# Instructing Hierarchical Tasks to Robots by Verbal Commands

Péter Telkes1, Alexandre Angleraud1 and Roel Pieters1 1Cognitive Robotics
group, Unit of Automation Technology and Mechanical Engineering, Tampere
University, 33720, Tampere, Finland<EMAIL_ADDRESS>
###### Abstract

Natural language is an effective tool for communication, as information can be
expressed in different ways and at different levels of complexity. Verbal
commands, utilized for instructing robot tasks, can therefor replace
traditional robot programming techniques, and provide a more expressive means
to assign actions and enable collaboration. However, the challenge of
utilizing speech for robot programming is how actions and targets can be
grounded to physical entities in the world. In addition, to be time-efficient,
a balance needs to be found between fine- and course-grained commands and
natural language phrases. In this work we provide a framework for instructing
tasks to robots by verbal commands. The framework includes functionalities for
single commands to actions and targets, as well as longer-term sequences of
actions, thereby providing a hierarchical structure to the robot tasks.
Experimental evaluation demonstrates the functionalities of the framework by
human collaboration with a robot in different tasks, with different levels of
complexity. The tools are provided open-source at
https://petim44.github.io/voice-jogger/

##  I Introduction

Speech and natural language is the most common modality for interaction
between humans, due to its expressive nature and its ability for dialogue.
Communication between humans and robots can benefit as well from such
functionalities, and, as a research field, efforts to achieve this have been
ongoing since several decades [1]. The main challenge in instructing robots by
verbal commands is how natural language can be easily grounded to the actions
of a robot. On the one hand this can be done by directly matching input
commands to a single robot action or target location in the world (i.e.,
lexical grounding), thereby effectively performing tele-operation. On the
other hand, learning-based approaches can generalize input commands to desired
outcomes in the world, from Large Language Models (LLM), possibly assisted by
visual perception (vision-language pre-training, or VLP).

While lexical grounding provides a one-to-one connection from commands to
actions, learning-based models do not provide this, making them less reliable
to achieve desired outcomes.

Figure 1: Instructing robots by speech requires fine-grained verbal commands
(left) that directly refer to actions and targets in the workspace. This work
presents methods that extend verbal command instructions to hierarchical
tasks.

In both cases basic robot functionalities are needed that act in the world,
such as movement primitives in world space, gripper actions and higher-level
hierarchical tasks that execute multiple actions in sequence. Most
importantly, the grounding of (verbal) natural language commands to robot
actions requires manual selection of what robot actions are needed for what
tasks. While LLMs and VLP can help in extracting relevant information from
language and the scene, they cannot alone solve this grounding problem.

In this work, we present a framework for verbal commands as tool to instruct
tasks to robots, including automatic speech recognition and the grounding of
commands to robot actions and targets in the work space (see Fig. 1). Single
and multiple command phrases from natural language enable different
functionalities to be included such as task hierarchies and task
concatenation, without explicitly relying on high-level task planners. These
capabilities are of particular importance to ensure smooth and efficient
collaboration between human and robot, as continuously using fine-grained
language commands is time-consuming (see Fig. 1). Course-grained commands
enable high-level functionalities and long-term sequences of actions as
demonstrated by our work with two hierarchical tasks for gear assembly. Our
developments and tools are integrated in a robotic system and numerous
experiments with single commands and long-term tasks demonstrate the results.

The contributions of this work are as follow:

  * •

Framework for instructing robots by verbal commands with basic robot
functionalities

  * •

Grounding of commands to robot actions and targets in the scene

  * •

High-level functionalities to support task hierarchies and concatenation

  * •

Experimental demonstration of the framework with representative tasks

##  II Related Work

###  II-A Human-Robot Collaboration

Collaboration between human and robot can be supported in many ways by
different technologies [2]. While physical interaction (e.g., hand-guiding or
learning by demonstration) and visual perception (e.g., for object and gesture
detection) are popular modalities [3], also audio perception in the form of
natural language and speech is common [1]. Success of the collaboration
depends on different factors, such as technical quality of the tools, and how
communication between human and robot is enabled [4]. In context to this work,
the collaboration between human and robot has the purpose to provide robot
instructions via speech. This means that communication is only directed from
human to robot, and human verbal commands (i.e., natural language) need to be
recognized and interpreted to be utilized for robot task execution. Several
earlier works have developed capabilities that enable robots to act on voice
commands [1], with different levels of complexity and even with additional
sensor modalities to support the understanding of the world and tasks within
it.

The related work therefore covers the state of the art in natural language
processing and further divides robotics approaches in two directions;
lexically-grounded methods and learning-based methods (see Table I).

###  II-B Natural language processing

Speech recognition and natural language processing (NLP) are two related
fields, which both have been propelled forward recently due to the application
of deep learning models [5, 6]. For this work, we focus on NLP for the
understanding of text and assume that a correct recognition of speech (by e.g.
Vosk [7]) provides natural language (words or sentences) in textual format.
The interpretation of language by Large Language Models (LLM), such as BERT
[8] and PaLM [9], have demonstrated capabilities including language
understanding and generation, reasoning, and code-related tasks. As such, to
enable robot instructions, they still need the explicit grounding of verbal
commands to actions of the robot or objects in the scene.

###  II-C Lexical-grounding of robot commands

Commanding robots from natural language by lexical grounding has been
demonstrated for single verbal commands [10], command pairs [11] and short
phrases [12]. In most of these cases, words to be recognized and tasks to
executed are predefined, limiting the applicability of the methods to new
situations and new tasks. Verbal commands have also been utilized to aid in
hand-guided task programming [13], thereby effectively enabling multi-modal
robot programming. A list of supported single word commands can be commanded
and, separately, different parameters can be set via an app to adapt different
parameters for robot motion and skills. Speech interfaces have also been
utilized for modification of robot skills [14], utilizing bi-directional and
high-level communication between human and robot. Interaction scenarios are
developed that show long human commands and instructions to modify robot
knowledge. In [15] the learning of complex hierarchical tasks from natural
language instruction is demonstrated, utilizing hierarchical task networks
(HTNs) and queries from robot to human in case tasks are unknown. Similarly,
in [16] behavior trees are utilized to learn and perform novel complex tasks.
As such, both these approaches focused on the planning of complex tasks and
the learning of task hierarchies and behaviors. Complex natural language
instructions have also been utilized for planning in linear temporal logic
formulation [17], with incomplete world knowledge [18] and for the ordering
[19] and allocation [20] of tasks to be executed.

TABLE I: Comparison between two approaches of utilizing speech in robotics |  Lexical grounding |  Learning-based
---|---|---
Approach |  Match words directly to robot actions and targets in the workspace |  Use high-level natural language instructions to learn robot actions in the workspace
Commands |  Fine-grained |  Course-grained
Vocabulary |  Limited |  Expressive
Actions |  Short-time |  Mid- to long-term
Response |  Fast |  Slow
References |  Single commands [12, 13] |  CLIPort [21]
|  Command pairs [11] |  PaLM-E [22]
|  Command sentences [10] |  Interactive language [23]
|  Hierarchical tasks [15, 16] |
|  This work |

What all of the mentioned approaches have in common is that verbal commands
directly result in robot actions, or indirectly through task plans. As hand-
crafted algorithms extract relevant words from input, this means approaches
are typically limited to a predefined vocabularies and action plans.

###  II-D Learning-based robot commands

Natural language as input for robot tasks has seen recent interest by
generating task policies that are language-conditioned [24]. Even more recent
are the inclusion of other modalities, in addition to natural language, to
establish the link between words and perception [25, 21, 26], for tasks such
as robot object manipulation and motion trajectory generation. While these
methods prove well-capable of transforming natural commands to robot actions,
the question remains whether such vision-and-language pretraining improves
lexical grounding [27]. This embodiment issue is addressed in [22] and [28]
which directly incorporate real-world continuous sensor modalities into a LLM
(PaLM) to plan and execute long horizon tasks, i.e., robot planning on table-
top and object manipulation and navigation in office kitchen environments,
respectively. Another example is embodied BERT (emBERT) [29], which grounds
language instructions against visual observations and actions to take in an
environment, with the aim of home and office navigation. Finally, real-time
(i.e., 5Hz) interaction by speech is presented in [23], which demonstrates the
instruction of multiple robots on table-top setting at the same time.

For both lexical grounding and learning-based approaches, still a connection
needs to be made to robot actions and the speech commands. While this
connection can be learned, as demonstrated by learning-based approaches, only
a person can verify whether the learned grounding is correct. In case of high-
performance or safety critical tasks (e.g., industry or search-and-rescue, as
compared to household actions by a service robot) such uncertainty might not
be acceptable. Our lexical-grounding approach is human-coordinated and aims to
enable basic and hierarchical tasks to be commanded from human to robot, with
both fine and course-grained commands.

##  III Approach

Our approach divides robot instructions into basic and hierarchical commands
(see Table II), enabling both low- and high-level commands to be instructed at
the same time. New target poses and tasks can be defined at any time with any
combination of basic and hierarchical commands.

###  III-A Grounding of commands

Grounding of verbal commands is done by connecting individual words to actions
or targets, when they are detected from automatic speech recognition. Robot
actions can be named with any spoken word and multiple words can refer to the
same action. This similarly applies to robot pose targets (command save) and
new defined (hierarchical) tasks (command record). Verbal commands are also
grounded to robot motion in case of single direction motion (command
<direction> <value>) and other robot actions (home, open, close, rotate,
etc.).

###  III-B Basic commands

All relevant commands needed to enable a functional robot are referred to as
basic commands. This includes commands to start and stop the robot system and
individual motion actions of the different components (single direction step
motion, continuous point-to-point motion, gripper motion, etc.). Depending on
the functionality of the command, different instruction combinations can be
given as arguments, as described in Table II. Basic commands can be utilized
at any point during robot operation and can be used for hierarchical tasks in
short- or long-term form.

###  III-C Hierarchical task commands

Hierarchical tasks offer the ordering of robot actions in different ways (see
Table II). For example, task recording (initiated by command record,
terminated by command finish) enables task hierarchies to be assigned by the
human operator. In its simplest form these contain a list of actions and
commands that are executed in sequence, as described by few included template
commands (i.e., pick, place, stack, push). Task concatenation is enabled by
the and command, by appending following commands to a list. Combining multiple
robot motion actions is enabled by the then command, which computes the spoken
motion phrases to one single motion command outcome. For example, multiple
directional commands in sequence will simply be added up as a robot pose
target (i.e., down then down then left 200 mm will result in a single robot
pose translation target with the spoken offsets). Task repetition is enabled
by the commands again and repeat, given suitable input for executing the
motion commands (i.e., list of robot poses and number of repetitions).

TABLE II: List of available basic and hierarchical commands for instructing robot motion and tasks.  Basic commands |  Explanation
---|---
start robot |  Initialize robot
stop robot |  Shut-down robot
set mode step |  Set robot to move in steps
set mode continuous |  Set robot to move continuous
save position <string> |  Save robot pose as <string>
position <string> |  Move robot to pose <string>
<direction> <value> |  Move robot <value> {up, down, left, right, back, front}
stop execution |  Stop robot motion
step size <value> |  Set robot motion step size
open |  Open gripper
close |  Close gripper
rotate <value> |  Rotate gripper by <value> deg.
home |  Move robot to home position
Hierarchical commands |  Explanation
record <string> |  Start recording commands as <string>
finish |  Stop recording command sequence
task <string> |  Execute command sequence <string>
repeat <task> <string> |  Repetition of <task>, continuously or specified number of times by <string>
repeat <task> <list> |  Repetition of <task>, until <list> of positions is empty
pick <string> |  pick object from pose <string>
place <string> |  place object to pose <string>
stack <string> <value> |  Pick and place on pose <string> with vertical offset <value>
push <string> <direction> |  Planar motion at pose <string> starting from <direction>

New (hierarchical) commands can be easily defined, utilizing existing commands
and by providing desired arguments. Several typical robot manipulation
commands, such as pick-and-place, wiping and polishing have been developed
this way and are demonstrated individually and as integrated in two assembly
use cases in Section IV.

###  III-D Robot Control Architecture

The architecture for controlling the robot by verbal commands includes modules
for command detection, robot motion planning and robot control as follows.
Command detection recognizes speech commands, adds these to a queue and
publishes them as suitable messages over ROS topics. This means that the
system is continuously listening to audio and adds new robot tasks when they
are received. A high priority topic is used for the stop command, which allows
to bypass the queue and executes a stopping motion immediately, with suitable
motion profile. In case of teaching positions or task hierarchies, the system
stores them locally in custom XML format with commanded name, to be used when
requested. The motion planning module relies on MoveIt2 and is continuously
waiting for new motion commands from the command topics. Finally, robot
control utilizes the Franka Control Interface (FCI) to achieve real-time
execution of control actions.

##  IV Results and Discussion

###  IV-A Hardware and implementation

The experimental setup includes a Franka Emika robot with parallel gripper.
Robot motion control utilizes Moveit2 motion planner for generating work space
motion and ROS (ROS.org) handles all communication. Speech of a person is
recorded by a standard microphone (16kHz audio signal), transmitted to the
main PC and passed through a voice activity detector [30], to identify if the
individual audio packet has human voice or not. A time-delayed filter is
applied on the running stream of audio packets to consider the natural pause
in human speech while speaking. For speech recognition, Vosk [7] is utilized,
which is trained on The People’s Speech dataset [31]. The model is configured
by filtering out unnecessary words, which are unsuitable for robot
instructions, and by including words and sentence instructions that reflect
the content of speech to be expected. All packages, tools and videos are open-
source available to replicate our work at https://petim44.github.io/voice-
jogger/.

###  IV-B Speech recognition

The accuracy of speech recognition was tested in a clean and noisy environment
with a non-native English speaker (see Table III). In the clean case, verbal
commands are recognized without any failures. In the noisy environment the
audio contains a background noise similar to a busy restaurant with other
people speaking, which deteriorates the detection rate slightly. Speech
recognition is also evaluated in the use cases for single and hierarchical
tasks, which resulted in a high success rate (see Table V).

TABLE III: Speech recognition experiment results Command | Clean | Noisy
---|---|---
start robot | 10/10 | 10/10
stop robot | 10/10 | 10/10
move up | 10/10 | 10/10
move down | 10/10 | 10/10
move left | 10/10 | 10/10
move right | 10/10 | 08/10
move back | 10/10 | 10/10
move front | 10/10 | 09/10
stop execution | 10/10 | 10/10
step size <value> | 10/10 | 10/10
open tool | 10/10 | 07/10
close tool | 10/10 | 07/10
rotate tool <value> | 10/10 | 10/10
save position <value> | 10/10 | 10/10
load position <value> | 10/10 | 10/10
home | 10/10 | 08/10

###  IV-C Single tasks

Single tasks refer to short robot actions, which by definition, can have
internal hierarchical structure. This is demonstrated by a pick and place
action as it is composed into hierarchical tasks by verbal instructions (see
Table IV). From the basic and hierarchical commands available, first robot
end-effector poses are defined (save position <value>) in combination with
intermediate robot motion and gripper actions (e.g., move, close, open), see
left column in Table II. As a result, these robot poses are utilized for
motion commands and included in a recorded hierarchical task, see right column
in Table II. This effectively means that from an initial low-level set of
commands (i.e., 27 verbal commands), human instructions can be narrowed down
to 19 commands and, finally, a single task command, to achieve the same robot
action outcome.

Table V describes the results of several single tasks that are commanded by
verbal instructions. During the preparation phase specific robot poses and
motion sequences are defined as required for the tasks, in few single and
multiple commands. During the execution phase robot actions utilize these
poses and motions for specific objects and target applications. Repetition of
single tasks towards automated applications is possible as well, by specifying
the task and how often it should be repeated. In case suitable, offsets in
motion commands can be included to achieve more complex operations, such as
wiping and polishing a larger surface. Fig. 4 depicts several snapshots of the
single tasks.

TABLE IV: A pick and place task composed into hierarchical tasks by verbal instructions. The original query required 27 commands and, with tasks composed to replace step-wise instructions, the query is narrowed down to 19 commands and, finally, a single task command.  Commands | Task | Commands | Task
---|---|---|---
left 300 mm |  }5*[pick] |  position pick |  }15*[one]
down 300 mm |  | step size low |
down |  |  record one |
front |  |  down ($\times 2$) |
down |  | back |
save position pick |  | down |
step size low |  | close |
down ($\times 3$) |  | step size medium |
back |  | up |
close |  |  Position place |
step size medium |  }3*[place] | step size low |
up |  |  down ($\times 3$) |
left 500 mm |  | open |
back 200 mm |  |  position place |
save position place |  | home |
step size low |  | finish |
down ($\times 4$) |  |  |
open |  |  |
step size high |  |  |
up |  |  |
home |  |  |

###  IV-D Hierarchical tasks

To demonstrate further the capabilities of verbal command instructions,
experiments were also performed with longer-term tasks with complex assembly
steps. This included the assembly of a helical gear system with six assembly
steps and a planetary gear system with nine assembly steps. In both cases
human verbal instructions guided the assembly procedure and, in the case of
the planetary gear, also contained three manual and collaborative tasks.
Similar to single tasks, a preparation phase records specific robot poses and
motion sequences, by hand-guiding and verbal commands. The execution phase
entails the actual assembly task, coordinated by the person with verbal
commands. Table V describes the results of these hierarchical tasks averaged
over ten trials each.

#### Helical gear assembly

The helical gear consists of four parts (two gears, front and back plate) and
requires six steps for assembly, executed by the robot. This includes five
pick-and-place actions with different locations and one pushing action to
connect the two gears together. Fig. 4 depicts several snapshots of the
helical gear assembly task.

#### Planetary gear assembly

The planetary gear consists of seven major parts (housing, sun gear, three
planet gears and top plate, excluding bolts, nuts and small gears) and
requires nine major steps for assembly, six executed by the robot and three by
a human operator. This includes five pick-and-place actions with different
locations, one holding action and three human assembly actions (i.e., fixing
the housing with four bolts, fixing planet gears on the sun gear and fixing
the top plate with four bolts). Fig. 4 depicts several snapshots of the
planetary gear assembly task.

In the preparation phase of both assembly cases four poses are demonstrated to
the robot, by utilizing both hand-guiding and verbal commands. In addition, in
the preparation and execution phases, a high number of speech commands are
used (see Table V), as the assembly steps require fine adjustments before
executing the assembly steps.

TABLE V: Results for speech commanded tasks. The preparation phase aims to define poses and record task sequences, by both verbal commands and hand-guiding. The execution phase entails the actual task, coordinated by the person with verbal commands. Results for both hierarchical tasks were obtained from ten trials. | Single tasks | Hierarchical tasks
---|---|---
| Pick | Place | Pick & Place | Wipe | Polish |  | Helical gear
---
assembly
| Planetary gear
---
assembly
Preparation phase |
Prerecorded poses | 1 | 1 | 3 | 3 | 2 | 4 | 4
| No. of commands
---
(single, multiple)
| 12
---
(8/4)
| 11
---
(6/5)
| 27
---
(16/11)
| 22
---
(16/6)
| 5
---
(2/3)
| 44
---
(15/21)
| 34
---
(11/20)
| Speech recognition succes rate
---
| 12/12
---
100%
| 11/11
---
100%
| 27/27
---
100%
| 22/22
---
100%
| 5/5
---
100%
| 36/44
---
82%
| 31/34
---
91%
Duration | 0:45 | 0:30 | 1:28 | 1:26 | 0:36 | 6:12 | 3:14
Execution phase |
No. of automated steps | 1/1 | 1/1 | 2/2 | 1/1 | 1/1 | 6/6 | 6/8
| No. of commands
---
(single/multiple)
| 6
---
(4/2)
| 6
---
(4/2)
| 17
---
(11/6)
| 6
---
(2/3)
| 3
---
(1/2)
| 55
---
(21/26)
| 65
---
(25/28)
| Speech recognition succes rate
---
| 6/6
---
100%
| 6/6
---
100%
| 17/17
---
100%
| 5/6
---
83%
| 3/3
---
100%
| 46/55
---
83%
| 53/65
---
82%
Duration | 0:23 | 0:22 | 1:06 | 2:20 | 0:30 | 4:18 | 8:50

(a) (b) (c) (d)

Figure 2: Snapshots of verbally commanded single tasks: before (a) and after
(b) the wiping task, picking (c) and placing (d).

(e) (f) (g) (h) (i) (j)

Figure 3: Snapshots of the verbally commanded helical gear assembly task: (a)
pick back plate with pick part, (b) pick gear with pick part, (c) place gear
with place gear three, (d) push gears together with forward ninety, (e) pick
both gears with close and (f) place top plate with down three.

(a) (b) (c) (d) (e) (f) (g) (h)

Figure 4: Snapshots of the planetary gear assembly task: (a) place motor with
hold table distance thirty, (b) pick housing with pick part, (c) place housing
on motor with stack table distance fifty, (d) pick sun gear with pick part,
(e) operator fixes planet gears on sun gear, (f) place sun gear in housing
with hold table distance one hundred, (g) place top plate on housing with
place top (h) operator fixes top plate to housing.

###  IV-E Discussion

This work demonstrated how verbal commands can instruct robot actions to
achieve single and hierarchical tasks. The approach allows for the fast
generation of robot actions and tasks, directly instructed by a person and
without the need for robot programming or training a model (see Table I and
[21, 22]). Compared to offline programming, benefits can be especially
identified when changes (e.g., pick-up location, motion sequence) need to be
made during or after task execution. Speech commands allow for direct
adaptation of robot motion parameters, as compared to the more traditional
software practices (i.e., code compilation and evaluation). In addition, the
functionalities for task concatenation enable long-term sequences of actions
to be scheduled, without explicit task planners and without complex structures
for task dependencies.

Natural language interaction has the benefit of expressivity [4], as tasks can
be instructed by commands selected by the operator in any desired level of
granularity. This means that accurate motions involving contact can be done
more careful and under supervision of the operator (see Fig. 3(f)), leading to
real-time interaction between human and robot. To ensure a responsive system,
speech recognition is continuously listening for input commands and all tasks
can be interrupted and overwritten by new tasks, as decided by the human
operator.

Limitations of our work can be identified as well, and relate to the use of
only speech for instructing commands, requiring human actions and feedback for
successful robot tasks. For example, objects need to be placed on known or
instructed locations and fine-grained verbal instructions are needed for
delicate and contact manipulation actions. Vision as additional modality would
provide a solution to these limitations, either as single perception model for
object detection [2] or as visual-speech model that combines speech commands
with low-level action sequences [23].

One additional limitation is that no formal planner is utilized for
constructing a task plan. While this keeps the sequencing of hierarchical
tasks simple, it also means that pre-conditions or dependencies between tasks
are not included. Moreover, analysis on whether a (hierarchical) task sequence
is suitable or optimal needs to be done by the human operator.

Failure cases of our work were mostly caused by the limitations in automatic
speech recognition (see Table III and V). While current state-of-the-art
language models are very robust, still disturbances occurred in case of verbal
accents and words that are similar in sound (e.g., ’four’ and ’for’, ’tool’
and ’two’). These issues were avoided by intentional clear speech
pronunciations and excluding words from speech recognition that were often
mis-recognized. As can be seen in the videos, few times commands had to be
repeated to achieve correct speech recognition.

Future work will investigate how Large Language Models (LLMs) can be utilized
to assist in interpreting high-level verbal commands, such that more natural
sentences can be instructed to the robot, instead of a predefined vocabulary.
In addition, LMMs could provide additional functionalities, such as generating
hierarchical tasks or complete assembly sequences automatically in the most
effective and efficient manner.

##  V CONCLUSION

This work presented a framework for instructing robot action commands by human
natural language. Instructions can be both fine-grained and provide the
functionalities for single and hierarchical tasks, as specified by a human
operator in real-time. Instructing verbal commands includes a preparation
phase were all required robot poses are demonstrated to the robot, utilizing
both hand-guiding and verbal commands. In addition, hierarchical tasks can be
composed to combine and concatenate robot actions. The execution phase entails
the actual assembly task, coordinated by the person with verbal commands.
Experiments with several single and hierarchical assembly tasks demonstrate
that verbal commands can replace traditional robot programming techniques, and
provide a more expressive means to assign robot actions and enable human-robot
collaboration. To provide more natural language for instructions, future work
will investigate how Large Language Models can be utilized in the framework.

## Acknowledgements

Project funding was received from European Union’s Horizon 2020 research and
innovation programme, grant agreement no. 871252 (METRICS). The authors thank
all students in the Robotics Project Work course at Tampere University that
contributed to the developments of the speech tool: Wajeh Ahmad, Ara Jo, Omar
Hassan, Tuomas Kiviö, Mikko Kulju, Waleed Rafi, Niklas Sorri and Kalle
Tanninen.

## References

  * [1] S. Tellex, N. Gopalan, H. Kress-Gazit, and C. Matuszek, “Robots that use language,” _Annual Review of Control, Robotics, and Autonomous Systems_ , vol. 3, no. 1, pp. 25–55, 2020.
  * [2] V. Villani, F. Pini, F. Leali, and C. Secchi, “Survey on human–robot collaboration in industrial settings: Safety, intuitive interfaces and applications,” _Mechatronics_ , vol. 55, pp. 248–266, 2018.
  * [3] S. Gross and B. Krenn, “A communicative perspective on human–robot collaboration in industry: Mapping communicative modes on collaborative scenarios,” _International Journal of Social Robotics_ , pp. 1–18, 2023\.
  * [4] M. Marge _et al._ , “Spoken language interaction with robots: Recommendations for future research,” _Computer Speech & Language_, vol. 71, p. 101255, 2022.
  * [5] D. Yu and L. Deng, _Automatic speech recognition_. Springer, 2016, vol. 1.
  * [6] D. W. Otter, J. R. Medina, and J. K. Kalita, “A survey of the usages of deep learning for natural language processing,” _IEEE Transactions on Neural Networks and Learning Systems_ , vol. 32, no. 2, pp. 604–624, 2021.
  * [7] Alpha Cephei, “Vosk Speech Recognition Toolkit,” https://github.com/alphacep/vosk-api, 2023.
  * [8] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “BERT: Pre-training of deep bidirectional transformers for language understanding,” _arXiv preprint arXiv:1810.04805_ , 2018.
  * [9] A. Chowdhery _et al._ , “PaLM: Scaling language modeling with pathways,” _arXiv preprint arXiv:2204.02311_ , 2022.
  * [10] M. Stenmark and J. Malec, “Describing constraint-based assembly tasks in unstructured natural language,” _IFAC Proceedings Volumes_ , vol. 47, no. 3, pp. 3056–3061, 2014.
  * [11] A. Angleraud, A. Mehman Sefat, M. Netzev, and R. Pieters, “Coordinating shared tasks in human-robot collaboration by commands,” _Frontiers in Robotics and AI_ , vol. 8, 2021.
  * [12] A. C. Tenorio-Gonzalez, E. F. Morales, and L. Villasenor-Pineda, “Teaching a robot to perform tasks with voice commands,” in _Mexican International Conference on Artificial Intelligence_. Springer, 2010, pp. 105–116.
  * [13] T. B. Ionescu and S. Schlund, “Programming cobots by voice: A human-centered, web-based approach,” _Procedia CIRP_ , vol. 97, pp. 123–129, 2021.
  * [14] T. Frasca, B. Oosterveld, M. Chita-Tegmark, and M. Scheutz, “Enabling fast instruction-based modification of learned robot skills,” in _AAAI Conference on Artificial Intelligence_ , vol. 35, no. 7, 2021, pp. 6075–6083.
  * [15] G. Suddrey, C. Lehnert, M. Eich, F. Maire, and J. Roberts, “Teaching robots generalizable hierarchical tasks through natural language instruction,” _IEEE Robotics and Automation Letters_ , vol. 2, no. 1, pp. 201–208, 2016\.
  * [16] G. Suddrey, B. Talbot, and F. Maire, “Learning and executing re-usable behaviour trees from natural language instruction,” _IEEE Robotics and Automation Letters_ , vol. 7, no. 4, pp. 10 643–10 650, 2022.
  * [17] A. Boteanu, T. Howard, J. Arkin, and H. Kress-Gazit, “A model for verifiable grounding and execution of complex natural language instructions,” in _IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_ , 2016, pp. 2649–2654.
  * [18] D. Nyga _et al._ , “Grounding robot plans from natural language instructions with incomplete world knowledge,” in _Conference on robot learning_ , 2018, pp. 714–723.
  * [19] P. Pramanick, H. B. Barua, and C. Sarkar, “Decomplex: Task planning from complex natural instructions by a collocating robot,” in _IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_ , 2020, pp. 6894–6901.
  * [20] J. K. Behrens, K. Stepanova, R. Lange, and R. Skoviera, “Specifying dual-arm robot planning problems through natural language and demonstration,” _IEEE Robotics and Automation Letters_ , vol. 4, no. 3, pp. 2622–2629, 2019\.
  * [21] M. Shridhar, L. Manuelli, and D. Fox, “Cliport: What and where pathways for robotic manipulation,” in _Conference on Robot Learning_ , 2022, pp. 894–906.
  * [22] D. Driess _et al._ , “PaLM-E: An embodied multimodal language model,” _arXiv preprint arXiv:2303.03378_ , 2023.
  * [23] C. Lynch _et al._ , “Interactive language: Talking to robots in real time,” _arXiv preprint arXiv:2210.06407_ , 2022.
  * [24] J. Luketina _et al._ , “A survey of reinforcement learning informed by natural language,” _arXiv preprint arXiv:1906.03926_ , 2019.
  * [25] L. Shao, T. Migimatsu, Q. Zhang, K. Yang, and J. Bohg, “Concept2robot: Learning manipulation concepts from instructions and human demonstrations,” in _Robotics: Science and Systems (RSS)_ , 2020.
  * [26] P. Sharma _et al._ , “Correcting robot plans with natural language feedback,” _arXiv preprint arXiv:2204.05186_ , 2022.
  * [27] T. Yun, C. Sun, and E. Pavlick, “Does vision-and-language pretraining improve lexical grounding?” 2021.
  * [28] M. Ahn _et al._ , “Do as i can, not as i say: Grounding language in robotic affordances,” 2022.
  * [29] A. Suglia, Q. Gao, J. Thomason, G. Thattai, and G. Sukhatme, “Embodied BERT: A transformer model for embodied, language-guided visual task completion,” _arXiv preprint arXiv:2108.04927_ , 2021.
  * [30] Silero Team, “Silero VAD: pre-trained enterprise-grade Voice Activity Detector (VAD), Number Detector and Language Classifier,” https://github.com/snakers4/silero-vad, 2021.
  * [31] D. Galvez _et al._ , “The People’s Speech: A Large-Scale Diverse English Speech Recognition Dataset for Commercial Usage,” _arXiv preprint arXiv:2111.09344_ , 2021.


# Semantic Analysis of Subexponential Modalities in Distributive Non-
commutative Linear Logic

Daniel Rogozin University College London
London, UK<EMAIL_ADDRESS>
###### Abstract

In this paper, we consider the full Lambek calculus enriched with
subexponential modalities in a distributive setting. We show that the
distributive Lambek calculus with subexponentials is complete with respect to
its Kripke frames via canonical extensions. In this approach, we consider
subexponentials as ${\bf S4}$-like modalities and each modality is interpreted
with a reflexive and transitive relation similarly to usual Kripke semantics.

###### keywords:

Lambek calculus, subexponentials, canonical extensions, residuated lattices

##  1 Introduction

Substructual logics are a kind of non-classical logic lacking some of the
structual rules: weakening, contraction and exchange, we refer the reader to
[24] to have a more systematic introduction. The Lambek calculus is a logic
with none of these rules, which was introduced initially by Lambek for
modelling natural language grammar by proof-theoretic means [19]. From an
algebraic point of view, the Lambek calculus is the logic of residuated
semigroups, which are also connected to other substructural logics such as
relevant or linear logics, see e.g. [21, 27].

The $!$-modality originally comes from linear logic, where $!$ introduces
lacking structural rules in a restricted way, see [12, 13]. Such modal
enrichments were also discussed in the context of resource management in
computation based on linear types, see [18, 28]. Enrichments of the Lambek
calculus with the $!$-modality can be also motivated in terms of mathematical
lingustics, see [17].

The polymodal expansion of the full Lambek calculus (that is, the Lambek
calculus with additive connectives and constants) with subexponentials has
been introduced by Kanovich, Kuznetsov, Scedrov and Nigam [16] to generalise
linear logical frameworks used for linear authorisation logics and concurrent
programming languages [20]. At the moment, subexponential modalities in non-
commutative linear logics have not been analysed comperehensively in terms of
semantics and the current known results are rather about proof-theoretic and
computability aspects, see [16].

In this paper, we show the completeness result for subexponential expansions
of distributive non-commutative linear logic. As far as subexponential
$!$-modalites obey some formal properties of the $\Box$-modality from
classical modal logic, we consider them as Kripke-style necessity operators as
in usual Kripke semantics of modal logic:

$\mathcal{M},x\models\>!\varphi$ iff $\forall
y\>(xRy\Rightarrow\mathcal{M},y\models\varphi)$

Generally, the full Lambek calculus is non-distributive, so the law of the
form

$\varphi\land(\psi\lor\theta)\vdash(\varphi\land\psi)\lor(\varphi\land\theta)$

is not valid unless we require this principle as an extra-axiom. So we
consider the distributive full Lambek calculus enriched with subexponentials
in terms of Kripke semantics with the standard truth definition for
disjunction and conjunction. Notice that the truth definition for conjuction
along with the Kripke-style definition for $!$ semantically imply that the
following principals should be valid as well, so we take them as the
additional axioms:

  * •

$\top\>\vdash\>!\top$,

  * •

$!\varphi\>\land\>!\psi\vdash\>!(\varphi\land\psi)$.

We prove that the distributive full Lambek calculus with subexponential
modalities is Kripke complete by showing that the corresponding variety of
distributive residuated lattices with modal operators is closed under
canonical extensions and conclude Kripke-completeness from canonicity. In
turn, canonical extensions are a kind of completions in algebraic logic
initially introduced for the purposes of extending Stone representation
theorem for Boolean algebras with operators [15], but canonical extensions
were further generalised for bounded (non)distributive lattices with operators
[9, 11, 4].

##  2 The full Lambek calculus and subexponentials

In this section, we recall the reader some preliminary notions related to the
full Lambek calculus, see [6, Chapter 2] for details.

Assume we have a fixed countable set of propositional variables
$\operatorname{PV}=\\{p_{i}\>|\>i<\omega\\}$. The set of formulas is generated
by the following grammar:

$\varphi::=\bot\>|\>\top\>|\>{\bf
1}\>|\>p\>|\>(\varphi\bullet\varphi)\>|\>(\varphi\setminus\varphi)\>|\>(\varphi/\varphi)\>|\>(\varphi\lor\varphi)\>|\>(\varphi\land\varphi)$

A family of substructural modal logics we are going to consider further enrich
the full Lambek calculus, which is defined the following way.

######  Definition 1.

_The full Lambek calculus_ is defined as the smallest set of pairs of formulas
$\varphi\vdash\psi$ that contains the following axioms and is closed under the
inference rules:

  * •

$p\vdash\top$

  * •

$\bot\vdash p$

  * •

$p\bullet(q\bullet r)\dashv\vdash(p\bullet q)\bullet r$

  * •

$p\dashv\vdash{\bf 1}\bullet p\dashv\vdash p\bullet{\bf 1}$

  * •

$p_{i}\vdash p_{1}\lor p_{2}$, for $i=1,2$

  * •

$p_{1}\land p_{2}\vdash p_{i}$, for $i=1,2$

$\varphi\vdash\psi$ $\varphi\vdash\theta$ $\varphi\vdash\psi\land\theta$

$\varphi\vdash\psi$ $\theta\vdash\tau$
$\varphi\bullet\theta\vdash\psi\bullet\tau$

$\varphi\bullet\psi\vdash\theta$ $\psi\vdash\varphi\setminus\theta$

$\varphi\vdash\theta$ $\psi\vdash\theta$ $\varphi\lor\psi\vdash\theta$

$\varphi(p)\vdash\psi(p)$ $\varphi[p:=\theta]\vdash\psi[p:=\theta]$

$\varphi\bullet\psi\vdash\theta$ $\psi\vdash\theta/\varphi$

$\varphi\vdash\psi$ $\psi\vdash\varphi$ $\varphi\vdash\theta$

The _distributive_ full Lambek calculus extends the full Lambek calculus with
the following extra axiom

$p\land(q\lor r)\vdash(p\land q)\lor(p\land r)$.

Algebraically, the full Lambek calculus is the logic of residuated lattices.
_A residuated lattice_ is an algebra
$\mathcal{L}=(L,\wedge,\vee,\cdot,\cdot,\setminus,/,\top,\bot,\epsilon)$ such
that $(L,\wedge,\vee,\top,\bot)$ is a bounded lattice, $(L,\cdot,\epsilon)$ is
a monoid and $\setminus$ and $/$ are binary operations (_residuals_) such that
for all $a,b,c\in L$:

$b\leq a\setminus b\leftrightarrow a\cdot b\leq c\leftrightarrow a\leq c/b$

A residuated lattice is _distributive_ if its lattice reduct is distributive.

Note that the classes of distributive and non-distributive residuated lattices
are known to be varieties [14, Lemma 2.3], so both of these classes are
equationally axiomatisable. Thus the (distributive) full Lambek calculus is
complete with respect to the variety of (distributive) residuated lattices.
Standarly, each pair of formulas $\varphi\vdash\psi$ corresponds to an
algebraic inequality $\varphi\leq\psi$, which is, in turn, an algebraic
identity of the form $\varphi\wedge\psi=\psi$.

To enrich the full Lambek calculus with a family of modalities
$(!_{i})_{i\in\Sigma}$ for some $\Sigma\neq\emptyset$, we extend the grammar
of formulas as follows:

$!_{i}\varphi$ is a formula for each $i\in\Sigma$ whenever $\varphi$ is a
formula.

Let $(I,\preceq)$ be a preorder. A _subexponential signature_ is a tuple
$\Sigma=(I,\preceq,W,E,C)$, where $W,E,C$ are upward closed subsets of $I$
such that $W\cap C\subseteq E$. Each of these distinguished subsets
corresponds to those modalities that emulate a particular structural rule.
That is, subexponentials introducing the weakining rule are indexed by
elements from $W$ and the same for the exchange and contraction rules. The
requirement $W\cap C\subseteq E$ means that if a subexponential allows
weakining and exchanging, then it also allows contracting, see [16].

Originally, the Lambek calculus with subexponentials was introduced in [16] as
follows.

######  Definition 2.

_The full Lambek calculus with subexponential modalities_ over a signature
$\Sigma$, denoted as $\operatorname{SMALC_{\Sigma}}$, is the defined by the
following axioms and inference rules, for all $i,j,k\in\Sigma$:

  1. 1.

The axioms and inference rules of the full Lambek calculus,

  2. 2.

$!_{i}p\>\vdash\>p$ and $!_{i}p\>\vdash\>!_{i}!_{i}p$,

  3. 3.

$!_{i}p\>\bullet\>!_{j}q\vdash\>!_{k}(p\bullet q)$ such that $k\succeq i,j$,

  4. 4.

${\bf 1}\vdash\>!_{i}{\bf 1}$,

  5. 5.

$!_{i}p\>\bullet q\vdash\>!_{i}p\bullet q\>\bullet\>!_{i}p$ and
$q\>\bullet\>!_{i}p\vdash\>!_{i}p\bullet q\>\bullet\>!_{i}p$, for $i\in C$,

  6. 6.

$!_{i}p\vdash{\bf 1}$, for $i\in W$,

  7. 7.

$!_{i}p\bullet\>q\dashv\vdash\>q\>\bullet\>!_{i}p$ for $i\in E$,

  8. 8.

From $\varphi\vdash\psi$ infer $!_{i}\varphi\>\vdash\>!_{j}\psi$ whenever
$j\preceq i$.

######  Remark 3.

We also can drop the ${\bf S4}$-axioms and all results of this paper will be
preserved in that case. However, these axioms are important as the promotion
and dereliction principles from linear logic and modal enrichments of the
Lambek calculus.

######  Remark 4.

In [16], $\operatorname{SMALC_{\Sigma}}$ was introduced as a Gentzen-style
sequent calculus, but our version is more Hilbert-like. One can show that both
formalisms are equivalent standardly.

######  Remark 5.

Instead of the traditional contraction axiom of the form
$!_{i}\>p\vdash\>!_{i}p\>\bullet\>!_{i}p$, we use its more general version
introduced in [16], which is called _non-local contraction_.

As we have already discussed in the first section, we extend
$\operatorname{SMALC_{\Sigma}}$ with additional axioms to define the
distributive version of the full Lambek calculus with subexponentials.

######  Definition 6.

_The distributive full Lambek calculus with subexponential modalities_ over a
subexponential signature $\Sigma$, denoted as
$\operatorname{DSMALC_{\Sigma}}$, extends $\operatorname{SMALC_{\Sigma}}$ with
the following extra axioms, for each $i\in\Sigma$:

  1. 1.

$p\land(q\lor r)\vdash(p\land q)\lor(p\land r)$,

  2. 2.

$!_{i}p\>\land\>!_{i}q\vdash\>!_{i}(p\land q)$,

  3. 3.

$\top\vdash\>!_{i}\top$.

For algebraic semantics, we define $\Sigma$-algebras for a subexponential
signature $\Sigma$ by rewriting the axioms of $\operatorname{DSMALC_{\Sigma}}$
as algebraic inequalities.

######  Definition 7.

Let $\Sigma$ be a subexponential signature, a $\Sigma$-algebra is an algebra
$\mathcal{R}=(R,\wedge,\vee,\cdot,\setminus,/,(!_{i})_{i\in\Sigma},\\\
\top,\bot,\epsilon)$ such that, for each $i,j,k\in\Sigma$:

  1. 1.

$(R,\wedge,\vee,\cdot,\setminus,/,\top,\bot,\epsilon)$ is a distributive
residuated lattice,

  2. 2.

$!_{i}$ preserves finite infima, $\epsilon\leq\>!_{i}\epsilon$ and
$!_{i}!_{i}a=\>!_{i}a\leq a$ for all $a\in\mathcal{R}$,

  3. 3.

If $k\preceq i,j$, then $!_{i}a\>\cdot\>!_{j}b\leq\>!_{k}(a\cdot b)$ for all
$a,b\in\mathcal{R}$,

  4. 4.

$!_{i}a\>\cdot b\leq\>!_{i}a\cdot b\>\cdot\>!_{i}a$ and
$b\>\cdot\>!_{i}a\leq\>!_{i}a\cdot b\>\cdot\>!_{i}a$, for $a,b\in\mathcal{R}$
and $i\in C$,

  5. 5.

$!_{i}a\leq\epsilon$, for $a\in\mathcal{R}$ and $i\in W$,

  6. 6.

$!_{i}a\cdot\>b=\>b\>\cdot\>!_{i}a$ for $a,b\in\mathcal{R}$ and $i\in E$.

Let $\mathcal{R}$ be a $\Sigma$-algebra, an algebraic interpretation is a
function $[\\![.]\\!]:\operatorname{PV}\to\mathcal{R}$ that commutes with
connectives usually. Clearly that the class of all $\Sigma$-algebras is a
variety and $\operatorname{DSMALC_{\Sigma}}$ is complete with respect to that
class. We denote the free algebra with $\omega$ generators from the variety of
$\Sigma$-algebras as $F_{\Sigma}$. The fact that $\varphi\vdash\psi$ is
provable in $\operatorname{DSMALC_{\Sigma}}$ iff
$F_{\Sigma}\models[\\![\varphi]\\!]\leq[\\![\psi]\\!]$ can be thought as
folklore from universal algebra.

##  3 Kripke semantics

In this subsection, we introduce relational semantics for
$\operatorname{DSMALC_{\Sigma}}$ in the fashion of Routley-Meyer models for
relevant logic (see [25, 26]) and well as for other substructural logics (see
[2]).

First of all, we define ternary frames for the distributive full Lambek
calculus.

######  Definition 8.

A ternary Kripke frame is a structure $F=(W,\leq,R,\mathcal{O})$ where
$R\subseteq W^{3}$, $\mathcal{O}\subseteq W$ and $(W,\leq)$ is a poset such
that for all $u,v,w,u^{\prime},v^{\prime},w^{\prime}$:

  * •

$\exists x\in W\>Ruwx\>\&\>Rxu^{\prime}v^{\prime}\Leftrightarrow\exists y\in
W\>Rwu^{\prime}y\>\&\>Ruyv^{\prime}$,

  * •

$Ruvw\>\&\>u^{\prime}\leq u\Rightarrow Ru^{\prime}vw$,

  * •

$Ruvw\>\&\>v^{\prime}\leq v\Rightarrow Ruv^{\prime}w$

  * •

$Ruvw\>\&\>w\leq w^{\prime}\Rightarrow Ruvw^{\prime}$,

  * •

$O$ is an upward closed subset such that $\forall o\in O\>Rvow\Leftrightarrow
Rovw$ and $v\leq w\Rightarrow\exists o\in O\>Rvow$,

One can associate a distributive residuated lattice with every ternary Kripke
frame. Given a ternary frame $\mathcal{F}=(W,\leq,R,\mathcal{O})$, its dual
_complex algebra_ is the algebra
$\operatorname{Cm}(\mathcal{F})=(\operatorname{Up}(W),\cap,\cup,\cdot,\setminus,/,\mathcal{O},\emptyset,W)$
where:

  1. 1.

$\operatorname{Up}(\mathcal{F})$ is the set of all upward closed subsets of
$\operatorname{Up}(W)$,

  2. 2.

$A\cdot B=\\{w\in W\>|\>\exists u,v\in W\>Ruvw\>\&\>u\in A\>\&\>v\in B\\}$,

  3. 3.

$A\setminus B=\\{w\in W\>|\>\forall u,v\in W\>Ruwv\>\&\>u\in A\Rightarrow v\in
B\\}$,

  4. 4.

$A/B=\\{w\in W\>|\>\forall u,v\in W\>Rwuv\>v\in B\Rightarrow u\in A\\}$.

It is readily checked that $\operatorname{Cm}(\mathcal{F})$ is well-defined
since if $A,B$ are upward closed, so are $A\cdot B$, $A\setminus B$ and $A/B$.
Also, using the definition of a ternary frame, one can show that
$\operatorname{Cm}(\mathcal{F})$ is indeed a distributive residuated lattice.

######  Definition 9.

Let $\mathcal{F}$ be a ternary frame and let $\Sigma$ be a subexponential
signature. A _$\Sigma$ -frame_ is an expansion of $\mathcal{F}$ with a family
of binary relations $(R_{i})_{i\in\Sigma}$ such that for all $i,j,k\in\Sigma$:

  1. 1.

For all $i\in\Sigma$, $(\mathcal{F},R_{i})$ is a preorder,

  2. 2.

For all $u,v,w,w^{\prime}\in\mathcal{F}$, if $k\preceq
i,j,Ruvw\>\&\>wR_{k}w^{\prime}$, then $\exists x,y\in
W\>Rxyw^{\prime}\>\&\>uR_{i}x\>\&\>vR_{j}y$,

  3. 3.

$i\preceq j$ implies $R_{j}\subseteq R_{i}$,

  4. 4.

$\mathcal{O}\subseteq[R_{i}]\mathcal{O}$.

  5. 5.

For $A,B\in\operatorname{Up}(\mathcal{F})$ one has $[R_{i}]A\cdot
B=B\cdot[R_{i}]A$ whenever $i\in E$,

  6. 6.

For $A,B\in\operatorname{Up}(\mathcal{F})$ one has $[R_{i}]A\cdot
B\subseteq[R_{i}]A\cdot B\cdot[R_{i}]A$ and
$B\cdot[R_{i}]A\subseteq[R_{i}]A\cdot B\cdot[R_{i}]A$ whenever $i\in C$,

  7. 7.

For $A\subseteq\operatorname{Up}(\mathcal{F})$ one has
$[R_{i}]A\subseteq\mathcal{O}$ whenever $i\in W$.

where $[R_{i}]A=\\{u\in W\>|\>\forall w\in W\>(uR_{i}w\Rightarrow u\in A)\\}$
for $i\in\Sigma$.

Given a $\Sigma$-frame
$\mathcal{F}=(W,\leq,(R_{i})_{i\in\Sigma},R,\mathcal{O})$, the _complex
algebra_ of $\mathcal{F}$ is the algebra
$\operatorname{Cm}(\mathcal{F})=(\operatorname{Up}(W),\cap,\cup,\cdot,\setminus,/,([R_{i}])_{i\in\Sigma},\mathcal{O},\emptyset,W)$.
Frames and complex algebras are connected with each other as usual in duality
theory:

######  Proposition 10.

Let $\mathcal{F}$ be a $\Sigma$-frame, then $\operatorname{Cm}(\mathcal{F})$
is a $\Sigma$-algebra.

###### Proof.

Follows from the definition of a $\Sigma$-frame. ∎

######  Definition 11.

_A Kriple model_ is a tuple $\mathcal{M}=(\mathcal{F},\vartheta)$ where
$\mathcal{F}$ is a modal ternary Kripke frame and
$\vartheta:\operatorname{PV}\to\operatorname{Up}(W,\leq)$ is a valuation map.
The truth definition is inductive:

  * •

$\mathcal{M},w\models p$ iff $w\in\vartheta(p)$,

  * •

$\mathcal{M},w\not\models\bot$,

  * •

$\mathcal{M},w\models\top$,

  * •

$\mathcal{M},w\models{\bf 1}$ iff $w\in O$,

  * •

$\mathcal{M},w\models\>!_{i}\varphi$ iff $\forall u\in
R_{i}(w)\>\mathcal{M},u\models\varphi$,

  * •

$\mathcal{M},w\models\varphi\lor\psi$ iff $\mathcal{M},w\models\varphi$ or
$\mathcal{M},w\models\psi$,

  * •

$\mathcal{M},w\models\varphi\land\psi$ iff $\mathcal{M},w\models\varphi$ and
$\mathcal{M},w\models\psi$,

  * •

$\mathcal{M},w\models\varphi\bullet\psi$ iff $\exists u,v\in
W\>Ruvw\>\&\>\mathcal{M},u\models\varphi\>\&\>\mathcal{M},v\models\psi$,

  * •

$\mathcal{M},w\models\varphi\setminus\psi$ iff $\forall u,v\in
W\>Ruwv\>\&\>\mathcal{M},u\models\varphi\Rightarrow\mathcal{M},v\models\psi$,

  * •

$\mathcal{M},w\models\psi/\varphi$ iff $\forall u,v\in
W\>Rwuv\>\&\>\mathcal{M},u\models\varphi\Rightarrow\mathcal{M},v\models\psi$,

  * •

$\mathcal{M},w\models\varphi\vdash\psi$ iff $\mathcal{M},w\models\varphi$
implies $\mathcal{M},w\models\psi$.

As usual, $\mathcal{M}\models\varphi\vdash\psi$ iff
$\mathcal{M},w\models\varphi\vdash\psi$ for each $w\in\mathcal{M}$.

Let $\mathcal{F}$ be a $\Sigma$-frame, then
$\mathcal{F}\models\varphi\vdash\psi$ iff
$(\mathcal{F},\vartheta)\models\varphi\vdash\psi$ for each $\vartheta$. Let
$T$ be a set of sequents, then $\mathcal{F}\models T$ iff
$\mathcal{F}\models\varphi\vdash\psi$ for each $\varphi\vdash\psi\in T$.

Given a model
$\mathcal{M}=(W,\leq,R,(R_{i})_{i\in\Sigma},\mathcal{O},\vartheta)$, define
its truth set $[\\![\varphi]\\!]=\\{w\in
W\>|\>\mathcal{M},w\models\varphi\\}$. One can show that the set of truth sets
form a subalgebra of the complex algebra $\operatorname{Cm}(\mathcal{F})$,
where $\mathcal{F}$ is the underlying $\Sigma$-frame of a model $\mathcal{M}$.
To be more precise, the following fact standardly holds:

######  Proposition 12.

Let $\mathcal{M}=(W,\leq,R,(R_{i})_{i\in I},\mathcal{O},\vartheta)$ be a
model, then:

  1. 1.

$[\\![p]\\!]=\vartheta(p)$,

  2. 2.

$[\\![\bot]\\!]=\emptyset$,

  3. 3.

$[\\![\top]\\!]=W$,

  4. 4.

$[\\![{\bf 1}]\\!]=\mathcal{O}$,

  5. 5.

$[\\![\varphi\lor\psi]\\!]=[\\![\varphi]\\!]\cup[\\![\psi]\\!]$

  6. 6.

$[\\![\varphi\land\psi]\\!]=[\\![\varphi]\\!]\cap[\\![\psi]\\!]$,

  7. 7.

$[\\![\varphi\bullet\psi]\\!]=[\\![\varphi]\\!]\cdot[\\![\psi]\\!]$,

  8. 8.

$[\\![\varphi\setminus\psi]\\!]=[\\![\varphi]\\!]\setminus[\\![\psi]\\!]$

  9. 9.

$[\\![\varphi/\psi]\\!]=[\\![\varphi]\\!]/[\\![\psi]\\!]$,

  10. 10.

$[\\![!_{i}\varphi]\\!]=[R_{i}][\\![\varphi]\\!]$ for each $i\in\Sigma$,

  11. 11.

$\mathcal{M}\models\varphi\vdash\psi$ iff
$[\\![\varphi]\\!]\subseteq[\\![\psi]\\!]$.

######  Theorem 13.

(Soundness)

Let $\mathcal{F}$ be a $\Sigma$-frame and $\vartheta$ a valuation, then
$\mathcal{F}\models\operatorname{DSMALC}_{\Sigma}$.

###### Proof.

We check only subexponential axioms, for the rest of the axioms and inference
rules, the proof is similar to [26, Theorem 2].

Let $w\in\mathcal{F}$ and $i,j,k\in\Sigma$.

  1. 1.

We show that
$\mathcal{M},w\models\>!_{i}p\>\bullet\>!_{j}q\vdash\>!_{k}(p\bullet q)\in L$.

Assume that $\mathcal{M},w\models\>!_{i}p\>\bullet\>!_{j}q$. Then there are
$u,v\in\mathcal{F}$ such that $Ruvw$, $\mathcal{M},u\models\>!_{i}p$ and
$\mathcal{M},v\models\>!_{j}q$. Take any $w^{\prime}\in R_{k}(w)$. Then $Ruvw$
and $wR_{k}w^{\prime}$ imply that there are $x,y\in\mathcal{F}$ such that
$Rxyw^{\prime}$, $uR_{i}x$ and $vR_{j}y$, so $\mathcal{M},x\models p$ and
$\mathcal{M},y\models q$, and, thus, $\mathcal{M},w^{\prime}\models p\bullet
q$ and $\mathcal{M},w\models\>!_{k}(p\bullet q)$.

  2. 2.

$\mathcal{M},w\models{\bf 1}\vdash\>!_{i}{\bf 1}$ follows from the condition
$[R_{i}]\mathcal{O}=\mathcal{O}$.

  3. 3.

Assume that $\mathcal{M},w\models\>!_{i}p\>\bullet\>q$ and $i\in E$, then, by
Proposition 12, $w\in[R_{i}][\\![p]\\!]\cdot[\\![q]\\!]$. But $i\in E$, then
$w\in[\\![q]\\!]\cdot[R_{i}][\\![p]\\!]$, then
$\mathcal{M},w\models\>q\>\bullet\>!_{i}p$.

  4. 4.

Assume that $\mathcal{M},w\models\>!_{i}p\>\bullet\>q$ and $i\in C$, then, by
Proposition 12, $w\in[R_{i}][\\![p]\\!]\cdot[\\![q]\\!]$, so, as far as $i\in
C$, $w\in[R_{i}][\\![p]\\!]\cdot[\\![q]\\!]\cdot[R_{i}][\\![p]\\!]$, then
$\mathcal{M},w\models\>!_{i}p\>\bullet\>q\>\bullet\>!_{i}p$. Another non-local
contraction case can be proved similarly.

  5. 5.

$\mathcal{M},w\models\>!_{i}p\>\vdash\>{\bf 1}$ for $i\in W$ follows from the
condition $[R_{i}][\\![p]\\!]\subseteq\mathcal{O}$.

Notice that validity of axioms $!_{i}p\>\vdash\>p$ and
$!_{i}p\>\vdash\>!_{i}!_{i}p$ follows from the condition that each $R_{i}$ is
reflexive and transitive, the proof is completely identical to [3, Proposition
3.30 and Proposition 3.31]. ∎

Also, we connect validity in $\Sigma$-frames and their complex algebras as
follows:

######  Proposition 14.

Let $\mathcal{F}$ be a $\Sigma$-frame, then
$\mathcal{F}\models\varphi\vdash\psi$ iff
$\operatorname{Cm}\models[\\![\varphi]\\!]\leq[\\![\psi]\\!]$.

###### Proof.

Follows from Proposition 12. ∎

##  4 Canonical extensions

In this section, we introduce canonical extensions of $\Sigma$-algebras for an
arbitrary subexponential signature $\Sigma$ and then use them to show Kripke
completeness of $\operatorname{DSMALC}_{\Sigma}$ with respect to its frames.
Alternatively, we can show that $\operatorname{DSMALC}_{\Sigma}$ is complete
with respect to Kripke semantics by constructing the canonical model of prime
theories, but we preferred this more algebraic approach to show both Kripke
completeness and canonicity of the corresponding variety of algebras.

Our approach is based on canonical extensions of distributive bounded lattice
expansions in the fashion of [9] and [10]. We recall some underlying notions
first of all.

Let $\mathcal{L}$ be a (complete) lattice and $a\in\mathcal{L}$, then $a$ is
_completely join-irreducible_ if $a=\bigvee\limits_{i\in I}a_{i}$ implies that
there is $j\in I$ such that $a=a_{j}$. Completely meet-irreducibeles are
defined dually. $\mathcal{J}^{\infty}(\mathcal{L})$
($\mathcal{M}^{\infty}(\mathcal{L})$) is the set of all completely join-
irreducible (meet-irreducible) elements.

Note that $\mathcal{J}^{\infty}(\mathcal{L})$ and
$\mathcal{M}^{\infty}(\mathcal{L})$ are order-isomorphic, the isomorphism
$\kappa:\mathcal{J}^{\infty}(\mathcal{L})\to\mathcal{M}^{\infty}(\mathcal{L})$
is defined as $\kappa:j\mapsto\bigvee(-\uparrow j)$, see, e.g., [9, Theorem
2.3].

Let us define canonical extensions for bounded distributive lattices first and
then extend it for required expansions. Recall that a complete distributive
lattice $\mathcal{L}$ is called _perfect_ if:

  1. 1.

$\mathcal{L}$ is completely distributive, that is, for every doubly indexed
family $(a_{i,j})_{i\in I,j\in J}$ of $\mathcal{L}$, one has:

$\bigwedge\limits_{i\in I}\bigvee\limits_{j\in
J}a_{i,j}=\bigvee\limits_{f:J\to I}\bigwedge\limits_{i\in I}a_{i,f(i)}$

  2. 2.

and every $a\in\mathcal{L}$ can be expressed as

$a=\bigvee\\{j\in\mathcal{J}^{\infty}(\mathcal{L})\>|\>j\leq
x\\}=\bigwedge\\{m\in\mathcal{M}^{\infty}(\mathcal{L})\>|\>x\leq m\\}$.

A _canonical extension_ of a bounded distributive lattice $\mathcal{L}$ is a
perfect lattice $\mathcal{L}^{\sigma}$ such that
$\mathcal{L}\hookrightarrow\mathcal{L}^{\sigma}$ and such that the compactness
property holds:

  * •

If $S,T\subseteq\mathcal{L}$ such that $\bigwedge S\leq\bigvee T$ in
$\mathcal{L}^{\sigma}$, then there are finite subsets $S^{\prime}\subseteq S$
and $T^{\prime}\subseteq T$ such that $\bigwedge S^{\prime}\leq\bigvee
T^{\prime}$ in $\mathcal{L}$.

It is known that every bounded distributive lattice has a unique canonical
extension, which is, in fact, based on the Stone-Priestley representation
[22].

To define canonical extensions for $\Sigma$-algebras, we combine techiques for
residuated lattices and distributive modal algebras developed in [8] and [10].
Let
$\mathcal{R}=(R,\vee,\wedge,\cdot,\setminus,/,(!_{i})_{i\in\Sigma},0,1,\epsilon)$
be a $\Sigma$-algebra, then $\mathcal{A}$ is _perfect_ if the following holds:

  1. 1.

The lattice reduct is perfect as a distributive lattice,

  2. 2.

$!$ preserves all infima,

  3. 3.

$\cdot$ is completely additive, that is, it preserve all suprema in both
arguments,

  4. 4.

The residuals $\setminus:\mathcal{R}\times\mathcal{R}\to\mathcal{R}$ and
$/:\mathcal{R}\times\mathcal{R}\to\mathcal{R}$ are complete in the following
sense: $\setminus$ preserves all infima in the first argument and all suprema
in the second one and $/$ preserves all suprema in the first agrument and all
infima in the second one.

Given a $\Sigma$-algebra
$\mathcal{R}=(R,\wedge,\vee,\cdot,\setminus,/,(!_{i})_{i\in\Sigma},0,1,\epsilon)$,
the canonical extension of its lattice reduct $\mathcal{R}^{\sigma}$ can be
extended to the canonical extension of a $\Sigma$-algebra. First of all, we
define the sets of _filter_ and _ideal_ elements of the canonical extension
$\mathcal{R}^{\sigma}$ as

$\mathcal{F}(\mathcal{R}^{\sigma})=\\{x\in\mathcal{R}^{\sigma}\>|\>\text{$x$
is a meet of elements from $\mathcal{R}$}\\}$,

$\mathcal{I}(\mathcal{R}^{\sigma})=\\{x\in\mathcal{R}^{\sigma}\>|\>\text{$x$
is a join of elements from $\mathcal{R}$}\\}$.

First of all, we define residuals, modalities and the product operation for
filter and ideal elements and then extend them for arbitrary elements of the
canonical extension $\mathcal{R}^{\sigma}$ of the lattice reduct of
$\mathcal{R}$. Given $x,x^{\prime}\in\mathcal{F}(\mathcal{R}^{\sigma})$ and
$y,y^{\prime}\in\mathcal{I}(\mathcal{R}^{\sigma})$, then:

  * •

$x\cdot^{\sigma}x^{\prime}=\bigwedge\\{a\cdot a^{\prime}\>|\>x\leq
a\in\mathcal{R}\>\&\>x^{\prime}\leq a^{\prime}\in\mathcal{R}\\}$,

  * •

$x\setminus^{\pi}y=\bigvee\\{a\setminus b\>|\>x\leq a\in\mathcal{R}\ni b\leq
y\\}$,

  * •

$!_{i}^{\sigma}x=\bigwedge\\{!_{i}a\>|\>x\leq a\in\mathcal{R}\\}$,
$i\in\Sigma$.

So we define the canonical extesion $\mathcal{R}^{\sigma}$ of $\mathcal{R}$ as
the algebra
$\mathcal{R}^{\sigma}=(R^{\sigma},\cdot^{\sigma},\setminus^{\pi},/^{\pi},(!_{i}^{\sigma})_{i\in\Sigma},\epsilon)$
such that for all $a,b\in\mathcal{R}^{\sigma}$:

  * •

$a\cdot^{\sigma}b=\bigvee\\{a^{\prime}\cdot^{\sigma}b^{\prime}\>|\>\mathcal{F}(\mathcal{R}^{\sigma})\ni
a^{\prime}\leq a\>\&\>\mathcal{F}(\mathcal{R}^{\sigma})\ni b^{\prime}\leq
b\\}$,

  * •

$a\setminus^{\pi}b=\bigwedge\\{a^{\prime}\setminus^{\pi}b^{\prime}\>|\>a\geq
a^{\prime}\in\mathcal{I}(\mathcal{R}^{\sigma})\>\&\>b\geq
b^{\prime}\in\mathcal{F}(\mathcal{R}^{\sigma})\\}$,

  * •

$!_{i}^{\sigma}a=\bigvee\\{!^{\sigma}a^{\prime}\>|\>\mathcal{F}(\mathcal{R}^{\sigma})\ni
a^{\prime}\leq a\\}$, $i\in\Sigma$.

The definition of $/^{\pi}$ is right-to-left symmetric to $\setminus^{\pi}$.

######  Theorem 15.

The variety of all $\Sigma$-algebras is canonical, that is, it is closed under
canonical extensions.

###### Proof.

The fact that the canonical extension of the lattice reduct is a perfect
distributive lattice is by Jonsson, see [7, Theorem 2.3]. The canonical
extension of the residuated lattice reduct of $\mathcal{R}$ is also a perfect
residuated lattice, see [8, Proposition 5]. The canonical extension of the $!$
modal operator preserves arbitrary infima as it was shown in [10, Lemma 2.21].
The items we have got to check are the following, for $i,j,k\in\Sigma$ and for
$a,b\in\mathcal{R}^{\sigma}$:

  1. 1.

$!^{\sigma}_{i}a\>\cdot^{\sigma}\>!^{\sigma}_{j}b\leq\>!_{k}^{\sigma}(a\cdot^{\sigma}b)$
whenever $k\preceq i,j$,

  2. 2.

$\epsilon\leq!^{\sigma}_{i}\epsilon$

  3. 3.

$!^{\sigma}_{i}a\>\cdot^{\sigma}b=b\>\cdot^{\sigma}\>!^{\sigma}_{i}a$ whenever
$i\in E$,

  4. 4.

$!^{\sigma}_{i}a\>\cdot^{\sigma}b\leq\>!^{\sigma}_{i}a\cdot^{\sigma}b\>\cdot^{\sigma}\>!^{\sigma}_{i}a$
and
$b\>\cdot^{\sigma}\>!^{\sigma}_{i}a\leq\>!^{\sigma}_{i}a\cdot^{\sigma}b\>\cdot^{\sigma}\>!^{\sigma}_{i}a$
whenever $i\in C$,

  5. 5.

$!^{\sigma}_{i}a\>\leq\epsilon$ whenever $i\in W$.

We check only the first and fourth items, the rest of them are shown
similarly.

  1. 1.

Fix $i,j,k\in\Sigma$ such that $k\preceq i,j$. First of all, take any
$x,y\in\mathcal{F}(\mathcal{R}^{\sigma})$. Observe that

$!^{\sigma}_{i}x\>\cdot^{\sigma}\>!^{\sigma}_{j}y=\bigwedge\\{!_{i}\>x^{\prime}\cdot\>!_{j}y^{\prime}\>|\>x\leq
x^{\prime}\in\mathcal{R}\>\&\>y\leq y^{\prime}\in\mathcal{R}\\}$

which follows from that fact $\cdot^{\sigma}$ is order-preserving and both
$!^{\sigma}_{i}$ and $!^{\sigma}_{j}$ preserve arbitrary infima and from the
definition of a filter element. Then we have:

$\begin{array}[]{lll}!^{\sigma}_{i}x\>\cdot^{\sigma}\>!^{\sigma}_{j}y=&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{By the definition of $\cdot^{\sigma}$}\\\
\>\>\>\>\>\>\>\>\bigwedge\\{!_{i}\>x^{\prime}\cdot\>!_{j}y^{\prime}\>|\>x\leq
x^{\prime}\in\mathcal{R}\>\&\>y\leq y^{\prime}\in\mathcal{R}\\}\leq&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{By the axiom of $\Sigma$-algebras}\\\
\>\>\>\>\>\>\>\>\bigwedge\\{!_{k}(x^{\prime}\cdot\>y^{\prime})\>|\>x\leq
x^{\prime}\in\mathcal{R}\>\&\>y\leq y^{\prime}\in\mathcal{R}\\}=&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{By the definition of $!_{k}^{\sigma}$}\\\
\>\>\>\>\>\>\>\>\bigwedge!_{k}^{\sigma}\\{x^{\prime}\cdot\>y^{\prime}\>|\>x\leq
x^{\prime}\in\mathcal{R}\>\&\>y\leq y^{\prime}\in\mathcal{R}\\}=&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{$!_{k}^{\sigma}$ commutes with all infima}\\\
\>\>\>\>\>\>\>\>!_{k}^{\sigma}\bigwedge\\{x^{\prime}\cdot\>y^{\prime}\>|\>x\leq
x^{\prime}\in\mathcal{R}\>\&\>y\leq y^{\prime}\in\mathcal{R}\\}=&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{By the definition of $\cdot^{\sigma}$}\\\
\>\>\>\>\>\>\>\>!_{k}^{\sigma}(x\>\cdot^{\sigma}y)\end{array}$

Now we show it for arbitrary elements of $\mathcal{R}^{\sigma}$. So take any
$a,b\in\mathcal{R}^{\sigma}$, then, using monotonicity of $!_{k}^{\sigma}$ and
the observation above, we have:

$\begin{array}[]{lll}!^{\sigma}_{i}a\>\cdot^{\sigma}\>!^{\sigma}_{j}b=&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{By the definition of $!^{\sigma}_{i}$,
$!^{\sigma}_{j}b$ and $\cdot^{\sigma}$ }&&\\\
\>\>\>\>\>\>\>\>\bigvee\\{!^{\sigma}_{i}x\>\cdot^{\sigma}\>!^{\sigma}_{j}y\>|\>a\geq
x\in\mathcal{F}(\mathcal{R}^{\sigma})\>\&\>b\geq
y\in\mathcal{F}(\mathcal{R}^{\sigma})\\}\leq&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{The observation above}&&\\\
\>\>\>\>\>\>\>\>\bigvee\\{!^{\sigma}_{k}(x\cdot^{\sigma}y)\>|\>a\geq
x\in\mathcal{F}(\mathcal{R}^{\sigma})\>\&\>b\geq
y\in\mathcal{F}(\mathcal{R}^{\sigma})\\}\leq&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{$!^{\sigma}_{k}$ is order-preserving}&&\\\
\>\>\>\>\>\>\>\>!^{\sigma}_{k}\bigvee\\{x\cdot^{\sigma}y\>|\>a\geq
x\in\mathcal{F}(\mathcal{R}^{\sigma})\>\&\>b\geq
y\in\mathcal{F}(\mathcal{R}^{\sigma})\\}=&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{The definition of $\cdot^{\sigma}$}&&\\\
\>\>\>\>\>\>\>\>!^{\sigma}_{k}(a\cdot^{\sigma}b)\end{array}$

  2. 2.

Now let us show that
$!^{\sigma}_{i}a\>\cdot^{\sigma}b\leq\>!^{\sigma}_{i}a\cdot^{\sigma}b\>\cdot^{\sigma}\>!^{\sigma}_{i}a$
for $i\in C$ and $a,b\in\mathcal{R}^{\sigma}$. Let us show this inequality for
filter elements first. Let $x,y\in\mathcal{F}(\mathcal{R}^{\sigma})$, then

$\begin{array}[]{lll}!^{\sigma}_{i}x\>\cdot^{\sigma}y=&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{The definition of filter elements and
$!^{\sigma}_{i}$}&&\\\ \>\>\>\>\>\>\>\>\bigwedge\\{!_{i}x^{\prime}\>|\>x\leq
x^{\prime}\in\mathcal{R}\\}\>\cdot^{\sigma}\bigwedge\\{y^{\prime}\in\mathcal{R}\>|\>y\leq
y^{\prime}\\}=&&\\\ \>\>\>\>\>\>\>\>\>\>\>\>\text{The definition of
$\cdot^{\sigma}$}&&\\\
\>\>\>\>\>\>\>\>\bigwedge\\{!_{i}x^{\prime}\>\cdot\>y^{\prime}\>|\>x\leq
x^{\prime}\in\mathcal{R},y\leq y^{\prime}\in\mathcal{R}\\}\leq&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{The axioms of $\Sigma$-algebras}&&\\\
\>\>\>\>\>\>\>\>\bigwedge\\{!_{i}x^{\prime}\>\cdot\>y^{\prime}\>\cdot\>!_{i}x^{\prime}\>|\>x\leq
x^{\prime}\in\mathcal{R},y\leq y^{\prime}\in\mathcal{R}\\}=&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{The definition of $\cdot^{\sigma}$}&&\\\
\>\>\>\>\>\>\>\>\bigwedge\\{!_{i}x^{\prime}\>|\>x\leq
x^{\prime}\in\mathcal{R}\\}\>\cdot^{\sigma}\bigwedge\\{y^{\prime}\in\mathcal{R}\>|\>y\leq
y^{\prime}\\}\cdot^{\sigma}\bigwedge\\{!_{i}x^{\prime}\>|\>x\leq
x^{\prime}\in\mathcal{R}\\}=&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{$!^{\sigma}_{i}$ preserves all infima}&&\\\
\>\>\>\>\>\>\>\>!^{\sigma}_{i}\bigwedge\\{x\>|\>x\leq
x^{\prime}\in\mathcal{R}\\}\>\cdot^{\sigma}\bigwedge\\{y\in\mathcal{R}\>|\>y\leq
y^{\prime}\\}\>\cdot^{\sigma}!^{\sigma}_{i}\bigwedge\\{x\>|\>x\leq
x^{\prime}\in\mathcal{R}\\}=&&\\\ \>\>\>\>\>\>\>\>\>\>\>\>\text{The definition
of $\cdot^{\sigma}$}&&\\\
\>\>\>\>\>\>\>\>!^{\sigma}_{i}x\>\cdot^{\sigma}y\>\cdot^{\sigma}\>!^{\sigma}_{i}x\end{array}$

So, for arbitrary $a,b\in\mathcal{R}^{\sigma}$ we have:

$\begin{array}[]{lll}{!}^{\sigma}_{i}a\>\cdot^{\sigma}b=&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{The definition of $\cdot^{\sigma}$}&&\\\
\>\>\>\>\>\>\>\>\bigvee\\{!^{\sigma}_{i}x\>\cdot^{\sigma}y\>|\>a\geq
x\in\mathcal{F}(\mathcal{R}^{\sigma}),y\geq\in\mathcal{F}(\mathcal{R}^{\sigma})\\}\leq&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{The observation above}&&\\\
\>\>\>\>\>\>\>\>\bigvee\\{!^{\sigma}_{i}x\>\cdot^{\sigma}y\>\cdot^{\sigma}!^{\sigma}_{i}x\>|\>a\geq
x\in\mathcal{F}(\mathcal{R}^{\sigma}),y\geq\in\mathcal{F}(\mathcal{R}^{\sigma})\\}=&&\\\
\>\>\>\>\>\>\>\>\>\>\>\>\text{The definition of $\cdot^{\sigma}$ and
$!^{\sigma}_{i}$}&&\\\
\>\>\>\>\>\>\>\>!^{\sigma}_{i}a\cdot^{\sigma}b\>\cdot^{\sigma}\>!^{\sigma}_{i}a\end{array}$

∎

In the previous section, we defined complex algebras of $\Sigma$-frames, so
let us define dual structures of perfect $\Sigma$-algebras. Let $\Sigma$ be a
subexponential signature and let
$\mathcal{A}=(A,\bigvee,\bigwedge,\cdot,\setminus,/,(!_{i})_{i\in
I},0,1,\epsilon)$ be a $\Sigma$-algebra, its _atom structure_ (that is, the
frame of completely join-irreducible elements) is the structure of the form
$\operatorname{At}(\mathcal{A})=(\mathcal{J}^{\infty}(\mathcal{A}),\leq_{\delta},R,(R_{i})_{i\in\Sigma},\uparrow\epsilon)$,
where $\leq_{\delta}$ is dual order on $\mathcal{J}^{\infty}(\mathcal{R})$
and:

  * •

$R$ is a ternary relation on $\mathcal{J}^{\infty}(\mathcal{A})$ such that
$Rabc$ iff $a\cdot b\leq_{\delta}c$,

  * •

$R_{i}$ is a binary relation on $\mathcal{J}^{\infty}(\mathcal{A})$ such that
$aR_{i}b$ iff $!_{i}\kappa(a)\leq_{\delta}\kappa(b)$ for each $i\in\Sigma$.

The definition of perfect $\Sigma$-algebras guarantee that
$\operatorname{At}(\mathcal{A})$ is a well-formed Kripke-frame.
$\operatorname{At}$ and $\operatorname{Cm}$ are connected with each other as
follows:

######  Lemma 16.

Let $\mathcal{R}$ be a perfect $\Sigma$-algebra, then
$\mathcal{R}\cong\operatorname{Cm}(\operatorname{At}(\mathcal{R}))$.

###### Proof.

We give only a proof for products, the modal part is similar to [10,
Proposition 2.25], the argument for products residuals is similar to [5, Lemma
6.10]. The isomorphism itself is established with the map
$\eta:a\mapsto\\{b\in\mathcal{J}^{\infty}(\mathcal{R})\>|\>a\leq_{\delta}b\\}$.
That is, we extend Raney representation of perfect distibutive lattices [23].
∎

Finally, using canonical extensions for $\Sigma$-algebras and duality between
perfect $\Sigma$-algebras and $\Sigma$-frames.

######  Theorem 17.

$\operatorname{DSMALC}_{\Sigma}$ is complete with respect to Kripke semantics.

###### Proof.

Suppose that $\varphi\nvdash\psi$, then $F_{\Sigma}\not\models\varphi\leq\psi$
by algebraic completeness, then
$F_{\Sigma}^{\sigma}\not\models\varphi\leq\psi$ since $F_{\Sigma}^{\sigma}$
contains $F_{\Sigma}$ as a subalgebra, $F_{\Sigma}^{\sigma}$ is a perfect
$\Sigma$-algebra as it was shown in Theorem 15. By Lemma 16,
$\operatorname{Cm}(\operatorname{At}(F_{\Sigma}^{\sigma}))\not\models\varphi\leq\psi$
and therefore
$\operatorname{At}(F_{\Sigma}^{\sigma})\not\models\varphi\vdash\psi$ by
Proposition 14. ∎

######  Remark 18.

$\operatorname{At}(F_{\Sigma}^{\sigma})$, the atom structure of the canonical
extension of the Lindenbaum-Tarksi algebra in the variety of all
$\Sigma$-alebras can be thought as the _canonical frame_ of
$\operatorname{DSMALC}_{\Sigma}$.

## References

  * [1]
  * [2] Gerard Allwein & J Michael Dunn (1993): _Kripke models for linear logic_.  The Journal of Symbolic Logic 58(2), pp. 514–545, 10.2307/2275217.
  * [3] Alexander V. Chagrov & Michael Zakharyaschev (1997): _Modal Logic_.  Oxford logic guides 35, Oxford University Press.
  * [4] Willem Conradie & Alessandra Palmigiano (2019): _Algorithmic correspondence and canonicity for non-distributive logics_.  Annals of Pure and Applied Logic 170(9), pp. 923–974, 10.1016/j.apal.2019.04.003.
  * [5] Nikolaos Galatos (2003): _Varieties of residuated lattices_.  Vanderbilt University.
  * [6] Nikolaos Galatos, Peter Jipsen, Tomasz Kowalski & Hiroakira Ono (2007): _Residuated lattices: an algebraic glimpse at substructural logics_.  Elsevier.
  * [7] Mai Gehrke (1994): _Bounded distributive lattices with operators_.  Math. Japonica 40(2), pp. 207–215.
  * [8] Mai Gehrke (2021): _Topological Duality and Algebraic Completions_.  In: Hiroakira Ono on Substructural Logics, Springer, pp. 143–184, 10.1007/978-3-030-76920-8_4.
  * [9] Mai Gehrke & Bjarni Jónsson (2004): _Bounded distributive lattice expansions_.  Mathematica Scandinavica, pp. 13–45, 10.7146/math.scand.a-14428.
  * [10] Mai Gehrke, Hideo Nagahashi & Yde Venema (2005): _A Sahlqvist theorem for distributive modal logic_.  Annals of pure and applied logic 131(1-3), pp. 65–102, 10.1016/j.apal.2004.04.007.
  * [11] Mai Gehrke & Jacob Vosmaer (2011): _A view of canonical extension_.  In: Logic, Language, and Computation: 8th International Tbilisi Symposium on Logic, Language, and Computation, TbiLLC 2009, Bakuriani, Georgia, September 21-25, 2009. Revised Selected Papers 8, Springer, pp. 77–100, 10.1007/978-3-642-22303-7_6.
  * [12] Jean-Yves Girard (1987): _Linear logic_.  Theoretical computer science 50(1), pp. 1–101, 10.1016/0304-3975(87)90045-4.
  * [13] Jean-Yves Girard & Yves Lafont (1987): _Linear logic and lazy computation_.  In: TAPSOFT’87: Proceedings of the International Joint Conference on Theory and Practice of Software Development Pisa, Italy, March 23–27, 1987 2, Springer, pp. 52–66, 10.1007/BFb0014972.
  * [14] Peter Jipsen & Constantine Tsinakis (2002): _A survey of residuated lattices_.  In: Ordered Algebraic Structures: Proceedings of the Gainesville Conference Sponsored by the University of Florida 28th February—3rd March, 2001, Springer, pp. 19–56, 10.1007/978-1-4757-3627-4_3.
  * [15] Bjarni Jónsson & Alfred Tarski (1951): _Boolean algebras with operators. Part I_.  American journal of mathematics 73(4), pp. 891–939, 10.2307/2372123.
  * [16] Max Kanovich, Stepan Kuznetsov, Vivek Nigam & Andre Scedrov (2019): _Subexponentials in non-commutative linear logic_.  Mathematical Structures in Computer Science 29(8), pp. 1217–1249, 10.1017/S0960129518000117.
  * [17] Max Kanovich, Stepan Kuznetsov & Andre Scedrov (2021): _The multiplicative-additive Lambek calculus with subexponential and bracket modalities_.  Journal of Logic, Language and Information 30, pp. 31–88, 10.1007/s10849-020-09320-9.
  * [18] Yves Lafont (1988): _The linear abstract machine_.  Theoretical computer science 59(1-2), pp. 157–180, 10.1016/0304-3975(88)90100-4.
  * [19] Joachim Lambek (1958): _The mathematics of sentence structure_.  The American Mathematical Monthly 65(3), pp. 154–170, 10.1080/00029890.1958.11989160.
  * [20] Vivek Nigam & Dale Miller (2009): _Algorithmic specifications in linear logic with subexponentials_.  In: Proceedings of the 11th ACM SIGPLAN conference on Principles and practice of declarative programming, pp. 129–140, 10.1145/1599410.1599427.
  * [21] Hiroakira Ono (2003): _Substructural logics and residuated lattices—an introduction_.  Springer, 10.1007/978-94-017-3598-8_8.
  * [22] Hilary A Priestley (1970): _Representation of distributive lattices by means of ordered Stone spaces_.  Bulletin of the London Mathematical Society 2(2), pp. 186–190, 10.1112/blms/2.2.186.
  * [23] George N Raney (1952): _Completely distributive complete lattices_.  Proceedings of the American Mathematical Society 3(5), pp. 677–680, 10.2307/2032165.
  * [24] Greg Restall (2002): _An introduction to substructural logics_.  Routledge, 10.4324/9780203016244.
  * [25] Richard Routley & Robert K Meyer (1972): _The semantics of entailment: II_.  Journal of philosophical logic, pp. 53–73, 10.1007/BF00649991.
  * [26] Takahiro Seki (2003): _A Sahlqvist theorem for relevant modal logics_.  Studia Logica 73, pp. 383–411, 10.1023/A:1023335229747.
  * [27] Alasdair Urquhart (1996): _Duality for algebras of relevant logics_.  Studia Logica, pp. 263–276, 10.1007/BF00370149.
  * [28] Philip Wadler (1993): _A taste of linear logic_.  In: Mathematical Foundations of Computer Science 1993: 18th International Symposium, MFCS’93 Gdańsk, Poland, August 30–September 3, 1993 Proceedings 18, Springer, pp. 185–210, 10.1007/3-540-57182-5_12.


# Exact time-dependent solutions of a Fisher-KPP-like equation obtained with
nonclassical symmetry analysis

Scott W McCue  Bronwyn H Bradshaw-Hajek  Matthew J Simpson  School of
Mathematical Sciences, Queensland University of Technology, Brisbane,
Australia.  UniSA STEM, University of South Australia, Mawson Lakes SA 5095,
Australia.

###### Abstract

We consider a family of exact solutions to a nonlinear reaction-diffusion
model, constructed using nonclassical symmetry analysis. In a particular
limit, the mathematical model approaches the well-known Fisher-KPP model,
which means that it is related to various applications including cancer
progression, wound healing and ecological invasion. The exact solution is
mathematically interesting since exact solutions of the Fisher-KPP model are
rare, and often restricted to long-time travelling wave solutions for special
values of the travelling wave speed.

###### keywords:

Fisher-Kolmogorov; Nonclassical symmetry analysis; Reaction-diffusion;
Population biology.

##  1 Introduction

Reaction-diffusion models are commonplace in applied mathematics. These
involve one or more parabolic partial differential equations (pdes) that, in
one dimension, are written as

$\frac{\partial u}{\partial t}=\frac{\partial}{\partial
x}\left(D(u)\frac{\partial u}{\partial x}\right)+R(u).$ (1)

We discuss possible forms for $D(u)$ and $R(u)$ shortly, but note that for the
purposes of our study, one of the most popular examples of (1) is the Fisher-
KPP equation (in dimensionless form)

$\frac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial x^{2}}+u(1-u),$
(2)

[1, 2], which is used extensively. The overall theme of our work is to study a
family of exact solutions of a version of (1) that are closely related to
solutions of (2).

More generally, in reaction-diffusion models of the form (1), the nonlinear
reaction term $R(u)$ is used to model reproduction in ecology [3, 4] or
proliferation in cell biology [7, 5]. A common form for this reaction term is
(in dimensional variables) the logistic growth term $R(u)=\lambda u(1-u/K)$,
where $\lambda$ is the reproduction rate and $K$ is the carrying capacity.
Indeed, this is the reaction term in (2). A feature of the logistic term is
that the per capita growth rate $\lambda(1-u/K)$ linearly decreases to zero at
$u=K$. Other qualitatively similar monostable reaction terms $R(u)$ exist that
retain key properties including a single local maximum and $R(0)=R(K)=0$ [8,
4]. In this sense, logistic growth falls into a class of qualitatively similar
reaction terms.

Forms for nonlinear diffusion $D(u)$ in (1) vary depending on the application,
although linear diffusion is the most commonly used form (as in the Fisher-KPP
model (2)). One option is to take the nonlinear diffusion to be $D(u)=u^{n}$,
where $n>0$ [5, 6]. Of particular relevance to our study here, in the context
of modelling biological cells, is where $D(u)$ is a decreasing function of $u$
[9].

Much attention has been devoted to deriving exact solutions to versions of
(1). Travelling wave solutions to the Fisher-KPP equation were first presented
by Ablowitz and Zepatella [10] and then others [11, 12]. Here, a number of
travelling wave solutions exist [13, 14, 15]. Periodic solutions have been
found [16, 17], and nonclassical symmetry solutions have been constructed [18,
19] (these solutions can also be found using Painlevé analysis [20, 21]).
Fewer solutions exist when the diffusivity is nonconstant (and the reaction
term is nonzero). Nonclassical symmetry solutions for particular forms of
$D(u)$ and $R(u)$ have been presented in Refs [25, 26, 24, 22, 23].

Here we report on a family of exact solutions to (1),

$u(x,t)=\beta\left[\mathrm{exp}\left(\ln\left(1+\frac{u_{0}}{\beta}\right)\mathrm{e}^{-\beta
t}\cos\left(\sqrt{1+\beta}\,x\right)\right)-1\right],\quad
0<x<\frac{\pi}{2\sqrt{1+\beta}},$ (3)

with

$D(u)=\frac{\beta}{u+\beta},\quad
R(u)=\beta(1-u)\ln\left(1+\frac{u}{\beta}\right).$ (4)

The form of $R(u)$ and $D(u)$ in (4) is such that (1) is analogous to (2), as
indicated in Fig. 1.

Figure 1: Nonlinear diffusion $D(u)$ and source term $R(u)$. Functions are
given for $\beta=0.1$ (blue), $\beta=1$ (orange) and $\beta=10$ (yellow).

Here, $R(u)$ is a monostable reaction term with zeros at $u=0$ and $1$, while
$D(u)$ is a positive decreasing function of $u$ for all $u>0$. In fact, this
same precise $D(u)$ has been used to study cell migration [9]. Therefore the
exact solutions (3) fall into the same class of solutions to (2) and we
discuss this connection in some detail.

The content of the letter is as follows. We summarise in Section 2.1 the
nonclassical symmetry analysis that leads to (3). In Section 2.2 we explain
how the solutions describe population extinction. Limits of large and small
$\beta$ are treated briefly in Section 2.3, noting the connection with (2),
while a further illustrative example is provided in Section 2.4. We close in
Section 3 with a discussion.

##  2 Derivation and interpretation

###  2.1 Nonclassical symmetry solution of the nonlinear reaction-diffusion
equation

Classical Lie point symmetry analysis, first introduced by Sophus Lie,
provides a systematic way to search for the invariant quantities in a
differential equation, where we seek transformations that leave the equation
of interest invariant. These classical symmetries can lead to the well-known
travelling wave solutions or scale-invariant solutions (for more detail see
for example [27, 28]). Galaktionov, et al. [29] provided the first complete
classical symmetry classification for equations of type (1).

The nonclassical symmetry (or Q-conditional) method was first introduced by
Bluman and Cole [30], where we again seek transformations that leave the
equation of interest invariant, but we also require that the invariant surface
condition be satisfied. This can sometimes result in additional symmetries
that cannot be found using the classical method. If nonclassical symmetries
can be found, they can be used in the same way as those found using the
classical method. That is, the differential equation can be simplified and an
analytic solution may be constructed. The most complete nonclassical symmetry
analysis of equations of type (1) is given in [25, 26].

To construct (3) we use of a nonclassical symmetry admitted by equation (1)
whenever $R(u)$ and $D(u)$ are related by [25, 26]

$R(u)=\left(\alpha-\displaystyle\frac{\beta}{D(u)}\right)\int_{u^{*}}^{u}\,D(u^{\prime})\,\textrm{d}u^{\prime},$
(5)

where $u^{*}$ is one of the zeros of $R(u)$ (here we choose $u^{*}=0$) and
$\alpha$ and $\beta$ are constants. Equation (1) can be reduced to the
Helmholtz equation $\Psi_{xx}+\alpha\Psi=0$, where $\Psi(x)$ is related to
$u(x,t)$ by

$\displaystyle\int_{u^{*}}^{u}\,D(u^{\prime})\,\textrm{d}u^{\prime}={\rm
e}^{-\beta t}\Psi.$ (6)

When $\alpha=k^{2}>0$, solutions to the Helmholtz equation can be written in
terms of sine and cosine functions,

$\Psi(x)=c_{1}\sin kx+c_{2}\cos kx,$ (7)

where $c_{1}$ and $c_{2}$ are constants.

A pair of nonlinear reaction and diffusion terms that satisfy relationship (5)
and are analogous to the Fisher-KPP equation are

$D(u)=\displaystyle\frac{\beta(1-b)}{\alpha(u-b)},\qquad
R(u)=\beta(1-u)\ln\left(1-\frac{u}{b}\right).$ (8)

The solution to equation (1) with $D(u)$ and $R(u)$ given by (8) can be found
by rearranging transformation (6) to obtain

$u(x,t)=b\left[1-\exp\left(\displaystyle\frac{\alpha}{\beta(b-1)}{\rm
e}^{-\beta t}\Psi(x)\right)\right].$ (9)

###  2.2 One-parameter family of solutions

To reduce the number of parameters in (7) and (9), we are motivated by the
linear problem [31]

$\frac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial x^{2}}+u,\quad
0<x<L,$ (10) $\frac{\partial u}{\partial x}=0\quad\mbox{on}\quad x=0,$ (11)
$u=0,\quad\mbox{on}\quad x=L,$ (12) $u(x,0)=f(x),\quad 0<x<L,$ (13)

for which there is an exact solution found using separation of variables. For
large time, this exact solution behaves like $u\sim
A_{1}\mathrm{e}^{-(\pi^{2}/4L^{2}-1)t}\cos\left(\pi x/2L\right)$, where
$A_{1}$ is related to the initial condition. In the derivation of (10)-(13)
from a dimensional system, the length-scale and time-scale of the original
physical problem are related to the dimensional diffusion coefficient and
growth rate parameter in the usual way so that the pde (10) does not contain
any parameters. Apart from the initial condition (13), the only parameter in
(10)-(13) is $L$. The final key point to note about (10)-(13) is that, because
the eigenvalue is $-\pi^{2}/4L^{2}+1$, clearly the solution of (10)-(13) grows
without bound if $L>\pi/2$ while it decays (goes extinct) if $L<\pi/2$.

Returning to (1), the dynamics of the solution becomes much clearer if we
choose the diffusion and reaction functions so that $D(u)\rightarrow 1$ and
$R(u)\sim u$ as $u\rightarrow 0$. In other words, for small population density
$u$, we want our nonlinear system to behave close to the linear system
(10)-(13). As a result, we choose $b=-\beta$, $\alpha=1+\beta$. Further, to
match the boundary conditions (11)-(12), we choose $c_{1}=0$. Finally, to
ensure $u(0,0)=u_{0}$, say, we choose
$c_{2}=-\beta\ln\left(1+u_{0}/\beta\right)$.

Before moving on, it is worth noting we are free to replace $t$ with $t-t_{0}$
in (3), since the original pde (1) is invariant under translations in time.
However, the term $\ln(1+u_{0}/\beta)\mathrm{e}^{\beta t_{0}}$ is a constant,
so changing $t_{0}$ is equivalent to redefining $u_{0}$. Therefore it is only
worth keeping one of $u_{0}$ or $t_{0}$. We choose to keep the former.

With a single parameter $\beta$, we can illustrate the exact solutions with
representative values of $\beta$. For example, we show in Fig. 2(a)-(c) the
solutions (3) for $\beta=0.1$, $1$ and $10$ (solid black curves), where the
arrow indicates increasing time. There are qualitatively similar features in
each case. For example, the solutions have the property $\partial u/\partial
x=0$ at $x=0$, corresponding to no flux at the left boundary. Further, the
solutions each have $u=0$ at $x=\pi/2\sqrt{1+\beta}$, which is a Dirichlet
condition at the right boundary. Thus, physically speaking, there is a loss of
mass at the right boundary and indeed $u$ continues to decay until the
population becomes extinct as $t\rightarrow\infty$. Note the domain is
decreasing in size as $\beta$ increases. Also included in Fig. 2(a)-(c) are
numerical solutions (green dashed), computed using finite differences with a
no-flux and Dirichlet conditions at the left and right boundaries,
respectively, with the details included on GitHub. Clearly there is a very
good match, confirming the derivation of the exact solution. We return to Fig.
2 shortly.

Figure 2: Exact time-dependent solutions of Equation (1). (a)-(c) Exact (solid
black) and numerical (dashed green) solutions for $\beta=0.1,1$ and $10$,
respectively, with the arrows showing the direction of increasing $t$.
Profiles in (a)-(c) are given at $\beta t=0,2,4,6$ and $8$, $\beta
t=0,0.2,0.5,1$ and $2$, and $\beta t=0,1,2,5$ and $10$, respectively. Profiles
in (d)-(f) compare numerical solutions for $u(x,0)=1$ with the exact solutions
with $u_{0}$ chosen so that the initial mass is the same in each case and we
see that the late-time solutions match reasonably well. In (d)-(f) profiles
are compared at $\beta t=0,4,6$ and $8$, $\beta t=0,1,2$ and $3$, and $\beta
t=0,0.1,0.2$ and $0.3$. Numerical solutions are obtained using the method of
lines where the $x$ variable is discretised on a uniform mesh with 201 grid
points.

###  2.3 Asymptotic limits

The regime $\beta\gg 1$ is interesting since

$D(u)\sim 1-\frac{u}{\beta},\quad R(u)\sim
u(1-u)+\frac{u^{2}(1-u)}{2\beta},\quad\mbox{as}\quad\beta\rightarrow\infty.$

That is, for large $\beta$, to leading order the diffusion term is constant
and the reaction term is logistic. Therefore, the reaction-diffusion equation
(1) with (4) is a close approximation of the Fisher-KPP equation (2). In this
limit, from (3) we have $u\sim u_{0}\mathrm{e}^{-\beta
t}\cos\left(\sqrt{1+\beta}x\right)$ on $0<x<(\pi/2\sqrt{1+\beta})$ as
$\beta\rightarrow\infty$. Note the domain length continues to shrink as
$\beta$ increases.

In the limit $\beta\rightarrow 0^{+}$, we have
$\pi/(2\sqrt{1+\beta})\rightarrow\pi/2^{-}$, so the domain for the exact
solution (3) is roughly $0<x<\pi/2$. For $\beta\ll 1$, the nonlinear
diffusivity $D(u)$ is not close to being constant, as we can see from Fig.
1(a). While the solution $u(x,t)$ decays very slowly for $\beta\ll 1$,
ultimately for $t\gg 1/\beta$ the population density becomes small and
$u\sim\beta\ln\left(1+u_{0}/\beta\right)\mathrm{e}^{-\beta
t}\cos\left(\sqrt{1+\beta}x\right)$ in this regime.

###  2.4 Illustrative example

As is common with exact solutions that come from symmetry analysis, we are not
free to choose our initial condition. However, in our case we are free to
choose $u_{0}$ to approximate a practical example. Suppose we employ the
obvious initial condition from a mathematical modelling perspective, namely
$u=1$ for $0<x<L$. In this case the initial mass of the solution is $L$. With
this in mind, we choose $u_{0}$ in our exact solution (3) so the initial mass
of the solution is $\pi/(2\sqrt{1+\beta})$ (remembering that
$L=\pi/(2\sqrt{1+\beta})$ for our solution). By integrating (3) at $t=0$ from
$x=0$ to $0<x<\pi/(2\sqrt{1+\beta})$, we arrive at the nonlinear algebraic
equation

$\dfrac{1}{\beta}+1=J_{0}\left(\textrm{ln}\left(\dfrac{\beta+u_{0}}{\beta}\right)\right)+L_{0}\left(\textrm{ln}\left(\dfrac{\beta+u_{0}}{\beta}\right)\right),$
(14)

where $J_{0}(x)$ is the Bessel function of the first kind and $L_{0}(x)$ is a
modified Struve function of the first kind. For a given $\beta$, we can solve
(14) for $u_{0}$. The result of these calculations is illustrated in Fig.
2(d)-(f). Here, numerical solutions (green dashed) computed for the physically
interesting initial condition are compared to the exact solution with $u_{0}$
found from (14). While there is no match for small time, we argue that the
exact solutions (3) provide reasonably good approximations for the numerical
solutions for intermediate to large times.

##  3 Discussion

In this article we have analysed a one-parameter family of exact solutions (3)
of the reaction-diffusion model (1), where the nonlinear diffusion and
reaction terms are given by (4). To put this work into context, all of these
exact solutions are qualitatively similar to solutions of the Fisher-KPP model

$\frac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial
x^{2}}+u(1-u),\quad 0<x<L,$ (15) $\frac{\partial u}{\partial
x}=0\quad\mbox{on}\quad x=0,$ (16) $u=0,\quad\mbox{on}\quad x=L,$ (17)
$u(x,0)=f(x),\quad 0<x<L,$ (18)

provided $L<\pi/2$. Indeed, for the one-parameter family of exact solutions
(3)-(4) and all solutions to the Fisher-KPP model (15)-(18), we have

$u\sim U_{1}\mathrm{e}^{-(\pi^{2}/4L^{2}-1)t}\cos\left(\frac{\pi
x}{2L}\right)\quad\mbox{as}\quad t\rightarrow\infty,$ (19)

where $L$ is: any constant $0<L<\pi/2$ for the Fisher-KPP model; related to
$\beta>0$ for our exact solutions (3) via $L=\pi/(2\sqrt{1+\beta})$. Further,
the constant $U_{1}$ in (19) is: related to $f(x)$ in (18) in Fisher-KPP in a
complicated way; or given by $U_{1}=\beta\ln\left(1+u_{0}/\beta\right)$ for
our exact solutions (3). Therefore, in this sense, the exact solutions (3) are
perfectly sensible and consistent with previous understanding of these types
of reaction-diffusion models (including the extinction property $L<\pi/2$, for
example [32, 33, Li2022]).

There is the usual downside that come from symmetry analysis, which is that
the initial condition of the exact solution is embedded. We are not free to
change the initial condition to suit the physical application or experimental
data, for example. Regardless, the initial condition for (3), namely

$u(x,0)=\beta\left[\mathrm{exp}\left(\ln\left(1+\frac{u_{0}}{\beta}\right)\cos\left(\sqrt{1+\beta}\,x\right)\right)-1\right],$
(20)

is a decreasing function with properties $u(0,0)=u_{0}$ and $u(L,0)=0$, which
is certainly reasonable, provided $u_{0}=\mathcal{O}(1)$. Of course, what
makes the exact solutions (3) exceptional and worth recording is not whether
they match a particular initial condition, but rather that fully explicit
time-dependent solutions to nonlinear reaction-diffusion models are very rare.
Further, apart from explicitly showing how the solution evolves, these
formulae can also be used as benchmarks for numerical simulations, for example
in courses for numerical methods in nonlinear pdes.

We close by mentioning that the particular nonclassical symmetry used to
construct solution (9) is valid in any number of dimensions and in any
coordinate system [35], so that solutions to equation (1) with nonlinear
diffusivity and reaction given by (8) could be constructed in
$\mathbb{R}^{n}$. For example, in radially symmetric coordinates in
$\mathbb{R}^{2}$, solution (9) will still be valid, however the solution of
the Helmholtz equation, $\Psi(r)$, will be written in terms of Bessel
functions (in $\mathbb{R}^{3}$ the solution will be in terms of spherical
Bessel functions [23]). The higher dimensional analogue of many of the other
results presented here can also be produced, for example, the solution will
decay (and become extinct) if $L<\lambda/\sqrt{1+\beta}$ where $\lambda$ is
the first zero of the Bessel function, $J_{0}(x)$.

Acknowledgements. This work is supported by the Australian Research Council
(DP200100177, DP200102130).

Supplementary Data. Supplementary material is available at GitHub.

## References

  * [1] RA Fisher. The wave of advance of advantageous genes. Annals of Eugenics. 7 (1937) 355–369.
  * [2] AN Kolmogorov, PG Petrovskii, NS Piskunov. A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. Moscow University Mathematics Bulletin. 1 (1937) 1–26.
  * [3] V Melica, S Invernizzi, G Caristi. Logistic density-dependent growth of an Aurelia aurita polyps population. Ecological Modelling. 291 (2014) 1–5.
  * [4] MJ Simpson, AP Browning, DJ Warne, OJ Maclaren, RE Baker. Parameter identifiability and model selection for sigmoid population growth models. Journal of Theoretical Biology. 535 (2022) 110998.
  * [5] JA Sherratt, JD Murray. Models of epidermal wound healing. Proceedings of the Royal Society of London: Series B. 241 (1990) 29–36.
  * [6] SW McCue, W Jin, TJ Moroney, K-Y Lo, S-E Chou, MJ Simpson. Hole-closing model reveals exponents for nonlinear degenerate diffusivity functions in cell biology. Physica D: Nonlinear Phenomena. 398 (2019) 130–140.
  * [7] ST Vittadello, SW McCue, G Gunasingh, NK Haass, MJ Simpson. Mathematical models for cell migration with real–time cell cycle dynamics. Biophysical Journal. 114 (2018) 1241–1253.
  * [8] W Jin, CJ Penington, SW McCue, MJ Simpson. Stochastic simulation tools and continuum models for describing two-dimensional collective cell spreading with universal growth functions. Physical Biology. 13 (2016) 056003.
  * [9] AQ Cai, KA Landman, BD Hughes. Multi-scale modeling of a wound-healing cell migration assay. Journal of Theoretical Biology. 245 (2007) 576–594.
  * [10] M Ablowitz, A Zeppetella. Explicit solutions of Fisher’s equation for a special wave speed. Bulletin of Mathematical Biology. 41 (1979) 835–840.
  * [11] P Kaliappan. An exact solution for travelling waves of $u_{t}=Du_{xx}+u-u^{k}$. Physica D: Nonlinear Phenomena. 11 (1983) 368–374.
  * [12] McCue SW, El–Hachem M, Simpson MJ. Exact sharp–fronted travelling wave solutions of the Fisher–KPP equation. Applied Mathematics Letters. 114 (2021) 106918.
  * [13] Y Kametaka. On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type. Osaka Journal of Mathematics. 13 (1976) 11–66.
  * [14] HP McKean. Nagumo’s equation. Advances in Mathematics. 4 (1970) 209–223.
  * [15] J Rinzel. Neutrally stable travelling wave solutions of nerve conduction equations. Journal of Mathematical Biology. 2 (1975) 205–217.
  * [16] GA Carpenter. Periodic solutions of nerve impulse equations. Journal of Mathematical Analysis and Applications. 58 (1977) 152–173.
  * [17] S Hastings. The existence of periodic solutions to Nagumo’s equations. The Quarterly Journal of Mathematics. 25 (1974) 369–378.
  * [18] Arrigo DJ, Hill JM, Broadbridge P. Nonclassical symmetry reductions of the linear diffusion equation with a nonlinear source. IMA Jounal of Applied Mathematics. 52 (1994) 1–24.
  * [19] PA Clarkson, EL Mansfield. Symmetry reductions and exact solutions of a class of nonlinear heat equations. Physica D 70 (1994) 250–288.
  * [20] R Conte. Universal invariance properties of Painlevé analysis and Bäcklund transformation in nonlinear partial differential equations. Physics Letters A. 134 (1988) 100–104.
  * [21] ZX Chen, BY Guo. Analytic solutions of the Nagumo equation. IMA Journal of Applied Mathematics. 48 (1992) 107–115.
  * [22] Bradshaw-Hajek BH, Broadbridge P. Analytic solutions for calcium ion ferilisation waves on the surface of eggs. Mathematical Medicine and Biology: A Journal of the IMA. 36 (2019) 549–562.
  * [23] Bradshaw-Hajek BH, Broadbridge P. An analytic solution for a Gompertz-like reaction-diffusion model for tumour growth. The Role of Metrics in the Theory of Partial Differential Equations. 1 (2020) 127–136.
  * [24] Broadbridge P, Bradshaw-Hajek BH. Exact solutions for logistic reaction-diffusion equations in biology. Zeitschrift für angewandte Mathematik und Physik. 67 (2016) 1–13.
  * [25] Arrigo DJ, Hill JM. Nonclassical symmetries for nonlinear diffusion and absorption. Studies in Applied Mathematics. 94 (1995) 21–39.
  * [26] JM Goard, P Broadbridge. Nonclassical symmetry analysis of nonlinear reaction–diffusion equations in two spatial dimensions. Nonlinear Analysis: Theory, Methods & Applications. 26 (1996) 735–754.
  * [27] PJ Olver. Applications of Lie Group to Differential Equations, AMS, No. 107, Springer, New York (1982).
  * [28] NH Ibragimov. CRC Handbook of Lie Group Analysis of Differential Equations, Volume I: Symmetries, Exact Solutions, and Conservation Laws, CRC Press, Boca Raton (1994).
  * [29] VA Galaktionov, VA Dorodnitsyn, GG Elenin, SP Kurdyumov, AA Samarskii. A quasilinear heat equation with a source: peaking, localization, symmetry exact solutions, asymptotics, structures. Journal of Soviet Mathematics. 41 (1988) 1222–1292.
  * [30] GW Bluman, JD Cole. General similarity solution of the heat equation. Journal of Mathematics and Mechanics. 18 (1969) 1025–1042.
  * [31] JG Skellam. Random dispersal in theoretical populations. Biometrika. 38 (1951) 196–218.
  * [32] E Bradford, JR Philip. Stability of steady distributions of asocial populations dispersing in one dimension. Journal of Theoretical Biology. 29 (1970) 13–26.
  * [33] M El–Hachem, SW McCue, W Jin, Y Du, MJ Simpson. Revisiting the Fisher-Kolmogorov-Petrovsky-Piskunov equation to interpret the spreading–extinction dichotomy. Proceedings of the Royal Society A. 475 (2019) 20190378.
  * [34] Li, ST Johnston, PR Buenzli, P van Heijster, MJ Simpson. Extinction of bistable populations is affected by the shape of their initial spatial distribution. Bulletin of Mathematical Biology. 84 (2022) 21.
  * [35] P Broadbridge, BH Bradshaw-Hajek, D Triadis. Exact non-classical symmetry solutions of Arrhenius reaction-diffusion. Proceedings of the Royal Society A. 471 (2015) 20150580.