# Unusual heat transport of the Kitaev material Na2Co2TeO6: putative quantum
spin liquid and low-energy spin excitations

Xiǎochén Hóng,${}^{1,2,}^{,}$111These authors contributed
<EMAIL_ADDRESS>Matthias Gillig,1,∗ Richard Hentrich,1,∗
Weiliang Yao,3 Vilmos Kocsis,1 Arthur R. Witte,1,4 Tino Schreiner,1 Danny
Baumann,1 Nicolás Pérez,1 Anja U. B. Wolter,1 Yuan
<EMAIL_ADDRESS>Bernd Büchner,1,4,6 and Christian
<EMAIL_ADDRESS>1Leibniz-Institute for Solid State and
Materials Research (IFW-Dresden), 01069 Dresden, Germany
2Fakult$\ddot{a}$t für Mathematik und Naturwissenschaften, Bergische
Universit$\ddot{a}$t Wuppertal, 42097 Wuppertal, Germany
3International Center for Quantum Materials, School of Physics, Peking
University, 100871 Beijing, China
4Institute of Solid State and Materials Physics and Würzburg-Dresden Cluster
of Excellence $ct.qmat$, Technische Universit$\ddot{a}$t Dresden, 01062
Dresden, Germany
5Collaborative Innovation Center of Quantum Matter, 100871 Beijing, China
6Center for Transport and Devices, Technische Universit$\ddot{a}$t Dresden,
01069 Dresden, Germany

###### Abstract

We studied the field dependent thermal conductivity ($\kappa$) of Na2Co2TeO6,
a compound considered as the manifestation of the Kitaev model based on the
high-spin $d^{7}$ Co2+ ions. We found that in-plane magnetic fields beyond a
critical value $B_{c}\approx$ 10 T are able to drastically enhance $\kappa$ at
low temperatures, resulting in a double-peak structure of $\kappa(T)$ that
closely resembles the behavior of $\alpha$-RuCl3. This result suggests that
heat transport in Na2Co2TeO6 is primarily phononic, and it is strongly
affected by scattering from magnetic excitations that are highly tunable by
external fields. Interestingly, for magnetic fields
$\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}$ (i.e., along the zigzag
direction of the Co-Co bonds), there is an extended field range which
separates the long-range magnetic order for $B\leq B_{c}\approx 10$ T and the
partially spin-polarized gapped high-field phase for $B\gtrsim 12$ T. The low-
energy phonon scattering is particularly strong in this field range,
consistent with the notion that the system becomes a quantum spin liquid with
prominent spin fluctuations down to energies of no more than 2 meV.

###### pacs:

not needed

A compass model of the effective spin$-1/2$ two-dimensional honeycomb lattice,
known as the Kitaev model, is an emerging research field Kitaev ; JvdB ;
Trebst ; JPCM . Topological quantum spin liquids (QSL) with Majorana fermions
as excitations are guaranteed as its ground states Kitaev . To accommodate
pseudospin$-1/2$ magnetism and bond-dependent exchange couplings in real
materials, heavy transition metal compounds in the low-spin $d^{5}$ electron
configuration are preferred TakagiRev ; JK . Guided by theoretical predictions
JK ; Rau ; ChenG , Ir4+ oxides and Ru3+ chloride have been studied extensively
Takagi ; Na213 ; a213 ; b213 ; g213 ; Cu213 ; Ag213 ; H213 ; CuLi213 ; CaoHB ;
Baenitz . Those studies imply that in real materials, the Heisenberg
interactions ($J$) and off-diagonal terms ($\Gamma$) have to be also taken
into account in the Hamiltonian. Thus, instead of a pure Kitaev QSL, the
ground states of these compounds are usually some form of magnetic order JPCM
. Nevertheless, the Kitaev physics is still expected to be prominent in
$proximate$ Kitaev materials as long as the Kitaev term ($K$) is relatively
large Perkins ; Pollmann .

Very recently, $\alpha$-RuCl3 became the most representative proximate Kitaev
material. Regardless of its zigzag ordered ground state in zero field,
evidences of Kitaev-Heisenberg excitations (i.e. the exotic magnetic
excitations associated with the prominence of Kitaev interactions) have been
reported by diverse experimental tools Raman ; NS-NM ; NS-NP ; NS-PRL ; NS-S ;
Loidl ; THz ; Wellm ; Richard ; LeeMY ; Shibauchi ; MiaoH . More
interestingly, its magnetic order can be melted by a moderate in-plane field
Baek ; Sears ; Anja ; Richard ; LeeMY ; WangZ , and the sought-after half-
integer quantization of the thermal Hall conductivity was claimed in a narrow
field range above this critical point Kasahara . In the high field limit,
$\alpha$-RuCl3 further enters into a partially-polarized phase with a spin
excitation gap which opens up linearly in field HF ; Richard ; Baek ; Anja ;
WangZ .

The rich temperature-field phase diagram of the $JK\Gamma$ model suggested by
$\alpha$-RuCl3 is exciting, in particular due to the putative QSL phase
surrounding the quantum critical point that separates the long-range magnetic
order from the partially field-polarized phase (see Fig. 1). However, it
remains to be clarified, which of the features in the phase diagram are
related to the $JK\Gamma$ model, and which of them are barely material-
specific properties. The latter is a particularly complex issue since
$\alpha$-RuCl3 is sensitive to disorder and suffers from stacking faults and
twinnings Vojta ; Coldea ; Yamashita . In this regard, it is useful if a
sibling Kitaev material can be coordinatively investigated and compared to
$\alpha$-RuCl3. Recently, two theoretical papers simultaneously proposed
another route to realize the Kitaev model in $d^{7}$ cobaltates LiuHM ; Motome
. Demonstrating Kitaev interactions in the high-spin $d^{7}$ systems not only
extends the terrain of candidate Kitaev compounds, but also introduces a
mechanism to strongly reduce the $J$ interactions, thus intensifying the
dominance of the $K$ term LiuHM ; Motome . In Ref. LiuHM , Na2Co2TeO6 was
explicitly pointed out to be such candidate. Stimulated by these predictions,
the overlooked Co-based frustrated materials again came into focus. Some
surprising results have already been reported in several honeycomb and
triangular Co-compounds since then ZRDd ; ZRDk ; ZRDt ; SunXF ; Trump ; Stock
; MaJ ; Park ; ChenWJ .

Figure 1: (Color online). Schematic sketch of the possible phase diagram of
the $JK\Gamma$ pseudospin-1/2 model, shaped by the studies mainly on
$\alpha$-RuCl3 JPCM ; TakagiRev . In-plane magnetic field can tune the zero-
field zigzag antiferromagnetic order into the high-field (partially) polarized
state with a gap which opens up linearly. A $Z_{2}$ QSL state is frequently
claimed to exist in a small field window at low temperature TakagiRev ; NS-NP
; Kasahara . The Kitaev-Heisenberg paramagnons are expected to survive up to
high temperatures comparable to the Kitaev interaction strength NS-S ; NS-NM ;
NS-NP ; Raman ; Richard ; JPCM .

In this letter, we report the thermal conductivity $\kappa$ of Na2Co2TeO6
single crystals from room temperature down to 6 K, with in-plane magnetic
fields $B$ up to 15 T. We unveil a strong impact of B on $\kappa$ at low
temperature: (i) beyond a magnetic field of $B_{c}\approx$ 10 T, $\kappa$
increases more than one order of magnitude, and (ii) the $\kappa$($T$) curves
at high fields exhibit a double-peak structure. Remarkably, this peculiar
$\kappa$($T,B$) profile is striking similar to that of $\alpha$-RuCl3 Richard
. Thus, our data strongly suggest a commonality in the physics of Na2Co2TeO6
and $\alpha$-RuCl3. We conclude in particular, that $\kappa$($T,B$) is
essentially phononic, where the predominant phonon scattering is caused by
Kitaev-Heisenberg-type excitations, similar to those of $\alpha$-RuCl3.

Figure 2: (Color online). (a) Temperature dependence of the thermal
conductivity of two Na2Co2TeO6 single crystals without magnetic field. The
heat current was applied in two orthogonal directions, parallel to the
armchair ($\kappa_{A}$) and zigzag ($\kappa_{Z}$) edges, respectively. Insert
shows the derivative of the $\kappa_{A}$($T$) curve around the magnetic
ordering temperature. (b) The ratio between $\kappa_{Z}$($T$) and
$\kappa_{A}$($T$). The dashed region indicates the three-dimensional
magnetically ordered state ChenWJ . The dotted line highlights the peak
temperature $T^{peak}$.  Figure 3: (Color online). Thermal conductivity of
Na2Co2TeO6 crystals in various magnetic fields applied along the two in-plane
high-symmetry directions. As the sketches show, the field was applied parallel
to the thermal gradient. (a) Fixed-field $\kappa$($T$) curves with
$\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}^{*}$. The field values (in
T) are indicated by the numbers next to each curve. Note that the $T$ axis is
plotted in log scale for clarity. (b) Magnified view of the low-temperature
part to show the low-field curves clearly. (c) Magnified view around the
magnetic transition $T_{N}\approx$ 26 K note . The zero-field $T_{N}$ is
indicated by the black arrow. (d) $\kappa$($B$) isotherms at selected
temperatures for $\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}^{*}$.
(e), (f), (g) and (h) show the same aspects for sample Z with
$\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}$. The slight increase of
$\kappa$ by suppressing the magnetic order is highlighted by the light green
region as an example for the 25 K isotherm in (h).

High-quality Na2Co2TeO6 single crystals were grown by a modified flux method
Yao . Two regular bar-shaped samples of $5\times 1\times 0.1$ mm3 were
employed in the thermal transport study. Due to the well-defined geometry, we
estimate the geometric error of $\kappa$ to be less than 10%. The longest
edges of the crystals were set parallel to the armchair (
$\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}^{*}$, Sample A) and zigzag (
$\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}$, Sample Z) directions of the Co-Co
bonds, respectively. Steady-state thermal conductivity measurements were
performed with the standard four point geometry. One side of the sample was
glued directly to the heat sink, and the other side was thermally excited by a
chip heater. The temperature gradient $\nabla T$ generated along the long edge
of the sample was measured by a differential AuFe/Chromel-P thermocouple which
had been calibrated carefully in magnetic field. A custom-built high-vacuum
low-noise probe provides the controlled heat sink temperature from 300 K to 6
K with a stability of about 0.1 mK. Magnetic fields up to 15 T were generated
by a commercial superconducting magnet. Fields were applied parallel to the
thermal gradient.

The temperature dependent thermal conductivity $\kappa$($T$) of both samples
in zero magnetic field are shown in Fig. 2(a). At first glance, these
$\kappa$($T$) curves resemble that of a conventional phononic heat conductor
unaffected by novel excitations Berman : here, $\kappa_{p}$($T$) $\approx
C_{p}\nu_{p}l_{p}$, with the phononic specific heat $C_{p}$, velocity
$\nu_{p}$, and mean free path $l_{p}$. At low temperature, $\kappa_{p}$
follows the specific heat, since $\nu_{p}$ and $l_{p}$ are practically
temperature independent, and thus rapidly grows with $T$. Towards higher $T$,
the reduction of the phonon mean free path $l_{p}$ by the sample-independent
phonon Umklapp processes dictates a $1/T$-like tail of $\kappa$($T$). A single
broad peak of $\kappa_{p}$($T$) is thus typically present at around 1/10 or
less of the Debye temperature $T^{peak}\lessapprox\Theta_{D}/10$ Berman .
However, a closer inspection of the results found such trivial interpretation
impeachable. The observed $T^{peak}\approx 45$ K seems abnormally high
considering $\Theta_{D}\approx$ 280 K which can be evaluated from the specific
heat data Yao . Additionally, as shown in Fig. 2(b), the
$\kappa_{Z}/\kappa_{A}$ ratio is essentially flat around $T^{peak}$, but drops
sharply at much lower $T$, suggesting $T^{peak}$ is not the result of a
canonical competition between growing $C_{p}$ and falling $l_{p}$ due to the
Umklapp scattering. Furthermore, the development of a three-dimensional long-
range magnetic order (see Ref. ChenWJ ) indeed has an impact on the
$\kappa$($T$) curves. The pertinent anomaly, albeit weak, manifests itself
clearly in the $T$-derivative of $\kappa$($T$) as shown in the inset of Fig.2.
For common magnets, including $\alpha$-RuCl3, a sharp upturn of $\kappa$($T$)
is expected below the magnetic transition temperature $T_{N}$, due to the
suppression of the phonon-magnon scattering in the magnetically ordered state
ZRDk ; Richard ; LeeMY . Notably, we find $\kappa$($T$) of Na2Co2TeO6 to
continue decreasing below $T_{N}$. In our scenario of phonon-dominated heat
transport which will be elaborated next, this difference may be attributed to
a substantially greater ratio between $k_{B}T_{N}$ ($T_{N}\sim 26$ K) and the
magnetic excitation gap ($\triangle\sim 1$ meV) of Na2Co2TeO6 than that of
$\alpha$-RuCl3 ($T_{N}\sim 7$ K and $\triangle\sim 2$ meV) ChenWJ ; NS-PRL ;
note .

Our above notion of predominant phonon heat transport, which is limited at low
$T$ by magnetic scattering, is further supported by the in-plane magnetic
field effects on $\kappa$. As depicted in Fig. 3, the $\kappa$($T$,$B$)
profiles are qualitatively similar between the two samples under B applied in
orthogonal in-plane directions, and also between Na2Co2TeO6 and $\alpha$-RuCl3
Richard . Magnetic fields have a negligible impact on the $\kappa$($T$) curves
above 50 K, but changes them profoundly at lower temperatures. The
$\kappa$($T$) curves soar in high fields ($B>$ 10 T), and display a double-
peak structure, resembling the peculiar features of $\alpha$-RuCl3 Richard .
The same holds for the low-temperature $\kappa$($B$) isotherms which increase
dramatically above a critical field $B_{c}\sim$ 10 T. The strong similarity of
our data to the findings for $\alpha$-RuCl3 therefore corroborates our above
notion of predominant phonon heat transport with a characteristic scattering
due to the Kitaev-Heisenberg excitations. These peculiar features presented by
$\kappa$($T$) and $\kappa$($B$) at high fields can straightforwardly be
explained by restoring the phonon conductivity via opening a gap for the
Kitaev-Heisenberg excitations in the partially-polarized high-field phase,
exactly the same mechanism established for explaining the mentioned features
in $\alpha$-RuCl3 Richard .

Figure 4: (Color online). Color-contour representation of the $T$ derivative
of $\kappa$($T,B$) with field (a)
$\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}^{*}$ and (b)
$\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}$. The magnetic transitions
were determined by thermodynamic probes note , and are represented by the blue
points. The evolution of the onset temperature $T_{N}$ ($\ast$) of three-
dimensional magnetic order is nearly the same for both B directions. A field-
induced canting reversal ($\times$) is found for
$\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}^{*}$ but cannot be
detected for $\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}$ Yao ; note .
There is one additional transition (+) recognizable only for
$\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}^{*}$ at low $T$ note . The
lines are guides to the eye. The gap energy $\hbar$$\omega_{0}$/$k_{B}$ in the
partially-polarized phase extracted from the Callaway fit of the data (see
text) are shown as black circles (squares) to the right ordinates.

These results are presented in a more intuitive way in Fig. 4 as a color
contour plot of $\delta\kappa$/$\delta$$T$, against the magnetic phase
boundaries extracted from the magnetization measurements note . Three main
regions (I, II and III) can be assigned by referring to the established
conclusions of $\alpha$-RuCl3 Richard . $Region$ $I$ represents the (canted)
antiferromagnetic order at low-temperatures and low-fields, enclosed by the
$T_{N}$($B$) line Yao . As mentioned above, the phase boundary at $T_{N}$ is
barely visible in $\kappa$($T$) curves, and the curves bend downwards in the
ordered phase. Instead of gapping out the Kitaev-Heisenberg excitations like
$\alpha$-RuCl3 Richard , spin-phonon scattering is enhanced by entering this
phase. Notably, an intermediate field region labeled as $Region$ $I^{*}$ can
be recognized below about 10 K for both B directions. It is featured by an
enhancement of $\kappa$, seen more clearly in the low $T$ $\kappa$($T$) curves
(Fig. 3(b) and 3(f)) and the $T=7$ K $\kappa$($B$) isotherms (Fig. 3(d) and
3(h)). It is obvious that the more complex magnetic phases detected via
thermodynamic studies for
$\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}^{*}$ confer more features
to the $\kappa$($B$) curve. We are aware that an intermediate-field phase was
also reported in recent $\alpha$-RuCl3 studies, but it only exists for a
certain B direction (corresponding to
$\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}$ in our representation)
Balz . One explanation for the features of Region I∗ is that magnetic texture
changes inside the canted AFM ordered state, which reduces the spin-phonon
scattering. Of course, it is also possible that additional transport channels
of magnons are activated inside Region I∗. Further studies on this region are
desired.

$Region$ $II$ is constituted by a narrow field range above $B_{c}\approx 10$ T
before the system enters the higher field phase (Region III). This Region II
is smoothly connected to the broader region at higher temperatures which is
dominated by the Kitaev-Heisenberg paramagnons. It is of particular interest
whether the ground state of this phase is a QSL as is conjectured for
$\alpha$-RuCl3 Kasahara . Indeed, while our data for
$\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}^{*}$ suggest a very narrow
width of a possible QSL state in Region II, the data for
$\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}$ leave a quite large range
of approximately 10 $-$ 11.5 T for it.

Finally, $Region$ $III$ is the high-field phase where the $\kappa$($T$) curves
acquire a double peak structure and $\kappa$($B$) exhibits a strong increase.
For $\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}^{*}$, Region III seems
to set in for $B>B_{c}\approx 10$ T, whereas for
$\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}$, our data suggest this
phase to appear at $B\gtrsim 11.5$ T. Both features imply the opening of a gap
for the spin excitations in high fields, connected to the partially polarized
phase HF ; Richard ; Baek ; Anja ; WangZ ; MV ; Winter . The inflection point
of $\kappa$($T$), $T_{min}$, can serve as a rough gauge of the energy of the
phonon-scattering magnetic modes Richard ; HF . $T_{min}$ manifests itself as
the zero contour line in Fig. 4. One can also see the tendency in Fig. 3 that
$T_{min}$ shifts to higher temperature in larger field.

We analyzed the high-field data based on the Callaway model as performed in
$\alpha$-RuCl3 Richard ; note . The extracted gap sizes
($\hbar\omega_{0}/k_{B}$) of magnetic excitations are summarized to the right
ordinates of Fig. 4. It is clear that fields applied in the two directions
open this gap roughly linearly. The $\hbar\omega_{0}/k_{B}$ $vs$ $B$ slopes
for the two field directions are slightly different, and both of them are much
larger than that of $\alpha$-RuCl3 Richard ; Anja ; Baek . In analogy to
$\alpha$-RuCl3 and according to the results of the theory and calculations for
the $JK\Gamma$ model, one might speculate that $\hbar\omega_{0}$ can be
identified with a van Hove singularity near the $\Gamma$ point Anja ; Winter ;
MV . We expect that our experimentally determined parameters of Na2Co2TeO6
will prove essential for a theoretical construction of its magnon spectrum and
for finding its place in the $JK\Gamma$ parameter space.

It is interesting to apply the information gained for Region III with respect
to the spin excitation gap to Region I and II, in particular at zero field. As
exhibited in Fig. 4, $T_{min}$ is related to the gap size by $\alpha
T_{min}=\hbar\omega_{0}/k_{B}$ with $\alpha\approx 3$ note . Since there is no
$T_{min}$ resolved in the zero field $\kappa$($T$) curves down to 6 K, we can
use this information to estimate that strong spin excitations exist at least
down to an energy scale of 2 meV, deep inside the magnetically ordered phase
in zero field. Besides, we stress that for
$\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}$, $\kappa$($T$) at
$B\approx B_{c}$ is virtually identical to that at zero field down to the
lowest $T$ (see Fig. 3(f)). This implies a similarly small energy scale for
the spin excitations in Region II, which underpins the possible realization of
a field-driven $U$(1) QSL state Hickey .

To summarize, we presented the thermal conductivity of a new Kitaev material
Na2Co2TeO6, and explained the data as phononic heat transport strongly
scattered by magnetic excitations. In-plane magnetic field confers different
ground states to Na2Co2TeO6. Strong scattering survives below $T_{N}$ in the
low-field magnetically ordered state Region I. A linearly opened excitation
gap is extracted from the unusual double-peak $\kappa$($T$) curves in the
high-field partially polarized state Region III. In between them is the most
exciting state Region II (and perhaps also Region I∗), where strong low-energy
spin fluctuations are present despite the absence of magnetic order.
Especially for $\textbf{B}\mathbin{\\!/\mkern-5.0mu/\\!}\textbf{a}$, we found
a relatively wide field window suitable for a QSL state. Overall, our results
support the conjecture that Na2Co2TeO6 is another materialization of the
Kitaev model. All the interesting results of $\alpha$-RuCl3 deserve seeking
for in Na2Co2TeO6, that will promote the understanding of proximate Kitaev
materials.

We would like to thank Vladislav Kataev, Christoph Wellm, and Xenophon Zotos
for fruitful discussions, and thank Juliane Scheiter for technical support.
This work has been supported by the Deutsche Forschungsgemeinschaft through
SFB 1143 (project-id 247310070), the Würzburg-Dresden Cluster of Excellence on
Complexity and Topology in Quantum Matter-ct.$qmat$ (EXC 2147, Project No.
390858490). This work has further been supported by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (Grant Agreement No. 647276-MARS-ERC-2014-CoG). Work at Peking
University was supported by the NSF of China under Grant No. 11888101, and by
the NBRP of China under Grant No. 2018YFA0305602.

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Utterance rewriting aims to recover coreferences and omitted information from the latest turn of a multi-turn dialogue. Recently, methods that tag rather than linearly generate sequences have proven stronger in both in- and out-of-domain rewriting settings. This is due to a tagger's smaller search space as it can only copy tokens from the dialogue context. However, these methods may suffer from low coverage when phrases that must be added to a source utterance cannot be covered by a single context span. This can occur in languages like English that introduce tokens such as prepositions into the rewrite for grammaticality. We propose a hierarchical context tagger (HCT) that mitigates this issue by predicting slotted rules (e.g., “besides ”) whose slots are later filled with context spans. HCT (i) tags the source string with token-level edit actions and slotted rules and (ii) fills in the resulting rule slots with spans from the dialogue context. This rule tagging allows HCT to add out-of-context tokens and multiple spans at once; we further cluster the rules to truncate the long tail of the rule distribution. Experiments on several benchmarks show that HCT can outperform state-of-the-art rewriting systems by $\sim$2 BLEU points.

§ INTRODUCTION

Modeling dialogue between humans and machines has long been an important research direction with high commercial value. Such tasks include dialogue response planning [Li et al., 2016], question answering [Reddy, Chen, and Manning, 2019] and semantic parsing in conversational settings [Yu et al., 2019]. Recent advances in deep learning and language model pre-training have greatly improved performance on many sentence-level tasks. However, machines are often challenged by coreference, anaphora, and ellipsis that are common in longer form conversations. Utterance rewriting [Kumar and Joshi, 2017, Su et al., 2019, Elgohary, Peskov, and Boyd-Graber, 2019] has been proposed to resolve these references locally by editing dialogues turn-by-turn to include past context. This way, models need only focus on the last rewritten dialogue turn. Self-contained utterances also let models leverage sentence-level semantic parsers [Zhang et al., 2019] for dialogue understanding.

Turn     Utterance

$1$     Why did Federer withdraw from the tournament?

$2$     He injured his back in yesterday's match.

$3$     Did he have any other injuries?

$3^\ast$     Did Federer have any other injuries besides his back?

Sample dialogue for utterance rewriting. Turns $1$–$2$ are the context, $3$ is the source, and $3^\ast$ is the target. The target is a context-independent rewrite of the source.

Much past work [Pan et al., 2019, Elgohary, Peskov, and Boyd-Graber, 2019] on this task frames it as a standard sequence-to-sequence (seq-to-seq) problem, applying RNNs or Transformers. This approach requires re-predicting tokens shared between source and target utterances. To ease the redundancy, models may include a copy mechanism [Gu et al., 2016, See, Liu, and Manning, 2017] that supports copying source tokens besides drawing from a separate vocabulary. Yet generating all target tokens from scratch remains a burden. Perhaps as a result, these early efforts do not generalize well between data domains. For instance, Hao et al., 2021 report a drop of 28 BLEU after transfer between two Chinese rewriting datasets [Su et al., 2019, Pan et al., 2019].

To exploit overlaps between source and target utterances, later work converts rewrite generation into source editing through sequence tagging. This tagging vastly simplifies the learning problem: predict a few fixed-length tag sequences, each with a small vocabulary. Hao et al., 2021 propose a system that predicts edit actions to (i) keep or delete a source token and (ii) optionally add a context span before the token. They rewrite Chinese datasets, where most targets can be covered by adding at most one context span per source token. Unfortunately, their single-span tagger is too brittle to insert out-of-context tokens or a series of non-contiguous spans, leading to low target phrase coverage.

Instead of separating edit action tagging from span insertion, Liu et al., 2020 predict a word-level edit matrix between context-source pairs. In contrast to [Hao et al., 2021], this approach can add arbitrary non-contiguous context phrases before each source token. Though it may cover more target phrases, an edit matrix involves $\mathcal{O}(m)$ times more tags than a sequence for $m$ context tokens. Its flexibility also makes it easier to produce ungrammatical outputs, since any subset of context tokens can be added to the source. Finally, Huang et al., 2021 combine a source sequence tagger with an LSTM-based decoder. However, reverting back to a seq-to-seq approach introduces the same large search space issue that sequence tagging was designed to avoid.

To predict added phrases, we would like to keep the small search space of a span predictor while extending it to (i) non-contiguous context spans and (ii) tokens missing from the context altogether. For (i), we first build a multi-span tagger (MST) that can autoregressively predict several context spans per source token. We use a syntax-guided method to automatically extract multi-span labels per target phrase. We further propose a hierarchical context tagger (HCT) that predicts a slotted rule per added phrase before filling the slots with spans. The slotted rules are learnt from training data and address (ii) since they can include out-of-context tokens (e.g., determiners and prepositions). By conditioning a multi-span predictor on a small set of slotted rules, HCT can achieve higher phrase coverage than MST. By first planning rules and then realizing their slots, HCT dramatically enhances the performance gains of MST.[Our code is available at <https://github.com/lisjin/hct>.]

MST reaches the best baseline performance for an open-domain query rewriting task and is competitive on simpler benchmarks in English and Chinese [Su et al., 2019]. HCT achieves state-of-the-art results two of the three benchmarks. It improves by 1.9 and 2.7 BLEU on  [Elgohary, Peskov, and Boyd-Graber, 2019] and  [Tseng et al., 2021]. In terms of robustness, it outperforms strong domain adaptation baselines on .

§ MULTI-SPAN TAGGER

Our initial model balances between high coverage of target strings and a small model search space. RaST [Hao et al., 2021] achieves the latter by allowing only one context span to be added before each source token. In contrast, RUN [Liu et al., 2020] sacrifices search space for higher coverage by letting any subset of the context string replace or precede a source span. We first propose a multi-span tagger (MST) that expands RaST's coverage and shrinks RUN's search space.

As seen in Figure <ref>, MST is composed of an action tagger on a source sequence and a semi-autoregressive span predictor over context utterances. It takes two token sequences as inputs: source $x = (x_1, \ldots, x_n)$ and context $c = (c_1, \ldots, c_m)$. For each source token, the action tagger decides whether or not to keep it—deleted tokens can later be replaced with context spans from the span predictor. In parallel, the multi-span predictor generates a variable-length sequence of context spans to insert before each source token.

§.§.§ Encoder
We adopt BERT [Devlin et al., 2019] as our encoder. The tokens from context utterances $c$ are concatenated with source $x$ and fed into the encoder:

\begin{align}
\label{eq:mst-enc}
\mathbf{E}_c; \mathbf{E}_x = \text{BERT}(c; x),
\end{align}

where $\mathbf{E}_c \in \mathbb{R}^{m \times d}$ and $\mathbf{E}_x \in \mathbb{R}^{n \times d}$ are the resulting $d$-dimensional contextualized embeddings. This way, global information from $c$ and $x$ is encoded into both $\mathbf{E}_c$ and $\mathbf{E}_x$.

§.§.§ Action Tagger
MST tags source token $x_i$ with a keep or delete action by linearly projecting its embedding $\mathbf{e}_i \in \mathbb{R}^d$, the $i$th row of $\mathbf{E}_x$:

\begin{align}
\label{eq:mst-act}
p(a_i \mid x_i) = \mathrm{Softmax}(\mathbf{W}_a\mathbf{e}_i),
\end{align}

where $\mathbf{W}_a \in \mathbb{R}^{2 \times d}$ is a learned parameter matrix.

[every node/.style=inner sep=.3em,align=center,
gra/.style=fill=gray!25,rounded corners=2pt,font=,
rnd/.style=draw,semithick,rounded corners=2pt,font=,
cir/.style=draw,semithick,circle,inner sep=.1em,font=]
[rnd,text width=5em] (x0) BERT;
[gra,below of=x0,yshift=.5em] (x1) $c\smash{;}\,x$;
[gra,above left of=x0,yshift=.5em] (x3) $\mathbf{E}\smash{_x}$;
[gra,above right of=x0,yshift=.5em] (x4) $\mathbf{E}\smash{_c}$;
[rnd,above of=x3,anchor=east,draw=cb-yellow,fill=cb-yellow!15] (x5) Action Tagger;
[rnd,above of=x3,anchor=west,draw=cb-red,fill=cb-red!15] (x6) Span Predictor;
[cir,left of=x0,anchor=east,xshift=-1em] (x8) 1;
[cir,above of=x3,yshift=1.5em] (x9) 2;

[above=.5em of x0.north] (aux);
[->] (x0) – (aux) -| (x3);
[->] (aux) -| (x4);
[->] (x1.north) edge (x0.south);
[->] (x3.north) edge (x5.south);
[->] (x3.north) edge (x6.south);
[->] (x4.north) edge (x6.south);

[every node/.style=inner sep=.3em,align=center,
gra/.style=fill=gray!25,rounded corners=2pt,font=,
rnd/.style=draw,semithick,rounded corners=2pt,font=,
cir/.style=draw,semithick,circle,inner sep=.1em,font=]
[rnd,text width=5em] (x0) BERT;
[gra,below of=x0,yshift=.5em] (x1) $c\smash{;}\,x$;
[gra,above left of=x0,yshift=.5em] (x3) $\mathbf{E}\smash{_x}$;
[gra,above right of=x0,yshift=.5em] (x4) $\mathbf{E}\smash{_c}$;
[rnd,above of=x3,anchor=east,draw=cb-yellow,fill=cb-yellow!15] (x5) Action Tagger;
[rnd,above of=x3,anchor=west,draw=cb-blue,fill=cb-blue!15] (x6) Rule Tagger;
[rnd,above=4em of x4,xshift=.7em,draw=cb-red,fill=cb-red!15] (x7) Span Predictor;
[cir,left of=x0,anchor=east,xshift=-1em] (x8) 1;
[cir,above of=x3,yshift=1.5em] (x9) 2;
[cir,below of=x7,xshift=.8em,yshift=1.45em] (x10) 3;

[above=.5em of x0.north] (aux);
[->] (x0) – (aux) -| (x3);
[->] (aux) -| (x4);
[->] (x1.north) edge (x0.south);
[->] (x3.north) edge (x5.south);
[->] (x3.north) edge (x6.south);
[->,bend right=2.25em] (x3.north) edge ($(x7.south east) - (0.2em, 0em)$);
[->] (x6) edge (x7.south);
[->] (x4.north) edge ($(x7.south east) - (0.2em, 0em)$);

Architecture comparison.

§.§.§ Span Predictor
The span predictor outputs at most $l$ spans $\{s_{ij}\}_{j \leq l}$ from context $c$ to insert before each source token $x_i$. It predicts these spans autoregressively; the $j$th span $s_{ij}$ depends on all previous spans $\{s_{ij^\prime}\}_{j^\prime < j}$ at position $i$:

\begin{align}
\label{eq:mst-sp}
p(s_{ij} \mid c, x_i, j) = \text{MST}_s(c, x_i, \{s_{ij^\prime}\}_{j^\prime < j}).
\end{align}

We model generation of span $s_{ij}$ as predicting its start and end indices in context $c$. These two indices are captured through separate distributions over positions of $c$, given token $x_i$. Specifically, we apply additive attention [Bahdanau, Cho, and Bengio, 2015] to let source embedding $\mathbf{e}_i$ attend to all context embedding rows of $\mathbf{E}_c$. For example, the $j$th start index at source position $i$ is predicted as

\begin{align}
\label{eq:sp-sing}
p(s_{ij}^\uparrow \mid c, x_i, j) = \text{Attn}^\uparrow(\mathbf{E}_c, \mathbf{u}_{ij}),
\end{align}

where the $\uparrow$ indicates the start index distribution. The end index ($\downarrow$) is analogous in form. The joint probability of all spans $\{s_{ij}\}_{j \leq l}$ at source index $i$, denoted by $s_i$, is

\begin{align}
\label{eq:sp-full}
p(s_i \mid c, x_i) &= \prod_{j = 1}^l p(s_{ij} \mid c, x_i),
\end{align}

\[p(s_{ij} \mid c, x_i) = p(s_{ij}^\uparrow \mid c, x_i, j)p(s_{ij}^\downarrow \mid c, x_i, j).\]

Because $s_{ij}$ depends on past spans indexed by $j^\prime < j$, the span predictor is considered semi-autoregressive for each source index $i$. Span prediction continues until either $j = l$ or $s^\uparrow_{ij}$ is a stop symbol (i.e., $0$), which can be predicted at $j = 0$ for an empty span. A span index at step $j$ depends on the attention distribution over context tokens at step $j - 1$:

\begin{align}
\mathbf{u}_{ij} &= \mathrm{ReLU}\left(\mathbf{W}_u\left[\hat{\mathbf{u}}_{ij}; \mathbf{u}_{i(j - 1)}\right]\right),\label{eq:sp-attn1}\\
\hat{\mathbf{u}}_{ij} &= \sum_{k \in [1, m]} \alpha_{i(j - 1)k} \cdot \mathbf{e}_k,\label{eq:sp-attn2}
\end{align}

where $\alpha_{i(j - 1)k}$ is the attention coefficient[Details can be found in Appendix <ref>.] between $c_k$ (embedded by $\mathbf{e}_k$) and $x_i$ and $\mathbf{W}_u \in \mathbb{R}^{d \times 2d}$. Similar to the notion of coverage in machine translation [Tu et al., 2016], this helps maintain awareness of past attention distributions.

[every node/.style=font=,inner sep=0.1em],

why     did     federer     withdraw     $\ldots$     ?     [SEP]     he     injured     his     back     …

1     2     3     4     5–7     8     9     10     11     12     13     14–18

[left=0.25em of x1] (x0) Context $c$;
[below=4.7em of x0] (x4) Rewrite $x^\ast$;
[below=3.7em of x0] (x3) Source $x$;
[below=1em of x1,anchor=north] (x2)

did     he     have     any     other     injuries                 ?

did     federer     have     any     other     injuries     besides     his     back      ?

3                         12     13

[on background layer]
[draw=none,fill=gray!12,rounded corners=2pt] (-3.4,-2.55) rectangle (-2.4,-.75);

[baseline=(c1.north),start chain=1 going below,start chain=2 going below,start chain=3 going below,start chain=4 going below,node distance=-.04em]
[every node/.style=semithick,inner sep=.45em,
in 1,2
[draw=cb-yellow,fill=,on chain=1] (a) ;

[lbl,left=.1em of a1] Action;
[lbl,right of=a1] ;
[lbl,right of=a2] ;
in 3,4
[on chain=1] (a) ;

in 1,2,3,4
[draw=cb-blue,fill=,on chain=2,right=5.5em of a,yshift=1em] (b) ;
[lbl,right of=b] ;
[draw=cb-red,fill=,on chain=3,above=3.5em of b,xshift=-4em] (c) ;
[lbl,right of=c] ;
[draw=cb-red,fill=,on chain=4,above=3.5em of b,xshift=4em] (d) ;
[lbl,right of=d] ;

[lbl,left=0.1em of b2] Rule;
[lbl,right=0.05em of b1] $\varnothing$;
[lbl,right=0.05em of b2] ;
[lbl,right=0.05em of b3] besides ;
[lbl,left=0.1 of c3] Span start;
[lbl,left=0.1 of d3] Span end;

HCT sample run. After encoding, every source token $x_i$ is tagged with an action $a_i$ and rule $r_i$. Context span(s) for $x_i$ are then predicted based on $r_i$ and cross-attention between source and context embeddings. For instance, $x_3 = \text{``he''}$ is tagged with $a_2 = \mathrm{\textit{delete}}$ and rule $r_2 = \mathrm{\bla{}}$. The slot in $r_2$ is filled with the $c$ span $s_2 = (3,3)$ that contains “federer”.

§.§.§ Optimization
MST is trained to minimize cross-entropy over gold actions $a$ and spans $s$:

\begin{align}
\label{eq:xc-loss}
\mathcal{L}_e &= -\sum_{i = 1}^n\log p(a_i \mid x_i)p(s_i \mid c, x_i).
\end{align}

Since MST runs in parallel over source tokens, output sequences may be disjointed. MST optimizes sentence-level BLEU under an RL objective [Chen and Cherry, 2014] to encourage more fluent predictions. Besides minimizing cross-entropy in Eq. <ref>, MST maximizes similarity between gold $x^\ast$ and sampled $\hat{x}$ as reward term $w$:

\begin{align}
\label{eq:rl-loss}
\mathcal{L}_r &= -\Delta(\hat{x}, x^\ast)\log p(\hat{x} \mid c, x) = -w\log p(\hat{x} \mid c, x)
\end{align}

where $\Delta$ denotes sentence-level BLEU. In practice, we use the greedily decoded $\hat{x}_g$ as a baseline reward relative to $\hat{x}$ to reduce variance in REINFORCE:

\begin{align*}
w &= \Delta(\hat{x}, x^\ast) - \Delta(\hat{x}_g, x^\ast).
\end{align*}

This encourages the model to generate both accurate and fluent sentences. To further stabilize gradient updates, we follow Kiegeland and Kreutzer, 2021 by scaling the rewards by the minimum ($\min$) and maximum ($\max$) $r$ values in the current batch: $r = (r - \min) / (\max - \min)$. This provides batch-level normalization across rewards.

The final loss is a weighted average over cross-entropy and RL losses in Eqs. <ref> and <ref>:

\begin{align}
\label{eq:fin-loss}
\mathcal{L} = (1 - \lambda)\mathcal{L}_e + \lambda\mathcal{L}_r,
\end{align}

where $\lambda$ is a scalar weight empirically set to $0.5$.

§ HIERARCHICAL CONTEXT TAGGER

While MST supports more flexible context span insertion, it can not recover tokens that are missing from the context (e.g., prepositions). We propose a hierarchical context tagger (HCT) that uses automatically extracted rules to fill this gap.

As shown in Figure <ref>, HCT adopts the BERT encoder (Eq. <ref>) and action tagger (Eq. <ref>) from MST. In addition, HCT includes a rule tagger that chooses which (possibly empty) slotted rule to insert before each source token. HCT is composed of two levels: both action and rule taggers run in parallel at the first level, then the second level span predictor takes tagged rules as inputs. Since the span predictor fills in a known number of slots per rule, it no longer needs to produce stop symbols as it does in MST (Eq. <ref>).

§.§.§ Rule Tagger
The rule tagger linearly projects the embedding of source token $x_i$ to choose a rule to insert before it:

\begin{align}
p(r_i \mid x_i) = \mathrm{Softmax}(\mathbf{W}_r\mathbf{e}_i),
\end{align}

where $\mathbf{W}_r$ parameterizes a rule classifier of $p$ rules that includes the null rule $\varnothing$ for an empty insertion. The rule set is automatically extracted from training data.

§.§.§ Span Predictor
The second-level span predictor expands rule $r_i$ containing $k \ge 1$ slots into spans $s_i = (s_{i1}, \ldots, s_{ik})$:

\begin{align}
p(s_{ij} \mid c, x_i, r_i, j) &= \text{HCT}_2(c, x_i, r_i, \{s_{ij^\prime}\}_{j^\prime < j}),
\end{align}

where $1 \leq j \leq k$. Unlike MST, HCT learns rule-specific slot embeddings to anchor each span to a rule $r_i$. Instead of conditioning spans $s_i$ on all tokens $x$ and rules $r$, we find it sufficent to restrict it to a single $x_i$ and $r_i$.

To condition its span predictor on tagged rules, HCT learns contextualized rule embeddings using the same input token BERT encoder. Slots at the same relative position across rules are represented by the same special slot token. For example, the rule “ and ” is assigned the tokens (, , ), whereas the rule “” is simply . Embeddings of these  tokens are learned from scratch and allow relative positional information to be shared across rules. A special  token is prepended to a rule's token sequence before applying the BERT encoder, and its embedding is used to represent the rule.

We bias context-source attention (Eq. <ref>) on a rule embedding by updating the query embedding $\mathbf{e}_i$ as

\begin{align*}
\mathbf{e}_i = \mathrm{ReLU}\left(\mathbf{W}_c\left[\mathbf{e}_i; \mathbf{r}_i\right]\right),
\end{align*}

where $\mathbf{W}_c \in \mathbb{R}^{d \times 2d}$ is a learned projection matrix. Eq. <ref> can then be replaced by

\begin{align}
p(s_{ij}^\uparrow \mid c, x_i, r_i, j) = \text{Attn}^\uparrow\left(\mathbf{E}_c; \mathbf{u}_{ij}\right).
\end{align}

We note that HCT's nested phrase predictor can also be seen as learning a grammar over inserted rules. Each source token is preceded by a start symbol that can be expanded into some slotted rule. Rules come from a fixed vocabulary and take the form of a sequence of terminal tokens and/or slots (e.g., “ by ” or “in ”). In contrast, slots are nonterminals that can only be rewritten as terminals from the context utterances (i.e., spans). While slotted rules are produced from start symbols in a roughly context-free way—conditioned on the original source tokens—terminal spans within a rule are not. Spans in the same rule are predicted autoregressively to support coherency of successive spans.

§.§.§ Optimization
Following Eq. <ref>, HCT's loss is:

\begin{align}
\label{eq:xc-loss2}
\mathcal{L}_e &= -\sum_{i = 1}^n\log p(a_i \mid x_i)p(r_i \mid x_i)p(s_i \mid c, x_i, r_i),
\end{align}

where $p(s_i \mid c, x_i, r_i)$ is analogous to Eq. <ref>. HCT optimizes the same RL objective as MST by replacing $p(\hat{x} \mid c, x)$ in Eq. <ref> with $p(\hat{x} \mid c, x, r)$. Its total loss is the same as Eq. <ref>.

[every node/.style=font=,inner sep=.25em]
[column sep=,row sep=2em,matrix of nodes]
(x0) $x^\ast$;     (x1) [BOS];     (x2) the;     (x3) puppy;     (x4) sleeps;     (x5) well;     (x6) at;     (x7) night;     (x8) now;     (x9) .;

(y0) $x$;     (y1) [BOS];     (y2) it;     (y3) sleeps;     (y4) well;     (y5) ,;     (y6) mostly;     (y7) at;     (y8) night;     (y9) .;

(x1) – (y1);
(x5) – (y4);
(x4) – (y3);
(x6) – (y7);
(x7) – (y8);
(x9) – (y9);

Token-aligned sentences to illustrate LCS usage.

sn edges/.style=for tree=parent anchor=south, child anchor=north,align=center,base=bottom,inner sep=.2em,
redst/.style=for tree=text=cb-red,for descendants=edge=cb-red

sn edges [S [NP,redst [DT [the]] [NN [puppy]]] [VP [sleep …,roof]]]

sn edges [S [NP [PRP [we]]] [VP [VBD [adopt]] [NP [DT [a]] [NN,redst [puppy]]]]]

Subtree of missing target (left) and that of the context (right). Note that both strings are lemmatized.

Examples of automatic action and span labeling.

§ AUTOMATIC LABEL AND RULE EXTRACTION

Since most datasets for utterance rewriting do not provide alignments between target phrases and context spans, we apply methods to extract them automatically. The LCS algorithm extracts actions and context-target alignments for span insertions. A bottom-up syntactic method attempts to align phrases that LCS failed to find context spans for. While these alignment methods are shared by MST and HCT, only HCT depends on rule extraction.

§.§ LCS Action and Phrase Alignment

To annotate actions and phrases added to source $x$, we compute the longest common subsequence (LCS) between $x$ and target utterance $x^\ast = (x_1, \ldots, x_l)$. The LCS algorithm relies on dynamic programming and runs in time $\mathcal{O}(nl)$. We traverse token alignment pairs in LCS n-grams from left to right, extracting tags based their positions in $x^\ast$ and $x$. Figure <ref> shows an example alignment. Concretely, if the alignment pairs are

* adjacent in both sequences: keep the aligned tokens (e.g., “sleeps well”).

* only adjacent in $x$: keep the aligned tokens (e.g., “night .”), add the phrase between pairs in $x^\ast$ (“now”).

* only adjacent in $x^\ast$: keep the aligned tokens (e.g., “well at”) and delete tokens between pairs in $x$ (“, mostly”).

* adjacent in neither $x$ nor $x^\ast$: keep the aligned tokens (e.g., “[BOS] sleeps”) and delete tokens between pairs in $x$ (“it”), add those between pairs in $x^\ast$ (“the puppy”).

This produces the desired actions $a$ and a list of unaligned target phrases that map to context spans. Phrases that can be found as single context spans are kept, while the remaining ones are searched for in the next step.

§.§ Syntax-Guided Phrase Alignment

Unaligned phrases are common in datasets like , where only $\sim$42% of dialogue utterances are free from such phrases. We extract context spans that partially cover the phrases for additional model supervision during training. To do so, we lemmatize [Manning et al., 2014] the unaligned phrase and the context, then try to align syntactic spans of the target phrase to context spans. For example, “the puppy” may be missing from the context but “puppy” aligns to a span (Figure <ref>, right). Each target span must perfectly match some context span, and we find the set of aligned spans with highest coverage of the unaligned target phrase. In HCT, the target tokens that remain unaligned to a context span can be covered by tokens in an extracted rule.

Searching only for syntactic spans lets us avoid naïvely exploring all possible splits of the target phrase. Span-level instead of purely token-level alignment (similar to RUN) also limits how many insertions the model must predict per phrase. Furthermore, we align the lemmatized target phrase and context strings to ignore inflection changes. This way, target tokens inflected differently from those in the context (e.g., “sleeps” and “slept”) map to the same lemma (e.g., “sleep”) and can still align.

Target alignment to context spans

[1]
Target subtree $ttr$, context string $c$
Context spans $sps = [s_1, \ldots, s_c]$
DescendTree$ttr, i, k, c$
Find $n$-length span $s^\prime$ of $c$ matching $ttr[i, k]$
$s^\prime, n \gets \Call{AlignSpan}{ttr, i, k, c}$
$sps, n\_sum \gets [\,], 0$
$(i2, k2) \in ttr.children$
$ch\_sps, n2 \gets \Call{DescendTree}{ttr, i2, k2, c}$
$sps \gets sps \oplus ch\_sps$
$n\_sum \gets n\_sum + n2$
$n\_sum > n$ Adopt best spans of children
$sps, n\_sum$
$[s^\prime], n$
$sps, n \gets \Call{DescendTree}{ttr, 0, |ttr.leaves|, c}$

For each missing phrase, we parse [Kitaev, Cao, and Klein, 2019] the constituency tree of its target utterance $x^\ast$, then isolate the smallest subtree $ttr$ covering it (e.g., left subtree of Figure <ref>). Next, we iterate over subtree $ttr$ to find the context spans $sps$ that most overlap with the target phrase (Alg. <ref>). We first try to match the current target constituent's span $[i, k]$ to a span in context $c$ (line 2), where $n$ is the character count of the matching spans' strings—$0$ if no match is found. We then find the best spans $ch\_sps$ covering the constituent's children (lines 3–7). If their summed match lengths $n\_sum$ is greater than $n$, then the children's spans replace the constituent's (lines 8–10). The result is a list of the best-matching context spans per unaligned target phrase.

\[
\begin{array}{l *{7}c}
\toprule
& \text{B}_1 & \text{B}_2 & \text{B}_4 & \text{R}_1 & \text{R}_2 & \text{R}_\text{L}\\
\midrule
\text{Pro-Sub} & 60.4 & 55.3 & 47.4 & 73.1 & 63.7 & 73.9\\
\text{Ptr-Gen} & 67.2 & 60.3 & 50.2 & 78.9 & 62.9 & 74.9\\
\text{RUN} & \mathbf{70.5} & 61.2 & 49.1 & 79.1 & 61.2 & 74.7\\
\text{RaST} & 53.5 & 47.6 & 38.1 & 62.7 & 50.5 & 61.9\\
\text{MST} & 66.6 & 59.9 & 48.7 & 79.5 & 64.1 & 79.0\\
\text{HCT} & 68.7 & \mathbf{62.3} & \mathbf{52.1} & \mathbf{80.0} & \mathbf{66.5} & \mathbf{79.4}\\
\bottomrule
\end{array}
\]

BLEU-$n$ ($\text{B}_n$) and ROUGE-$n/\text{L}$ ($\text{R}_{n/\text{L}}$) on . Pro-Sub, Ptr-Gen, and RUN results are drawn from their respective works.

§.§ Rule Extraction and Clustering

We extract rules from the single- or multi-span alignments from the previous two subsections, where each span maps to a rule slot. Recall that slotted rules can (i) bind together multiple spans in the same target phrase and (ii) include out-of-context target tokens. For (i), rules consist entirely of slots (e.g., “ ” for $k = 2$), resembling glue rules introduced in machine translation [Chiang, 2005]. Rules in (ii) arise when a phrase maps to non-contiguous context spans; the uncovered target sub-phrases are left as-is in the rule (e.g., “the” in rule “the ”).

Some rules extracted from the training set may be rare, existing in the long tail of the rule distribution. To concentrate the rule vocabulary on higher frequency rules, we cluster lexically similar rules and map low-frequency rules to high-frequency ones (e.g., “in addition to ” $\to$ “in addition ”). Though this adds noise to low-frequency rules, we assert that the cost is outweighed by reducing the rule vocabulary size.

We apply affinity propagation [Frey and Dueck, 2007] clustering on the negative token-level LCS distance of rules. We use LCS rather than Levenshtein distance to disallow token substitutions that can unevenly affect short rules. A rule slot counts as a single token, and we normalize the LCS distance to fall between 0 and 1. Affinity propagation is ideal as it does not require tuning the number of clusters. Briefly, it iteratively clusters rules based on their suitability with exemplars, or representative points. The suitability of a point $i$ and exemplar $k$ depends both on other exemplars $k^\prime$ of the same point and other points $i^\prime$ of the same exemplar.

After clustering, we further filter the rule clusters based on the proportion of data points that each cluster covers. If a representative rule falls below a preset threshold (e.g., 0.5% of the points), then all rules in the cluster are labeled with the glue rules that match their slot counts.

§ EXPERIMENTS

We evaluate HCT on several multi-turn utterance rewriting datasets—two in English and one in Chinese.

§.§ Setup

§.§.§ Datasets
[Elgohary, Peskov, and Boyd-Graber, 2019] is derived from  [Choi et al., 2018], a question answering dialogue dataset about Wikipedia articles. The first two utterances are usually an article's title and section heading. Next is an exchange between a student, who asks questions to learn about the text, and a teacher, who answers questions by providing text excerpts.

[Tseng et al., 2021] includes conversations between a system and user on multiple domains (e.g., calling or news). Notably, it is about half the size of  and contains much shorter dialogues. Furthermore, the majority of its dialogues require no edits, in contrast to . Its relative simplicity may reflect the task-oriented rather than information-seeking nature of its dialogues.

\[
\begin{array}{l *{8}c}
\toprule
& \multicolumn{2}{c}{\text{Calling}} & \multicolumn{2}{c}{\text{Messag.}} & \multicolumn{2}{c}{\text{Music}} & \multicolumn{2}{c}{\text{All}}\\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}\cmidrule(lr){6-7}\cmidrule(lr){8-9}
& \text{B}_4 & \text{EM} & \text{B}_4 & \text{EM} & \text{B}_4 & \text{EM} & \text{B}_4 & \text{EM}\\
\midrule
\text{Joint} & 84.3 & \mathbf{77.7} & 83.1 & 68.8 & 61.8 & 40.9 & 80.4 & 69.3\\
\text{RaST} & 83.3 & 75.2 & 82.0 & 69.1 & 66.1 & 44.6 & 80.2 & 68.5\\
\text{MST} & 82.9 & 73.6 & 81.4 & 64.7 & 64.4 & 40.2 & 81.2 & 65.8\\
\text{HCT} & \mathbf{86.3} & 76.3 & 84.0 & \mathbf{69.8} & \mathbf{71.8} & \mathbf{47.3} & 83.1 & \mathbf{69.7}\\
\hspace{.75em}\text{-RL} & 86.1 & 75.9 & \mathbf{84.2} & 68.6 & 70.7 & 45.6 & \mathbf{84.1} & 69.2\\
\bottomrule
\end{array}
\]

BLEU-4 ($\text{B}_4$) and exact match accuracy (EM) on . Only three of the six domains are shown. The “-RL” line ablates BLEU rewards under an RL objective.

[Su et al., 2019] contains Chinese conversational dialogues crawled from social media platforms. It is comparable in size to , yet virtually all of its targets differ from the source utterances. Since Chinese lacks inflection and has fewer function words than English, the majority of phrases ($87.57\%$ of  dialogues) added to  sources can map to single context spans.

§.§.§ Baselines
We compare against results reported in the following baseline systems. RaST [Hao et al., 2021] can be seen as a special case of MST, where the maximum number of rule slots $l = 1$. Pro-Sub is a simple baseline from Elgohary, Peskov, and Boyd-Graber, 2019 that replaces the first pronoun in a source with the main entity of the dialogue (e.g., article title). Ptr-Gen is the LSTM-based hybrid pointer generator network of See, Liu, and Manning, 2017. RUN is the edit matrix tagging approach by Liu et al., 2020. It has an LSTM-based encoder and applies a semantic segmentation network to tag the edit matrix. Joint is the coreference resolution and query rewriting model from Tseng et al., 2021.

§.§.§ Evaluation
We measure utterance rewriting quality via widely used automatic metrics: BLEU, ROUGE, and EM. The BLEU [Papineni et al., 2002] score between the predicted and gold target utterances is reported on all datasets. For  and , we add BLEU-$n$ and ROUGE-$n$ for $n \in \{1,2\}$ to evaluate smaller n-grams, in addition to ROUGE-L [Lin and Hovy, 2002]. EM refers to string-level exact match accuracy and is the least forgiving.

§.§.§ Implementation details
We use the uncased and Chinese variants of  [Wolf et al., 2020] as the respective encoders for English and Chinese experiments. Adam is used to optimize the full model with a learning rate of 1e-5 for  and  and 2e-5 for . We apply early stopping after 15 epochs once the development set BLEU-4 stops growing three epochs in a row.

§.§ Results

As shown in Table <ref>, HCT achieves the highest score on nearly all query rewriting metrics. It surpasses the previous top model, Ptr-Gen, by margins of 1.9 BLEU-4 and 4.5 ROUGE-L. By extension, it exceeds the edit matrix and sequence tagger baselines of RUN, RaST, and MST. We note that adding adding multi-span prediction between RaST and MST provides a sizeable performance boost: 10.6 BLEU-4 and 17.1 ROUGE-L. Adding hierarchy to MST via HCT's rule predictor gives a boost of 3.4 BLEU-4. The expressivity of multi-span tagging and structure of rule prediction appear to boost rewrite quality.

\[
\begin{array}{l *{8}c}
\toprule
& \multicolumn{2}{c}{\text{Calling}} & \multicolumn{2}{c}{\text{Messag.}} & \multicolumn{2}{c}{\text{Music}} & \multicolumn{2}{c}{\text{All}}\\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}\cmidrule(lr){6-7}\cmidrule(lr){8-9}
& \text{B}_4 & \text{EM} & \text{B}_4 & \text{EM} & \text{B}_4 & \text{EM} & \text{B}_4 & \text{EM}\\
\midrule
\text{RaST} & 84.7 & 74.9 & 80.5 & 66.7 & 61.9 & 39.5 & 81.6 & 67.1\\
\text{MST} & 84.1 & 74.0 & 79.6 & 65.0 & 62.6 & 39.5 & 81.7 & 66.0\\
\text{HCT} & \mathbf{86.0} & \mathbf{75.9} & \mathbf{83.4} & \mathbf{69.1} & \mathbf{63.2} & \mathbf{40.9} & \mathbf{82.8} & \mathbf{68.4}\\
\bottomrule
\end{array}
\]

BLEU-4 ($\text{B}_4$) and EM on  for models trained on the calling domain only.

Table <ref> summarizes rewriting results for multi-domain coreference utterances. Scores are grouped by the main domains—calling, messaging, and music—that respectively cover about 54%, 20%, and 14% of the dataset. Overall, HCT outperforms the Joint baseline of Tseng et al., 2021 by 2.7 BLEU-4. Within the smaller messaging and music domains, HCT scores 1 and 6.4 EM points higher than Joint. The trend holds to a lesser degree for the RaST baseline as well. Thus, HCT generalizes even better on lower-data domains than the simpler baselines of RaST and MST. This suggests that HCT's slotted rules improve data coverage without hurting model robustness across domains.

We also use  as a testbed for domain adaptation experiments. In particular, we focus on zero-shot generalization of models on unseen data domains. We train models on the calling domain only and evaluate on the full dataset that includes five unseen domains. Besides the three shown in Table <ref>, the full  dataset also contains news, reminders, and weather domains. Since RaST generalizes well on cross-domain Chinese utterance rewriting, it is useful for domain adaptation evaluation. HCT generalizes especially well on the messaging and music domains; compared to RaST, it gains 2.9 BLEU-4 and 2.4 EM points for the former and 1.3 BLEU-4 and 1.4 EM for the latter. On the full dataset, HCT achieves a 1.3 higher EM score than RaST. This is especially surprising given that HCT was trained on rules extracted only from the calling domain; its ability to generalize these rules to other domains is promising.

As a final benchmark, we consider how well HCT can rewrite utterances in a different language like Chinese. The  dataset is a challenging setting for HCT since over 90% of its dialogues can be covered using single-span insertions from the context; this is likely due to linguistic differences between Chinese and English. Thus, the better expressivity offered by a multi-span predictor only benefits a small proportion of the examples. However, Table <ref> shows that HCT scores 1.5 BLEU-4 and 1.9 EM points above RaST. Compared to RUN, it achieves a 2.2 higher ROUGE-L score. RUN may score higher on metrics over small n-grams due to its addition of a fixed set of unseen tokens to each context, improving target token coverage.

\begin{array}{l *{7}c}
\toprule
& \text{B}_1 & \text{B}_2 & \text{B}_4 & \text{R}_1 & \text{R}_2 & \text{R}_\text{L} & \text{EM}\\
\midrule
\text{RUN} & \mathbf{93.5} & \mathbf{90.9} & \mathbf{85.5} & \mathbf{95.8} & \mathbf{90.3} & 91.3 & 65.1\\
\text{RaST} & 90.5 & 88.3 & 83.6 & 94.1 & 88.9 & 92.9 & 63.4\\
\text{MST} & 92.9 & 90.1 & 84.2 & 94.7 & 88.9 & 93.1 & 62.2\\
\text{HCT} & 92.7 & 90.2 & 85.1 & 94.4 & 89.3 & \mathbf{93.5} & \mathbf{65.3}\\
\bottomrule
\end{array}

BLEU, ROUGE, and EM metrics on .

[xlabel=Lower bound per-rule phrase coverage (%),ylabel=EM,
legend pos=south west,width=0.495,height=0.275,
cycle list/Dark2-3,every node near coord/.style=font=,
+[mark=x,semithick,nodes near coords,
visualization depends on=value val ,
every node near coord/.append style=label=[yshift=-2.65em]$(\val)$]
x	y	val
0.225	24.5	43
0.5	24.9	24
0.75	24.0	19
1.1	23.8	13

Effect of rule clustering threshold parameter on EM for . Rule count is shown in parentheses.

§.§ Analysis

Here we analyze how properties of rule vocabularies affect HCT's rewriting performance. We also examine a few generated rewrites to compare model behavior.

1          9      by      17     for

2      's     10      than      18     in

3           11      than       19     in

4       '     12     aside from      20     of

5       's     13     besides      21     other than

6       's      14     besides       22     other than

7            15     besides        23     the

8        's     16     by      24     to

HCT rule vocabulary for .

1          5           9     okay

2      's     6       's     10     song

3      ,      7     of      11     the

4      ,  ,     8     of       12     the

For .

§.§.§ Rule vocabulary
An ideal rule vocabulary will maximize phrase coverage while minimizing rule quantity. Recall that after clustering rules extracted from the data, we filter out rare rules that fall below a certain frequency threshold. Higher thresholds filter rules more aggressively; the points from left to right in Figure <ref> map to 43, 24, 19, and 13 rules. Based on the slight bump in EM between 43 and 24 rules, too many low-frequency rules can be harmful. Yet EM drops dramatically after reducing vocabulary size below 24. Although we did not fully tune it in experiments, our chosen threshold of 0.5% looks reasonable.

Table <ref> lists 24 extracted rules from  for more insight into their properties. Generally, we notice the use of possessive form (e.g., “'s”), prepositions, and determiners. The rules also show redundancy as multiple groups of rules have the same tokens but differing slot counts; we note that HCT's BERT-encoded rule embeddings help it generalize within groups. The rule vocabularies for  (Table <ref>) and  are comparable to 's. Comparing  and  rules, we observe that former's cover more prepositions phrases such as “besides” and “other than”. This trait reflects domain differences;  dialogues are longer and require more ellipsis resolution, while those in  often involve short user requests to a system. However, there are shared rules between the two datasets (e.g., “ 's”, “of ”, and “the ”) that show shared patterns in missing target phrases.

§.§.§ Semantic role labeling evaluation
Since source and target strings in  and  often show high overlap, models may easily achieve high scores on n-gram matching metrics such as BLEU and ROUGE. We follow Hao et al., 2021 by augmenting these metrics with evaluation based on semantic role labeling (SRL). As SRL can be seen as a shallow version of semantic parsing, semantically similar predictions to the target should share SRL annotations. We apply state-of-the-art SRL models [Che et al., 2021, Shi and Lin, 2019] to extract the predicate-argument structure from predictions and targets on  and . We then compute precision, recall, and F1 scores of the model-predicted predicate-argument spans. Table <ref> shows that HCT achieves significantly higher F1 scores than RaST and MST. Interestingly, the simpler RaST model outperforms MST in terms of this span-based SRL evaluation.

§ RELATED WORK

§.§.§ Iterative refinement
Instead of generating text in a single pass, iterative approaches repeatedly edit an output sequence. This applies to post-editing in machine translation [Novak, Auli, and Grangier, 2016, Xia et al., 2017, Grangier and Auli, 2018], allowing humans to refine hypotheses with a machine's help. These approaches have encoder-decoder architectures with two decoders. The first decoder attends over the source string to output a hypothesis, while the second attends to both the source and hypothesis to refine the latter. In contrast, the second level of HCT depends only on the initial source token and predicted rule embeddings; it does not re-encode the source after inserting predicted rules.

§.§.§ Retrieval-based editing
Another approach that constrains the search space of text generation is the retrieve-then-edit framework. A model is trained to retrieve a prototype output, then edit it to match the input context [Hashimoto et al., 2018, Wu et al., 2019]. The intent is to encourage diverse outputs that remain grammatical and coherent. While we have similar motivations, the preceding work edits prototypes using attention-based seq-to-seq models. By limiting prediction length to that of the source string, our rule tagger and span predictor greatly reduce search space in comparison.

§.§.§ Coreference resolution
The utterance rewriting task often requires coreference resolution, which has a long history of machine learning approaches. Recent neural models have eliminated parsers or hand-engineered algorithms, instead scoring span or mention pairs using distributed representations [Clark and Manning, 2016, Lee et al., 2017]. Pretraining methods like SpanBERT [Joshi et al., 2020] refine such span representations. Wu et al., 2020 extend SpanBERT to frame coreference resolution as query-based span prediction. We go beyond coreferences by resolving ellipses as well.

\[
\begin{array}{l *{6}c}
\toprule
& \multicolumn{3}{c}{\text{\mud{}}} & \multicolumn{3}{c}{\text{\rew{}}}\\
\cmidrule(lr){2-4}\cmidrule(lr){5-7}
& \text{P} & \text{R} & \text{F1} & \text{P} & \text{R} & \text{F1}\\
\midrule
\text{RaST} & 82.9 & 83.4 & 83.1 & \mathbf{83.8} & 76.6 & 80.0\\
\text{MST} & 82.9 & 80.4 & 81.6 & 80.7 & 76.7 & 78.6\\
\text{HCT} & \mathbf{85.1} & \mathbf{84.2} & \mathbf{84.7} & 83.6 & \mathbf{79.8} & \mathbf{81.7}\\
\bottomrule
\end{array}
\]

SRL evaluation on  and .

§ CONCLUSION

This work proposed a hierarchical context tagger for simple, high coverage utterance rewriting. We first demonstrated the benefit of a multi-span tagger that generates spans autoregressively at each source position. Next, we showed how HCT improves flexibility of MST's span predictor by first predicting slotted rules, then replacing a fixed number of slots with spans. While the two systems apply the same action tagger for source token editing, HCT conditions its span predictor on rules for more constrained phrase insertion. We also described automatic labeling methods using LCS and syntactic alignment, in addition to a clustering technique for rule extraction. Experiments revealed that HCT surpasses several state-of-the-art utterance rewriting systems by large margins. Furthermore, HCT generalizes better than simpler model variants lacking a rule tagger on unseen data domains.

HCT has applications to text editing tasks with a high overlap between source and target sequences. These tasks are plentiful and include sentence fusion, grammar correction, and prototype editing for machine translation. Leveraging syntactic structures to further restrict tagger and span predictor search space is an intriguing future direction.

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§ SEMI-AUTOREGRESSIVE SPAN ATTENTION

To predict start index $s_{ij}^\uparrow$ in Eq. <ref>, we compute attention between source token $x_i$ and all context tokens semi-autoregressively. Instead of using the same source token embedding across spans to attend to context tokens, we use span-level embeddings $\mathbf{u}_{ij}$ (Eq. <ref>), which depends on the below components.

* $\hat{\mathbf{u}}_{ij}$: A mixture of context token embeddings weighted by the previous span's attention distribution. Attention coefficients $\alpha_{i(j - 1)k}$ (Eq. <ref>) over context tokens indexed by $k$ represent the attention distribution at span $j - 1$.

* $\mathbf{u}_{i(j - 1)}$: The previous span's embedding for source $x_i$.

Note that at $j = 0$, the previous span's attention distribution is initialized as uniform over all context tokens. In addition, $\mathbf{u}_{i(j - 1)}$ is initialized to $\mathbf{e}_i$ from the source encoder.