly reveal reasoning path in commonsense scenarios, our first
step is to classify a word into _Mental_ or _Physical_. We provide clear
definitions of these two categories and a simple criterion for judging them.
Active learning algorithm is implemented to fine-tune a BERT model, reducing
the required annotation resources. The BERT model automatically infers which
class an adjective belongs to. We release the inferred tags publicly to
facilitate future research. We also compare our result with SentiWordNet, and
find the notions of _Mental/Physical_ is different from _Subjective/Objective_
in sentiment analysis. Many _Objective_ adjectives bear mental functionalities
under MPC definition.

Future research works focus on designing more fine-grained tags and training
models to automatically infer them over words. Links are built between tags in
a manual way or machine-learning style, to form an applicable reasoning graph.
We hope to translate what humans know about the world and about themselves
into graph that will improve the intelligence of machines. Large language
models (LLMs) provide powerful techniques to extract data patterns in nature
language, which makes it possible to perfectly associate words with all kinds
of human-designed tags. However, at the level of reasoning, relying on LLMs is
not necessarily feasible, and painstaking manual work may be essential.

## Limitations

Although the model by ENTROPY achieves acceptable F1 scores, there’s still a
lot of room for improvement of classification precision and recall. For
example, use more annotated words for fine-tuning, or try other deep learning
algorithms. We leave this optimization in the future after we verify the whole
research plan becomes feasible and the classification performance is a
bottleneck for commonsense reasoning ability.

As a word has different meanings in different contexts, the best granularity
for MPC is gloss level rather than lexical level. That’s to say, use each
piece of gloss text as BERT input instead of merging all glosses of a word
into one piece of text. Then, the output shows if a gloss belongs to _Mental_
or _Physical_. However, lexical level facilitate the development of reasoning
graph as there’s no need to consider context. We will change to gloss level by
the time it’s verified that context becomes a bottleneck and it should be
integrated into reasoning graph.

## Acknowledgements

We appreciate valuable suggestions for this work from every reviewer.

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##  Appendix A Word examples in MPC task

Table 7 shows seven adjectives with their classes in MPC task. From these
examples, we could see clear difference in definition texts between _Mental_
and _Physical_ classes. Therefore, it’s possible to fine-tune a high-accuracy
BERT for MPC task.

Word | Class | Definition
---|---|---
interested | _Mental_ |  | having or showing interest; especially curiosity or fascination or concern;
---
involved in or affected by or having a claim to or share in;
angry | _Mental_ |  | feeling or showing anger; (of the elements) as if showing violent anger;
---
severely inflamed and painful;
clever | _Mental_ |  | showing self-interest and shrewdness in dealing with others;
---
mentally quick and resourceful;
lazy | _Mental_ |  | moving slowly and gently; disinclined to work or exertion;
---
molecular | _Physical_ |  | relating to or produced by or consisting of molecules;
---
relating to simple or elementary organization;
blue | _Physical_ |  | of the color intermediate between green and violet;
---
having a color similar to that of a clear unclouded sky;
automated | _Physical_ |  | operated by automation;
---
Table 7:  Examples of words and their classes in MPC task. Definitions are
provided by WordNet. From these examples, we could see clear difference in
definition texts between two classes.

##  Appendix B Precision and Recall

Table 8 shows averaged precision and recall after 3,4,5 iterations of each
strategy. For _Mental_ class, ENTROPY achieves the highest precision around
0.8 and CAL has the highest recall above 0.9. For _Physical_ class, CAL
achieves the highest precision above 0.9 and ENTROPY has the highest recall
around 0.9.

(a) _Mental_ Precision

Iteration | ENTROPY | CORESET | CAL | Random
---|---|---|---|---
3 | 0.70 | 0.62 | 0.53 | 0.54
4 | 0.81 | 0.63 | 0.51 | 0.72
5 | 0.79 | 0.59 | 0.54 | 0.69

(b) _Mental_ Recall

Iteration | ENTROPY | CORESET | CAL | Random
---|---|---|---|---
3 | 0.54 | 0.79 | 0.74 | 0.73
4 | 0.65 | 0.81 | 0.87 | 0.57
5 | 0.64 | 0.82 | 0.94 | 0.61

(c) _Physical_ Precision

Iteration | ENTROPY | CORESET | CAL | Random
---|---|---|---|---
3 | 0.77 | 0.86 | 0.81 | 0.79
4 | 0.82 | 0.87 | 0.88 | 0.79
5 | 0.82 | 0.88 | 0.94 | 0.79

(d) _Physical_ Recall

Iteration | ENTROPY | CORESET | CAL | Random
---|---|---|---|---
3 | 0.87 | 0.73 | 0.64 | 0.60
4 | 0.91 | 0.73 | 0.53 | 0.88
5 | 0.90 | 0.68 | 0.55 | 0.83

Table 8: Averaged precision and recall of _Mental_ and _Physical_ after 3,4,5
iterations.