# Quantifying Maximum Actuator Degradation for a Given
$\mathcal{H}_{2}/\mathcal{H}_{\infty}$ Performance with Full-State Feedback
Control

Hrishav Das  Aerospace Engineering
Indian Institute of Technology Madras
Chennai, India
<EMAIL_ADDRESS>Eliot Nychka  Aerospace Engineering
Texas A&M University
College Station, TX, USA
<EMAIL_ADDRESS>Raktim Bhattacharya  Aerospace Engineering
Texas A&M University
College Station, TX, USA
<EMAIL_ADDRESS>
###### Abstract

In this paper, we address the issue of quantifying maximum actuator
degradation in linear time invariant dynamical systems. We present a new
unified framework for computing the state-feedback controller gain that meets
a user-defined closed-loop performance criterion while also maximizing
actuator degradation. This degradation is modeled as a first-order filter with
additive noise. Our approach involves two novel convex optimization
formulations that concurrently determine the controller gain, maximize
actuator degradation, and maintain the desired closed-loop performance in both
the $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ system norms. The results are
limited to open-loop stable systems. We demonstrate the application of our
results through the design of a full-state feedback controller for a model
representing the longitudinal motion of the F-16 aircraft.

##  I Introduction

This paper explores quantifying maximum actuator degradation in linear
dynamical systems, a critical issue in engineering control systems. Actuator
degradation results from wear, environmental influences, or aging components,
leading to diminished performance and, in severe cases, system failure. In
aerospace engineering, for example, the health of control systems is vital for
the safe and efficient operation of aircraft; actuator degradation could
significantly compromise flight control, endangering safety. Similar issues
also exist in other engineering fields and become essential in safety-critical
applications. Hence, developing methodologies for assessing and quantifying
maximum actuator degradation is critical for quantifying robustness.

###  I-A Literature Survey

In the current state of the art, there are generally two approaches to
addressing faults or degradation in systems and their controllers: enhancing
the robustness and fault tolerance of controllers by integrating degradation
estimates, or developing improved models for actuator degradation.

Papers such as [1, 2, 3, 4, 5] look at methods related to enhancing controller
robustness and fault tolerance by incorporating degradation estimates.
Specifically, [1] presents a fault-tolerant control design that utilizes
actuator health information to improve system reliability and reduce reliance
on more sensitive actuators. The approach involves solving a fault-tolerant
tracking problem using a Linear Matrix Inequality (LMI) framework while taking
into account actuator health. Zhang et.al. [2] introduces an algorithm that
models actuator degradation using the Weiner Diffusion method. This
degradation is then incorporated as a separate state variable in the so called
Degradation Informed State System (DISS) within a controller. The objective is
to optimize tracking performance while minimizing degradation. The controller
tracks a reference trajectory of physical states such as position,
orientation, and degradation quantity, employing a Model Predictive Control
(MPC) approach, although other optimization-based methods could also be used.
Notably, the paper also addresses the concept of ‘maximum degradation’, which
is stochastically computed from historical data on actuators and their health
to impose constraints on the MPC. In [3], the primary objective is to
introduce a controller for electric vehicles that enhances speed tracking,
reliability, and lifespan. The approach is applied to a power converter that
operates a permanent magnet synchronous motor (PMSM). The paper presents a
method that employs an extended $\mathcal{H}_{\infty}$ design framework
alongside reliability models to mitigate thermal stress on the power
electronic Insulated Gate Bipolar Transistors (IGBTs). The results demonstrate
that the proposed control strategies are more effective in real-world
scenarios compared to existing state-of-the-art methods. Shen et. al. [4]
presents a high-level Model Reference Adaptive Control (MRAC) approach to
generate desired torques for tracking specific trajectories and orientations
while addressing disturbances. These torques serve as input to a robust
control allocation scheme that apportions control effort, taking into account
practical constraints on each control input and inaccuracies in fault
estimation. This study addresses uncertainties in actuator effectiveness and
fault detection and diagnostics, without focusing on estimating the fault or
calculating the maximum allowable fault. In [5], a method to design a robust
and optimal controller for open-loop unstable systems with actuator redundancy
is proposed. Primarily the system outlines the LQR method as a ‘robust Linear
Quadratic (LQ) regulator’ and shows it to be equivalent to a traditional LQ
regulator.

A considerable amount of research focuses on developing models for actuator
degradation. For instance, [6, 7] models actuator degradation by adjusting its
influence (modifying the B matrix) and discusses a method for estimating this
degradation. In [8] various degradation models are explored that can be
selected based on the actuator’s degradation state. It also presents a
technique for adjusting the desired state input to the controller to prevent
saturation or exceeding limits in degraded actuators.

Nguyen et al. [9] and Si et al. [10] incorporate the concept of Remaining
Useful Life (RUL) in their degradation modeling formulations. The work of
Nguyem et.al. employs a statistical approach, considering discrete time
impacts where each impact contributes to degradation within a probabilistic
range. The RUL of actuators is determined by simulating the state until a
specified precision criterion is no longer met (indicating failure). The work
of Si et. al. integrates the degradation process model of the actuator with
the system’s state transition model and utilizes a particle filter algorithm
to estimate both the hidden degradation state and the system state.

There has been limited work on quantifying worst-case or maximum degradation
of actuators. Few papers [9, 10, 2] come closest to addressing the broader
issue of computing maximum actuator degradation for an acceptable closed-loop
performance. There has been no prior work, to the best of our knowledge, that
computes the maximum actuator degradation for a given closed-loop performance.

###  I-B Key Contributions

We introduce a novel unified framework to compute the state-feedback
controller gain that satisfies a user-specified closed-loop performance
criterion while maximizing actuator degradation. The degradation is
represented as a first-order filter with additive noise. We propose two new
convex optimization formulations that simultaneously determine the controller
gain, maximize actuator degradation, and ensure the desired closed-loop
performance in both the $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ system
norms. The key contributions are detailed in theorems 1 and 2. This is the
first time a joint formulation is presented in the
$\mathcal{H}_{2}/\mathcal{H}_{\infty}$ framework. However, these results are
limited to open-loop stable systems.

##  II Preliminaries

The aircraft can generally experience actuator faults that can be classified
as either total or partial faults. In actuators, a total fault is a complete
loss of control action, and examples include a stuck or floating actuator,
which produces no controllable actuation. They can occur due to breakage, cut
or burnt wiring, short-circuit presence of a foreign body in the actuator,
etc. Partially failed actuators produce only a part of the normal actuation
and correspond to reduced rate and magnitude limits. These faults can result
from hydraulic or pneumatic leakage, increased resistance, a fall in the
supply voltage, etc.

In this paper, complete and partial faults are modeled as shown in Fig.1,
where a loss in the actuator rate is modeled using a low-pass filter,
$F(s):=\frac{\gamma_{u}}{s/\omega_{c}+1}$, with cutoff frequency
$\omega_{c}\geq 0$. The biases and inaccuracies are modeled using an additive
noise $w_{a}(t)$ and is assumed to be a norm-bounded signal
$\|w_{a}(t)\|_{2}\leq\gamma_{a}$. The control magnitude loss is modeled as a
constraint on $\gamma_{u}$ the dc gain of the actuator. Therefore, with
parameters $\omega_{c},\gamma_{a}$ and $\gamma_{u}$, we can capture a rich
class of actuator faults commonly encountered in control systems.

Fault
---
Filter $F(s)$
$+$ Control
---
Signal
$u(t)$
Faulty
---
Control
Signal
$\hat{u}(t)$
Actuator
---
Noise
$w_{a}(t)$
$x_{F}(t)$ Figure 1: Modeling of a faulty actuator.

For systems with multiple inputs, the dynamics of the filter can be written as

$\displaystyle\dot{\boldsymbol{x}}_{F}(t)$
$\displaystyle=-\textbf{diag}\left(\boldsymbol{\omega}_{c}\right)\boldsymbol{x}_{F}(t)+\textbf{diag}\left(\boldsymbol{\omega}_{c}\right)\boldsymbol{u}(t),$
(1a) $\displaystyle\hat{\boldsymbol{u}}(t)$
$\displaystyle=\boldsymbol{x}_{F}(t)+\boldsymbol{w}_{a}(t),$ (1b)

where $n_{u}$ dimensional vectors $\boldsymbol{x}_{F}$, $\boldsymbol{u}$,
$\boldsymbol{w}_{a}$, and $\hat{\boldsymbol{u}}$ are the filter states,
control signals, actuator noises, and faulty control signals, respectively.

We consider the open-loop LTI system,

$\displaystyle\dot{\boldsymbol{x}}(t)$
$\displaystyle=\boldsymbol{A}\boldsymbol{x}(t)+\boldsymbol{B}_{u}\boldsymbol{u}(t)+\boldsymbol{B}_{d}\boldsymbol{d}(t),$
(2a) $\displaystyle\boldsymbol{z}(t)$
$\displaystyle=\boldsymbol{C}_{z}\boldsymbol{x}(t),$ (2b)

where $\boldsymbol{x}\in^{n_{x}}$ is the state vector,
$\boldsymbol{u}\in^{n_{u}}$ is the control vector, $\boldsymbol{d}\in^{n_{d}}$
is the exogeneous disturbance signal, and $\boldsymbol{z}\in^{n_{z}}$ is the
controlled variable.

$\boldsymbol{F}(s)$$+$$\boldsymbol{P}(s)$$\boldsymbol{K}$$\boldsymbol{z}(t)$$\boldsymbol{w}_{a}(t)$$\boldsymbol{d}(t)$$\boldsymbol{u}(t)$$\boldsymbol{x}_{F}(t)$$\boldsymbol{\hat{u}}(t)$$\boldsymbol{x}(t)$
Figure 2: Closed-loop system with full-state feedback controller and faulty
actuator.

This paper considers a full-state feedback formulation, i.e.,
$\boldsymbol{u}:=\boldsymbol{K}\boldsymbol{x}$. Therefore, the closed-loop
system with the faulty actuators is shown in Fig.2, and its dynamics is given
by

$\begin{bmatrix}\dot{\boldsymbol{x}}\\\
\dot{\boldsymbol{x}}_{F}\end{bmatrix}=\begin{bmatrix}\boldsymbol{A}&\boldsymbol{B}_{u}\\\
\underbrace{\textbf{diag}\left(\boldsymbol{\omega}_{c}\right)\boldsymbol{K}}_{:=\boldsymbol{V}}&-\textbf{diag}\left(\boldsymbol{\omega}_{c}\right)\end{bmatrix}\begin{bmatrix}\boldsymbol{x}\\\
\boldsymbol{x}_{F}\end{bmatrix}\\\
+\begin{bmatrix}\boldsymbol{B}_{d}&\boldsymbol{B}_{u}\\\
\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}\begin{bmatrix}\boldsymbol{d}\\\
\boldsymbol{w}_{a}\end{bmatrix},$ (3a) $\displaystyle\boldsymbol{z}$
$\displaystyle=\begin{bmatrix}\boldsymbol{C}_{z}&\boldsymbol{0}\end{bmatrix}\begin{bmatrix}\boldsymbol{x}\\\
\boldsymbol{x}_{F}\end{bmatrix}.$ (3b)

The problem is determining $\boldsymbol{K}$ that bounds the
$\mathcal{H}_{2}/\mathcal{H}_{\infty}$ norm of the closed-loop transfer
function from $\begin{bmatrix}\boldsymbol{d}\\\
\boldsymbol{w}_{a}\end{bmatrix}\to\boldsymbol{z}$ while maximizing actuator
faults. In this paper, we propose that the actuator fault is maximized when
$\|\boldsymbol{w}_{a}(t)\|_{2}$ is maximized, and the actuation rate and the
actuator’s DC gain are minimized. The latter two are achieved by minimizing
$\|\boldsymbol{\omega}_{c}\|_{2}$ and
$\textbf{tr}\left[\boldsymbol{V}^{T}\boldsymbol{V}\right]$.

Thus, the problem is reduced to designing a control law that meets the
specified closed-loop performance with the maximum actuator noise and minimum
control rate and magnitude.

######  Remark 1.

In (2b), we assume that $\boldsymbol{z}(t)$ is not dependent on
$\boldsymbol{u}(t)$, which is necessary to solve problem in a convex
optimization framework. Our future work will explore formulations to remove
this restriction.

######  Remark 2.

The results presented in the paper apply to open-loop stable systems, due to
the assumption of a particular structure for the Lyapunov function, which
results in a convex formulation.

##  III Technical Results

We will use the following standard notations:

  * •

n is the space of $n$-dimensional vectors,

  * •

${}_{+}^{n}$ is the space of $n$-dimensional vectors with real non-negative
entries,

  * •

m×n is the space of $m\times n$ dimensional vectors with real entries,

  * •

$\mathbb{S}^{n}_{+}$ is the space of $n\times n$ symmetric positive semi-
definite matrices with real entries,

  * •

vectors and matrices are represented in bold font, and

  * •

scalars are represented in normal font.

######  Theorem 1.

For the system in (3), the state-feedback gain that guarantees
$\left\|\begin{bmatrix}\boldsymbol{d}\\\
\boldsymbol{w}_{a}\end{bmatrix}\to\boldsymbol{z}\right\|_{\infty}\leq\gamma$
for the system in (3), with maximum actuator noise, and minimum control
magnitude and rate, is given by the following convex optimization problem,

$\displaystyle\min_{\boldsymbol{\kappa}_{a},\boldsymbol{\omega}_{c},\gamma_{\boldsymbol{x}_{F}},\boldsymbol{V},\boldsymbol{Y},\boldsymbol{Q}}\lambda_{a}\left\|\boldsymbol{\kappa}_{a}\right\|_{2}+\lambda_{\boldsymbol{\omega}_{c}}\left\|\boldsymbol{\omega}_{c}\right\|_{2}+\lambda_{\boldsymbol{x}_{F}}\gamma_{\boldsymbol{x}_{F}},$
subject to
$\displaystyle\begin{bmatrix}\boldsymbol{P}+\boldsymbol{P}^{T}&\begin{bmatrix}\boldsymbol{Y}\boldsymbol{B}_{d}&\boldsymbol{Y}\boldsymbol{B}_{u}\\\
\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}&\begin{bmatrix}\boldsymbol{C}_{z}^{T}\\\
\boldsymbol{0}\end{bmatrix}\\\\[5.69054pt]
\ast&\gamma\begin{bmatrix}\boldsymbol{W}_{d}^{-2}&\boldsymbol{0}\\\
\boldsymbol{0}&\textbf{diag}\left(\boldsymbol{\kappa}_{a}\right)\end{bmatrix}&\begin{bmatrix}\boldsymbol{0}\\\
\boldsymbol{0}\end{bmatrix}\\\ \ast&\ast&\gamma\boldsymbol{I}\end{bmatrix}\leq
0,$ $\displaystyle\begin{bmatrix}\boldsymbol{Q}&\boldsymbol{V}^{T}\\\
\boldsymbol{V}&\boldsymbol{I}\end{bmatrix}\geq
0,\textbf{tr}\left[\boldsymbol{Q}\right]\leq\gamma_{\boldsymbol{x}_{F}},$
$\displaystyle\boldsymbol{\kappa}\in_{+}^{n_{u}},\boldsymbol{\omega}_{c}\in_{+}^{n_{u}},\gamma_{\boldsymbol{x}_{F}}\in_{+},$
$\displaystyle\boldsymbol{V}\in^{n_{u}\times
n_{x}},\boldsymbol{Y}\in\mathbb{S}_{+}^{n_{x}},\boldsymbol{Q}\in\mathbb{S}_{+}^{n_{z}},$

with

$\boldsymbol{P}:=\begin{bmatrix}\boldsymbol{Y}\boldsymbol{A}&\boldsymbol{Y}\boldsymbol{B}_{u}\\\
\boldsymbol{V}&-\textbf{diag}\left(\boldsymbol{\omega}_{c}\right)\end{bmatrix},$

where $\lambda_{a}$, $\lambda_{\boldsymbol{\omega}_{c}}$, and
$\lambda_{\boldsymbol{x}_{F}}$ are user-defined weights, and
$\boldsymbol{W}_{d}$ is a given scaling matrix for $\boldsymbol{d}(t)$. The
controller gain can be recovered as
$\boldsymbol{K}:=\textbf{diag}\left(\boldsymbol{\omega}_{c}\right)^{-1}\boldsymbol{V}$.

###### Proof.

We introduce diagonal matrix weights $\boldsymbol{W}_{a}$ and
$\boldsymbol{W}_{d}$ such that

$\displaystyle\boldsymbol{w}_{a}(t):=\boldsymbol{W}_{a}\boldsymbol{\bar{w}}_{a}(t),$
$\displaystyle\boldsymbol{d}(t):=\boldsymbol{W}_{d}\boldsymbol{\bar{d}}(t),$
(4)

where
$\left\|\boldsymbol{\bar{w}}(t)\right\|_{2}=\left\|\boldsymbol{\bar{d}}(t)\right\|_{2}=1$,
$\boldsymbol{W}_{d}$ is known, and $\boldsymbol{W}_{a}$ is unknown. Therefore,
the closed-loop system in terms of the normalized exogeneous signals is given
by

$\displaystyle\left[\begin{array}[]{c}\dot{\boldsymbol{x}}\\\
\dot{\boldsymbol{x}}_{F}\\\ \hline\cr\boldsymbol{z}\end{array}\right]=$ (8)
$\displaystyle\left[\begin{array}[]{cc|cc}\boldsymbol{A}&\boldsymbol{B}_{u}&\boldsymbol{B}_{d}\boldsymbol{W}_{d}&\boldsymbol{B}_{u}\boldsymbol{W}_{a}\\\
\textbf{diag}\left(\boldsymbol{\omega}_{c}\right)\boldsymbol{K}&-\textbf{diag}\left(\boldsymbol{\omega}_{c}\right)&\boldsymbol{0}&\boldsymbol{0}\\\
\hline\cr\boldsymbol{C}_{z}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\end{array}\right]\left[\begin{array}[]{c}\boldsymbol{x}\\\
\boldsymbol{x}_{F}\\\ \hline\cr\boldsymbol{\bar{d}}\\\
\boldsymbol{\bar{w}}_{a}\end{array}\right].$ (16)

Suppose $\boldsymbol{G}(s)$ is the transfer function from
$\begin{bmatrix}\boldsymbol{\bar{d}}\\\
\boldsymbol{\bar{w}}_{a}\end{bmatrix}\to\boldsymbol{z}$ as defined by (16),
then the Bounded Real Lemma [11, 12, 13, 14] states
$\left\|\boldsymbol{G}(s)\right\|_{\infty}\leq\gamma$ is equivalent to the
inequality

$\displaystyle\begin{bmatrix}\boldsymbol{M}_{11}^{T}\boldsymbol{X}+\boldsymbol{X}\boldsymbol{M}_{11}&\boldsymbol{X}\boldsymbol{M}_{12}&\boldsymbol{M}_{21}^{T}\\\
\ast&-\gamma\boldsymbol{I}&\boldsymbol{M}_{22}^{T}\\\
\ast&\ast&-\gamma\boldsymbol{I}\end{bmatrix}\leq 0,$ (17)

for $\boldsymbol{X}\geq 0$, where $\boldsymbol{M}_{ij}$ are defined from the
matrix partitions in (16). The matrix inequality in (17), is not convex due to
the products of the unknowns $\boldsymbol{\omega}_{c}$, $\boldsymbol{K}$,
$\boldsymbol{W}_{a}$, and $\boldsymbol{X}$.

Introducing
$\boldsymbol{V}:=\textbf{diag}\left(\boldsymbol{\omega}_{c}\right)\boldsymbol{K}$
and assuming a special structure for $\boldsymbol{X}$,

$\boldsymbol{X}:=\begin{bmatrix}\boldsymbol{Y}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{I}\end{bmatrix},$ (18)

then

$\boldsymbol{X}\boldsymbol{M}_{11}=\begin{bmatrix}\boldsymbol{Y}\boldsymbol{A}&\boldsymbol{Y}\boldsymbol{B}_{u}\\\
\boldsymbol{V}&-\textbf{diag}\left(\boldsymbol{\omega}_{c}\right)\end{bmatrix},$

which is linear in $\boldsymbol{Y}$, $\boldsymbol{\omega}_{c}$, and
$\boldsymbol{V}$. The controller gain can be recovered as
$\boldsymbol{K}:=\textbf{diag}\left(\boldsymbol{\omega}_{c}\right)^{-1}\boldsymbol{V}$.

The term $\boldsymbol{X}\boldsymbol{M}_{12}$ can be written as

$\boldsymbol{X}\boldsymbol{M}_{12}=\begin{bmatrix}\boldsymbol{Y}\boldsymbol{B}_{d}&\boldsymbol{Y}\boldsymbol{B}_{u}\\\
\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}\textbf{diag}\left(\boldsymbol{W}_{d},\boldsymbol{W}_{a}\right).$

Therefore, the quadratic term corresponding to
$\boldsymbol{X}\boldsymbol{M}_{12}$ in the Schur complement of (17) can be
expressed as
$\begin{bmatrix}\boldsymbol{Y}\boldsymbol{B}_{d}&\boldsymbol{Y}\boldsymbol{B}_{u}\\\
\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}\textbf{diag}\left(\boldsymbol{W}^{2}_{d},\boldsymbol{W}^{2}_{a}\right)\begin{bmatrix}\boldsymbol{Y}\boldsymbol{B}_{d}&\boldsymbol{Y}\boldsymbol{B}_{u}\\\
\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}^{T}$. Defining,

$\displaystyle\boldsymbol{W}^{2}_{a}$
$\displaystyle:=\textbf{diag}\left(\boldsymbol{\kappa}_{a}\right)^{-1},$

the matrix inequality in (17) can be written as

$\begin{bmatrix}\boldsymbol{P}+\boldsymbol{P}^{T}&\begin{bmatrix}\boldsymbol{Y}\boldsymbol{B}_{d}&\boldsymbol{Y}\boldsymbol{B}_{u}\\\
\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}&\begin{bmatrix}\boldsymbol{C}_{z}^{T}\\\
\boldsymbol{0}\end{bmatrix}\\\\[5.69054pt]
\ast&-\gamma\begin{bmatrix}\boldsymbol{W}_{d}^{-2}&\boldsymbol{0}\\\
\boldsymbol{0}&\textbf{diag}\left(\boldsymbol{\kappa}_{a}\right)\end{bmatrix}&\begin{bmatrix}\boldsymbol{0}\\\
\boldsymbol{0}\end{bmatrix}\\\
\ast&\ast&-\gamma\boldsymbol{I}\end{bmatrix}\leq 0,$

which is convex in $\boldsymbol{Y},\boldsymbol{V},\boldsymbol{\omega}_{c}$,
and $\boldsymbol{\kappa}_{a}$, where

$\boldsymbol{P}:=\begin{bmatrix}\boldsymbol{Y}\boldsymbol{A}&\boldsymbol{Y}\boldsymbol{B}_{u}\\\
\boldsymbol{V}&-\textbf{diag}\left(\boldsymbol{\omega}_{c}\right)\end{bmatrix}.$

The DC gain matrix of the actuator is minimized, by minimizing a upper bound
of $\textbf{tr}\left[\boldsymbol{V}^{T}\boldsymbol{V}\right]$, i.e., minimize
$\gamma_{\boldsymbol{x}_{F}}$ such that
$\boldsymbol{V}^{T}\boldsymbol{V}\leq\boldsymbol{Q}$ and
$\textbf{tr}\left[\boldsymbol{Q}\right]\leq\gamma_{\boldsymbol{x}_{F}}$ and
$\boldsymbol{Q}\geq 0$. The constraint on $\boldsymbol{V}$ can be expressed as
the following LMI using Schur complement,

$\begin{bmatrix}\boldsymbol{Q}&\boldsymbol{V}^{T}\\\
\boldsymbol{V}&\boldsymbol{I}\end{bmatrix}\geq 0.$

The actuator noise is maximized by minimizing
$\left\|\boldsymbol{\kappa}_{a}\right\|_{2}$, the control rate is minimized by
minimizing $\left\|\boldsymbol{\omega}_{c}\right\|_{2}$, and the control
magnitude is minimized by minimizing $\gamma_{\boldsymbol{x}_{F}}$. Therefore,
the cost function for the optimization is

$\min_{\boldsymbol{\kappa}_{a},\boldsymbol{\omega}_{c},\gamma_{\boldsymbol{x}_{F}},\boldsymbol{V},\boldsymbol{Y}\geq
0,\boldsymbol{Q}\geq
0}\lambda_{a}\left\|\boldsymbol{\kappa}_{a}\right\|_{2}+\lambda_{\boldsymbol{\omega}_{c}}\left\|\boldsymbol{\omega}_{c}\right\|_{2}+\lambda_{\boldsymbol{x}_{F}}\gamma_{\boldsymbol{x}_{F}},$

where $\lambda_{a}$, $\lambda_{\boldsymbol{\omega}_{c}}$, and
$\lambda_{\boldsymbol{x}_{F}}$ are user-defined weights. ∎

We next present the result for $\mathcal{H}_{2}$ optimal closed-loop
performance.

######  Theorem 2.

For the system in (3) with $\boldsymbol{D}_{d}=0$, the state-feedback gain
that guarantees $\left\|\begin{bmatrix}\boldsymbol{d}\\\
\boldsymbol{w}_{a}\end{bmatrix}\to\boldsymbol{z}\right\|_{2}\leq\gamma$ for a
given $\gamma\geq 0$, with maximum actuator noise, and minimum control
magnitude and rate, is given by the following convex optimization problem,

$\displaystyle\min_{\boldsymbol{\kappa}_{a},\boldsymbol{\omega}_{c},\gamma_{\boldsymbol{x}_{F}},\boldsymbol{V},\boldsymbol{Y},\boldsymbol{Q}_{1},\boldsymbol{Q}_{2}}\lambda_{a}\left\|\boldsymbol{\kappa}_{a}\right\|_{2}+\lambda_{\boldsymbol{\omega}_{c}}\left\|\boldsymbol{\omega}_{c}\right\|_{p}+\lambda_{\boldsymbol{x}_{F}}\gamma_{\boldsymbol{x}_{F}},$
subject to
$\displaystyle\begin{bmatrix}\boldsymbol{P}+\boldsymbol{P}^{T}&\begin{bmatrix}\boldsymbol{Y}\boldsymbol{B}_{d}&\boldsymbol{Y}\boldsymbol{B}_{u}\\\
\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}\\\
\ast&-\begin{bmatrix}\boldsymbol{W}_{d}^{-2}&\boldsymbol{0}\\\
\boldsymbol{0}&\textbf{diag}\left(\boldsymbol{\kappa}_{a}\right)\end{bmatrix}\end{bmatrix}\leq
0$
$\displaystyle\begin{bmatrix}\boldsymbol{Q}_{1}&\boldsymbol{C}_{z}&\boldsymbol{0}\\\
\ast&\boldsymbol{Y}&0\\\ \ast&\ast&\boldsymbol{I}\end{bmatrix}\geq
0,\;\begin{bmatrix}\boldsymbol{Q}_{2}&\boldsymbol{V}^{T}\\\
\boldsymbol{V}&\boldsymbol{I}\end{bmatrix}\geq 0,$
$\displaystyle\textbf{tr}\left[\boldsymbol{Q}_{1}\right]\leq\gamma,\textbf{tr}\left[\boldsymbol{Q}_{2}\right]\leq\gamma_{\boldsymbol{x}_{F}},\boldsymbol{\kappa}\in_{+}^{n_{u}},\boldsymbol{\omega}_{c}\in_{+}^{n_{u}},\gamma_{\boldsymbol{x}_{F}}\in_{+},$
$\displaystyle\boldsymbol{V}\in^{n_{u}\times
n_{x}},\boldsymbol{Y}\in\mathbb{S}_{+}^{n_{x}},\boldsymbol{Q}_{1}\in\mathbb{S}_{+}^{n_{z}},\boldsymbol{Q}_{2}\in\mathbb{S}_{+}^{n_{x}},$

with

$\boldsymbol{P}:=\begin{bmatrix}\boldsymbol{Y}\boldsymbol{A}&\boldsymbol{Y}\boldsymbol{B}_{u}\\\
\boldsymbol{V}&-\textbf{diag}\left(\boldsymbol{\omega}_{c}\right)\end{bmatrix},$

where $\lambda_{a}\geq 0$, $\lambda_{\boldsymbol{\omega}_{c}}\geq 0$, and
$\lambda_{\boldsymbol{x}_{F}}\geq 0$ are user-defined weights, and
$\boldsymbol{W}_{d}$ is a given scaling matrix for $\boldsymbol{d}(t)$. The
controller gain can be recovered as
$\boldsymbol{K}:=\textbf{diag}\left(\boldsymbol{\omega}_{c}\right)^{-1}\boldsymbol{V}$.

###### Proof.

The condition for bounding the $\mathcal{H}_{2}$ norm from
$\begin{bmatrix}\boldsymbol{\bar{d}}\\\
\boldsymbol{\bar{w}}_{a}\end{bmatrix}\to\boldsymbol{z}$ is given by
$\boldsymbol{X}\boldsymbol{M}_{11}+\boldsymbol{M}^{T}_{11}\boldsymbol{X}+\boldsymbol{X}\boldsymbol{M}_{12}\begin{bmatrix}\boldsymbol{W}_{d}^{2}&0\\\
0&\boldsymbol{W}^{2}_{a}\end{bmatrix}\boldsymbol{M}^{T}_{12}\boldsymbol{X}\leq
0$, and
$\textbf{tr}\left[\boldsymbol{M}_{21}\boldsymbol{X}^{-1}\boldsymbol{M}^{T}_{21}\right]\leq\gamma$
[15], which for the problem considered are the following LMIs

$\displaystyle\begin{bmatrix}\boldsymbol{P}+\boldsymbol{P}^{T}&\begin{bmatrix}\boldsymbol{Y}\boldsymbol{B}_{d}&\boldsymbol{Y}\boldsymbol{B}_{u}\\\
\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}\\\
\ast&-\begin{bmatrix}\boldsymbol{W}_{d}^{-2}&\boldsymbol{0}\\\
\boldsymbol{0}&\textbf{diag}\left(\boldsymbol{\kappa}_{a}\right)\end{bmatrix}\end{bmatrix}\leq
0,$
$\displaystyle\begin{bmatrix}\boldsymbol{Q}_{1}&\boldsymbol{C}_{z}&\boldsymbol{0}\\\
\ast&\boldsymbol{Y}&0\\\ \ast&\ast&\boldsymbol{I}\end{bmatrix}\geq
0,\textbf{tr}\left[\boldsymbol{Q}_{1}\right]\leq\gamma.$

Similar to theorem 1,
$\textbf{tr}\left[\boldsymbol{V}^{T}\boldsymbol{V}\right]\leq\gamma_{\boldsymbol{x}_{F}}$
is equivalent to $\begin{bmatrix}\boldsymbol{Q}_{2}&\boldsymbol{V}^{T}\\\
\boldsymbol{V}&\boldsymbol{I}\end{bmatrix}\geq 0$ and
$\textbf{tr}\left[\boldsymbol{Q}_{2}\right]\leq\gamma_{\boldsymbol{x}_{F}}$.
Maximum actuator degradation is achieved by minimizing
$\lambda_{a}\left\|\boldsymbol{\kappa}_{a}\right\|_{2}+\lambda_{\boldsymbol{\omega}_{c}}\left\|\boldsymbol{\omega}_{c}\right\|_{2}+\lambda_{\boldsymbol{x}_{F}}\gamma_{\boldsymbol{x}_{F}}$,
where $\lambda_{a}$, $\lambda_{\boldsymbol{\omega}_{c}}$, and
$\lambda_{\boldsymbol{x}_{F}}$ are user-defined weights. ∎

######  Remark 3.

In both theorem 1 and 2, the minimum control rate is given by the elements
$\boldsymbol{\omega}_{c}$, the minimum control magnitude is characterized by
$\gamma_{\boldsymbol{x}_{F}}$, and the maximum actuator noise is
characaterized by the scaling
$\textbf{diag}\left(1/\sqrt{\boldsymbol{\kappa}}\right)$, where the square-
root and the inverse are defined to be element-wise.

Therefore, the optimal $\boldsymbol{\omega}_{c}$,
$\gamma_{\boldsymbol{x}_{F}}$, and $\boldsymbol{\kappa}$ quantities maximum
actuator degradation (as defined above) for which a state-feedback controller
is able to achieve the user specified-performance in terms of the closed-loop
systems $\mathcal{H}_{2}$ or $\mathcal{H}_{\infty}$ norms.

##  IV Examples

###  IV-A F-16 Flight Control Application

We apply the results to the design of a full-state feedback controller for a
model of the longitudinal motion of the F-16 aircraft. The model is based on
the aerodynamics from the NASA Technical Report 1538 [16]. The nonlinear model
is trimmed at $10000$ $ft$ at $900$ $ft/s$ for steady-level flight. The
longitudinal states are $\theta$ (pitch angle), $V_{t}$ (total velocity),
$\alpha$ (angle-of-attack), $q$ (pitch rate); with trim value
$\bar{\theta}=5.95^{\circ}$, $\bar{V}_{t}=900\;ft/s$,
$\bar{\alpha}=5.95^{\circ}$, $\bar{q}=7.85\;deg/s$. The control variables are
$T$ (thrust), $\delta_{e}$ (elevator angle), and $\delta_{lef}$ (leading
flap); with trim values $\bar{T}=10461.84\;lb$,
$\bar{\delta}_{e}=-3.82^{\circ}$, and $\delta_{lef}=12.42^{\circ}$. We assume
the aircraft experiences disturbance in the angle-of-attack $\alpha$, with
$W_{d}=0.01$.

The open-loop system in (2) is given by the following system matrices

$\displaystyle\boldsymbol{A}:=\begin{bmatrix}0.0&0.0&0.0&1.0\\\
-32.1699&-0.0358&-131.646&-3.1099\\\ 0.0&-0.0002&-1.5333&0.9281\\\
0.0&0.0003&-4.6719&-1.9076\end{bmatrix},$
$\displaystyle\boldsymbol{B}_{u}:=\begin{bmatrix}0.0&0.0&0.0\\\
0.0016&0.0525&0.1574\\\ -0.0&-0.0031&0.0008\\\
0.0&-0.4503&-0.0614\end{bmatrix},$
$\displaystyle\boldsymbol{B}_{d}:=\begin{bmatrix}0\\\ 1\\\ 0\\\
0\end{bmatrix}.$

The control objective is to keep perturbations in the flight-path angle
$\theta-\alpha$ and $V_{t}$ to be small due to the disturbance in $V_{t}$, in
the $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ sense. Therefore,

$\displaystyle\boldsymbol{C}_{z}:=\boldsymbol{W}_{z}\begin{bmatrix}1&0&-1&0\\\
0&1&0&0\end{bmatrix},\boldsymbol{D}_{d}:=\begin{bmatrix}0\\\ 0\end{bmatrix},$

where
$\boldsymbol{W}_{z}:=\textbf{diag}\left(\begin{bmatrix}11.46&0.1\end{bmatrix}\right)$,
scales $\boldsymbol{z}$ such that the desired closed-loop performance, defined
by the $\mathcal{H}_{2}$ or $\mathcal{H}_{\infty}$ norm of the transfer
function from $\begin{bmatrix}\boldsymbol{\bar{d}}\\\
\boldsymbol{\bar{w}}_{a}\end{bmatrix}\to\boldsymbol{z}$, is bounded by one.

Results from theorems 1 and 2, will quantify the maximum actuator degradation
in thrust ($T$), elevator ($\delta_{e}$), and leading-edge flap
($\delta_{lef}$).

###  IV-B Simulation Results

The numerical values for the minimum actuator cut-off frequencies and dc
gains, and maximum actuator noises are shown in table I, and are plotted in
Fig.3, Fig.4, and Fig.5.

|  $\boldsymbol{\omega}^{\ast}_{c}$ (rad/s) | $\|\boldsymbol{x}\to x_{F_{i}}\|^{\ast}_{\infty}$ | $1/\sqrt{\kappa_{i}^{\ast}}$
---|---|---|---
Actuator | $\mathcal{H}_{2}$ | $\mathcal{H}_{\infty}$ | $\mathcal{H}_{2}$ | $\mathcal{H}_{\infty}$ | $\mathcal{H}_{2}$ | $\mathcal{H}_{\infty}$
$T$ | 0.0033 | 2.4381 | 2.4945 | 0.0207 | 0.7157 | 0.0402
$\delta_{e}$ | 12.9757 | 16079.2678 | 16.1579 | 0.1707 | 0.0246 | 0.0014
$\delta_{lef}$ | 2.0541 | 2033.7134 | 17.8508 | 0.4800 | 0.0468 | 0.0027
TABLE I: Parameters that quantify maximum actuator degradation for a given
closed-loop performance ($\gamma$ = 0.5).

as

Figure 3: Minimum actuator cutoff frequencies (rad/s). Figure 4: Minimum
actuator dcgain. Figure 5: Maximum actuator noise scaling. Figure 6: Time
Response to Gust, $d(t)=15\text{wn}(t)+\sin(0.075t)$, where $\text{wn}(t)$ is
Gaussian white noise.

Table 1 indicates that the control rate for thrust is sigificantly lower than
that for the elevator and leading-edge flap, with the elevator requiring a
higher actuation rate compared to the leading-edge flap. Furthermore, the
actuation rate resulting from the $\mathcal{H}_{2}$ formulation is
substantially lower than that from the $\mathcal{H}_{\infty}$ formulation.
This discrepancy arises because the exogenous inputs are presumed to be broad-
band, and the $\mathcal{H}_{\infty}$ formulation attenuates disturbances
across this broad range. Employing a frequency-weighted model for the
exogenous signal is anticipated to yield improved results and will be explored
in our future work.

The control magnitude, which is quantified by the $\mathcal{H}_{\infty}$ norm
of the actuator dynamics, is significantly higher for the $\mathcal{H}_{2}$
closed-loop performance than the $\mathcal{H}_{\infty}$ closed-loop
performance. For the $\mathcal{H}_{2}$ case, the elevator’s and the leading-
edge-flap’s magnitude are comparable and significantly higher than thrust
magnitude. For the $\mathcal{H}_{\infty}$, we see similar trend, but orders of
magnitude smaller.

The final two columns of Table 1 present the noise scaling for actuator
disturbances. It is noted that the noise levels for the elevator and the
leading-edge flap are relatively low, indicating that these actuators require
high precision to achieve the specified closed-loop performance. Additionally,
the noise levels for the $\mathcal{H}_{\infty}$ case are substantially lower
than those for the $\mathcal{H}_{2}$ case, a disparity attributed to
considering broad-band disturbances in the problem.

Based on these observations, we see that the $\mathcal{H}_{2}$ performance
requires lower control rate and higher control magnitude with less precisions,
and the $\mathcal{H}_{\infty}$ performance requires much higher control rate
and lower control magnitude with high precisions.

The time responses to the disturbance for both the open-loop system and the
closed-loop system with actuator degradations are shown in Fig.6. It is
observed that the $\mathcal{H}_{2}$ optimal controller exhibits superior
disturbance attenuation compared to the $\mathcal{H}_{\infty}$ controller.
This outcome aligns with expectations, as $\left\|z\right\|_{\infty}$ is
bounded in the former case, while $\left\|z\right\|_{2}$ is bounded in the
latter.

##  V Summary & Conclusions

In this paper, we investigate the quantification of maximum actuator
degradation in linear dynamical systems for a given closed-loop performance.
We introduce a novel unified framework for computing the state-feedback
controller gain, which meets a user-specified closed-loop performance
criterion while maximizing actuator degradation. This degradation is modeled
as a first-order filter with additive noise. We present two new convex
optimization formulations that simultaneously determine the controller gain,
maximize actuator degradation, and ensure the desired closed-loop performance
in both the $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ system norms. The
results are limited to open-loop stable systems.

We apply our findings to the design of a full-state feedback controller for
the longitudinal motion of the F-16 aircraft. Our observations indicate that
the $\mathcal{H}_{2}$ performance requires a lower control rate and higher
control magnitude with less precision, while the $\mathcal{H}_{\infty}$
performance demands a higher control rate and lower control magnitude with
high precision. The $\mathcal{H}_{\infty}$ outcomes are conservative due to
the broad-band disturbance models. We aim to extend this to frequency-weighted
input-output formulations in future work.

## References

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# ReCo: Reliable Causal Chain Reasoning via Structural Causal Recurrent Neural
Networks

Kai Xiong1111This work was conducted during the internship of Kai Xiong at
Huawei Cloud Xiao Ding1222Corresponding Author Zhongyang Li2 Li Du1 Ting Liu1
Bing Qin1 Yi Zheng2 Baoxing Huai2
1Research Center for Social Computing and Information Retrieval
Harbin Institute of Technology, China
2Huawei Cloud, China
{kxiong, xding, ldu, tliu<EMAIL_ADDRESS>
{lizhongyang6, zhengyi29<EMAIL_ADDRESS>
###### Abstract

Causal chain reasoning (CCR) is an essential ability for many decision-making
AI systems, which requires the model to build reliable causal chains by
connecting causal pairs. However, CCR suffers from two main transitive
problems: threshold effect and scene drift. In other words, the causal pairs
to be spliced may have a conflicting threshold boundary or scenario. To
address these issues, we propose a novel Reliable Causal chain reasoning
framework (ReCo), which introduces exogenous variables to represent the
threshold and scene factors of each causal pair within the causal chain, and
estimates the threshold and scene contradictions across exogenous variables
via structural causal recurrent neural networks (SRNN). Experiments show that
ReCo outperforms a series of strong baselines on both Chinese and English CCR
datasets. Moreover, by injecting reliable causal chain knowledge distilled by
ReCo, BERT can achieve better performances on four downstream causal-related
tasks than BERT models enhanced by other kinds of knowledge.

##  1 Introduction

Causal chain reasoning aims at understanding the long-distance causal
dependencies of events and building reliable causal chains. Here, _reliable_
means that events in the causal chain can naturally occur in the order of
causal evolution within some circumstance based on the commonsense Roemmele et
al. (2011). Causal chain knowledge is of great importance for various
artificial intelligence applications, such as question answering Asai et al.
(2019), and abductive reasoning Du et al. (2021a). Many studies focus on the
reliability of causal pair knowledge but ignore that of causal chain
knowledge, especially in the natural language processing (NLP) community.

Previous works mainly acquire causal chain knowledge by first extracting
precise causal pairs from text with rule-based Heindorf et al. (2020); Li et
al. (2020) or neural-based Ding et al. (2019); Zhang et al. (2020) methods,
then connecting these causal pairs into causal chains based on the textual or
semantic similarity between events. However, this straightforward approach may
bring some transitive problems Johnson and Ahn (2015), leading to unreliable
causal chains, which would hinder causal-enhanced models to get higher
performances. For example, given a cause event: _“playing basketball”_ , and
two candidate effect events: _“gets a technical foul”_ and _“gets a red card”_
, an unreliable causal chain (_“playing basketball”_ $\rightarrow$_“dispute”_
$\rightarrow$_“gets a red card”_) would mislead the model to choose the less
plausible effect _“gets a red card”_.

Figure 1: Causal chains with (a) threshold effect and (b) scene drift
problems, which can be estimated by the contradictions of threshold and scene
factors in the contexts, respectively.

Among these transitive problems Johnson and Ahn (2015), threshold effect and
scene drift are the most two salient ones. As shown in Figure 1 (a), given two
causal pairs (A causes B, and B causes C), the threshold effect problem is
that the influence of A on B is not enough for B to cause C. We can notice
that, _“swimming in the sea”_ can only result in tens of milliliters of _“salt
water intake”_ , while _“dehydration”_ is caused by hundreds of milliliters of
_“salt water intake”_. Therefore, _“salt water intake”_ conditioned on
_“swimming in the sea”_ cannot lead to _“dehydration”_. Similarly, as shown in
Figure 1 (b), the scene drift problem means that A $\rightarrow$ B and B
$\rightarrow$ C would not happen within the same specific scene. These two
_“dispute”_ events are wrongly joined together by their surface forms.
_“Dispute”_ that happened in a video game scene cannot lead to _“gets a red
card”_ in a football match scene. Therefore, we find that the threshold effect
and scene drift problems are caused by the contradictions between the
threshold factors and between the scene factors, respectively.

To address these two issues, in ReCo, we first build a structural causal model
(SCM) Pearl (2009) for each causal chain, and the SCM introduces exogenous
variables to represent the threshold and scene factors of the causal pairs
within the causal chain. Then, we conduct an exogenous-aware conditional
variational autoencoder (EA-CVAE) to implicitly learn the semantic
representations of exogenous variables according to the contexts of the causal
pairs. Subsequently, we devise a novel causal recurrent neural network named
SRNN to estimate the contradictions between the exogenous variables by
modeling the semantic distance between them. Finally, we present a task-
specific logic loss to better optimize ReCo.

Extensive experiments show that our method outperforms a series of baselines
on both Chinese and English CCR datasets. The comparative experiments on
different lengths of the causal chains further illustrate the superiority of
our method. Moreover, BERT Devlin et al. (2019) injected with reliable casual
chains distilled by ReCo, achieves better results on four downstream causal-
related tasks, which indicates that ReCo could provide more effective and
reliable causal knowledge. The code is available on https://github.com/Waste-
Wood/ReCo.

Figure 2: (a) Constructing SCM based on an antecedent causal chain and a
causal pair. (b) If there is threshold effect or scene drift problem, then
$U_{xy}$ would contradict $U_{yx}$. And it is worth discussing the threshold
effect problem when scenes are consistent.

##  2 Background

###  2.1 Problem Definition

In this paper, the CCR task is defined as a binary classification problem.
Specifically, input a reliable antecedent causal chain
($x_{1}\rightarrow\cdots\rightarrow x_{n}$) and a causal pair
($x_{n}\rightarrow x_{n+1}$), the model needs to output whether the causal
chain $x_{1}\rightarrow\cdots\rightarrow x_{n}\rightarrow x_{n+1}$ is reliable
or not.

###  2.2 Structural Causal Model

Structural Causal Model (SCM) was proposed by Pearl (2009), which is a
probabilistic graph model that represents causality within a single system.
SCM is defined as an ordered triple $<U,V,E>$, where $U$ is a set of exogenous
variables determined by external (implicit) factors of the system. $V$ is a
set of endogenous variables determined by internal (explicit) factors of the
system. $E$ is a set of structural equations, each structural equation
represents the probability of an endogenous variable with the variables in $U$
and $V$.

Figure 3: (a) The overall architecture of ReCo. (b) The detailed structure of
EA-CVAE. (c) The detailed structure of SRNN which is a kind of recurrent
neural networks.

As shown in Figure 2 (a), given two causal pairs ($x\rightarrow y$ and
$y\rightarrow z$), they can be connected into a causal chain ($x\rightarrow
y\rightarrow z$). We construct an SCM for this causal chain. Events ($x,y$ and
$z$) are the endogenous variables, and exogenous variables ($U_{xy}$,
$U_{yz}$) contain the threshold and scene factors of the causal pairs. Each
structural equation represents the probability of an endogenous variable in
$V=\\{x,y,z\\}$ (eg. $P$($y$|$x,U_{xy}$)). And as shown in Figure 2 (b), if
the causal chain possesses threshold effect or scene drift problem, there are
contradictions between $U_{xy}$ and $U_{yz}$.

##  3 Method

###  3.1 Overview

In this paper, we devise ReCo to estimate the reliability of the input causal
chain. Figure 3 (a) shows the architecture of ReCo, which consists of four
components: (1) an encoder to encode causal events and their contexts into
dense vectors; (2) an exogenous-aware CVAE to capture the exogenous variables
with contexts; (3) an SRNN to understand the causal chain along the direction
of causality in the constructed SCM and solve the two transitive problems with
two designed estimators; (4) a predictor to predict the existence of the two
transitive problems and the reliability of the causal chain.

###  3.2 Encoder

Given a reliable antecedent causal chain ($X_{1}\rightarrow\cdots\rightarrow
X_{4}$) and a causal pair ($X_{4}\rightarrow X_{5}$), the inputs of ReCo are a
causal chain ($X_{1}\rightarrow\cdots\rightarrow X_{5}$) with 5 events and
their 4 corresponding contexts ($C_{1},\cdots,C_{4}$). $C_{i}$ denotes the
context of the causal pair $X_{i}\rightarrow X_{i+1}$. We first construct an
SCM for each causal chain, which introduces exogenous variables
$U=\\{U_{1},\cdots,U_{4}\\}$ to represent the threshold and scene factors of
the causal pairs. The endogenous variables are the events
$X=\\{X_{1},\cdots,X_{5}\\}$ in the causal chain. Then we use BERT to encode
the input events and contexts.

Specifically, we concatenate the events and their contexts into two sequences:
_[CLS] $X_{1}$ [SEP] $X_{2}$ [SEP] $X_{3}$ [SEP] $X_{4}$ [SEP] $X_{5}$ [SEP]_,
and _[CLS] $C_{1}$ [SEP] $C_{2}$ [SEP] $C_{3}$ [SEP] $C_{4}$ [SEP]_. The final
hidden states of _[SEP]_ tokens are set as the initial representations of the
corresponding events and contexts. Then we scale them to $m$-dimension.
Finally, we acquire event embeddings
$H_{X}=\\{h_{1},h_{2},h_{3},h_{4},h_{5}\\}$ and context embeddings
$H_{C}=\\{h_{1}^{C},h_{2}^{C},h_{3}^{C},h_{4}^{C}\\}$, where
$h_{i},h_{i}^{C}\in\mathbb{R}^{m}$ denote the $i$-th event and context,
respectively.

###  3.3 Exogenous-aware CVAE

Since the exogenous variables are hard to explicitly capture and CVAE has
shown its ability to implicitly estimate variables Chen et al. (2021); Du et
al. (2021a). Thus, we devise an EA-CVAE to capture the exogenous variables
based on each causal pair and its corresponding contexts.

The EA-CVAE takes a causal pair and its corresponding context as inputs, and
outputs the distribution of the exogenous variable. For example, as shown in
Figure 3 (b), given a causal pair $h_{i}\rightarrow h_{i+1}$ and the
corresponding context $h_{i}^{C}$, we first concatenate
$h_{i},h_{i+1},h_{i}^{C}$ into $V_{i}=[h_{i};h_{i+1}]\in\mathbb{R}^{2m}$ and
$V^{\prime}_{i}=[h_{i};h_{i}^{C};h_{i+1}]\in\mathbb{R}^{3m}$. Hereafter,
$V_{i}$ and $V^{\prime}_{i}$ are fed into the linear layers to estimate the
mean and standard deviation values of the exogenous variable distribution:

$\displaystyle\mu_{i}=W_{1}V_{i}+b_{1},$ (1)
$\displaystyle\sigma_{i}=\text{exp}(W_{2}V_{i}+b_{2}),$
$\displaystyle\mu^{\prime}=W_{3}V_{i}^{\prime}+b_{3},$
$\displaystyle\sigma_{i}^{\prime}=\text{exp}(W_{4}V_{i}^{\prime}+b_{4}),$

where $W_{1},W_{2}\in\mathbb{R}^{2m\times m}$ and
$W_{3},W_{4}\in\mathbb{R}^{3m\times m}$ are trainable parameters. The size of
the multivariate normal distribution is set as $m$. Finally, we obtain two
multivariate normal distributions $\mathcal{N}_{i}(\mu_{i},\sigma_{i}^{2})$
and $\mathcal{N}_{i}^{\prime}(\mu_{i}^{\prime},\sigma^{\prime 2}_{i})$.

After that, we conduct reparameterization trick to sample exogenous variables
from $\mathcal{N}_{i}(\mu_{i},\sigma_{i}^{2})$ and
$\mathcal{N}_{i}^{\prime}(\mu_{i}^{\prime},\sigma^{\prime 2}_{i})$. First, we
sample a value $\epsilon$ from the standard normal distribution
$\mathcal{N}(\mathbf{0},\mathbf{I})$, and then we obtain the representation
$u_{i}\in\mathbb{R}^{m}$ of the exogenous variable $U_{i}$ based on
$\epsilon$:

$\displaystyle
u_{i}=\begin{cases}\mu_{i}^{\prime}+\epsilon\times\sigma^{\prime}_{i}\quad\text{training},\\\
\mu_{i}+\epsilon\times\sigma_{i}\quad\text{prediction}.\end{cases}$ (2)

Hereafter, for each causal pair, we get the representation of its
corresponding exogenous variable and obtain
$u=\\{u_{1},u_{2},u_{3},u_{4}|u_{i}\in\mathbb{R}^{m}\\}$. In the training
stage, $u_{i}\in u$ is sampled from
$\mathcal{N}_{i}^{\prime}(\mu_{i}^{\prime},\sigma^{\prime 2}_{i})$. While in
the prediction stage, $u_{i}$ is sampled from
$\mathcal{N}_{i}(\mu_{i},\sigma_{i}^{2})$. Thus, contexts are not required as
inputs in the prediction stage.

Finally, we obtain the representations of the endogenous and exogenous
variables in the SCM.

###  3.4 Structural Causal Recurrent Neural Networks

We propose SRNN to measure the reliability of the causal chain, and estimate
the two transitive problems by measuring the semantic distance between the
exogenous variables. As shown in Figure 3 (c), the SRNN consists of the
following five components. The input of the SRNN in the first recurrent step
is a quintuple $<h_{1},h_{2},h_{3},u_{1},u_{2}>$.

##### Scene Drift Estimator

We design this component to estimate the scene drift problem between two
exogenous variables:

$\displaystyle\alpha_{1}=\sigma(W_{m1}u_{1}+b_{m1}-W_{m2}u_{2}-b_{m2}),$ (3)

where $\alpha_{1}\in\mathbb{R}^{m}$ is the measurement of the scene drift
problem. $W_{m1},W_{m2}\in\mathbb{R}^{m\times m}$ are trainable parameters,
and $\sigma$ is the sigmoid function.

##### Hidden Gate

Hidden gate is used for aggregating the information within the endogenous
variables for the next recurrent step of the SRNN and estimating the threshold
effect problem:

$\displaystyle h_{2}^{\prime}=\tanh{(W_{h}[h_{1};h_{2}]+b_{h})},$ (4)
$\displaystyle h_{3}^{\prime}=\tanh{(W_{h}[h_{2};h_{3}]+b_{h})},$ (5)

where $h_{2}^{\prime},h_{3}^{\prime}\in\mathbb{R}^{m}$ are the aggregated
endogenous variables, and $W_{h}\in\mathbb{R}^{2m\times m}$ is a trainable
parameter.

##### Threshold Effect Estimator

Since the threshold effect problem can be discussed iff the scene is
consistent, and threshold factors are event-specific, we can estimate the
threshold effect problem with the endogenous and exogenous variables based on
the result of the scene drift estimator.

$\displaystyle\beta_{1}\\!=\\!\sigma(W_{\beta}([u_{2};h_{3}^{\prime}]\\!-\\![u_{1};h_{2}^{\prime}])\\!\odot\\!(1-\alpha_{1})),$
(6)

where $\beta_{1}\in\mathbb{R}^{m}$ estimates whether the threshold effect
problem exists, and $W_{\beta}\in\mathbb{R}^{2m\times m}$ is a trainable
parameter.

##### Exogenous Gate

$u_{1}$ contradicts $u_{2}$ if there is threshold effect or scene drift
problem. We can learn the contradiction of $u_{1}$ on $u_{2}$ by:

$E_{1}=\tanh(W_{e}(u_{2}+\frac{\alpha_{1}+\beta_{1}}{2}\odot u_{1})+b_{e}),$
(7)

where $E_{1}\in\mathbb{R}^{m}$ is the representation of the contradiction of
$u_{1}$ on $u_{2}$, and $W_{e}\in\mathbb{R}^{m\times m}$ is a trainable
parameter. If there are not threshold effect and scene drift problems,
$\alpha_{1}$ and $\beta_{1}$ are equal to 0, and $E_{1}$ is close to $u_{2}$.

##### Output Gate

For the inputs of the next recurrent step of the SRNN, we compose $u_{1}$ and
$E_{1}$ into $u_{2}^{\prime}$.

$\displaystyle u_{2}^{\prime}=\tanh(W_{o}[u_{1};E_{1}]+b_{o}),$ (8)

where $u_{2}^{\prime}\in\mathbb{R}^{m}$ is the aggregated exogenous variable,
and $W_{o}\in\mathbb{R}^{m\times m}$ is a trainable parameter.

Finally, we denote $<h_{2}^{\prime},h_{3}^{\prime},h_{4},u_{2}^{\prime},$
$u_{3}>$ as the input to the next recurrent step of the SRNN.

###  3.5 Problems and Reliability Predictor

After the SRNN, we can obtain the final output
$<\alpha_{3},\beta_{3},h_{4}^{\prime},h_{5}^{\prime},E_{3}>$. First, we can
measure the existence of the threshold effect and scene drift problems based
on $\beta_{3}$ and $\alpha_{3}$, respectively:

$\displaystyle P^{T}$ $\displaystyle=\text{Softmax}(W_{T}\beta_{3}+b_{T}),$
(9) $\displaystyle P^{S}$
$\displaystyle=\text{Softmax}(W_{S}\alpha_{3}+b_{S}),$

where
$P^{T}=[P_{0}^{T};P_{1}^{T}],P^{S}=[P_{0}^{S};P_{1}^{S}]\in\mathbb{R}^{2}$ are
the probability distributions of threshold effect and scene drift problems,
respectively. The subscript $0$ and $1$ of $P^{T}$ and $P^{S}$ denote the non-
existence and existence probabilities of the corresponding problems.
$W_{S},W_{T}\in\mathbb{R}^{m\times 2}$ are trainable parameters. Therefore, we
can explain why this causal chain breaks according to $P^{T}$ and $P^{S}$.

Finally, we can measure the reliability of the causal chain as follows:

$\displaystyle P^{1}$
$\displaystyle=\tanh(W_{1}[h_{4}^{\prime};u_{3}]+b_{1}),$ (10) $\displaystyle
P^{2}$ $\displaystyle=\tanh(W_{2}[h_{5}^{\prime};E_{3}]+b_{2}),$
$\displaystyle P^{C}$
$\displaystyle=\text{Softmax}(W_{C}[P^{1};P^{2}]+b_{C}),$

where $P^{1},P^{2}\in\mathbb{R}^{m}$ are the intermediate parameters,
$P^{C}=[P_{0}^{C};P_{1}^{C}]\in\mathbb{R}^{2}$ is the probability distribution
of the reliability of the causal chain $X_{1}\rightarrow\cdots\rightarrow
X_{5}$, and $P_{0}^{C},P_{1}^{C}\in\mathbb{R}^{1}$ denote the probabilities
that the causal chain is unreliable and reliable, respectively.
$W_{1},W_{2}\in\mathbb{R}^{2m\times m}$ and $W_{C}\in\mathbb{R}^{m\times 2}$
are trainable parameters.

###  3.6 Optimizing with a Logic Loss

We design a logic loss to reduce the loss function from 4 parts to 3 parts.
For example, if the causal chain is reliable, the probabilities that the two
problems do not exist and the causal chain is reliable should be equal.
Therefore, the logic loss is:

$L_{\text{Logic}}=|\log(P^{T}_{0}\times P^{S}_{0})-\log(P^{C}_{1})|,$ (11)

where $P_{o}^{T}$ and $P_{0}^{S}$ are the probabilities that the threshold
effect and scene drift problems do not exist, and $P^{C}_{1}$ is the
probability that the causal chain is reliable. Moreover, if the causal chain
is unreliable due to the scene drift problem, the logic loss is
$L_{\text{Logic}}=|\log(P^{S}_{1}\times P^{T}_{0})-\log(P^{C}_{0})|$.

Finally, the loss function is denoted as:

$\displaystyle
L=L_{\text{Chain}}+\lambda_{1}L_{\text{Logic}}+\lambda_{2}L_{\text{kl}},$ (12)
$\displaystyle L_{\text{Chain}}=\text{CrossEntropy}(Y,P^{C}),$ $\displaystyle
L_{\text{kl}}=\sum_{i=1}^{4}\text{\text{KL}}(\mathcal{N}(\mu_{i},\sigma_{i}^{2})||\mathcal{N}(\mu_{i}^{\prime},\sigma_{i}^{\prime
2})),$

where $L_{\text{Chain}}$ is the loss of the causal chain reliability.
$L_{\text{Logic}}$ is the logic loss. $L_{\text{kl}}$ is the Kullback-Leibler
divergence loss Hershey and Olsen (2007) of the EA-CVAE. $\lambda_{1}$ and
$\lambda_{2}$ are loss coefficients.

##  4 Experiments

###  4.1 CCR Datasets Construction

We choose the Chinese causal event graph CEG Ding et al. (2019) and English
CauseNet Heindorf et al. (2020) to obtain unlabeled causal chain reasoning
examples.

We first use Breadth-First Search on CEG and CauseNet to retrieve 2,911 and
1,400 causal chains with contexts, respectively. Each causal chain has 5
events, and no more than three events are overlapped between any two causal
chains.

Then, we label the causal chains through crowd-sourcing. Professional
annotators are asked to label the first causal relationship where the causal
chain breaks, and which problem (threshold effect or scene drift) causes this
break. Each chain will be labeled by three annotators, the Cohen’s agreement
scores are $\kappa$ = 78.21% and 75.69% for Chinese and English CCR datasets,
respectively.

We split the causal chains into different lengths of training examples
(Instance-3, Instance-4, Instance-5). If a causal chain of length 5 breaks at
the third causal relationship, 1 positive and 1 negative training examples are
constructed (positive: $X_{1}\rightarrow X_{2}\rightarrow X_{3}$; negative:
$X_{1}\rightarrow X_{2}\rightarrow X_{3}\rightarrow X_{4}$). The statistics of
the two CCR datasets are shown in Table 1. Refer to Appendix B for Chinese and
English CCR examples.

CCR | Train | Dev | Test
---|---|---|---
Zh | Chain | 2,131 | 290 | 490
Instance-3 | 2,131 | 290 | 490
Instance-4 | 1,552 | 207 | 324
Instance-5 | 1,077 | 139 | 188
Total | 4,760 | 636 | 1,002
En | Chain | 1,037 | 139 | 224
Instance-3 | 1,037 | 139 | 224
Instance-4 | 829 | 109 | 164
Instance-5 | 612 | 80 | 105
Total | 2,478 | 328 | 493
Table 1: Statistics of CCR datasets. Chain denotes the causal chains retrieved
from causal event graphs. Instance-3, Instance-4 and Instance-5 denote the
instance with chain lengths of 3, 4 and 5, respectively.

###  4.2 Baselines

We compare the performance of ReCo against a variety of sequence modeling
methods, and causal reasoning methods developed in recent years. In Embedding
and ExCAR, for a causal chain $X_{1}\rightarrow\cdots X_{n}$, we treat
$X_{1}\rightarrow\cdots X_{n-1}$ and $X_{n}$ as the cause and effect,
respectively.

##### Embedding

Xie and Mu (2019) measures word-level causality through causal embedding. We
choose the max causality score between cause and effect words, and apply a
threshold for prediction.

##### LSTM

Hochreiter and Schmidhuber (1997) is a recurrent neural network. We use BiLSTM
to represent the causal chains for binary classification.

##### BERT

Devlin et al. (2019); Cui et al. (2020) is pre-trained unsupervised with
massive unlabeled data. Specifically, we use BERT-base to represent the causal
chains for the reliability classification.

##### ExCAR

(Du et al., 2021b) introduces evidence events for explainable causal
reasoning. We introduce evidence events to the cause-effect pair for ExCAR
experiments.

##### CausalBERT

(Li et al., 2021) injects massive causal pair knowledge into BERT. CausalBERT
is used to represent the causal chain for experiments.

We use precision, recall, F1 score, and accuracy to measure the performance of
each method.

###  4.3 Training Details

For ReCo, we use the pre-trained BERT-base Devlin et al. (2019); Cui et al.
(2020) as the encoder to encode events and contexts. The batch size is set to
24, the dimension $m$ is 256, we choose Adam Kingma and Ba (2014) as the
optimizer with a learning rate of 1$e$-5. The loss coefficients $\lambda_{1}$
and $\lambda_{2}$ are 1 and 0.01, respectively. ReCo runs 50 epochs on two
Tesla-P100-16gb GPUs.

CCR | Methods | P | R | F1 | Acc %
---|---|---|---|---|---
Zh | Embedding | 61.30 | 82.75 | 70.43 | 58.18
LSTM | 63.64 | 83.58 | 72.26 | 61.38
BERT | 64.85 | 86.90 | 74.27 | 63.77
ExCAR | 63.97 | 86.57 | 73.57 | 62.57
CausalBERT | 64.53 | 87.23 | 74.19 | 63.47
ReCo (Ours) | 66.50 | 87.56 | 75.59 | 65.97
En | Embedding | 65.30 | 81.17 | 72.55 | 59.63
LSTM | 71.13 | 85.19 | 77.53 | 67.55
BERT | 72.75 | 84.88 | 78.35 | 69.17
ExCAR | 73.33 | 84.88 | 78.68 | 69.78
CausalBERT | 72.38 | 87.35 | 79.16 | 69.78
ReCo (Ours) | 74.03 | 87.96 | 80.39 | 71.81
Table 2: Overall results on the CCR test sets.

###  4.4 Overall Results

We implement Embedding, LSTM, BERT, ExCAR, CausalBERT and ReCo on both Chinese
and English CCR datasets. The overall results are shown in Table 2, from which
we can observe that:

(1) Comparing word-level method (Embedding) to event-level methods (LSTM,
BERT, ExCAR, CausalBERT and ReCo), event-level methods achieve absolute
advantages, which indicates that considering the causality between words and
ignoring the semantics of events is not better for CCR.

(2) Knowledge-enhanced methods (ExCAR and CausalBERT) achieve comparable
results to BERT. This is mainly because not all the evidence events in ExCAR
are reliable, and CausalBERT only possesses causal pair knowledge, making
ExCAR and CausalBERT struggle with the CCR tasks.

(3) ReCo outperforms BERT, ExCAR and CausalBERT in F1 score and accuracy,
which shows that the exogenous variables captured by the EA-CVAE are
significant to conducting CCR tasks and the SRNN is important to address the
two transitive problems. Moreover, the advantage of ReCo is mainly reflected
in precision, it is because capturing the threshold and scene factors is
effective to measure the transitive problems and estimate the reliability of
the causal chains.

(4) All six methods get lower precision scores on the Chinese CCR test set
than that on the English CCR test set. This is mainly because all events in
the Chinese CCR dataset are sentences, and most of the events in the English
CCR dataset consist of only one word, making the Chinese CCR task more
challenging and more complex.

Figure 4: Accuracy on (a) Chinese and (b) English CCR test sets categorized by
the lengths of the causal chains.

Moreover, we also compare ReCo with baselines on different lengths of causal
chains. Results are illustrated in Figure 4. We can observe that:

(1) Most of the methods perform worse as the chain gets longer. It indicates
that longer instances need stronger CCR ability.

Datasets | BERT${}_{\text{O}}$ | BERT${}_{\text{P}}$ | BERT${}_{\text{C}}$ | BERT${}_{\text{R}}$
---|---|---|---|---
Event StoryLine v0.9***Only the intra-sentence event pairs are kept for experiments and the train, dev, test sets are split randomly. We also ensure the cause event precedes the effect event. Caselli and Vossen (2017) (F1 %) | 66.84 | 68.08 | 69.05 | 70.66
BeCAUSE 2.1 Dunietz et al. (2017) (Accuracy %) | 79.17 | 81.94 | 83.33 | 83.80
COPA Roemmele et al. (2011) (Accuracy %) | 73.80 | 74.00 | 74.20 | 75.40
CommonsenseQA Talmor et al. (2019) (Accuracy %) | 54.71 | 54.87 | 55.04 | 55.12
Table 3: Overall results of causal knowledge injection. The evaluation metrics
are computed based on manually split test sets (Event StoryLine v0.9, BeCAUSE
2.1), official test (COPA) and dev (CommonsenseQA) sets.

(2) ReCo performs best on almost all instance levels of both CCR datasets.
This is mainly because the threshold and scene factors captured by the EA-CVAE
are important for CCR tasks, and the SRNN can properly capture the two
transitive problems by estimating the semantic distance between threshold
factors or scene factors. However, CausalBERT achieves the best performance on
the instance-5 of the English CCR test set. This is mainly because Instance-5
in the English CCR dataset might rely more on massive external causal pair
knowledge.

(3) Compared with the results on Instance-4, results drop more on the English
Instance-5 than that on the Chinese Instance-5. The reason is that conducting
CCR on the causal chains with five or more word-level events might need more
extra information to reason from the first event to the last event.

###  4.5 Causal Knowledge Injection

Methods | Accuracy %
---|---
BERT | 69.17
-w context | 69.37
ReCo | 71.81
-w/o EA-CVAE | 68.97
-w/o Problems Estimators | 70.18
-w/o Logic Loss | 70.18
Table 4: Overall results of the ablation study on the English CCR test set.
“w” and “w/o” denote “with” and “without”, respectively.

To further investigate the effectiveness of ReCo, we inject different kinds of
causal knowledge into BERT. Then following Du et al. (2022), we test the
causal-enhanced BERT models on four NLP benchmark datasets: a causal
extraction dataset Event StoryLine v0.9 Caselli and Vossen (2017), two causal
reasoning datasets BeCAUSE 2.1 Dunietz et al. (2017) and COPA Roemmele et al.
(2011), as well as a commonsense reasoning dataset CommonsenseQA Talmor et al.
(2019). To give a careful analysis, we inject causal knowledge into BERT in
the following four different ways (The details of knowledge injection can
refer to Appendix C):

$\bullet$ BERT${}_{\text{O}}$ injected with no external knowledge.

$\bullet$ BERT${}_{\text{P}}$ injected with causal pair knowledge.

$\bullet$ BERT${}_{\text{C}}$ injected with unfiltered causal chain knowledge.

$\bullet$ BERT${}_{\text{R}}$ injected with causal chain knowledge distilled
by ReCo.

The results are shown in Table 3, from which we can observe that:

(1) Methods (BERT${}_{\text{P}}$, BERT${}_{\text{C}}$, BERT${}_{\text{R}}$)
enhanced with causal knowledge outperform the original BERT
(BERT${}_{\text{O}}$) on all four tasks, which indicates that causal knowledge
can provide extra information to conduct causal-related tasks.

(2) Comparisons between causal chain knowledge enhanced methods
(BERT${}_{\text{C}}$, BERT${}_{\text{R}}$) and causal pair knowledge enhanced
method (BERT${}_{\text{P}}$) show that BERT${}_{\text{C}}$ and
BERT${}_{\text{R}}$ can push the model to a higher level than
BERT${}_{\text{P}}$ on all four tasks. The main reason is that causal chains
contain more abundant knowledge than causal pairs.

(3) Unfiltered causal chain knowledge enhanced method (BERT${}_{\text{C}}$)
performs worse than BERT${}_{\text{R}}$ injected with causal chain knowledge
distilled by ReCo. The main reason is that some unfiltered causal chains would
be unreliable due to the threshold effect or scene drift problem, which would
mislead the model to choose the wrong answer.

###  4.6 Ablation Study

We provide ablation studies to show the superiority and effectiveness of ReCo.
First, we provide the contexts of the causal pairs to BERT to prove the
advantages of the EA-CVAE and SRNN in ReCo. Second, we remove the EA-CVAE in
ReCo and set the contexts as the exogenous variables to investigate the effect
of the EA-CVAE. Third, we remove the extra supervised signals of the problem
estimators to study the effect of the problem estimators. Finally, we replace
the logic loss with cross-entropy losses to validate the effectiveness of the
logic loss. Overall results are shown in Table 4. From which we can find that:

(1) After providing contexts to BERT, the performance of BERT increases
slightly, which shows that there is effective information in the contexts to
conduct CCR tasks, but BERT cannot use it sufficiently. This proves that
properly utilizing information in the contexts is of great importance.

(2) After removing EA-CVAE, the performance of ReCo drops and is worse than
BERT. This is because the contexts are not proper estimations of the exogenous
variables and there is also noise in the contexts which has negative impacts
on ReCo.

(3) After removing the supervised signals of the problem estimators, RoCo
performs worse, it indicates the problem estimators supervised by the extra
problem labels are important to measure the existence of the two transitive
problems. Moreover, ReCo without the extra supervised signals outperforms
contexts-enhanced BERT, which indicates the EA-CVAE in ReCo can properly
estimate the exogenous variables with contexts, and the SRNN in ReCo plays an
important role in deeply understanding the causal chain.

(4) After replacing the logic loss with cross-entropy losses, the performance
of ReCo drops 1.63 in accuracy, which indicates that the logic constraints
applied by the logic loss can guide ReCo to better generalization.

###  4.7 Case Study

To intuitively investigate whether ReCo can discover the right problem when
the causal chain is unreliable, we provide an example made by ReCo. As shown
in Table 5, “ _bacteria_ ” in a cosmetic scene caused by “ _acne_ ” cannot
lead to “ _salmonellosis_ ”. ReCo gives the right label and the right problem
which causes the causal chain unreliable. Refer to Appendix E for more cases.

production of sebum$\rightarrow$ acne $\rightarrow$ bacteria $\rightarrow$
salmonellosis
---
ReCo Prediction | Unreliable
Scene Drift | True
Threshold Effect | False
Table 5: An example made by ReCo. ReCo makes the right prediction and gives
the reason why this chain breaks: _“salmonellosis”_ will not happen in the
scene where _“acne”_ causes _“bacteria”_.

##  5 Related Work

###  5.1 Causal Knowledge Acquisition

Causal knowledge is crucial for various artificial intelligence applications.
Many works Heindorf et al. (2020); Zhang et al. (2020) extract large-scale and
precise causal pairs through neural or symbolic ways. Hereafter, they connect
causal pairs into causal chains or graphs based on the textual or semantic
similarity between events Chang and Choi (2004); Li et al. (2020); Hashimoto
et al. (2014).

Luo et al. (2016) used linguistic patterns Chang and Choi (2004) to construct
CausalNet. Heindorf et al. (2020) built CauseNet from web resources. Rashkin
et al. (2018) constructed Event2mind and Sap et al. (2019) built Atomic both
through crowd-sourcing. Zhang et al. (2020) proposed a large-scale eventuality
knowledge graph called ASER. Li et al. (2020) built CausalBank to improve the
coverage of the causal knowledge base.

Previous studies mainly focused on extracting high-precision causal pairs,
while ignoring the transitive problems when connecting event pairs into causal
chains. We are trying to solve the two transitive problems in generating
reliable causal chains.

###  5.2 Causal Reasoning

Causal reasoning aims at grasping the causal dependency between cause and
effect, which consists of statistical-based and neural-based methods.

As for statistical-based methods, Gordon et al. (2011) measured PMI based on a
personal story corpus and then measured causality between words with PMI. Luo
et al. (2016) and Sasaki et al. (2017) introduced direction information into
causal strength index. Then they infer causality between events by combining
the causality of word pairs.

Many neural-based methods introduce the semantics of events to measure the
causality of causal pairs. Of late, Xie and Mu (2019) proposed to measure
word-level causality with an attention-based mechanism. Wang et al. (2019) and
Li et al. (2019) finetuned the pre-trained language model to resolve causal
reasoning task and achieve impressive results. Li et al. (2021) injected a
vast amount of causal pair knowledge into the pre-trained language model and
got a noticeable improvement in COPA Roemmele et al. (2011) causal reasoning
task. Du et al. (2021b) introduced evidence events to the causal pairs and
used a conditional Markov neural logic network to model the causal paths
between cause and effect events, to achieve stable and self-explainable causal
reasoning. Du et al. (2022) introduced general truth to event pair for
investigating explainable causal reasoning.

Most of the above causal reasoning studies focus on causal pair reasoning,
while we are trying to solve the reliable causal chain reasoning.

##  6 Conclusion

We explore the problem of causal chain reasoning and propose a novel framework
called ReCo to overcome the two main transitive problems of threshold effect
and scene drift. ReCo first constructs an SCM for each causal chain, the SCM
introduces exogenous variables to represent the threshold and scene factor of
the causal pairs, and then conducts EA-CVAE to implicitly learn the
representations of the exogenous variables with the contexts. Finally, ReCo
devises SRNN to estimate the threshold and scene contradictions across the
exogenous variables. Experiments show that ReCo can achieve the best CCR
performances on both Chinese and English datasets.

##  7 Acknowledgments

We would like to thank Bibo Cai and Minglei Li for their valuable feedback and
advice, and the anonymous reviewers for their constructive comments, and
gratefully acknowledge the support of the Technological Innovation “2030
Megaproject” - New Generation Artificial Intelligence of China
(2018AAA0101901), and the National Natural Science Foundation of China
(62176079, 61976073).

##  8 Limitations

There may be some possible limitations in this study. First, the threshold and
scene factors are hard to explicitly capture, which might hinder ReCo to
achieve higher performances. Second, due to the loss function possessing three
components and the nature of CVAE, it needs more attempts to reach convergence
in training. Third, due to the nature of the CEG, each causal pair consists of
only one context. Having multiple contexts for each causal pair would be
better to cover more conditions as well as capture the threshold effect and
scene drift problems more precisely. Moreover, it would be better to have
larger CCR datasets. Future research should be undertaken to explore a more
efficient and general model architecture as well as obtain larger Chinese and
English CCR datasets with higher agreements and multiple contexts.

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##  Appendix A CEG Construction

CEG (Chinese Event Graph) Ding et al. (2019) is a large-scale and open-domain
causal event graph, which consists of more than 1.6 million events and 3.6
million cause-effect edges. We list the steps of constructing CEG as follows:

1) Crawling news documents from the news websites (such Netease
news†††https://news.163.com/, Tencent news‡‡‡https://news.qq.com/, etc.).

2) Conducting causal pairs extraction through sequence labeling
($B_{\text{Cause}},I_{\text{Cause}},B_{\text{Effect}},I_{\text{Effect}},O$),
training data are annotated through crowdsourcing.

3) Event similarity computation through Jaccard similarity coefficient
Niwattanakul et al. (2013) and event clustering using a threshold.

4) Extracting common elements from events (make sure that at least a verb and
a noun are kept) in the same cluster to generalize events.

5) Connecting event pairs into causal chains and CEG.

##  Appendix B CCR Examples

Examples of Chinese and English CCR are shown in Table 6 and Table 7,
respectively. $2$ and $4$ in the labels represent the causal chain that will
meet problems at the second and the fourth causal relationship, respectively.
And the problem types are threshold effect and scene drift for Chinese and
English CCR examples, respectively.

Events |  |  A: 销量下滑
---
B: 市场竞争加剧
C: 深圳发展
D: 城市化进程快
E: 水源水质差
Contexts |  |  A $\rightarrow$ B: 销量下滑导致了终端市场竞
---
争加剧
B $\rightarrow$ C: 通信市场竞争加剧将有助于
深圳的通信设备业发展
C $\rightarrow$ D: 深圳的向西发展使得宝安的
城市化进程越来越快
D $\rightarrow$ E: 水源水质极差的原因是周边
城市化进程较快
Label | 2
Wrong Type | Threshold Effect
Table 6: An example in the Chinese CCR dataset. Events |  |  A: Tired at work
---
B: Relax
C: Playing games
D: Dispute
E: Sent off by a red card
Contexts |  |  A $\rightarrow$ B: Tired at work makes me need
---
to relax at weekends.
B $\rightarrow$ C: Tom wants to relax by playing
games.
C $\rightarrow$ D: Jack and Mike dispute because
of playing games.
D $\rightarrow$ E: David Silver gets a red card
because of the dispute with the referee.
Label | 4
Wrong Type | Scene Drift
Table 7: An example in the English CCR dataset.

##  Appendix C Causal Knowledge Injection

We use the English CCR dataset for different knowledge injections: causal pair
knowledge (BERT${}_{\text{P}}$), unfiltered causal chain knowledge
(BERT${}_{\text{C}}$), and causal chain knowledge distilled by ReCo
(BERT${}_{\text{R}}$). All the models are based on BERT-base Devlin et al.
(2019).

###  C.1 Knowledge Injection Settings

For BERT${}_{\text{P}}$, we split causal chains in English CCR datasets into
causal pairs, and for each causal pair, we randomly sample a cause event or
effect event from other causal pairs to obtain negative samples. The cause
together with the effect event will be concatenated and sent into the pre-
trained BERT, then we use the representation of _[CLS]_ token in the last
hidden state for binary classification.

For BERT${}_{\text{C}}$, we split causal chains in the English CCR dataset
into causal chains of length 2 to 5. As for negative samples, for each causal
chain, we sample an event from another causal chain to replace the first or
last event of the causal chain. The events in a causal chain will be
concatenated into a sequence and sent into the pre-trained BERT, then we use
the representation of _[CLS]_ token in the last hidden state for binary
classification.

For BERT${}_{\text{R}}$, we split causal chains filtered by ReCo into causal
chains of length 2 to 5. As for negative samples, for each causal chain, we
sample an event from another causal chain to replace the first or last event
of the causal chain. The events in a causal chain will be concatenated into a
sequence and sent into the pre-trained BERT, then we use the representation of
_[CLS]_ token in the last hidden state for binary classification.

###  C.2 Knowledge Injection Details

For all methods (BERT${}_{\text{P}}$, BERT${}_{\text{C}}$ and
BERT${}_{\text{R}}$), we use the base version of BERT Devlin et al. (2019).
The batch size is 36, and we use Adam Kingma and Ba (2014) optimizer with the
learning rate of 1$e$-5. All three models are pre-trained for 2 epochs.

###  C.3 Downstream Tasks Finetuning

Datasets | Train | Dev | Test
---|---|---|---
Event StoryLine v0.9 | 8,279 | 1,034 | 1,034
BeCAUSE 2.1 | 1,741 | 216 | 216
COPA | 450 | 50 | 500
CommonsenseQA | 9,741 | 1,221 | -
Table 8: Statistics of Event StoryLine v0.9 Caselli and Vossen (2017), BeCUASE
2.1 Dunietz et al. (2017), COPA Roemmele et al. (2011), CommonsenseQA Talmor
et al. (2019) datasets.

####  C.3.1 Dataset Settings

$\bullet$ Event StoryLine v0.9 Caselli and Vossen (2017) For the Event
StoryLine v0.9 dataset, we only keep the intra-sentence causal pairs and
ensure that the cause event precedes the effect event. Finally, we randomly
split the filtered causal pairs into train, dev, test sets.

$\bullet$ BeCAUSE 2.1 Dunietz et al. (2017) For the BeCAUSE 2.1 dataset, we
first extract event pairs from the annotated data, then we manually split the
event pairs into train, dev, test sets.

$\bullet$ COPA Roemmele et al. (2011) For the COPA dataset, for the reason
that COPA does not have a training set, we randomly sample 90% of the dev set
for training, the remaining 10% as the new dev set.

$\bullet$ CommonsenseQA Talmor et al. (2019) For the CommonsenseQA dataset, we
use the dev set for testing due to the test set of CommonsenseQA is a blind
set.

The statistics of the four datasets are shown in Table 8.

####  C.3.2 Finetuning

We finetune BERT${}_{\text{O}}$, BERT${}_{\text{P}}$, BERT${}_{\text{C}}$ and
BERT${}_{\text{R}}$ on the above four downstream tasks.

For Event StoryLine v0.9 and BeCAUSE 2.1, we concatenate the event pair into a
sequence and send it into the above four models, then the representation of
_[CLS]_ in the last hidden state is used for binary classification. We use F1
score and accuracy as the evaluation metrics of Event StoryLine v0.9 and
BeCAUSE 2.1, respectively.

For COPA and CommonsenseQA tasks, we concatenate the premise (question)
together with one of the hypotheses (alternatives) and feed it into all four
models, then we use the _[CLS]_ token in the last hidden state for
classification. We use accuracy as the evaluation metric for both COPA and
CommonsenseQA.

As for the finetuning settings of the above four models, the batch size is set
to 40, and we use Adam Kingma and Ba (2014) optimizer with the learning rate
of 1$e$-5. An early-stopping mechanism is applied for finetuning.

##  Appendix D Ablation Study

###  D.1 EA-CVAE

For investigating the importance of EV-CVAE in ReCO, we remove the EA-CVAE
component in ReCo, and for constructing the SCM Pearl (2009), we use the
contexts of the causal pairs as the exogenous variables in the SCM. Other
components of ReCo are not changed and the training settings are the same as
the original ReCo.

###  D.2 Problems Estimators

We devise two problem estimators to estimate threshold effect and scene drift
problems. For investigating the importance of these two problem estimators, we
remove the supervised signal (by removing $L_{\text{logic}}$ in the loss) of
Threshold and Scene Estimators in the SRNN, the parameters of the two problem
estimators are only tuned by the final reliability prediction task (note that
EA-CVAE are kept for training, and the tuning of Kullback-Leibler divergence
loss Hershey and Olsen (2007) will not change the parameters in the two
problem mechanisms). The model architecture and the training settings of this
setting are the same as the original ReCo.

###  D.3 Logic Loss

The logic loss is used to apply a logic constraint on the predictions of ReCo.
When the causal chain is reliable, both the threshold effect and scene drift
problems do not exist. Moreover, when the causal chain is unreliable, one of
the transitive problem should exist. For investigating the effect of the logic
loss, we replace the logic loss with two cross entropy losses. One of the
cross-entropy loss is conducted to supervise the threshold effect problem, and
the other is used to supervise the scene drift problem.

##  Appendix E Case Study

reading$\rightarrow$ myopia $\rightarrow$ problems $\rightarrow$ stress
---
ReCo Prediction | Unreliable
Scene Drift | False
Threshold Effect | True
Table 9: An example made by ReCo. Reco makes the right prediction and gives
the reason why this chain is unreliable. “ _Problems_ ” conditioned on “
_reading_ ” and “ _myopis_ ” is not enough to lead to “ _stress_ ”.

volume growth$\rightarrow$ revenue growth $\rightarrow$ improvement
$\rightarrow$ energy savings
---
ReCo Prediction | Unreliable
Scene Drift | True
Threshold Effect | False

Table 10: An example made by ReCo. Reco makes the right prediction and gives
the reason why this chain is unreliable. “ _Energy savings_ ” will not happen
in the scene of “ _volume growth_ ”$\rightarrow$“ _revenue growth_ ”
$\rightarrow$“ _improvements_ ”.

We provide another two English examples predicted by ReCo. The examples of
threshold effect and scene drift problems are shown in Table 9 and Table 10,
respectively.