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http://arxiv.org/abs/2404.15457v1 | 20240423190045 | Hidden in Plain Sight: Exploring the Intersections of Mental Health, Eating Disorders, and Content Moderation on TikTok | [
"Charles Bickham",
"Kia Kazemi-Nia",
"Luca Luceri",
"Kristina Lerman",
"Emilio Ferrara"
] | cs.SI | [
"cs.SI"
] |
[
Lennart E. Nacke
April 29, 2024
====================
Social media platforms actively moderate content glorifying harmful behaviors like eating disorders, which include anorexia and bulimia. However, users have adapted to evade moderation by using coded hashtags. Our study investigates the prevalence of moderation evaders on the popular social media platform TikTok and contrasts their use and emotional valence with mainstream hashtags. We notice that moderation evaders and mainstream hashtags appear together, indicating that vulnerable users might inadvertently encounter harmful content even when searching for mainstream terms.
Additionally, through an analysis of emotional expressions in video descriptions and comments, we find that mainstream hashtags generally promote positive engagement, while moderation evaders evoke a wider range of emotions, including heightened negativity. These findings provide valuable insights for content creators, platform moderation efforts, and interventions aimed at
cultivating a supportive online environment for discussions
on mental health and eating disorders.
Warning: This paper discusses eating disorders, which some may find distressing.
§ INTRODUCTION
This study explores the interplay between mental health and social media, using a case study of TikTok and eating disorders.
Eating disorders represent a complex mental health condition characterized by disruptions in eating and related behaviors.
These conditions, which include anorexia, bulimia, and binge eating disorder (BED) <cit.>, have alarming mental health impacts: one study reported that up to 23% of individuals diagnosed with BED in the US had attempted suicide, with 94% experiencing a lifetime of mental health problems <cit.>.
The COVID-19 pandemic further complicated the landscape of eating disorders <cit.>. During this period, there was a notable surge in symptomatology, accompanied by diminished access to treatment for individuals struggling with eating disorders <cit.>. This crisis highlighted the urgency of understanding the dynamic interplay between external stressors, mental health, and the manifestation of eating disorders.
TikTok, a platform for sharing short-form video content, has billions of users worldwide <cit.> and is especially popular among adolescents. While TikTok provides a creative outlet for many young people, it has also become a space for discussions about mental health and eating disorders <cit.>. The brevity and immediacy of TikTok videos, often accompanied by succinct descriptions containing hashtags,
present an opportunity to investigate how individuals engage with emotional issues, as well as broader questions about the role of social media in mental wellbeing, particularly in adolescents <cit.>.
This work explores the emotional landscape of TikTok videos related to eating disorders, focusing on the role of hashtags in the discoverability and organization of content.
While TikTok heavily moderates searches for harmful content that promotes or glorifies eating disorders <cit.>, users have adapted to evade moderation through the use of coded hashtags <cit.>. A study of TikTok by the Center for Countering Digital Hate (CCDH)[<https://counterhate.com/wp-content/uploads/2022/12/CCDH-Deadly-by-Design_120922.pdf#page=44.12>] found content promoting eating disorders relied on coded hashtags to avoid getting flagged by content moderation algorithms. These coded hashtags were based on misspellings, abbreviations, or references to the artist Ed Sheeran, whose name coincidentally begins with “ED" – an abbreviation commonly associated with eating disorders.
For the purpose of this study, we refer to such hashtags as moderation evaders. In this paper, we analyze the emotional expressions in content across different hashtags pertaining to eating disorders, including moderation evaders and mainstream hashtags.
* RQ1: Do moderation evaders co-occur with mainstream hashtags related to mental health and healthy living?
* RQ2: How do emotional expressions in TikTok video descriptions differ between content associated with mainstream hashtags and moderation evaders?
* RQ3: How do emotional expressions in TikTok video descriptions correlate with those in user comments, and how does this interaction differ across mainstream hashtags and moderation evaders?
Our study of the emotional expressions within video descriptions and user comments reveals systematic differences between mainstream hashtags and moderation evaders. While mainstream hashtags tend to promote an overall positive engagement, moderation evaders evoke a broader range of emotions, including heightened negativity. The disparity in emotional engagement suggests that moderation evading hashtags are used for spreading problematic content related to eating disorders. Moreover, since moderation evaders co-occur frequently with mainstream hashtags, this raises concerns about the potential exposure of users to harmful content, as moderation evaders often circumvent platform regulations.
Our findings underscore the complex emotional landscape of TikTok content related to eating disorders and emphasize the need for tailored moderation strategies and interventions to cultivate a supportive online environment conducive to discussions on mental health and eating disorders. In the subsequent sections, we present related works, our methodology, discuss our results, and draw meaningful conclusions that contribute to the evolving discourse on mental health and eating disorders in the digital age.
§ RELATED WORKS
§.§ Social Media and Body Image Concerns
The rise in social media use has fueled worries about its impact on negative body image concerns. Social comparison has been found to play a negative role in how people think about their body image <cit.>. Women and young girls especially tend to suffer from negative body image concerns <cit.>. Social media platforms that are based on photos are thought of to be more negative when it comes to body image since they tend to be more focused on physical appearance <cit.>. With photo-based platforms, there are more opportunities for people, especially women, to self-objectify, internalize appearance ideals, and compare themselves negatively. This can lead to many mental health risks that include but is not limited to unipolar depression, sexual dysfunction, and eating disorders <cit.>. Additionally, the use of photo-based platforms allows for images to be manipulated. Studies show that there has been a link between lower body image and self-esteem based on posting exposure to manipulated images <cit.>. Also, many people may not be aware of the images being manipulated, which can lead to a normalization of unrealistic body and beauty ideals <cit.>. Exposure to the thin ideal has been linked to an increase in body dissatisfaction, eating disorder symptoms,
and negative mood in women <cit.>. The presence of a negative self-body image in an individual is identified as a contributing factor to eating disorders <cit.>.
§.§ Social Media and Eating Disorders
Social media use has been linked with distorted eating over recent years <cit.>. With the rise of this trend, there have been efforts to prevent eating disorders. <cit.> showed that self-criticism intervention can be a strategy to address this need. Communication strategies have also been looked into to help the prevention of eating disorders <cit.>. <cit.> map the relationship between social media and eating disorders to an online radicalization process <cit.>. Exploring the concept of online radicalization within “pro-ana" communities, <cit.> highlights how social media platforms facilitate radicalized behaviors by creating echo chambers that can normalize distorted eating. The study emphasizes the importance of understanding and quantifying the impact of these online communities to develop strategies aimed at promoting better mental health. <cit.> details how easy it is for someone to “stumble" upon potentially harmful ED content without explicitly searching for it.
Specifically on TikTok, <cit.> analyzes the mainstream hashtag #edrecovery, while <cit.> analyzes the textual content, using mixed methods, on the hashtag #bedrecovery. <cit.> conducts content analysis on the hashtags #fitspiration and #thinspiration. <cit.> performed a comparative analysis on pro-recovery communities across five eating disorder hashtags: #anarecovery, #arfidrecovery, #bedrecovery, #miarecovery, and #orthorexiarecovery. <cit.> supports indirect connections between involvement with appearance/eating-related content on social media and eating disorder symptoms, with mediation through elevated exposure to recommended content in this category and increased levels of upward social media appearance comparisons.
The impact of social media and eating disorders on the youth have also been studied <cit.>. These studies emphasize the need for an understanding of the relationship between social media use and eating disorders among young adults and children.
§ METHODOLOGY DATA
§.§ Dataset
For the purpose of this research, we collected information about TikTok videos focusing on content associated with various issues related to body image, dieting, and eating disorders. To collect the data, we curated a set of hashtags that reflect the diversity of topics within the TikTok platform. See the Appendix for the full list of keywords. Spanning the timeframe from December 2016 to April 2023, the dataset has a total of 14,816 posts and 562,856 comments associated with these videos.
The dataset encompasses various aspects of TikTok videos with each entry containing textual information. Textual information is captured through the video description (Description) and a list of hashtags used in the video (Challenges). Additionally, the dataset includes the comments the videos received.
§.§ Hashtag Co-Occurrence Graph
A hashtag co-occurrence graph operates as a network that outlines the relationships among hashtags present in a series of social media posts. When two hashtags share an edge, it signifies their joint appearance in a post, and the weight of this edge reflects the frequency of these occurrences—essentially measuring how often they appeared together in our dataset.
§.§ Measuring Emotions
Textual content conveys signals related to emotions and feelings, encompassing both positive sentiments, such as joy and love, and negative ones like anger and disgust. In our approach, we employ an emotion detection tool inspired by SpanEmo <cit.>, named Demultiplexer (Demux) <cit.>. The tool takes the categories (in this case, emotions) as the first input sequence and the actual content as the second sequence. The contextual embeddings specific to each emotion contribute to deriving probabilities for individual emotions. Our application of Demux extends to every video description and comment. The emotions included were anger, anticipation, disgust, fear, joy, love, optimism, pessimism, sadness, surprise, trust, and none. For the purpose of this study, anticipation, pessimism, surprise, and trust were grouped into a single category named “other emotions" similar to <cit.>.
§ RESULTS
The methods detailed above were used to analyze our dataset and answer our research questions about the co-occurrences and emotional patterns across mainstream hashtags and moderation evaders.
§.§ RQ1: Co-Occurrence of Moderation Evaders and Mainstream Hashtags
To address our first research question, we created a hashtag co-occurrence graph of popular hashtags (that occurred at least 30 times), with 612 hashtags in total. Figure <ref> shows the hashtag co-occurrence network. The node size in this graph corresponds to the hashtag's PageRank centrality. We used the Louvain algorithm to identify clusters of highly interlinked nodes, revealing seven main communities. The central hashtags within these clusters are listed below, including those that are intentionally misspelled.
* Eating disorders (ED) recovery community with hashtags #edrecovery, #edrecvery (sic), #anorexiarecovery, …
* Healthy living community: #exercise, #workout, #healthyrecipes, …
* Body positivity/acceptance community: #bodypositivity, #fatacceptance, #selflove, …
* Juice-related content: #juicerecipes, #healthyjuice, #juicingforhealth, …
* Beauty and body positivity community: #thighs, #beautiful, #curvy, …
* Music promotion: JuiceWRLD-related content community which include #juicewrld #juicewrldmusic, …
* Miscellaneous: #water #aquarium
The largest cluster in this network, with 182 hashtags and over 1.9 million views per video, is devoted to ED recovery.
Table <ref> gives a full breakdown of the statistics for each cluster.
We identified 10 moderation evaders within the 612 hashtags, and all of them were located in the ED recovery community. These hashtags were: #edrecocery, #edsheeranrecoveryy, #edawarewness, #anarecvery, #edrecov, #anoreksja, #edtt, #edsheeran, #ednotsheeren, and #ana. Moderation evaders frequently co-occurred with mainstream hashtags like #fyp, #fearfood, #recovery, #ed, and #mentalhealthmatters. For instance, #edrecocery appeared 138 times in total, with 64 occurrences alongside #ed, while #anarecovry appeared 61 times in total, with 55 occurrences with #recovery. See table <ref> for full list.
The co-occurrence of mainstream hashtags with moderation evaders suggests that users who search for terms such as #ed or #recovery could be exposed to content containing moderation evaders.
The hashtag #anoreksja also co-occurred with #tw (trigger warning) over 20 times. Furthermore, the hashtag #wl (weight loss) co-occurred with #edtt and #ednotsheeren over 10 times. The associations of moderation evaders with weight loss (#wl) and trigger warnings (#tw) indicate potential triggering content that may be harmful to individuals struggling with eating disorders.
Findings and Remarks: Addressing RQ1, the analysis of TikTok hashtags revealed a prominent cluster related to ED recovery, with 182 hashtags with over 1.9 million views per video.
This contrasts with Twitter, where harmful content promoting eating disorders is far more common <cit.>. While this shows that TikTok provides a more positive, pro-recovery platform, the presence of moderation evaders within this cluster suggests that harmful content may be intentially obscured to avoid moderation. Moderation evaders are associated with hashtags weight loss (#wl) and trigger warnings (#tw), highlighting links to potentially problematic content in the context of eating disorders. Moreover, the fact that these moderation evaders are
linked to mainstream hashtags, such as #fyp and #recovery, suggests potential exposure to pro-eating disorder content for users searching for recovery-related content.
§.§ RQ2: Emotions in Video Descriptions
To address our second research question, we compared emotional expressions in the descriptions of TikTok videos tagged with mainstream hashtags and moderation evaders.
We first focused on videos that have been tagged with any of the ten most popular mainstream hashtags (based on the number of occurrences). Joy consistently emerged as the dominant emotion, followed closely by optimism. Among negative emotions, sadness was the most common. A substantial portion of posts expressed “no emotion”, potentially influenced by the succinct and hashtag-centric nature of TikTok video descriptions. See Figure <ref>a for the emotional analysis results for the mainstream hashtags.
Emotional analysis of the descriptions of videos tagged with any of the ten
moderation evaders that we previously identified
revealed them to be more emotional overall.
Optimism was the dominant emotion. Sadness and fear were also higher, suggesting a more negative emotional tone (Figure <ref>b).
Comparing emotional expressions of videos tagged with mainstream hashtags and moderation evaders revealed interesting patterns. On the one hand, mainstream hashtags tended to be more positive overall, with higher occurrences of joy and love. Moderation evaders displayed a more diverse emotional landscape, with notable expressions of negative sentiments such as fear and sadness, underscoring the potential risks associated with these hashtags. The identified emotional differences have implications for user exposure. Mainstream content, characterized by positive emotions, may contribute to a more uplifting user experience. On the other hand, the diverse emotional landscape of moderation evaders' content, including expressions of fear and sadness, suggests potential exposure to content with a more complex emotional impact, indicating both positive and potentially harmful messaging.
Figure <ref> lists sample video descriptions that express different emotions.
To better understand these differences, we conducted a manual review of the video content for posts containing moderation evaders. The videos containing the five moderation evaders with the highest number of occurrences (#edrecocery, #edsheeranrecoveryy, #edawarewness, #anarecvery, and #edrecov) often featured recovery content and positive messaging, providing tips on recovery approaches, highlighting the benefits of recovery, and addressing the toll of eating disorders.
Amongst the other five, the hashtags #anoreksja, #ednotsheeren, and #ana were not searchable, i.e., they are blocked when searching for them in the TikTok search bar, and links to mental health resources are provided instead (see Figure <ref> in the Appendix). It should be noted that the mainstream hashtag #edrecovery was blocked as well.
Furthermore, an examination of posts containing #edtt revealed that many of the videos included displays of eating disorders-related behaviors, dark humor, and an overall glorification of eating disorders.
Due to the co-occurrences of #edtt with more mainstream hashtags such as #edvent, #ed, and #fyp, this raises concerns about the potential exposure of users to harmful content of this nature. From the manual inspection of moderation evaders, we concluded they are not exclusively associated with harmful or negative content; instead, they encompass a variety of themes, including those focused on recovery and positive narratives related to eating disorders.
Findings and Remarks:In response to RQ2, we explored the emotional expressions in TikTok video descriptions, revealing differences between mainstream hashtags and moderation evaders. Mainstream content tends toward positivity, with joy and optimism prevalent, while moderation evaders' content shows a wider emotional range, including fear and sadness. A manual review of moderation evaders underscores the need to understand TikTok's emotional dynamics and associated risks across various hashtags while also showing that TikTok has increased its moderation efforts by blocking some of these hashtags.
§.§ RQ3: Emotions in Comments
Regarding RQ3, we investigated the relationship between emotional expressions in TikTok video descriptions and the emotions expressed in the user comments in response to these videos. Our analysis revealed that comments displayed a broader range of emotions compared to video descriptions themselves, with an increase in anger (Fig. <ref>).
In general, the emotional analysis of the video descriptions and comments of mainstream hashtags showed a consistent pattern. In posts tagged with recovery-related hashtags, both descriptions of videos and comments tended to be positive, with more expressions of joy, love, and optimism. This suggests that content associated with mainstream and recovery-related hashtags tends to foster positive engagement, creating a supportive and optimistic community atmosphere on TikTok. In comparison, the emotional analysis of video descriptions and comments associated with the moderation evaders reveals a clear emotional distinction. The user comments on these posts, while generally less emotionally charged than their corresponding video descriptions, still manifest a higher degree of joy, especially for the recovery-related hashtags.
Mainstream hashtags, while generally positive, exhibited slightly more anger in their comments. We also observed an increase in anger for the moderation evaders' posts comments in comparison to the video descriptions (see Fig. <ref>). The heightened expression of anger in comments, especially in moderation evaders' content, raises questions about the nature of discussions and interactions surrounding sensitive or controversial topics on TikTok. Figure <ref> lists sample comments that express different emotions.
We have observed that some of the moderation evaders' comments displayed less emotion than the video descriptions, specifically #edsheeran, #edrecocery, #edawarewness #anoreskja, #ana, #anarecovry, and #edtt. Also, the comments for the hashtags #edsheeran, #ednotsheeren, and #edtt expressed less positive emotions compared to the video descriptions.
Interestingly, when analyzing the video descriptions for the hashtag #ednotsheeren, it is evident that positive emotions such as joy and optimism are present but relatively subdued, with percentages ranging from 0% to 12.5%. Notably, the emotion love is not expressed in the video descriptions. On the contrary, negative emotions, specifically sadness, stand out at 8.33%. Moving to comments, the disparity between positive and negative emotions becomes more pronounced. Positive emotions in comments, including joy, love, and optimism, collectively amount to 9.61%, significantly lower than the corresponding video description percentage, which was 25%. In contrast, negative emotions in comments, particularly sadness at 21.47%, surpass the negative emotions in the video descriptions. This discrepancy suggests that the content under #ednotsheeren may not be resonating positively with the audience, as indicated by the diminished expression of positive emotions in both video descriptions and comments.
Also - for the hashtag #edtt - even though the video descriptions contained mostly positive emotions (with 50% being labeled with optimism), this did not mean that the viewers exhibited the same sentiment. For the comments, the negative emotions total 29.86% while the positive emotions come to 12.98%. Despite 50% of the posts being classified as optimism, it appears that viewers might not perceive the content in the same way. This suggests that there may be a gap between the positive message in the video descriptions and how the audience actually interprets it.
Findings and Remarks: For RQ3, we examined the relationship between emotional expressions in video descriptions and comments. We observed that comments on videos with mainstream hashtags exhibited more emotions than video descriptions themselves, while moderation evaders generally elicited fewer emotions. Furthermore, there was a noticeable increase in negative emotions expressed in comments for moderation evaders compared to mainstream hashtags. This hints that some moderation evaders could hide potentially problematic content.
§ DISCUSSION
We detected various TikTok communities related to eating disorders recovery, body positivity, and healthy living. These communities mostly contain content that is designed to be informational, inspirational, or uplifting in some shape or form; however, it should be noted that some of this well-intentioned content can have adverse effects as well. For example, content showing a person's body can encourage negative comparison and lead to viewers feeling poorly about their own bodies. Additionally, weight loss tips from unqualified TikTok users may be harmful to people struggling with eating disorders.
Prior work done by the CCDH<ref>
and <cit.> demonstrated that users within the ED community utilize misspellings, abbreviations, and the musical artist Ed Sheeran's name to avoid moderation. Within our dataset, we found 10 hashtags fitting this criterion, with each appearing in over 30 posts, all within the ED recovery/positivity community. These hashtags were found to co-occur with mainstream hashtags such as #fearfood, #mentalhealthmatters, and #recovery. This, combined with the fact that 1) the video descriptions with moderation evaders tend to evoke more negative emotions and 2) videos with some moderation evaders contain content that could promote ED behaviors, suggests that vulnerable users could potentially stumble upon harmful content even if they search for mainstream topics.
Nevertheless, the fact that some of these hashtags are blocked from the search page shows that TikTok has likely made an effort to moderate content that could potentially promote eating disorder behaviors, which may explain the overall positive tone of the content. The hashtag #edtt was one that was not blocked, though, and appeared to contain harmful content with “dark humor", ED-glorification, and ED-promoting advice. This is a hashtag that should be investigated further in future studies that analyze a larger amount of video content.
The emotional analysis for the video descriptions of the mainstream hashtags showed that they generally displayed a more balanced emotional distribution, representing a wide array of positive and neutral emotions. This could be because mainstream hashtags are often more visible to a broader audience which may lead users to adopt a more restrained and balanced emotional tone to appeal to a diverse viewership. On the other hand, moderation evaders' video descriptions tended to be more emotionally charged, with an increased presence of sadness, fear, and anger. This could suggest that those trying to avoid moderation are deliberately posting more negative content, believing that their content is less likely to be taken down.
The user comments on posts containing mainstream hashtags were more emotionally charged than their respective video descriptions. This could possibly be because users tend to be more careful and deliberate when sharing videos under popular hashtags, aiming to present a carefully curated image to a larger audience.
These comments revealed a similar pattern to that of the video descriptions – joy and optimism were often the most common emotions. In contrast, the user comments on posts containing the moderation evaders were less emotionally charged than their respective video descriptions. Despite being less emotionally charged overall, these comments contained a higher amount of joy compared to their video descriptions, especially for hashtags related to recovery. This could indicate potential community formation where individuals show support, particularly concerning eating disorder recovery. Amongst both the comments for the mainstream hashtags and moderation evaders, anger was more present also - albeit still at a minimal level.
The discovery of moderation evaders, their co-occurrences with more mainstream hashtags, and the persistence of harmful content despite some blocked hashtags underscore the ongoing challenges in content moderation. The emotional analysis indicates that posts containing mainstream hashtags tend to display a balanced emotional distribution in the video description and comments, possibly due to broader visibility, whereas moderation evader hashtags exhibit more emotionally charged content, emphasizing the need for targeted moderation efforts to mitigate negative emotional impacts.
§.§ Ethics Statement
All data used for this study is public and collected following
TikTok’s terms of service. In our analysis of TikTok videos and comments within these videos, no identifiable information related to any user has been included and analysis was carried out on aggregated data. These steps ensure that negative outcomes due to use of these data are minimized.
The authors declare no competing interests.
§.§ Limitations
It is crucial to acknowledge the limitations of our analysis. The prevalence of “no emotion” in our analysis may be influenced by the hashtag-centric nature of TikTok video descriptions, potentially impacting the accuracy of emotional expression detection. Additionally, our emotional analysis was mostly confined to the textual video descriptions and comments, overlooking the visual and audio elements of the videos. While we manually reviewed some of the videos containing specific hashtags, this inspection was not comprehensive, and future research should look into analyzing these elements further while incorporating AI techniques and statistical analysis.
§.§ Conclusion
Our results provide valuable insights into the emotional dynamics of TikTok content concerning mental health and eating disorders – specifically content containing moderation evader hashtags. We have found that moderation evaders do co-occur with mainstream hashtags and that the video descriptions/comments of posts containing the former tend to be more emotionally charged (including a higher amount of negative emotions such as fear and sadness). The findings have implications for content creators, platform moderation, and interventions aimed at fostering a supportive online environment for discussions on mental health and eating disorders. Future studies should continue to inspect moderation evaders and add onto our findings by taking a comprehensive look at the video content of posts in addition to their video descriptions and comments.
§ APPENDIX
Search Terms
thinspo, proana, proanatips, anatips, meanspo, fearfood, sweetspo, eatingdisorder, bonespo, promia, redbracetpro, m34nspo, fatspo, lowcalrestriction, edvent, WhatIEatInADay, Iwillbeskinny, thinspoa, ketodiet, skinnycheck, thighgapworkout, bodyimage, bodygoals, weightloss, skinnydiet, chloetingchallange, fatacceptance, midriff, foodistheenemy, cleanvegan, keto, cleaneating, intermittentfasting, juicecleanse, watercleanse, EDrecovery, bodypositivity, dietculture.
|
http://arxiv.org/abs/2404.16229v1 | 20240424220339 | Two qubit gate with macroscopic singlet-triplet qubits in synthetic spin-one chains in InAsP quantum dot nanowires | [
"Hassan Allami",
"Daniel Miravet",
"Marek Korkusinski",
"Pawel Hawrylak"
] | cond-mat.mes-hall | [
"cond-mat.mes-hall",
"quant-ph"
] |
Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5, Canada
Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5, Canada
Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5, Canada
Security and Disruptive Technologies, National Research Council, Ottawa, Canada K1A0R6
Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5, Canada
We present a theory of a two qubit gate with macroscopic singlet-triplet (ST) qubits in synthetic spin-one chains in InAsP quantum dot nanowires. The macroscopic topologically protected singlet-triplet qubits are built with two spin-half Haldane quasiparticles. The Haldane quasiparticles are hosted by synthetic spin-one chain realized in chains of InAsP quantum dots embedded in an InP nanowire, with four electrons each. The quantum dot nanowire is described by a Hubbard-Kanamori (HK) Hamiltonian derived from an interacting atomistic model. Using exact diagonalization and Matrix Product States (MPS) tools, we demonstrate that the low-energy behavior of the HK Hamiltonian is effectively captured by an antiferromagnetic spin-one chain Hamiltonian. Next we consider two macroscopic qubits and present a method for creating a tunable coupling between the two macroscopic qubits by inserting an intermediate control dot between the two chains. Finally, we propose and demonstrate two approaches for generating highly accurate two-ST qubit gates : (1) by controlling the length of each qubit, and (2) by employing different background magnetic fields for the two qubits.
Two qubit gate with macroscopic singlet-triplet qubits in synthetic spin-one chains in InAsP quantum dot nanowires
Pawel Hawrylak
April 29, 2024
==================================================================================================================
§ INTRODUCTION
There is currently interest in developing robust qubits with long coherence times for quantum information processing on various platforms. Examples of platforms include superconducting qubits based on macroscopic quantum states <cit.>, trapped ions <cit.>, quantum photonics <cit.>, and semiconductor spin qubits <cit.>. The semiconductor spin qubits approach is particularly attractive for its promise of seamless integration with electronic devices. An important objective of any design is to achieve robustness against quantum noise. To that end, some strategies include realizing topologically protected qubits <cit.>, using well-isolated qubits <cit.>, and encoding the qubit in composite structures <cit.>. A notable example of the latter is a qubit encoded in the singlet and triplet states of two electron spins in two gated lateral quantum dots <cit.>, a design that is protected against collective dephasing <cit.>. Another related approach is to encode the qubit in two complex states of a spin cluster, thereby reducing the chance of a bit-flip error <cit.>.
Following these ideas, it was proposed to encode the qubit in the low-energy macroscopic quantum states of a synthetic antiferromagnetic spin-one chain <cit.>. The low-energy spectrum of an antiferromagnetic spin-one chain consists of a singlet and a triplet separated by a gap from the rest of the spectrum <cit.>, an example of topological phases of matter <cit.>. The topologically protected low-energy spectrum can be understood in terms of two spin-half Haldane quasiparticles localized at the two ends of the chain. A synthetic system of InAs quantum dots in an InP nanowire with four electrons each was proposed to realize such a chain <cit.>, with the two spin-half quasiparticles resulting in a singlet-triplet Haldane (STH) qubit.
Previously, we have demonstrated how to realize the Haldane qubit in different physical systems <cit.>. In this work, we demonstrate how to couple two STH qubits and generate two-qubit gates, opening the path toward universal quantum computation with STH qubits. To study various microscopic and effective spin Hamiltonians constructed throughout the paper, we used the exact diagonalization (ED), and MPS tools <cit.> whenever the Hilbert space size was beyond the scope of ED.
The paper is organized as follows. We start in Section <ref> with a description of a single STH qubit realized in a chain of InAs quantum dots in an InP nanowire, for which we construct a Hubbard-Kanamori Hamiltonian and an effective spin-one Hamiltonian. In this section, we also briefly discuss how to generate all single-qubit gates using STH qubits. In Section <ref>, we construct a microscopic model for on-demand coupling of two STH qubits by an intermediate gated control quantum dot. Here we demonstrate that two coupled STH qubits, each made of two spin-half Haldane quasiparticles, behave akin to two electronic ST qubits. Then in Section <ref>, we discuss how to generate two-qubit gates using our proposed coupling scheme. Finally, in Section <ref>, we summarize the results of our work and discuss future directions.
§ SINGLET-TRIPLET QUBIT IN SYNTHETIC SPIN-ONE CHAIN
Let us start by describing the synthesis of an effective spin-one chain in an InAsP quantum dot nanowire system.
We consider an InP nanowire hosting a sequence of InAsP quantum dots. Experimental fabrication of such quantum dot nanowires has been successfully demonstrated, with theoretical investigations extending to atomic-scale details <cit.>.
Fig. <ref>(a) shows a schematic view of such a quantum dot nanowire.
§.§ Microscopic model of synthetic spin-one chain
The atomistic microscopic model of InAsP quantum dot has been developed already <cit.>. Despite hexagonal cross section and disorder due to low concentration of phosphor P atoms, the electronic states are grouped into shells similar to shells of a 2D harmonic oscillator, featuring m+1 orbitals corresponding to angular momentum m.
In Fig. <ref>(b), the conduction band shells of a dot are depicted schematically.
As the illustration shows, we populate each dot with four electrons, where two occupy the s-shell and the subsequent two occupy the p-shell.
We focus on these half-filled p-shell orbitals, which we call p_±, to construct a microscopic electronic Hamiltonian for the dot.
As we proceed to consider two such quantum dot nanowires, we designate the Hamiltonian describing dot i in nanowire A as H_A,i, given by
H_A,i = U (
(n_A,i,+ - 1)(n_A,i,- - 1) + ∑_α=±n_A,i,α,↑n_A,i,α,↓)
- 2W (s⃗_A,i,+·s⃗_A,i,- + 1/4n_A,i,+n_A,i,-)
+ δ/2∑_σ c^†_A,i,+,σc_A,i,-,σ
+ c^†_A,i,-,σc_A,i,+,σ,
where U denotes the Hubbard repulsion on each orbital, which has the same value as the direct Coulomb interaction between electrons on the two orbitals <cit.>, W is the Coulomb exchange between the electrons on two orbitals, and δ represents the splitting between the two orbitals induced by disorder and deviation from cylindrical symmetry.
All these terms are depicted graphically in Fig.<ref>(b). Note that there is no onsite energy for the orbitals, indicating that the energy is measured from the p_± levels. In writing (<ref>), we use
n_A,i,α,σ = c_A,i,α,σ^† c_A,i,α,σ and n_A,i,α = n_A,i,α,↑ + n_A,i,α,↓,
where c_A,i,α,σ denotes the annihilation operator of spin σ for orbital α at site i of the nanowire A. The spin operators are defined as
s_A,i,α^+ = c_A,i,α,↑^† c_A,i,α,↓, s_A,i,α^- = c_A,i,α,↓^† c_A,i,α,↑, and s_A,i,α^z = 1/2 (n_A,i,α,↑ - n_A,i,α,↓),
which then are used to build s⃗_A,i,+·s⃗_A,i,- = s_A,i,+^z s_A,i,-^z + 1/2 (s_A,i,+^+ s_A,i,-^- + s_A,i,+^- s_A,i,-^+).
The Hubbard repulsion U prevents double occupation of the orbitals, while the exchange W aligns the spin of the two electrons. Thus, unless the disorder δ is very large, the ground state of an isolated dot is a triplet. Consequently, in the low-energy limit, each dot can be deemed as effectively a spin-one object.
Next, we turn to the Hamiltonian describing the coupling between the p-shells of two adjacent dots in the nanowire A
H_A, i,i+1 = t∑_α,σ(
c_A,i,α,σ^† c_A,i+1,α,σ + h.c. )
+ V(n_A,i - 2)(n_A,i+1 - 2),
where t is hopping energy between the same orbitals and the same spins, and V is direct Coulomb energy. Here we neglect the possible hopping between different orbitals and different spins, due to the symmetry of the wire, and the negligible spin-orbit interaction. And we only keep direct Coulomb matrix element as the other allowed Coulomb matrix elements are negligibly small <cit.>.
In writing (<ref>), we also used the compact notation n_A,i = n_A,i,+ + n_A,i,-. These terms are graphically shown between the two dots in Fig. <ref>(c).
Now, in the spirit of the Hubbard model, one would anticipate that the inter-dot hopping combined with intra-dot repulsion results in an effective antiferromagnetic coupling between the dots. In fact, second-order perturbation theory shows that for weak inter-dot coupling, the low-energy behavior of two dots is the same as two spin-one objects, coupled antiferromagnetically by an exchange energy given by J_ eff=2t^2/(U+W-V) <cit.>.
Combining the intra-dot (<ref>), and inter-dot Hamiltonians (<ref>), for a chain, we obtain a Hubbard-Kanamori (HK) electronic Hamiltonian that describes a quantum dot nanowire system of length N as
H_A = ∑_i=1^N H_A,i + ∑_i=1^N-1 H_A,i,i+1.
Following the above discussion and results of exact diagonalization of a two quantum dot system with N=8 electrons, we find that in the limit of small inter-dot coupling, the HK Hamiltonian behaves like an antiferromagnetic spin-one chain <cit.>.
§.§ Spin-one chain:
a topological singlet-triplet qubit
Now, we proceed to quantitatively confirm that in the limit of weak inter-dot coupling, the HK Hamiltonian behaves like an antiferromagnetic spin-one chain described by a Heisenberg Hamiltonian
H_A = J_ eff∑_i=1^N-1S⃗_A,i·S⃗_A,i+1.
For this demonstration, we employ the HK parameters listed in Table <ref>, derived from an atomistic study <cit.>.
To compare the low-energy spectra of the HK and Heisenberg Hamiltonians, we pick chains of even length, whose ground state is always a singlet.
In Figure <ref> we present the results of our DMRG computation.
Fig. <ref>(a) presents the low-energy spectra of the HK and Heisenberg Hamiltonians, measured from the ground state, for increasing chain length N. As anticipated for an antiferromagnetic spin-one chain, the low-energy spectrum includes a singlet, a triplet, and a quintuplet.
Haldane showed <cit.> that this system possesses a topological gap. In particular, in the thermodynamic limit, the spin-one antiferromagnetic chain features a four-fold degenerate ground state comprising a singlet and three triplets, separated by a topological gap from the rest of the spectrum.
Fig. <ref>(a) shows the exponential drop of the singlet-triplet gap Δ with chain length, while the topological Haldane gap Γ converges to a constant value.
The spectrum of the HK closely mirrors that of the Heisenberg Hamiltonian, providing quantitative evidence that there exists a set of parameters for which the HK Hamiltonian behaves akin to a chain of spin-ones coupled by an effective antiferromagnetic exchange J_eff.
The gapped spectrum suggests that one can utilize the isolated singlet and triplet states of a Haldane chain to construct a robust macroscopic singlet-triplet qubit, provided that the operation temperature remains below the Haldane gap <cit.>.
We refer to such a qubit as a singlet-triplet Haldane (STH) qubit. Similar to a regular ST qubit <cit.>, one can envision an STH qubit as comprising two spin-half quasiparticles.
The two spin-half quasiparticles are the emerging fractional particles of the Haldane phase on the two ends of the chain, where the topological phase has an interface with the trivial phase outside the chain <cit.>.
As a way of visualizing these spin-half quasiparticles for a chain of length N=30, in Fig. <ref>(b) we show the expectation value of S_i^z, the z-component of spin operator on site i, over the state |T_+⟩, which is the lowest-energy state with S_ tot^z=1.
The spin-half objects are evident at the two ends of the chain for both the HK and Heisenberg Hamiltonian cases.
The nearly identical behavior of ⟨ S_i^z ⟩ in both cases provides further evidence of how closely the HK Hamiltonian mimics a spin-one Heisenberg chain in the low-energy regime.
§.§ Generating single-qubit gates
Before moving on to discuss how we couple two STH qubits to generate two-qubit gates, let us briefly describe how one can generate single-qubit gates using them.
Previously, it was shown that single-qubit operations on a STH qubit can be achieved by applying a local magnetic field on the first site of the chain <cit.>.
Here, we demonstrate that any local field that does not cover the entire chain is suitable for performing single-qubit operations, thereby relaxing the need for a spatially highly resolved and controllable magnetic field.
Consider the Heisenberg model of the qubit A, now with two sets of magnetic field as depicted in Fig. <ref>(d)
H_A = J_ eff∑_i=1^N-1S⃗_A,i·S⃗_A, i+1 +
b_A ∑_i=1^N_b S_A,i^z +
B_A S_ A, tot^z,
where b_A is a dynamic local magnetic field covering the first N_b sites, and B_A is a uniform background magnetic field covering the entire chain.
The computational basis of the STH qubit, much like regular ST qubits, consists of the singlet state |S⟩, which we take to be |0⟩, and the triplet state |T_0⟩ with S_tot^z=0, taken to be |1⟩. These states are separated from the rest of the spectrum by the Haldane gap Γ, which is of the order of 0.4 J_eff. Meanwhile, the gap between them, Δ, diminishes exponentially with the chain length (see Fig. <ref>(a)).
Turning on the local magnetic field b_A breaks the conservation of total spin S⃗_ tot^2, causing a mixing between |S⟩ and |T_0⟩, allowing for the generation of single-qubit gates. The uniform background magnetic field B_A serves to push the other two low-energy triplet states, |T±⟩, away from the computational basis, thereby reducing the errors stemming from spin-flipping noises.
As we discuss below, the background magnetic field plays a more substantial role in implementing two-qubit operations.
Note that since the local field operation still conserves S_ tot^z, it does not cause leakage to |T_±⟩ even in the absence of the uniform background magnetic field.
Since the local field breaks S⃗_ tot^2 conservation, it also mixes the computational basis with every other state in the S_ tot^z=0 subspace. But as long as b_A≪Γ, such leakages remain negligible, as all other states are separated from the computational basis by Γ.
In Fig. <ref> we demonstrate the generation of a Hadamard gate using a STH qubit of length N=10, where the local field is applied to the first N_b=5 sites.
For N=10 the singlet-triplet (ST) gap is Δ≈ 0.14 J_ eff, the Haldane gap is Γ≈ 0.76 J_ eff, and the matrix element of the local magnetic field between the singlet and the triplet is M_b≈ 0.67 b_A.
Choosing b_A such that M_b = Δ/2, one can generate a Hadamard gate by applying the local field for t_H = π/√(2)Δ duration <cit.>.
To generate Fig. <ref>, we keep the first 100 states of H_A(b_A=0) to form a new basis for a reduced Hilbert space. Subsequently, we construct the Hamiltonian including the local field. The time evolution is then computed using e^-i tH_A in the reduced Hilbert space.
In the top panel, solid(dashed) curves show the evolution of the qubit initialized in the |0⟩(|1⟩) state. It is visible that in both cases, at the end of the operation, the system is in a nearly equal superposition of the |0⟩ and |1⟩ states, as expected for a Hadamard gate.
Note that when the qubit is initialized in |1⟩, there exists a π phase difference between the two projections of |ψ(t)⟩, which is not visible in the plot, as we plot the square of the projections for visual clarity.
As mentioned, there is a small probability of the system leaking out of the computational basis. The leakage is quantified by the projection of the chain's state |ψ(t)⟩ outside the computational basis
ϵ_1 = 1 - ∑_i=0,1 |⟨ψ(t)|i⟩|^2.
As the bottom panel of Fig. <ref> shows, the leakage is well below 1% when the qubit is initialized in the |0⟩, and it reaches a maximum of approximately 1% when initialized in the |1⟩. This can be attributed to the smaller separation of the triplet state (|1⟩) from the rest of the spectrum above the Haldane gap.
Notice that to implement the Hadamard gate we only needed to control the local magnetic field. Furthermore, with a fixed Δ, we can also generate any phase gate by setting b_A=0. This implies that we can generate all the single-qubit gates required for universality by solely controlling the local magnetic field <cit.>.
§ COUPLING TWO SINGLET-TRIPLET HALDANE QUBITS
In this section, we first describe how to make a tunable coupling between two STH qubits, A and B, each realized in a quantum dot nanowire system. Then, using the effective spin Hamiltonian of the two coupled STH qubits, we compare their coupling with that of two coupled simple electron based ST qubits, each consisting of two spin-half particles.
§.§ Microscopic model of the inter-chain coupling
Consider two STH qubits, A and B, coupled by a tunable link, as depicted in Figure <ref>. The tunable link is made of another quantum dot, labeled C and referred to as the “control dot," which is gated by a gate tunable with applied voltage ε_c. As shown in Fig. <ref>, adjusting the gate voltage shifts the states of the control dot relative to its neighbors. When the p-shell of the control dot is out of resonance with their neighbors, the two ends of qubits A and B are effectively decoupled. Lowering ε_c, an effective coupling between the two ends of A and B develops, which, as we demonstrate, acts as a tunable effective antiferromagnetic spin coupling between the two ends of the two STH qubits.
The Hamiltonian describing the two coupled STH qubits is given by
H = ε_c n_C + H_C +
H_AC + H_BC + H_A + H_B,
where n_C is the electron number operator at the control dot, H_C describes the control dot without the detuning ε_c as in (<ref>), H_AC and H_BC describe the hopping and Coulomb interaction between the last dots of qubits A and B and the control dot as in (<ref>), and H_A and H_B are the HK Hamiltonians as in (<ref>), describing each qubit in isolation.
To demonstrate that the link, realized by the gated control dot, functions as an effective tunable spin coupling between the two ends of the two STH qubits, let us focus on the link and consider three dots populated by four electrons. Two electrons belong to qubit A and two electrons belong to a dot of qubit B. Using the HK parameters listed in Table <ref>, Figure <ref> shows the low-energy spectrum of the three-dot system as a function of ε_c.
Recalling that two antiferromagnetically coupled spin-ones form a singlet, a triplet, and a quintuplet, one can observe that once ε_c exceeds a critical value, the spectrum of the three-dot system resembles that of two antiferromagnetically coupled spin-ones. In this regime the changing singlet-triplet gap gives the value of the tunable effective spin coupling J_AB.
It is evident how this coupling quickly diminishes as ε_c increases, allowing the two ends of qubits A and B to be perceived as two decoupled spin-one objects for large enough ε_c. Therefore, by adjusting ε_c, we can control J_AB and effectively switch it on and off.
§.§ Spin model of two coupled STH qubits
Let us now consider the effective spin Hamiltonian of two coupled STH qubits, A and B, given by
H = H_A + H_B +
J_ABS⃗_A,N·S⃗_B,N,
where H_A and H_B are Heisenberg Hamiltonians, as described in (<ref>), representing qubits A and B in isolation. The last term describes how the last dots of qubit A and B are coupled by the tunable coupling J_AB.
As mentioned, the low-energy behavior of a single chain can be interpreted as two emerging spin-half quasiparticles coupled by an exchange coupling equal to the singlet-triplet gap, Δ. In this perspective, which is schematically illustrated inside Fig. <ref>, we anticipate that coupling qubits A and B by S⃗_A,N·S⃗_B,N will be analogous to coupling one spin-half quasiparticle of qubit A to a spin-half quasiparticle from qubit B.
On the other hand, two simple ST qubits, each composed of two spin-halfs, coupled in the same fashion can be described by
H = Δ (s⃗_A,1·s⃗_A,2 + s⃗_B,1·s⃗_B,2)
+ J_ABs⃗_A,2·s⃗_B,2,
where s⃗_X,i denotes the spin-half number i of qubit X, Δ is the coupling in each ST qubit, and J_AB couples the two ends of qubit A and B.
Since H conserves the total spin, the computational basis of the two simple ST qubits evolves in the subspace s_ tot^z=0, which is a six-dimensional space spanned by {|SS⟩, |ST_0⟩, |T_0S⟩, |T_0T_0⟩, |T_+T_-⟩, |T_-T_+⟩}, where each component represents the tensor product of a state from qubit A and a state from qubit B.
Notice how the two components |T_± T_∓⟩ do not belong to the computational basis.
In this basis and with this ordering H has the following structure
H=
[ H_QQ H_QQ; H_QQ H_QQ ],
with
H_QQ =
[ -Δ 0 0 χ J_AB; 0 0 χ J_AB 0; 0 χ J_AB 0 0; χ J_AB 0 0 Δ ],
H_QQ =
[ -χ J_AB χ' J_AB -χ' J_AB χ” J_AB; -χ J_AB -χ' J_AB χ' J_AB χ” J_AB ],
H_QQ =
[ Δ -χ” J_AB 0; 0 Δ -χ” J_AB ],
where the energy is measured from |ST_0⟩ state, and for H in (<ref>) we have χ=χ'=χ”=1/4.
In Appendix <ref>, we show how to analytically determine the spectrum of H comprising two singlets, three triplets, and one quintuplet. This spectrum is plotted in Fig. <ref> by dashed lines as a function of J_AB/Δ.
Now, considering that the low-energy spectrum of H in (<ref>) primarily consists of the low-energy singlets and triplets of the STH qubits A and B, and given that H also conserves total spin, its low-energy spectrum should be well-approximated in a similar six-dimensional basis, comprising the computational basis along with the two additional states |T_± T_∓⟩.
In Appendix <ref>, we provide a rigorous proof of why the projection of H onto this basis follows the structure shown in (<ref>). But for STH qubits, unlike the simple ST qubits, χ, χ', and χ” are not identical, and they generally vary depending on the size of the chain. For two STH qubits of length N=10, they are approximately given by χ≈ 0.34, χ' ≈ 0.30, and χ”≈ 0.26.
In Figure <ref>, we present the low-energy spectrum of two coupled STH qubits with a length of N=10, described by H in (<ref>), computed using exact diagonalization. The solid curves represent the energy levels of H measured from the ground state as a function of J_AB/Δ, where Δ is the singlet-triplet gap of a single chain, approximately given by 0.14 J_ eff.
It is evident that the spectrum, normalized to Δ, closely follows the spectrum of two simple ST electron spin based qubits described by (<ref>).
The color of the curves corresponds to the total spin of the respective state, which is consistent for the lowest six states in both systems. The S_ tot^z=0 subspace of two coupled simple ST qubits is of course six-dimensional, but for two coupled STH qubits described by H, the S_ tot^z=0 subspace is much larger, and contains many higher-energy states. However, these states are all separated from the six-dimensional low-energy spectrum by a Haldane gap. We display the first few of these higher-energy states as grey curves in Fig. <ref>.
§ GENERATING TWO-QUBIT GATES
Now that we have established how to couple two STH qubits, we proceed to demonstrate how to generate two-qubit gates using this coupling. It is well-known that for a quantum computer to be universal, it is sufficient to be able to generate all single-qubit gates, along with any two-qubit gate that is “locally equivalent" to the CNOT gate <cit.>. In Appendix <ref>, we describe the concept of locally equivalent two-qubit gates and demonstrate how one can generate a gate locally equivalent to CNOT using a Hamiltonian with the structure of H_QQ in (<ref>).
The primary challenge in generating two-qubit gates from ST qubits is apparent from the previous section, where we presented the spectrum of two coupled qubits. This challenge arises from the two undesired states |T_± T_∓⟩, which couple to the computational basis. For conventional ST qubits, Levy addressed this challenge by devising a unique sequence to produce the controlled Z (CZ) gate <cit.>, which is locally equivalent to the CNOT gate. In Levy's sequence, the undesired states fully decouple from the computational basis at the end of the sequence. Crucial to this sequence is (a) the application of one of the local magnetic fields on the coupled ends of the qubits and the other on the non-coupled one, and (b) the ability to decouple the spin-halfs comprising each ST qubit, that is to be able to set Δ=0 in our language. The first requirement can be fulfilled for STH qubits as well, as illustrated in Figure <ref>(a). However, the second requirement presents a serious challenge, as Δ remains fixed for a given chain length in STH qubits.
To address this challenge, we can leverage the fact that the singlet-triplet gap Δ in Haldane chains decreases exponentially with the chain length. Therefore, by connecting and disconnecting a few more dots at the end of each STH qubit, we can effectively switch Δ on and off. This may be achievable by incorporating an additional controllable link within each STH qubit, labeled J_L in Fig. <ref>(a). When J_L is activated, the chain becomes longer, turning Δ off, and when it is deactivated, Δ is turned on. It is important to carefully design the internal link of each qubit so that when J_L is activated, it accurately replicates the J_ eff coupling between other dots of the chain, ensuring a properly uniform Haldane chain of increased length.
Demonstrating a Levy sequence on two STH qubits with variable lengths, sufficiently long to effectively turn off Δ, poses a significant computational challenge, particularly in tracking the time evolution of a system with a massive Hilbert space. Additionally, designing and fabricating STH qubits with variable lengths could present its own set of challenges. Therefore, we propose a simpler scheme where Δ remains constant in each STH qubit. As we discuss below, this is achieved by introducing different background magnetic fields for the two coupled qubits.
§.§ Using different background magnetic fields
Introducing two different background magnetic fields for the two STH qubits modifies the spin Hamiltonian describing the coupled qubits to
H_AB = H + B_A S_A, tot^z + B_B S_B, tot^z,
where H, as given in (<ref>), describes the coupled qubits without background magnetic fields. The addition of the background magnetic fields further modifies the structure of H_QQ as
H_QQ =
[ Δ -χ” J_AB + B_AB 0; 0 Δ -χ” J_AB - B_AB ],
with B_AB = B_A - B_B.
This implies that when the two background magnetic fields are different and B_AB≠ 0, the two undesired states |T_± T_∓⟩ are displaced from the |T_0T_0⟩ component of the computational basis by B_AB.
The greater the B_AB, the lower the leakage of the computational basis into |T_± T_∓⟩.
However, it is important to be cautious about the higher-energy states, as they can come down due to the non-zero B_AB and potentially interfere with the computational basis, resulting in leakage.
The first of such states is |Q_-T_+⟩, comprising the quintuplet in the S_A, tot^z=-1 subspace and |T_+⟩ of qubit B. In the absence of coupling, |Q_-T_+⟩ is positioned Γ - B_AB above |T_0T_0⟩, where Γ is the Haldane gap, while |T_+T_-⟩ is B_AB above |T_0T_0⟩, and |T_-T_+⟩ is B_AB - 2Δ below |SS⟩. Therefore, minimal interference from the outside of the computational basis can be achieved around the point where Γ-B_AB= B_AB - 2Δ→ B_AB = Δ + Γ/2.
In Figure <ref>, the solid curves show the evolution of two STH qubits, each with a length of N=10, coupled by J_AB=Δ/2, and subjected to B_AB = Δ + Γ/2. To generate this plot, we use the low-energy product states as the basis of the reduced Hilbert space, formed by the products of states from S_A, tot^z=m and S_B, tot^z=-m. We progressively incorporate higher-energy states until the convergence of leakage, quantified by
ϵ_2 = 1-∑_i,j=0,1 |⟨ψ(t)|ij⟩|^2,
where |ψ(t)⟩ is the state of the coupled system at time t, and i and j go over the computational basis.
In this case, we included 338 product states in the basis.
As the leakage panel of Fig. <ref> shows, the maximum leakage is about 2%. But note that one can always reduce the leakage by choosing smaller J_AB.
In Appendix <ref>, we demonstrate that for any given J_AB, it is possible to create a gate locally equivalent to CNOT by applying the coupling J_AB twice, with a local transformation in between. We also show how to find the duration of the coupling pulses, denoted as t_c, for any J_AB. In Fig. <ref>, we use the t_c value obtained for J_AB=Δ/2 as the duration of the time evolution.
To produce the time evolution depicted in Fig. <ref>, we initialize the two-qubit system in the state |ψ(0)⟩ = (|00⟩ + |01⟩ + |11⟩)/√(3). We choose this initial state to ensure that the two-qubit system evolves through all four components of the computational basis.
We can see that being initialized in |ψ(0)⟩ and with the small leakage, the two-qubit system primarily evolves within the computational basis. The evolution appears to comprise two nearly independent rotations, one involving the components |00⟩ and |11⟩, and the other involving |01⟩ and |10⟩. This behavior arises from the structure of H_QQ in (<ref>), which couples |01⟩ and |10⟩ with ω_1 = χ J_AB, and |00⟩ and |11⟩ with ω_2 = √(ω_1^2 + Δ^2), thus resulting in a faster evolution of the two-qubit system among the latter components.
If we choose two very large background magnetic fields for the two qubits in opposite directions, we can effectively push away all the spin polarised states of the system from the computational basis. In this limit, the coupling effectively becomes an Ising coupling. Therefore, the only product states that can interfere with the computational basis are the pairs made of non-spin-polarised states, where each state belongs to the S_ tot^z=0 subspace of its own chain.
Among these states, the closest to the computational basis are |SQ_0⟩ and |Q_0S⟩, consisting of the nonmagnetic first quintuplet of one chain and the ground state singlet of the other chain. When the two chains are not coupled, these states are separated from the computational basis by Γ-Δ.
In this limit, where the coupling is effectively an Ising coupling, one can imagine that the leakage will be smaller for the same coupling energy J_AB.
To illustrate this point, we also present, with dashed curves, the time evolution of the same two STH qubits now coupled by an Ising interaction instead of the Heisenberg interaction of (<ref>).
It is evident that with an Ising interaction, the leakage is significantly reduced for the same J_AB. However, the problem with choosing two very large magnetic fields in the background of the qubits is that it transforms the computational basis into highly excited states. Therefore, such a scenario may not be stable enough for executing a long sequence of unitary operations.
The significantly reduced leakage observed with Ising coupling prompts speculation on effective methods to generate Ising coupling between two quantum dots without relying on strong background magnetic fields. This intriguing question remains open for future investigations.
§ CONCLUSION
Starting from a microscopic model, we investigated the coupling of two Singlet-Triplet Haldane( STH ) qubits realized in a quantum dot nanowire with four electrons in each dot through a gated control dot. By computing the energy spectrum of the fermionic system, we demonstrated that each quantum dot nanowire system behaves like a topological Haldane chain with Haldane quasiparticles at both ends. Furthermore, we established the equivalence between controlling the detuning of the control dot and controllable antiferromagnetic coupling between the ends of the two STH qubits. We showed that the low-energy behavior of two coupled STH qubits, effectively made of two spin-half Haldane quasiparticles, resembles that of two simple ST qubits, each made of two real spin-halfs.
Using the spin model, we focused on the issue of decoupling |T_+T_-⟩ states from the computational basis while generating two-qubit gates. We proposed that by using an internal link in each STH qubit, one can vary their length and effectively switch the singlet-triplet gap Δ on and off, thereby enabling the use of Levy sequence for generating two-qubit gates <cit.>.
Furthermore, we demonstrated that resolving the leakage problem without altering the length of the STH qubits is attainable by employing different background magnetic fields for the two qubits. Additionally, we showed that inter-chain Ising coupling could substantially decrease leakage, and discussed its potential realization through the use of very large magnetic fields in the background. We anticipate exploring alternative methods for achieving Ising coupling that do not rely on the use of large magnetic fields in the future.
Exploring other avenues for coupling these STH qubits, such as photon bus, remains an interesting direction for future research. We hope this work motivates future experimental work on macroscopic quantum states in semiconductor nanostructures.
We acknowledge NSERC Alliance Quantum Consortium PQS2D grant ALLRP/578466-2022, the QSP-078, AQC-004 and HTSN-341 projects of the Quantum Sensors, Applied Quantum Computing and On-chip integrated circuits based on 2D materials Programs at the National Research Council of Canada, University of Ottawa Research Chair in Quantum Theory of Materials, Nanostructures, and Devices, and Digital Research Alliance Canada with computing resources.
§ TWO COUPLED SIMPLE ST QUBITS
Consider H in (<ref>) that describes two coupled simple ST qubits. Since H conserves total spin, its spectrum consists of two singlets, three triplets, and one quintuplet. The quintuplet is |Q_TT⟩∝ 2|T_0T_0⟩+|T_+T_-⟩+|T_-T_+⟩, and its energy is given by E_Q_TT=Δ+J_AB/4. Next observe that one of the triplets is |T_ST+⟩∝|ST_0⟩+|T_0S⟩ and its energy is given by E_T_ST+= J_AB/4. This leaves us with two 2D subspaces, one for the singlets and the other for the remaining two triplets. Finding the eigenstates of 2D Hamiltonians is straightforward. We find that the two remaining triplets are |T_ST-⟩∝ C_1 (|ST_0⟩-|T_0S⟩) - J_AB(|T_+T_-⟩-|T_-T_+⟩), and |T_TT⟩∝ J_AB (|ST_0⟩-|T_0S⟩) + C_1(|T_+T_-⟩-|T_-T_+⟩), with C_1=Δ+√(Δ^2+J_AB^2); their energy is given by E_T_ST- = -1/2√(Δ^2 + J_AB^2) +Δ/2 - J_AB/4, and E_T_TT=1/2√(Δ^2 + J_AB^2) +Δ/2 - J_AB/4. In the limit of J_AB≪Δ, |T_ST-⟩ is primarily made of one singlet and one triplet, while |T_TT⟩ is primarily made of two triplets.
And finally we find that the two singlets are |S_TT⟩∝ C_2(|T_+T_-⟩+|T_-T_+⟩-|T_0T_0⟩)-3J_AB|SS⟩, and |S_SS⟩∝ J_AB(|T_+T_-⟩+|T_-T_+⟩-|T_0T_0⟩)+C_2|SS⟩, with C_2=2√(4Δ^2 - 2Δ J_AB + J_AB^2) + 4Δ-J_AB; their energy is given by E_S_TT = 1/2√(4Δ^2 - 2Δ J_AB + J_AB^2) - J_AB/4, and E_S_SS=-1/2√(4Δ^2 - 2Δ J_AB + J_AB^2) - J_AB/4. The ground state is |S_SS⟩ and in the limit of J_AB≪Δ, it is primarily made of two singlets, while |S_TT⟩ is primarily made of two triplets.
This spectrum is shown in Figure <ref> by dashed curves as a function of J_AB/Δ.
§ THE STRUCTURE OF COUPLING MATRIX
Here, we demonstrate that the coupling operator
H_AB = S⃗_A,N·S⃗_B,N
= S_A,N^z S_B,N^z + 1/2 (S_A,N^+ S_B,N^- + S_A,N^- S_B,N^+),
which appears in (<ref>), has the same structure as described in Eqs. (<ref> and <ref>) within the subspace spanned by {|SS⟩, |ST_0⟩, |T_0S⟩, |T_0T_0⟩, |T_+T_-⟩, |T_-T_+⟩}.
Before starting the proof, observe that we can write a product state of two chains in terms of their configurations as
|ij⟩=∑_p,qa_ipb_jq|p;q⟩,
where we are concerned with i,j∈{ S,T_0,T_+,T_-}, and p(q) are configurations of a single chain of length N like p = p_1⋯ p_N, with p_k∈{0,± 1}.
Let us also introduce inverted configuration p, for which p_k=-p_k for all k's.
Then since the Hamiltonian does not distinguish between ±ẑ directions, for all eigenstates of a single chain in its S_ tot^z=0 subspace, we have |a_ip|=|a_ip|.
§.§ The structure of H_QQ
For H_QQ block we need to show that the only non-zero matrix elements are those located on the anti-diagonal of H_QQ, corresponding to the terms proportional to J_AB in (<ref>).
Notice that in this block the only relevant terms of H_AB is the Ising part S_A,N^z S_B,N^z.
Then using orthonormality of the configurations we can write
⟨i'j'|S_A,N^zS_B,N^z|ij⟩=
∑_p,qp_Nq_N a_i'p^*a_ip b_j'q^*b_jq.
If i=i' or j=j' we have |a_ip|^2 or |b_jq|^2 in the summand of (<ref>).
But since the sum in (<ref>) goes over all p's and q's, and we have p_N|a_ip|^2 + p_N|a_ip|^2= q_N|b_jq|^2 + q_N|b_jq|^2 = 0, then ⟨i'j'|S_A,N^zS_B,N^z|ij⟩=0, unless i≠ i' and j≠ j', that is on the anti-diagonal of H_QQ.
Moreover, since H_QQ is Hermitian to complete the proof we just need to show
⟨SS|S_A,N^zS_B,N^z|T_0T_0⟩ = ⟨ST_0|S_A,N^zS_B,N^z|T_0 S⟩.
To see that, we realize that since the Hamiltonian is real, its eigenvectors are also real, therefore
⟨SS|S_A,N^zS_B,N^z|T_0T_0⟩ = ∑_p,qp_Nq_N a_Sp a_T_0 p b_S q b_T_0 q
= ⟨ST_0|S_A,N^zS_B,N^z|T_0 S⟩=χ.
§.§ The structure of H_QQ
Here, we derive the structure of the matrix elements of H_AB between |T_+T_-⟩ and the computational basis, and the procedure is the same for |T_-T_+⟩. Notice that the only term in H_AB, given in (<ref>), that produces a non-zero matrix element between |T_+T_-⟩ and the computational basis is 1/2 S_A,N^-S_B,N^+.
Let us start by making a few observations.
First, observe that we have S_ tot^±|T_0⟩=√(2)|T_±⟩, and S_ tot^±|S⟩ = S_ tot^±|T_±⟩=0, where S_ tot^± = ∑_n S_n^±.
Second, observe that using the commutation relation [S^+,S^-]=2S^z, and the first observation, we have ⟨S|S_N^∓|T_±⟩ = 1/√(2)⟨S|S_N^∓ S_ tot^±|T_0⟩=∓√(2)⟨S|S_N^z|T_0⟩.
Similarly we also have ⟨T_0|S_N^∓|T_±⟩ = 1/√(2)⟨T_±|S_ tot^± S_N^∓|T_±⟩=±√(2)⟨T_±|S_N^z|T_±⟩.
Here we suppressed the chain index A and B to stay general.
Then, using the second observation, we can write ⟨SS|H_AB|T_+T_-⟩=1/2⟨SS|S_A,N^-S_B,N^+|T_+T_-⟩=-⟨SS|S_A,N^zS_B,N^z|T_0T_0⟩=-χ, which gives us the first column of H_QQ in agreement with the form in (<ref>).
Next, for the next two columns of H_QQ, it is evident that we should have |⟨ST_0|H_AB|T_± T_∓⟩|=|⟨T_0 S|H_AB|T_∓ T_±⟩|, as the Hamiltonian doesn't have preferred direction, and as it is symmetric with respect to the two chains.
To complete the proof, we show that ⟨ST_0|H_AB|T_+T_-⟩=⟨T_0 S|H_AB|T_-T_+⟩=χ', and the other two matrix elements have the opposite sign.
As mentioned we can write |T_+T_-⟩ as
|T_+ T_-⟩ =∑_p,q a_p b_q |p;q⟩
→|T_- T_+⟩ =∑_p,q a_p b_q |q;p⟩.
Consequently, we have
1/2 S_A,N^-S_B,N^+|T_+ T_-⟩ = ∑_p,q a_p b_q |p';q'⟩
1/2 S_A,N^+S_B,N^- |T_- T_+⟩ = ∑_p,q a_p b_q |q';p'⟩,
where p' and q' are configurations that we obtain by flipping the last spin down and up in each chain respectively.
It is evident that (<ref>) and (<ref>) are the same up to switching between the A and B configurations, which implies that we have ⟨ST_0|S_A,N^-S_B,N^+|T_+T_-⟩=⟨T_0 S|S_A,N^+S_B,N^-|T_-T_+⟩, as we claimed.
Finally, for the last elements of H_QQ, using the observations we made above, we can write ⟨T_0T_0|H_AB|T_+T_-⟩=1/2⟨T_0T_0|S_A,N^-S_B,N^+|T_+T_-⟩=-⟨T_+T_-|S_A,N^zS_B,N^z|T_+T_-⟩=χ”, which proves full agreement with (<ref>).
§.§ The structure of H_QQ
The structure of H_QQ is simple.
It is clear that it should be proportional to the identity matrix, as H_AB cannot connect |T_+ T_-⟩ and |T_- T_+⟩, because such a process would require two spin flips in each chain. Moreover, since the Hamiltonian does not distinguish between the ±ẑ directions, we should have ⟨T_+ T_-|H_AB|T_+ T_-⟩ = ⟨T_- T_+|H_AB|T_- T_+⟩= -χ”, in agreement with (<ref>).
§ GENERATING CNOT GATE
Here we show how to generate a gate locally equivalent to CNOT using H_QQ in (<ref>).
We say two two-qubit gates U and U' are locally equivalent if U' = (L_A⊗ L_B) U (R_A⊗ R_B), where L's and R's are single-qubit gates acting on qubits A and B. It can be shown that if two two-qubit gates have the same canonical form α=(α_1,α_2,α_3) in the Weyl chamber π/4≥α_1 ≥α_2 ≥ |α_3| ≥ 0, then they are locally equivalent <cit.>.
There is a standard procedure for finding the canonical form of a two-qubit gate like U, as the following <cit.>: (1) Factor out a global phase U = e^iα_0U such that (U)=1. (2) Form V=M^†U M where
M = 1/√(2)[ 1 0 0 i; 0 i 1 0; 0 i -1 0; 1 0 0 -i ].
(3) Find λ's, the eigenvalues of V^TV. (4) Then the canonical form α is the unique solution of the following set of equations in the Weyl chamber
λ_1 = e^2i(α_1 - α_2 + α_3),
λ_2 = e^2i(α_1 + α_2 - α_3),
λ_3 = e^-2i(α_1 + α_2 + α_3),
λ_4 = e^-2i(α_1 - α_2 - α_3).
The canonical form of CNOT – along with many other common two-qubit gates – is α_ CNOT = (π/4,0,0).
Next, we show how to use H_QQ in (<ref>) to generate a gate with the same canonical form.
§.§ Arbitrary coupling value
Here we show how to find t_c such that for an arbitrary coupling J_AB, U_2QQ(t_c)=e^-it_c H_QQ (XI) e^-it_c H_QQ has a canonical form the same as α_ CNOT.
Using the procedure mentioned above, we observe that for any t, the canonical form of U_2QQ(t) has the form α_2QQ(t)=(ϕ,0,0), where
sin(ϕ) = abs( sin(ω_1 t)cos(ω_2 t)
+ sin(ω_2 t)/ω_2
(ω_1 cos(ω_1 t) + i Δsin(ω_1 t))),
cos(ϕ) = abs(cos(ω_1 t)cos(ω_2 t)
+i sin(ω_2 t)/ω_2
(Δcos(ω_1 t) + i ω_1 sin(ω_1 t))),
with ω_1 = χ J_AB, and ω_2 = √(ω_1^2 + Δ^2).
Therefore, to find t_c one needs to solve sin^2(ϕ(t_c)) = cos^2(ϕ(t_c)) = 1/2, which can be done numerically, for any given J_AB.
§.§ Special coupling values
It turns out that for a set of special coupling values J_AB, one can generate a gate locally equivalent to CNOT by one time application of H_QQ in (<ref>), as U_QQ(t_c)=e^-it_c H_QQ.
To see that, we observe that the canonical form of U_QQ(t) has the form α_QQ(t) = 1/2(ϕ_1+ϕ_2, ϕ_1-ϕ_2, 0), where ϕ_1=ω_1 t, and sin(ϕ_2) = ω_1/ω_2sin(ω_2 t).
Therefore, α_QQ can reach at the α_ CNOT point, if we have ϕ_1 + ϕ_2 = π/2 + m_1π, and ϕ_1 - ϕ_2=m_2π, with integer m's. This implies that ϕ_1 = (2m + 1)π/4, and that sin^2(ϕ_2)=1/2.
Finding t from ϕ_1 and plugging it in the expression for sin(ϕ_2) we find
2sin^2((2m+1)π/4ω_2/ω_1) = (ω_2/ω_1)^2.
For integers m>1, this equation has a set of solutions for ω_2/ω_1, from which the special values of J_AB for a given Δ can be obtained.
|
http://arxiv.org/abs/2404.15169v1 | 20240423160740 | A Note on Centralizers and Twisted Centralizers in Clifford Algebras | [
"E. R. Filimoshina",
"D. S. Shirokov"
] | math.RA | [
"math.RA",
"math-ph",
"math.MP",
"15A66, 11E88"
] |
A Note on Centralizers and Twisted Centralizers in Clifford Algebras]
A Note on Centralizers and Twisted Centralizers in Clifford Algebras
E. Filimoshina]Ekaterina Filimoshina
HSE University
Moscow 101000
Russia
filimoshinaek@gmail.com
D. Shirokov]Dmitry Shirokov
HSE University
Moscow 101000
Russia
and
Institute for Information Transmission Problems of the Russian Academy of Sciences
Moscow 127051
Russia
dm.shirokov@gmail.com
15A66, 11E88
Last Revised:
April 29, 2024
This paper investigates centralizers and twisted centralizers in degenerate and non-degenerate Clifford (geometric) algebras. We provide an explicit form of the centralizers and twisted centralizers of the subspaces of fixed grades, subspaces determined by the grade involution and the reversion, and their direct sums. The results can be useful for applications of Clifford algebras in computer science, physics, and engineering.
[
[
April 29, 2024
==================
§ INTRODUCTION
In this work, we consider degenerate and non-degenerate real and complex Clifford (geometric) algebras _p,q,r of arbitrary dimension and signature (in
the case of any complex Clifford algebra, we can take q = 0).
Degenerate Clifford algebras have applications in physics <cit.>, geometry <cit.>, computer vision and image processing <cit.>, motion capture and robotics <cit.>, neural networks and machine learning <cit.>, etc.
Several recent works on Clifford algebras use the notion of centralizers and twisted centralizers in _p,q,r <cit.>.
We call a centralizer of a set in _p,q,r a subset of all elements of _p,q,r that commute with all elements of this set.
A twisted centralizer of a set in _p,q,r is a subset of such multivectors that their projections onto the even ^(0)_p,q,r and odd ^(1)_p,q,r subspaces commute and anticommute respectively with all elements of this set (see details in Section <ref>).
Centralizers and twisted centralizers of some particular sets in _p,q,r are used in literature for various purposes.
For example, the recent paper <cit.> finds an explicit form of the twisted centralizer of the grade-1 subspace in _p,q,r and applies it in the construction of Clifford group equivariant neural networks.
The work <cit.> uses the explicit form of the same twisted centralizer when considering degenerate spin groups.
The book <cit.> finds an explicit form of the centralizer of the even subspace in the case of the non-degenerate Clifford algebra _p,q,0.
In the papers <cit.>, the centralizers and twisted centralizers of the even subspace and the grade-1 subspace are employed in consideration of Lie groups preserving the even and odd subspaces under the adjoint and twisted adjoint representations in the non-degenerate Clifford algebras _p,q,0.
The works <cit.> find an explicit form of these centralizers in the case of arbitrary _p,q,r.
In light of appearance of centralizers and twisted centralizers in _p,q,r in the recent papers,
we decided to investigate them in this note.
We concentrate on the centralizers and twisted centralizers of the subspaces of fixed grades ^k_p,q,r, k=0,1,…,n, the subspaces ^m_p,q,r, m=0,1,2,3, determined by the grade involution and the reversion, and their direct sums. In particular, we consider the centralizers and twisted centralizers of the even ^(0)_p,q,r and odd ^(1)_p,q,r subspaces. We find an explicit form of these centralizers and twisted centralizers in the case of arbitrary k=0,1,…,n and m=0,1,2,3.
We study the relations between the considered centralizers and the twisted centralizers.
This paper also considers the centralizers and twisted centralizers in the particular cases of the non-degenerate Clifford algebra _p,q,0 and the Grassmann algebra _0,0,n. Theorems <ref>, <ref>, <ref> and Lemmas <ref>, <ref>, <ref> are new.
The paper is structured as follows.
Section <ref> introduces all the necessary notation related to _p,q,r.
Section <ref> provides an explicit form of the centralizers and twisted centralizers of the subspaces ^k_p,q,r, k=0,1,…,n and considers the relations between them.
In Section <ref>, we write out all the considered centralizers and twisted centralizers in the particular cases _p,q,0 and _0,0,n and in the case of small k≤4.
Section <ref> provides an explicit form of the centralizers and twisted centralizers of the subspaces ^m_p,q,r, m=0,1,2,3, and their direct sums, in particular, the even and odd subspaces.
The conclusions follow in Section <ref>.
§ DEGENERATE AND NON-DEGENERATE CLIFFORD ALGEBRAS _P,Q,R
Let us consider the Clifford (geometric) algebra <cit.> (V)=_p,q,r, p+q+r=n≥1, over a vector space V with a symmetric bilinear form, where V can be real ^p,q,r or complex ^p+q,0,r. We use to denote the field of real numbers in the first case and the field of complex numbers in the second case. In this work, we consider both the case of the non-degenerate Clifford algebras _p,q,0 and the case of the degenerate Clifford algebras _p,q,r, r≠0.
We use Λ_r to denote the subalgebra _0,0,r, which is the Grassmann (exterior) algebra <cit.>.
The identity element is denoted by e, the generators are denoted by e_a, a=1,…,n. The generators satisfy the following conditions:
e_a e_b + e_b e_a = 2 η_abe, ∀ a,b=1,…,n,
where η=(η_ab) is the diagonal matrix with p times +1, q times -1, and r times 0 on the diagonal in the real case (^p,q,r) and p+q times +1 and r times 0 on the diagonal in the complex case (^p+q,0,r).
Let us consider the subspaces ^k_p,q,r of fixed grades k=0,…,n. Their elements are linear combinations of the basis elements e_A=e_a_1… a_k:=e_a_1⋯ e_a_k, a_1<⋯<a_k, labeled by ordered multi-indices A of length k, where 0≤ k≤ n. The multi-index with zero length k=0 corresponds to the identity element e. The grade-0 subspace is denoted by ^0 without the lower indices p,q,r, since it does not depend on the Clifford algebra's signature. We have ^k_p,q,r={0} for k<0 and k>n.
Let us use the following notation:
^≥ k_p,q,r := ^k_p,q,r⊕^k+1_p,q,r⊕⋯⊕^n_p,q,r,
^≤ k_p,q,r := ^0⊕^1_p,q,r⊕⋯⊕^k_p,q,r
for 0≤ k≤ n. For example, ^≥0_p,q,r=^≤ n_p,q,r=_p,q,r and ^≥ n_p,q,r=^n_p,q,r.
Consider such conjugation operations as grade involution and reversion. The grade involute of an element U∈_p,q,r is denoted by U and the reversion is denoted by U. These operations satisfy
UV=UV, UV=VU, ∀ U, V∈_p,q,r.
The grade involution defines the even ^(0)_p,q,r and odd ^(1)_p,q,r subspaces:
^(k)_p,q,r = {U∈_p,q,r: U=(-1)^kU}=⊕_j=k2^j_p,q,r, k=0,1.
We can represent any element U∈_p,q,r as a sum
U=⟨ U⟩_(0)+⟨ U⟩_(1), ⟨ U⟩_(0)∈^(0)_p,q,r, ⟨ U⟩_(1)∈^(1)_p,q,r.
We use the angle brackets ⟨·⟩_(l) to denote the operation of projection of multivectors onto the subspaces ^(l)_p,q,r, l=0,1.
For an arbitrary subset ⊆̋_p,q,r, we have
⟨⟩̋_(0) := ∩̋^(0)_p,q,r, ⟨⟩̋_(1) := ∩̋^(1)_p,q,r.
The grade involution and the reversion define four subspaces ^0_p,q,r, ^1_p,q,r, ^2_p,q,r, and ^3_p,q,r (they are called the subspaces of quaternion types 0, 1, 2, and 3 respectively in the papers <cit.>):
^k_p,q,r={U∈_p,q,r: U=(-1)^k U, U=(-1)^k(k-1)/2 U}, k=0, 1, 2, 3.
Note that the Clifford algebra _p,q,r can be represented as a direct sum of the subspaces ^k_p,q,r, k=0, 1, 2, 3, and viewed as ℤ_2×ℤ_2-graded algebra with respect to the commutator and anticommutator <cit.>.
To denote the direct sum of different subspaces, we use the upper multi-index and omit the direct sum sign. For instance, ^(1)24_p,q,r:=^(1)_p,q,r⊕^2_p,q,r⊕^4_p,q,r.
§ CENTRALIZERS AND TWISTED CENTRALIZERS OF THE SUBSPACES OF FIXED GRADES
Consider
the subset ^m_p,q,r of all elements of _p,q,r commuting with all elements of the grade-m subspace for some fixed m:
^m_p,q,r:={X∈_p,q,r: X V = V X, ∀ V∈^m_p,q,r}.
Note that ^m_p,q,r=_p,q,r for m<0 and m>n.
We call the subset ^m_p,q,r the centralizer (see, for example, <cit.>) of the subspace ^m_p,q,r in _p,q,r.
The center of the Clifford algebra _p,q,r is also the centralizer but of the entire Clifford algebra _p,q,r.
We denote the center of the degenerate and non-degenerate Clifford algebra _p,q,r by _p,q,r. It is well known (see, for example, <cit.>) that
_p,q,r=
{[ ^(0)_r⊕^n_p,q,r, ,; ^(0)_r, . ].
Similarly, consider
the set ^m_p,q,r:
^m_p,q,r:={X∈_p,q,r: X V = V X, ∀ V∈^m_p,q,r}.
Note that ^m_p,q,r=_p,q,r for m<0 and m>n.
We call the set ^m_p,q,r the twisted centralizer of the subspace ^m_p,q,r in _p,q,r.
The particular case ^1_p,q,r is considered in the papers <cit.>.
Note that the projections ⟨^m_p,q,r⟩_(0) and ⟨^m_p,q,r⟩_(0) of ^m_p,q,r and ^m_p,q,r respectively onto the even subspace ^(0)_p,q,r (<ref>) coincide by definition:
⟨^m_p,q,r⟩_(0)=⟨^m_p,q,r⟩_(0), ∀ m=0,1,…,n.
In the case m=0, we have
^0_p,q,r=_p,q,r, ^0_p,q,r={X∈_p,q,r: X=X}=^(0)_p,q,r.
In Theorem <ref>, we find explicit forms of the centralizers ^m_p,q,r and the twisted centralizers ^m_p,q,r of the subspaces of fixed grades for an arbitrary m=1,…,n.
To prove Theorem <ref>, let us prove auxiliary Lemmas <ref>, <ref>, and <ref>.
In Lemmas <ref> and <ref>, we use that any non-zero X∈_p,q,r has the following decomposition over a basis:
X=X_1+⋯+X_k, X_i=α_ie_A_i, α_i∈^×, i=1,…,k,
where A_i is an ordered multi-index, A_i≠ A_j for i≠ j, and each X_i is non-zero, i.e. α_i≠0.
For any even m, we have
⟨^m_p,q,r⟩_(1)⊆⟨^m+1_p,q,r⟩_(1).
In the case of even m<0 or m≥ n, we have ^m_p,q,r=^m+1_p,q,r=_p,q,r. For m=0, we have ⟨^0_p,q,r⟩_(1)=⟨^(0)_p,q,r⟩_(1)={0}, where we use (<ref>).
Let us consider the case 0<m<n.
Consider a non-zero element X∈^m_p,q,r∩^(1)_p,q,r, where m is even, and its
decomposition over a basis (<ref>).
For any fixed e_a_1… a_m∈^m_p,q,r, each summand X_i, i=1,…,k, contains at least one such e_x_i that x_i∈{a_1,…, a_m}, because otherwise we have X_i e_a_1… a_m=e_a_1… a_mX_i, so
Xe_a_1… a_m≠ e_a_1… a_mX,
and we get a contradiction.
Therefore, for any e_a_1… a_m+1∈^m+1_p,q,r,
Xe_a_1… a_m+1=±X_1e_a_1…x̌_1… a_m+1e_x_1±⋯±X_ke_a_1…x̌_k… a_m+1e_x_k,
where X_i contains e_x_i, i=1,…,k, the sign depends on the parity of the corresponding permutation, and the checkmarks indicate that the corresponding indices are missing. From (<ref>), we obtain
Xe_a_1… a_m+1=
± e_a_1…x̌_1… a_m+1X_1 e_x_1±⋯± e_a_1…x̌_k… a_m+1X_k e_x_k
=± e_a_1…x̌_1… a_m+1e_x_1 X_1±⋯± e_a_1…x̌_k… a_m+1e_x_kX_k=e_a_1… a_m+1 X,
where all the signs preceding the terms remain the same in (<ref>)–(<ref>),
since X_i∈^m_p,q,r∩^(1)_p,q,r and X_i contains e_x_i, i=1,…,k. This completes the proof.
For any odd m, we have
⟨^m_p,q,r⟩_(1)⊆⟨^m+1_p,q,r⟩_(1).
In the case m<0 or m≥ n, we have ^m_p,q,r=^m+1_p,q,r=_p,q,r.
Let us consider the case 0<m<n.
Consider a non-zero X∈^m_p,q,r∩^(1)_p,q,r, where m is odd, and its decomposition (<ref>).
For any fixed e_a_1… a_m∈^m_p,q,r, each summand X_i, i=1,…,k, contains at least one such e_x_i that x_i∈{a_1,… a_m}, because otherwise X_i e_a_1… a_m=e_a_1… a_mX_i, so
X e_a_1… a_m≠ e_a_1… a_m X,
and we get a contradiction.
Hence, for any e_a_1… a_m+1∈^m+1_p,q,r, we have
X e_a_1… a_m+1=±X_1e_a_1…x̌_1… a_m+1e_x_1±⋯±X_ke_a_1…x̌_k… a_m+1e_x_k,
where X_i contains e_x_i, i=1,…,k, the sign depends on the parity of the corresponding permutation, and the checkmarks indicate that the corresponding indices are missing. From (<ref>), we obtain
Xe_a_1… a_m+1=
± e_a_1…x̌_1… a_m+1X_1 e_x_1±⋯± e_a_1…x̌_k… a_m+1X_k e_x_k
=± e_a_1…x̌_1… a_m+1e_x_1 X_1±⋯± e_a_1…x̌_k… a_m+1e_x_kX_k=e_a_1… a_m+1 X,
where all the signs preceding the terms in (<ref>)–(<ref>) remain the same,
since X_i∈^m_p,q,r∩^(1)_p,q,r
and X_i contains e_x_i, i=1,…,k. This completes the proof.
Note that the more general statement than (<ref>) holds true:
^m_p,q,r⊆^m+1_p,q,r, ,
which follows from Theorem <ref> below and is provided in the formula (<ref>) of Remark <ref>.
Note that the statement (<ref>) can not be generalized in a similar way. For any even m, we have
⟨^m_p,q,r⟩_(0)⊆⟨^m+1_p,q,r⟩_(0), ,
(⟨^m_p,q,r⟩_(0)∖^n_p,q,r)⊆⟨^m+1_p,q,r⟩_(0), ,
which follows from Theorem <ref> below as well.
For any M∈^m_p,q,r, K∈^k_p,q,r, and L∈Λ^n-m_r, we have
(KL)M=
{[ M(KL) ;; M(KL) ].
Suppose m,k=02. We have
(KL)M=(LM)K=(ML)K=M(KL),
where we use that LM∈^n_p,q,r commutes with any even element, LM=ML, and LK=KL, since m and k are even respectively.
If m,k,n=12, then we again have (<ref>), since L is even and LM∈^n_p,q,r⊂_p,q,r is odd.
Consider the case m=12 and k=02.
If n is odd, then L is even. We get (<ref>) again and obtain (KL)M=M(KL), since both K and L are even.
If n is even, then L is odd and
(KL)M=(LM)K=(ML)K=M(KL)=M(KL).
Finally, suppose m,n=02 and k=12. We obtain
(KL)M=(LM)K=(ML)K=M(KL)=M(KL),
since L is even, ML∈^n_p,q,r is even, and it anticommutes with all odd elements, including K.
Note that
Λ^k_r ^m_p,q,r⊆^k+m_p,q,r, k≥ 1; Λ^0_r ^m_p,q,r = ^m_p,q,r.
Moreover, if at least one of k and m is even, then
X V = V X, ∀ X∈Λ^k_r, ∀ V∈^m_p,q,r.
If both k and m are odd, then
X V = V X, ∀ X∈Λ^k_r, ∀ V∈^m_p,q,r.
We use Remark <ref> in the proof of Theorem <ref>.
In Theorem <ref>, we find the centralizers and twisted centralizers for any m=1,…,n in the case r≠ n. The case of the Grassmann algebra _0,0,n is written out separately in Remark <ref> for the sake of brevity in the theorem statement.
Consider the case r≠ n.
* For an arbitrary even m, where n≥ m≥ 2, the centralizer has the form
^m_p,q,r=Λ^≤ n-m-1_r⊕⊕_k=12^m-3^k_p,q,0Λ^≥ n-(m-1)_r⊕⊕_k=02^m-2^k_p,q,0Λ^≥ n-m_r ⊕^n_p,q,r,
and the twisted centralizer is equal to
^m_p,q,r=⟨Λ_r^≤ n-m-1⊕⊕_k=12^m-3^k_p,q,0Λ^≥ n-(m-1)_r⊕⊕_k=02^m-2^k_p,q,0Λ^≥ n-m_r ⊕^n_p,q,r⟩_(0)
⊕⟨⊕_k=02^m-2^k_p,q,0Λ^≥ n-(m-1)_r
⊕⊕_k=12^m-1^k_p,q,0Λ^≥ n-m_r ⟩_(1).
* For an arbitrary odd m, where n≥ m ≥ 1, we have
^m_p,q,r=Λ_r^≤ n-m-1⊕⊕_k=12^m-2^k_p,q,0Λ_r^≥ n-(m-1)⊕⊕_k=02^m-1^k_p,q,0Λ_r^≥ n-m
and
^m_p,q,r = ⟨Λ_r^≤ n-m-1⊕⊕_k=12^m-2^k_p,q,0Λ^≥ n-(m-1)_r ⊕⊕_k=02^m-1^k_p,q,0Λ^≥ n-m_r ⟩_(0)
⊕⟨⊕_k=02^m-3^k_p,q,0Λ^≥ n-(m-1)_r⊕⊕_k=12^m-2^k_p,q,0Λ^≥ n-m_r⊕^n_p,q,r⟩_(1).
Let us prove (<ref>).
Namely, we prove that for any X∈_p,q,r and even m, where n≥ m≥ 2, the condition X e_a_1… a_m = e_a_1… a_m X for any basis element e_a_1… a_m∈^m_p,q,r is equivalent to the condition
X∈Λ^≤ n-m-1_r⊕⊕_k=12^m-3^k_p,q,0Λ^≥ n-(m-1)_r⊕⊕_k=02^m-2^k_p,q,0Λ^≥ n-m_r ⊕^n_p,q,r.
For any fixed a_1,…, a_m, we can always represent X as a sum of 2^m summands:
X=Y + e_a_1Y_a_1 + ⋯ + e_a_mY_a_m + e_a_1 a_2 Y_a_1 a_2 + ⋯ + e_a_1… a_mY_a_1… a_m,
where Y, Y_a_1, …, Y_a_1… a_m∈_p,q,r do not contain e_a_1,…, e_a_m. We get
X e_a_1… a_m=(Y + ⋯ + e_a_1… a_mY_a_1… a_m) e_a_1… a_m
= e_a_1… a_m( ∑^m_k=02 e_a_i_1… a_i_kY_a_i_1… a_i_k
-∑^m-1_k=12 e_a_i_1… a_i_kY_a_i_1… a_i_k),
where a_i_1,…,a_i_k∈{a_1,…,a_m},
a_i_1<⋯<a_i_k, the elements
e_A and Y_A with the multi-indices of zero length are the identity element e and Y respectively, and the minus sign precedes summands with e_a_i_1… a_i_k∈^(1)_p,q,r.
We get that the condition X e_a_1… a_m=e_a_1… a_mX is equivalent to
2e_a_1… a_m(∑^m-1_k=12 e_a_i_1… a_i_kY_a_i_1… a_i_k)=0.
The equation (<ref>) is equivalent to the system of 2^m-1 equations:
(e_a_1)^2 e_a_2… a_mY_a_1=0, …, (e_a_m)^2 e_a_1… a_m-1Y_a_m=0,
(e_a_1)^2 (e_a_2)^2 (e_a_3)^2 e_a_4… a_mY_a_1 a_2 a_3=0, …, (e_a_2)^2⋯(e_a_m)^2e_a_1Y_a_2… a_m=0.
Using (e_a_1)^2 e_a_2… a_mY_a_1=0 (<ref>), we get that if (e_a_1)^2≠0, i.e. a_1∈{1,…, p+q}, then Y_a_1=0, since Y_a_1 does not contain e_a_2,…,e_a_m. On the other hand, if each summand of X either contains only the non-invertible generators, or contains at least 1 invertible generator and at the same time does not contain less than m-1 generators, then the equation (e_a_1)^2 e_a_2… a_mY_a_1=0 is satisfied. Therefore, (e_a_1)^2 e_a_2… a_mY_a_1=0 is satisfied if and only if X has no summands containing at least 1 invertible generator and at the same time not containing m-1 or more of any generators.
Similarly, for any other odd k<m, using (e_a_1)^2 (e_a_2)^2 … (e_a_k)^2 e_a_k+1… a_mY_a_1 a_2 … a_k=0 (<ref>), we get Y_a_1 a_2 … a_k=0 if a_1,…, a_k∈{1,…,p+q}.
Moreover, the equation (e_a_1)^2 (e_a_2)^2 … (e_a_k)^2 e_a_k+1… a_mY_a_1 a_2 … a_k=0 is satisfied if and only if X has no summands containing at least k invertible generators and at the same time not containing m-k or more of any generators for any odd k≤ m.
This implies that for
X∈_p,q,r=Λ_r⊕^1_p,q,0Λ_r⊕⋯⊕^p+q_p,q,0Λ_r,
we finally obtain
X ∈ Λ_r⊕^1_p,q,0Λ^≥ n-(m-1)_r⊕^2_p,q,0Λ^≥ n-m_r⊕^3_p,q,0Λ^≥ n-(m-1)_r
⊕⋯⊕^m-3_p,q,0Λ^≥ n-(m-1)_r⊕^m-2_p,q,0Λ^≥ n-m_r⊕^n_p,q,r,
since for any fixed odd k and even k+1, we have the following condition on d for the subspaces ^k_p,q,0Λ^d_r and ^k+1_p,q,0Λ^d_r respectively: the number of not contained generators should be less than m-k, i.e. n-(k+d)<m-k, so d≥ n-(m-1) for ^k_p,q,0Λ^d_r, and n-(k+1+d)<m-k, thus, d≥ n-m for ^k+1_p,q,0Λ^d_r.
This completes the proof.
Let us prove (<ref>). Namely, let us prove that for any X∈_p,q,r and odd m, where n≥ m≥ 1, the condition Xe_a_1… a_m=e_a_1 … a_mX for any basis element e_a_1 … a_m∈^m_p,q,r is equivalent to the condition
X∈Λ_r^≤ n-m-1⊕⊕_k=12^m-2^k_p,q,0Λ_r^≥ n-(m-1)⊕⊕_k=02^m-1^k_p,q,0Λ_r^≥ n-m.
For any fixed a_1,…,a_m, we can represent X as a sum of 2^m summands (<ref>), where Y,…,Y_a_1… a_m∈_p,q,r do not contain e_a_1,…,e_a_m.
We obtain
Xe_a_1… a_m=(⟨ X⟩_(0)-⟨ X⟩_(1))e_a_1… a_m
= e_a_1… a_m(∑^m-1_k=02 e_a_i_1… a_i_kY_a_i_1… a_i_k
-∑^m_k=12 e_a_i_1… a_i_kY_a_i_1… a_i_k),
where a_i_1,…,a_i_k∈{a_1,…,a_m}, a_i_1<⋯<a_i_k, the elements
e_A and Y_A with the multi-indices of zero length are the identity element e and Y respectively, and the minus sign precedes summands with e_a_i_1… a_i_k∈^(1)_p,q,r.
We get that the equality Xe_a_1… a_m=e_a_1… a_mX is equivalent to the formula
2e_a_1… a_m(∑^m_k=12 e_a_i_1… a_i_kY_a_i_1… a_i_k)=0.
Similar to how it is done for the formula (<ref>) above, from the formula (<ref>), we get that it
is equivalent to the condition that X has no summands containing at least k invertible generators and at the same time not containing m-k or more of any generators for any odd k≤ m. So, for X∈Λ_r⊕^1_p,q,0Λ_r⊕⋯⊕^p+q_p,q,0Λ_r, we get
X ∈ Λ_r⊕^1_p,q,0Λ^≥ n-(m-1)_r⊕^2_p,q,0Λ^≥ n-m_r⊕^3_p,q,0Λ^≥ n-(m-1)_r
⊕⋯⊕^m-2_p,q,0Λ^≥ n-(m-1)_r⊕^m-1_p,q,0Λ^≥ n-m_r,
since, similarly to the proof of (<ref>) above, for any odd k, for ^k_p,q,0Λ^d_r, we have the condition n-(k+d)<m-k, i.e. d≥ n-(m-1), and for any ^k+1_p,q,0Λ^d_r, we get n-(k+1+d)<m-k, thus, d≥ n-m. This completes the proof.
Now we prove (<ref>). Suppose m is even and n≥ m≥ 2. Since ⟨^m_p,q,r⟩_(0)=⟨^m_p,q,r⟩_(0) (<ref>), we only need to prove
⟨^m_p,q,r⟩_(1)=⟨⊕_k=02^m-2^k_p,q,0Λ^≥ n-(m-1)_r
⊕⊕_k=12^m-1^k_p,q,0Λ^≥ n-m_r ⟩_(1).
First, we prove that the right set is a subset of the left one in (<ref>).
We have ^k_p,q,0Λ^n-m_r⊆^m_p,q,r for any odd k, where m-1≥ k≥ 1, and even n by Lemma <ref>.
Let us prove ^k_p,q,0Λ^≥ n-(m-1)_r⊆^m_p,q,r for any even k=0,…,m-2.
Note that n-(m-1)≥ 1; hence, we have Λ^≥ n-(m-1)_r^m_p,q,r⊆^≥ n-(m-1)+m_p,q,r=^n+1_p,q,r={0} and, similarly, ^m_p,q,rΛ^≥ n-(m-1)_r={0} by Remark <ref>. Therefore, for any k=0,…,n,
(^k_p,q,0Λ^≥ n-(m-1)_r) ^m_p,q,r= ^m_p,q,r (^k_p,q,0Λ^≥ n-(m-1)_r)={0}.
Thus,
^k_p,q,0Λ^≥ n-(m-1)_r⊆^m_p,q,r.
Let us prove that the left set is a subset of the right one in (<ref>).
Using ⟨^m_p,q,r⟩_(1)⊆⟨^m+1_p,q,r⟩_(1) for any even m by Lemma <ref> and applying (<ref>) proved above, we get
⟨^m_p,q,r⟩_(1)⊆⟨Λ^≤ n-(m+1)_r⊕⊕_k=02^m^k_p,q,0Λ_r^≥ n-(m+1)⊕⊕_k=12^m-1^k_p,q,0Λ_r^≥ n-m⟩_(1).
Now let us show that the inclusion above implies the inclusion of the left set in the right one in (<ref>).
The projection of ⟨^m_p,q,r⟩_(1) onto the subspace ⟨Λ^≤ n-(m+1)_r⟩_(1) equals zero, since for any odd basis element X∈Λ^≤ n-(m+1)_r there exists such an even basis element V∈^m_p,q,r that XV≠0 and XV=VX. For example, in the case n=r=4 and m=2, for X=e_1, we have V=e_23, and e_1e_23=e_23e_1≠ e_23e_1.
The projection of ⟨^m_p,q,r⟩_(1) onto ⟨^k_p,q,0Λ^n-m_r⟩_(1) equals zero for any even k, m and odd n by Lemma <ref>. The projection of ⟨^m_p,q,r⟩_(1) onto ⟨^k_p,q,0Λ^n-m-1_r⟩_(1) equals zero for any even k, where m≥ k>0, since for any basis elements K=e_a_1… a_k∈^k_p,q,0⊂^(0)_p,q,r and L∈Λ^n-m-1_r⊆^(1)_p,q,r, there exists such an even grade-m element M∈ e_a_1… a_k^m-k_p,q,r that LM≠0, LM=ML, and KM=MK, so we get (KL)M=M(KL)≠ M(KL), where we use Remark <ref>. For example, if n=6, p=k=2, and r=m=4, for K=e_12∈^2_2,0,0 and L=e_3∈Λ^1_4, there exists M=e_1245∈ e_12^2_2,0,4, such that (e_12e_3)e_1245=e_1245(e_12e_3)≠ e_1245(e_12e_3). Finally, the projection of ^m_p,q,r onto ^m_p,q,0Λ^≥ n-(m-1)_r={0} equals zero as well. Thus, we obtain (<ref>), and the proof is completed.
Finally, let us prove (<ref>). Suppose m is odd and n≥ m≥ 1. We have ⟨^m_p,q,r⟩_(0)=⟨^m_p,q,r⟩_(0) (<ref>), so we only need to prove
⟨^m_p,q,r⟩_(1)=⟨⊕_k=02^m-3^k_p,q,0Λ^≥ n-(m-1)_r⊕⊕_k=12^m-2^k_p,q,0Λ^≥ n-m_r⊕^n_p,q,r⟩_(1).
We obtain that the right set is a subset of the left one in (<ref>), using ^m_p,q,rΛ^≥ n-(m-1)_r={0}, Lemma <ref>, and ⟨^n_p,q,r⟩_(1)⊂_p,q,r.
Let us prove that the left set is a subset of the right one in (<ref>). Using ⟨^m_p,q,r⟩_(1)⊆⟨^m+1_p,q,r⟩_(1) by Lemma <ref> and applying (<ref>) proved above, we get
⟨^m_p,q,r⟩_(1) ⊆ ⟨Λ^≤ n-m-2_r⊕⊕_k=02^m-1^k_p,q,0Λ^≥ n-m-1_r
⊕⊕_k=12^m-2^k_p,q,0Λ^≥ n-m_r ⊕^n_p,q,r⟩_(1).
The projection of ⟨^m_p,q,r⟩_(1) onto the subspace ⟨Λ^≤ n-m-2_r⟩_(1) equals zero because for any odd X∈Λ^≤ n-m-2_r there exists such an odd V∈^m_p,q,r that XV≠0 and XV=-VX. For example, if n=r=4 and m=1, for X=e_1, we have V=e_2, and e_1e_2=-e_2e_1. The projection of ⟨^m_p,q,r⟩_(1) onto ^k_p,q,0Λ^n-m_r for any even k and odd m, n is zero by Lemma <ref>. The projection of ⟨^m_p,q,r⟩_(1) onto ⟨^k_p,q,0Λ^n-m-1_r⟩_(1) for any even k≤ m-1 equals zero because for any basis elements K=e_a_1… a_k∈^k_p,q,0⊆^(0)_p,q,r and L∈Λ^n-m-1_r⊆^(1)_p,q,r, there exists such an odd grade-m element M∈ e_a_1… a_k^m-k_p,q,r that LM≠0, LM=ML, and KM=MK, therefore, (KL)M=KML=M(KL)≠ M(KL). For example, if n=5, p=k=2, and r=m=3, for K=e_12∈^2_2,0,0 and L=e_3∈Λ^1_3, we can take M=e_124∈ e_12^1_2,0,3 and get (e_12e_3)e_124=-e_124(e_12e_3). Finally, ^m-1_p,q,0Λ^≥ n-(m-1)_r=^n_p,q,r. Thus, we obtain (<ref>), and the proof is completed.
Note that Theorem <ref> implies
^m_p,q,r⊆^m+2_p,q,r, ^m_p,q,r⊆^m+2_p,q,r, m=1,…, n-2;
^m_p,q,r⊆^m+1_p,q,r, ^m_p,q,r⊆^m+1_p,q,r, ;
^m_p,q,r⊆^m+2_p,q,r, .
Using (<ref>)–(<ref>), we get
^m_p,q,r⊆^4_p,q,r, ^m_p,q,r⊆^4_p,q,r, m=1,2,3.
If r≤ n-(m+1), then
^m_p,q,r=^1_p,q,r, ^m_p,q,r=^1_p,q,r, ;
^m_p,q,r=^2_p,q,r, ^m_p,q,r=^2_p,q,r, , m≠0.
We use these relations to prove Theorems <ref> and <ref>.
§ PARTICULAR CASES OF CENTRALIZERS AND TWISTED CENTRALIZERS
In this section, we consider the centralizers and twisted centralizers in the particular cases that are important for applications. In Remarks <ref> and <ref> below, we explicitly write out ^m_p,q,r and ^m_p,q,r, m=0,1,…,n, in the cases of the non-degenerate Clifford algebra _p,q,0 and the Grassmann algebra _0,0,n respectively. Note that in these special cases, the centralizers and twisted centralizers have a much simpler form than in the general case of arbitrary _p,q,r (Theorem <ref>).
In the particular case of the non-degenerate algebra _p,q,0, we get from Theorem <ref>
^m_p,q,0=
{[ _p,q,0, m=0; m=n m,n=12;; ^(0)_p,q,0, m=n m,n=02;; ^0n_p,q,0, m≠ 0,n m=02; m≠ n m,n=12;; ^0, m=12 n=02; ].
and
^m_p,q,0={[ _p,q,0, m=n, m,n=02;; ^(0)_p,q,0, m=0; m=n m,n=12;; ^0n_p,q,0, m≠ 0,n m,n=02;; ^0, m≠ n m=12; m≠ 0, m=02, n=12. ].
In the particular case of the Grassmann algebra _0,0,n=Λ_n, we obtain from Theorem <ref>
^0_0,0,n=Λ_n, ^0_0,0,n=Λ^(0)_n;
^m_0,0,n=Λ_n, ^m_0,0,n=Λ^(0)_n ⊕⟨Λ^≥ n-m+1_n⟩_(1), m=02, m≠0;
^m_0,0,n=Λ^(0)_n⊕⟨Λ^≥ n-m+1_n⟩_(1), ^m_0,0,n=Λ_n, m=12.
In Remark <ref> below, we explicitly write out the particular cases of Theorem <ref> and Remark <ref> in the case of small k≤ 4. We use these centralizers and twisted centralizers in Theorems <ref> and <ref>.
Note that some of these centralizers and twisted centralizers are considered, for instance, in the papers <cit.>.
The cases ^2_p,q,r (<ref>) and ^1_p,q,r (<ref>) are proved in detail, for example, in <cit.>. The other cases are presented for the first time.
We have:
^1_p,q,r=_p,q,r={[ Λ^(0)_r⊕^n_p,q,r, ,; Λ^(0)_r, , ].
^2_p,q,r=
{[ Λ_r⊕^n_p,q,r, r≠ n,; Λ_r, r=n, ].
^3_p,q,r={[ Λ^(0)_r⊕Λ^n-2_r⊕^1_p,q,0(Λ^n-3_r⊕Λ^n-2_r); ⊕^2_p,q,0Λ^n-3_r⊕^n_p,q,r, ,; ; Λ^(0)_r⊕Λ^n-1_r⊕^1_p,q,0Λ^≥ n-2_r⊕^2_p,q,0Λ^n-2_r, , ].
^4_p,q,r={[ Λ_r⊕^1_p,q,0(Λ^n-3_r⊕Λ^n-2_r)⊕^2_p,q,0(Λ^n-4_r⊕Λ^n-3_r)⊕^n_p,q,r, r≠ n,; Λ_r, r=n, ].
and:
^1_p,q,r=Λ_r,
^2_p,q,r=
{[ Λ^(0)_r⊕Λ^n_r⊕^1_p,q,0Λ^n-1_r, ,; Λ^(0)_r⊕Λ^n-1_r⊕^1_p,q,0Λ^n-2_r⊕^n_p,q,r, , r≠ n,; Λ^(0)_r⊕Λ^n-1_r, , r=n, ].
^3_p,q,r=Λ_r⊕^1_p,q,0Λ^≥ n-2_r⊕^2_p,q,0Λ^≥ n-3_r,
^4_p,q,r=
{[ Λ^(0)_r ⊕Λ^n-2_r⊕Λ^n_r⊕^1_p,q,0Λ^≥ n-3_r; ⊕^2_p,q,0Λ^≥ n-3_r⊕^3_p,q,0Λ^n-3_r, ,; ; Λ^(0)_r ⊕Λ^n-3_r⊕Λ^n-1_r⊕^3_p,q,0Λ^n-4_r; ⊕^2_p,q,0(Λ^n-4_r⊕Λ^n-3_r)⊕^n_p,q,r; ⊕^1_p,q,0(Λ^n-4_r ⊕Λ^n-3_r ⊕Λ^n-2_r), , r≠ n,; ; Λ^(0)_r ⊕Λ^n-3_r⊕Λ^n-1_r, , r=n. ].
In Remark <ref>, we consider how the centralizers and the twisted centralizers are related to the kernels of the adjoint and twisted adjoint representations.
Note that
^1×_p,q,r=(), ^1×_p,q,r=()
and
()⊆^m×_p,q,r, ()=Λ^(0)×_r⊆^m×_p,q,r, m=0,…,n,
where (), (), and () are the kernels of the adjoint representation and the twisted adjoint representations and respectively.
The adjoint representation :^×_p,q,r→(_p,q,r) acts on the group of all invertible elements as T↦_T, where
_T(U)=TU T^-1, U∈_p,q,r, T∈^×_p,q,r.
The twisted adjoint representation has been introduced in a particular case by Atiyah, Bott, and Shapiro in <cit.>. The representation :^×_p,q,r→(_p,q,r) acts on ^×_p,q,r as T↦_T with
_T(U)=TU T^-1, U∈_p,q,r, T∈^×_p,q,r.
The representation :^×_p,q,r→(_p,q,r) acts on ^×_p,q,r as T↦_T with
_T(U)=T⟨ U⟩_(0) T^-1+T⟨ U⟩_(1) T^-1, ∀ U∈_p,q,r, T∈^×_p,q,r.
See the details about , , , and their kernels, for example, in <cit.>.
§ CENTRALIZERS AND TWISTED CENTRALIZERS OF THE SUBSPACES DETERMINED BY THE GRADE INVOLUTION AND THE REVERSION
This section finds explicit forms of the centralizers and twisted centralizers of the subspaces ^m_p,q,r (<ref>), m=0,1,2,3, determined by the grade involution and the reversion and their direct sums. In particular, we consider the centralizers and twisted centralizers of the even ^(0)_p,q,r=^02_p,q,r and odd ^(1)_p,q,r=^13_p,q,r subspaces.
Let us consider the centralizers ^m_p,q,r and twisted centralizers ^m_p,q,r of the subspaces ^m_p,q,r (<ref>), m=0,1,2,3, in _p,q,r:
^m_p,q,r := {X∈_p,q,r: X V = V X, ∀ V∈^m_p,q,r}, m=0,1,2,3,
^m_p,q,r := {X∈_p,q,r: X V = V X, ∀ V∈^m_p,q,r}, m=0,1,2,3.
In Theorem <ref>, we prove that ^m_p,q,r and ^m_p,q,r coincide with some of the centralizers ^m_p,q,r and the twisted centralizers ^m_p,q,r of the subspaces of fixed grades, which are considered in Sections <ref> and <ref>.
We have
^m_p,q,r=^m_p,q,r, ^m_p,q,r=^m_p,q,r, m=1,2,3;
^0_p,q,r=^4_p,q,r, ^0_p,q,r=⟨^4_p,q,r⟩_(0).
The centralizers ^m_p,q,r and twisted centralizers ^m_p,q,r, m=1,2,3,4, are written out explicitly in Remark <ref> for the readers' convenience. In the formula (<ref>), we have
⟨^4_p,q,r⟩_(0)={[ Λ^(0)_r⊕^1_p,q,0Λ^n-2_r⊕^2_p,q,0Λ^n-3_r, ; Λ^(0)_r⊕^1_p,q,0Λ^n-3_r⊕^2_p,q,0Λ^n-4_r⊕^n_p,q,r, . ].
The inclusions ^m_p,q,r⊆^m_p,q,r, ^m_p,q,r⊆^m_p,q,r, m=1,2,3, and ^0_p,q,r⊆^4_p,q,r follow from ^k_p,q,r⊆^k_p,q,r, k=1,2,3, and ^4_p,q,r⊂^0_p,q,r respectively. We get ^0_p,q,r⊆^4_p,q,r∩^0_p,q,r=^4_p,q,r∩^(0)_p,q,r, using ^04_p,q,r⊆^0_p,q,r and ^0_p,q,r=^(0)_p,q,r (<ref>).
Let us prove ^m_p,q,r⊆^m_p,q,r and ^m_p,q,r⊆^m_p,q,r, m=1,2,3. Any basis element of ^k_p,q,r, k=1,2,3, can be represented as a product of one basis element of ^k_p,q,r and basis elements of ^4_p,q,r. Since ^m_p,q,r⊆^4_p,q,r and ^m_p,q,r⊆^4_p,q,r by the statement (<ref>) of Remark <ref>, we get ^m_p,q,r⊆^m_p,q,r and ^m_p,q,r⊆^m_p,q,r.
We obtain ^4_p,q,r⊆^0_p,q,r and ^4_p,q,r∩^(0)_p,q,r⊆^0_p,q,r because ^4_p,q,r⊆^0_p,q,r and ^4_p,q,r∩^(0)_p,q,r⊆^(0)_p,q,r=^0_p,q,r (<ref>) respectively and any basis element of ^0_p,q,r∖^0 can be represented as a product of basis elements of ^4_p,q,r.
Let us denote by ^km_p,q,r and ^km_p,q,r, k,m=0,1,2,3, the centralizers and the twisted centralizers respectively of the direct sums of the subspaces ^m_p,q,r (<ref>) in _p,q,r:
^km_p,q,r := ^k_p,q,r∩^m_p,q,r={X∈_p,q,r: X V = V X, ∀ V∈^km_p,q,r},
^km_p,q,r := ^k_p,q,r∩^m_p,q,r={X∈_p,q,r: X V = V X, ∀ V∈^km_p,q,r}.
Note that ^02_p,q,r, ^02_p,q,r, ^13_p,q,r, and ^13_p,q,r are the centralizers and the twisted centralizers of the even ^(0)_p,q,r and odd ^(1)_p,q,r subspaces respectively.
In Theorem <ref>, we find explicit forms of ^km_p,q,r and ^km_p,q,r, k,m=0,1,2,3.
We have
^01_p,q,r=^12_p,q,r=^13_p,q,r=_p,q,r, ^23_p,q,r=^2_p,q,r∩^3_p,q,r,
^02_p,q,r=^2_p,q,r, ^03_p,q,r=^3_p,q,r,
^12_p,q,r=^1_p,q,r∩^2_p,q,r, ^23_p,q,r=^2_p,q,r∩^3_p,q,r, ^13_p,q,r=^1_p,q,r,
^01_p,q,r=⟨^1_p,q,r⟩_(0), ^02_p,q,r=⟨^2_p,q,r⟩_(0), ^03_p,q,r=⟨^3_p,q,r⟩_(0).
The centralizers ^2_p,q,r, ^3_p,q,r, _p,q,r=^1_p,q,r and the twisted centralizer ^1_p,q,r are written out explicitly in Remark <ref>.
In the formulas (<ref>)–(<ref>), we have
^2_p,q,r∩^3_p,q,r=
{[ Λ^(0)_r⊕Λ^n-2_r⊕^n_p,q,r, ;; Λ^(0)_r⊕Λ^n-1_r⊕^1_p,q,0Λ^n-1_r⊕^2_p,q,0Λ^n-2_r, ; ].
^1_p,q,r∩^2_p,q,r =
{[ Λ^(0)_r⊕Λ^n_r, ;; Λ^(0)_r⊕Λ^n-1_r, ; ].
^2_p,q,r∩^3_p,q,r={[ Λ^(0)_r⊕Λ^n_r⊕^1_p,q,0Λ^n-1_r, ;; Λ^(0)_r⊕Λ^n-1_r⊕^1_p,q,0Λ^≥ n-2_r⊕^2_p,q,0Λ^n-2_r, ; ].
⟨^1_p,q,r⟩_(0)=Λ^(0)_r, ⟨^2_p,q,r⟩_(0) =
{[ Λ^(0)_r, ; ;; Λ^(0)_r⊕^n_p,q,r, ; ].
⟨^3_p,q,r⟩_(0) =
{[ Λ^(0)_r⊕^1_p,q,0Λ^n-2_r⊕^2_p,q,0Λ^n-3_r, ;; Λ^(0)_r⊕^1_p,q,0Λ^n-1_r⊕^2_p,q,0Λ^n-2_r, . ].
First, let us prove (<ref>) and (<ref>).
We get
^23_p,q,r=^2_p,q,r∩^3_p,q,r=^2_p,q,r∩^3_p,q,r
by Theorem <ref>.
For k=1,2,3, we obtain
^0k_p,q,r=^0_p,q,r∩^k_p,q,r=^4_p,q,r∩^k_p,q,r=^k_p,q,r,
using ^k_p,q,r=^k_p,q,r, ^0_p,q,r=^4_p,q,r by Theorem <ref> and ^k_p,q,r⊆^4_p,q,r by Remark <ref>. Since ^1_p,q,r=_p,q,r by Remark <ref>, we get ^01_p,q,r=_p,q,r. Similarly, for l=2,3, we obtain
^1l_p,q,r = ^1_p,q,r∩^l_p,q,r=^1_p,q,r∩^l_p,q,r=_p,q,r∩^l_p,q,r=_p,q,r,
where we use _p,q,r⊆^2_p,q,r and _p,q,r⊆^3_p,q,r by the formula (<ref>) of Remark <ref>.
Let us prove (<ref>). We get
^12_p,q,r=^1_p,q,r∩^2_p,q,r=^1_p,q,r∩^2_p,q,r, ^23_p,q,r=^2_p,q,r∩^3_p,q,r=^2_p,q,r∩^3_p,q,r,
and ^13_p,q,r=^1_p,q,r∩^3_p,q,r=^1_p,q,r∩^3_p,q,r=^1_p,q,r, using Theorem <ref> and ^1_p,q,r⊆^3_p,q,r by Remark <ref>.
Now we prove (<ref>). For k=1,2,3, we get
^0k_p,q,r = ^0_p,q,r∩^k_p,q,r=^4_p,q,r∩^(0)_p,q,r∩^k_p,q,r
= ^k_p,q,r∩^(0)_p,q,r=^k_p,q,r∩^(0)_p,q,r=⟨^k_p,q,r⟩_(0),
using Theorem <ref>, ^k_p,q,r⊆^4_p,q,r by Remark <ref>, and (<ref>).
The equalities for the centralizer ^02_p,q,r and the twisted centralizer ^02_p,q,r of the even subspace presented in the formulas (<ref>) and (<ref>) are proved in Lemma 3.2 <cit.> in the case of the degenerate and non-degenerate algebras _p,q,r. In the non-degenerate case _p,q,0, the set ^02_p,q,0 is considered, for example, in <cit.> and the set ^02_p,q,0 is considered in <cit.>. The other equalities in the formulas
(<ref>)–(<ref>) are presented for the first time.
§ CONCLUSIONS
In this work, we consider the centralizers and twisted centralizers in degenerate and non-degenerate Clifford algebras _p,q,r.
In Theorems <ref>, <ref>, and <ref>, we find an explicit form of the centralizers
and the twisted centralizers
^k_p,q,r, ^m_p,q,r, ^ml_p,q,r, ^k_p,q,r, ^m_p,q,r, ^ml_p,q,r
of the subspaces of fixed grades ^k_p,q,r, k=0,1,…,n, the subspaces ^m_p,q,r, m=0,1,2,3, determined by the grade involution and the reversion, and their direct sums ^ml_p,q,r, m,l=0,1,2,3.
In particular, we consider the centralizers and twisted centralizers of the even ^(0)_p,q,r and odd ^(1)_p,q,r subspaces.
The relations between ^k_p,q,r and ^k_p,q,r for different k are considered in Remark <ref>.
We also consider the relation between their projections ⟨^k_p,q,r⟩_(1) and ⟨^k_p,q,r⟩_(1) onto the odd subspace ^(1)_p,q,r in Lemmas <ref>, <ref> and Remark <ref>.
In the particular cases of the non-degenerate Clifford algebras _p,q,0 and the Grassmann algebras _0,0,n, the considered centralizers and the twisted centralizers have a simpler form than in the general case of arbitrary _p,q,r (see Remarks <ref> and <ref> respectively).
In the particular case of small k, the centralizers ^k_p,q,r and the twisted centralizers ^k_p,q,r have simple form as well and are written out in Remark <ref> for k≤ 4.
In the further research, we are going to use the explicit forms of the centralizers and the twisted centralizers presented in Theorems <ref>, <ref>, and <ref> to define and study several families of Lie
groups in _p,q,r. These groups preserve the subspaces ^m_p,q,r and their direct sums under the adjoint and twisted adjoint representations. These Lie groups can be considered as generalizations of Clifford and Lipschitz groups and are important for the theory of spin groups.
We hope that the explicit forms of centralizers and twisted centralizers can be useful for applications of Clifford algebras in physics <cit.>, computer science, in particular, for neural networks and machine learning <cit.>, image processing <cit.>, and in other areas.
§ ACKNOWLEDGEMENTS
The results of this paper were reported at the 13th International Conference on Clifford Algebras and Their Applications in Mathematical Physics, Holon, Israel, June 2023. The authors are grateful to the organizers and the participants of this conference for fruitful discussions.
The publication was prepared within the framework of the Academic Fund Program at HSE University (grant 24-00-001 Clifford algebras and applications).
Data availability Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Declarations
Conflict of interest The authors declare that they have no conflict of interest.
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|
http://arxiv.org/abs/2404.16492v1 | 20240425103224 | On the topology of concurrent systems | [
"Catarina Faustino",
"Thomas Kahl",
"Rodrigo Lopes"
] | cs.FL | [
"cs.FL",
"cs.LO",
"math.AT",
"55U10, 68Q85, 05E45"
] |
55U10, 68Q85, 05E45
On the topology of concurrent systems
Mathieu Isoard
=====================================
Higher-dimensional automata, i.e., pointed labeled precubical sets, are a powerful combinatorial-topological model for concurrent systems. In this paper, we show that for every (nonempty) connected polyhedron there exists a shared-variable system such that the higher-dimensional automaton modeling the state space of the system has the homotopy type of the polyhedron.
§ INTRODUCTION
As amply demonstrated in the literature, concepts and methods from algebraic topology can be profitably employed in concurrency theory, the field of computer science that studies systems of simultaneously executing processes, see, e.g., <cit.>. Several topological models of concurrency have been introduced by various authors, e.g., <cit.>. A particularly important combinatorial-topological model of concurrency is given by higher-dimensional automata <cit.>, which go back to <cit.>. It has been shown in <cit.> that higher-dimensional automata are more expressive than the principal traditional models of concurrency.
A higher-dimensional automaton (HDA) is a pointed precubical set (cubical set without degeneracy maps) with edge labeling such that opposite edges of cubes have the same label. The vertices of an HDA represent the states of a concurrent system, with the base vertex corresponding to the initial state. The labeled edges model the transitions of the system, and two- and higher-dimensional cubes express the independence of transitions: an n-cube in an HDA indicates that the n transitions starting from its origin are independent in the sense that they can occur in any order or even simultaneously without any observable difference.
A standard procedure for constructing an HDA model of a concurrent system is to first construct a transition system and then fill in empty squares and higher-dimensional cubes, see, e.g., <cit.>. To make this more precise, consider the example of Peterson's algorithm, a protocol designed to give two processes fair and mutually exclusive access to a shared resource <cit.>. Peterson's algorithm is based on three shared variables—namely, a variable t whose possible values are the process IDs, say 0 and 1, and two boolean variables b_0 and b_1. Process i executes the following protocol with four local states and four transitions:
* Set b_i to 1 to indicate the intention to enter the “critical section".
* Set t to 1-i to give priority to the other process.
* Wait until b_1-i = 0 or t = i, and then enter the critical section.
* Leave the critical section setting b_i to 0, and repeat the procedure from the beginning.
To start, all variables are set to 0. A global state of such a shared-variable system is a tuple whose components are local states of the processes and values of the variables. The transition system associated with the shared-variable system is a labeled directed graph whose vertices correspond to the global states that are actually visited during some execution of the system, and whose edges model the transitions between these global states.
The transitions starting from a given global state correspond to the actions that are enabled in that state. These actions are specified in the edge labels, indexed by the respective process IDs. The HDA model of the system is then constructed from the transition system as a kind of coskeleton, i.e., by suitably filling in empty cubes of dimensions ≥ 2, see Figure <ref> for the case of Peterson's algorithm.
It turns out that the topological analysis of HDAs provides information that is relevant from the point of view of computer science. Indeed, two executions of a concurrent system can be considered equivalent if and only if they can be modeled as directed paths that are homotopic in a directed sense, see, e.g., <cit.>. Additionally, the homology of an HDA model of a concurrent system contains global information about the independence of processes and components of the system <cit.>. Further connections between algebraic topology and concurrency theory are developed in <cit.>.
It turns out that the topological analysis of HDAs provides information that is relevant from the point of view of computer science. Indeed, two executions of a concurrent system can be considered equivalent if and only if they are modeled by directed paths that are homotopic in a directed sense, see, e.g., <cit.>. Additionally, the homology of an HDA contains global information about the independence of processes and components of the modeled concurrent system <cit.>. Further connections between algebraic topology and concurrency theory are developed in <cit.>.
The purpose of this paper is to show that the topology of an HDA model of a concurrent system can be arbitrarily complex. More precisely, we show that for every (nonempty) connected polyhedron there exists a shared-variable system whose HDA model has the same homotopy type as the polyhedron. This is similar in spirit to <cit.>, where it is shown that for every connected polyhedron there exists a PV-program (a particular kind of shared-variable system) whose execution space contains a connected component with the same homotopy type as the polyhedron. This paper is also related to <cit.>, where in particular it is shown that every polyhedron admits a cubical local partial order. In fact, the first step in the proof of our result is to show that the cubical barycentric subdivision of a simplicial complex can be constructed as a precubical set. This actually strengthens <cit.> because it shows that no further subdivision of the cubical barycentric subdivision is needed to equip a polyhedron with a cubical local partial order.
The paper is organized as follows. The precubical set corresponding to the cubical barycentric subdivision of a simplicial complex is constructed in Section <ref>. In Section <ref>, we turn this precubical set into an HDA, which we show to be an HDA model of the transition system given by its 1-skeleton. In the next section, we show that one can replace this HDA by a homotopy equivalent accessible one, i.e., an HDA where all states are reachable by a directed path from the initial state. In the last section, we then show that this accessible HDA is isomorphic to the HDA model of a shared-variable system.
§ THE SIMPLICIAL COMPLEX AND ITS CUBICAL BARYCENTRIC SUBDIVISION
Throughout this paper, we consider a connected abstract simplicial complex K with vertices 1, …, N and the associated polyhedron |K|, which we view as a subspace of the standard (N-1)-simplex Δ^N-1⊆^N. More precisely, we define |K| to be the subspace ⋃_σ∈ K |σ| ⊆Δ^N-1, where for a simplex σ∈ K, |σ| is the geometric simplex in ^N with vertices e_i = (0, …, 0, i1, 0, … 0), i ∈σ. In this section, we construct the cubical barycentric subdivision of K as a precubical set.
§.§ Precubical sets
Let us briefly recall some basic concepts about precubical sets. A precubical set is a graded set X = (X_n)_n ≥ 0 with face maps d^k_i X_n → X_n-1 (n>0, k∈{0,1}, i ∈{1, …, n}) satisfying the relations d^k_i∘ d^l_j= d^l_j-1∘ d^k_i (k,l ∈{0,1}, i<j). If x∈ X_n, we say that x is of degree or dimension n. The elements of degree n are called the n-cubes of X. The elements of degree 0 are also called the vertices of X, and the 1-cubes are also called the edges of X. Precubical sets form a category in which morphisms are morphisms of graded sets that are compatible with the face maps. A precubical subset of a precubical set X is a graded subset of X that is stable under the face maps. The tensor product of two precubical sets X and Y is the precubical set X⊗ Y given by
(X⊗ Y)_n = ∐_p+q = n X_p× Y_q
and
d_i^k(x,y) = {[ (d_i^kx,y), 1≤ i ≤ p,; (x,d_i-p^ky), p < i ≤ n, ]. (x,y) ∈ X_p× Y_q.
The geometric realization of a precubical set X is the quotient space
|X|=(∐ _n ≥ 0 X_n × [0,1]^n)/∼
where the sets X_n are given the discrete topology and the equivalence relation is generated by
(d^k_ix,u) ∼ (x,δ_i^k(u)), x ∈ X_n+1, u∈ [0,1]^n, i ∈{1, …, n+1}, k ∈{0,1}.
Here the map δ^k_i [0,1]^n → [0,1]^n+1 is defined by
δ^k_i(u_1, …, u_n) = (u_1, …, u_i-1,k,u_i, …, u_n).
The geometric realization is functorial. For a morphism of precubical sets f X → Y, the continuous map |f| |X| → |Y| is given by |f|([x,u])= [f(x),u]. The geometric realization of a precubical set is a CW complex. The nskeleton of |X| is the geometric realization of the precubical subset X_≤ n of X defined by (X_≤ n)_i = X_i for i ≤ n and (X_≤ n)_i = ∅ for i > n.
§.§ The precubical set P
The cubical barycentric subdivision of K is the precubical set P where the elements of P_n are pairs (τ, σ) of simplexes of K such that τ is a face of σ and σ∖τ has n elements, see Figure <ref> for a picture.
Considering the natural order on the set of vertices of K, the face maps of P are defined as follows: if σ∖τ = {w_1 < ⋯ < w_n}, we set
d^0_i(τ, σ) = (τ, σ∖{w_i} )
and
d^1_i(τ, σ) = (τ∪{w_i}, σ).
One easily checks that P is indeed a precubical set. The remainder of this section is devoted to the proof that |P|≈ |K|.
§.§ The map
Let (τ, σ) ∈ P_n, and suppose that
τ = {v_1 < ⋯ < v_r} σ∖τ = {w_1 < ⋯ < w_n}.
We decompose the standard n-cube [0,1]^n as the union of the n-simplexes
Δ_θ = {(t_1, …, t_n) ∈ [0,1]^n| t_θ(1)≥⋯≥ t_θ(n)}, θ∈ S_n
and define a continuous map f_τ, σ [0,1]^n → |K| by setting for an element (t_1, … , t_n) ∈Δ_θ,
f_τ, σ(t_1, … , t_n)
= 1 -t_θ(1)r∑_i=1^r e_v_i + t_θ(1) -t_θ(2)r+1(∑_i=1^r e_v_i + e_w_θ(1)) + ⋯
+ t_θ(n-1) -t_θ(n)r+n-1(∑_i=1^r e_v_i + ∑_i=1^n-1 e_w_θ(i)) + t_θ(n)r+n(∑_i=1^r e_v_i + ∑_i=1^n e_w_θ(i))
(see Figure <ref>). If σ = τ, this formula is to be interpreted in such a way that f_τ, σ(()) = 1/r∑_i=1^r e_v_i. Note that f_τ, σ(t_1, … , t_n) is a convex combination of barycenters of faces of |σ| and hence itself an element of |σ|. By the following fact, the proof of which is left to the reader, f_τ, σ is well defined:
Let θ, ψ∈ S_n and (t_1, …, t_n) ∈Δ_θ∩Δ_ψ. Then t_θ(i) = t_ψ(i) for all i ∈{1, …, n}. Moreover, if 1 ≤ i_1 < … < i_k = n are indices such that
t_θ(1) = … = t_θ(i_1) > t_θ(i_1+1) = … > t_θ(i_k-1+1) = … = t_θ(n),
then {θ(1), … , θ(i_s)} = {ψ(1), …, ψ(i_s)} for all s ∈{1, …, k}.
The next lemma shows that a well-defined continuous map f |P| → |K| is given by
f([(τ, σ),(t_1, …, t_n)]) = f_τ, σ(t_1, …, t_n), (τ, σ) ∈ P_n, (t_1, …, t_n) ∈ [0,1]^n.
Let (τ, σ) ∈ P_n (n ≥ 1), and suppose that τ = {v_1 < ⋯ < v_r} and σ∖τ = {w_1 < ⋯ < w_n}. Then
f_τ, σ∖{w_i} = f_τ, σ∘δ^0_i [0,1]^n-1→ |K| and f_τ∪{w_i}, σ = f_τ, σ∘δ^1_i [0,1]^n-1→ |K|.
We prove only the first equality and leave the similar proof of the second to the reader. Let us first note that
( σ∖{ w_i } ) ∖τ = { w_1 < … < w_i-1 < w_i+1 < … < w_n}
= {w̅_1 < … < w̅_i-1 < w̅_i < … < w̅_n-1}
with
w̅_j =
w_j, j<i,
w_j+1, j≥ i.
Let θ∈ S_n-1 and (t_1, …, t_n-1) ∈Δ_θ⊆ [0,1]^n-1. Then t_θ(1)≥…≥ t_θ(n-1), and so defining (x_1, …, x_n) = (t_1, …, t_i-1, 0, t_i, …, t_n-1), we will have that (x_1, …, x_n) ∈Δ_ψ⊆ [0,1]^n, with ψ∈ S_n defined by
ψ (j) =
θ(j), j<n, θ(j)<i,
θ(j) + 1, j<n, θ(j)≥ i,
i, j=n.
Indeed,
x_ψ (j) =
x_θ(j) = t_θ(j), j<n, θ(j)<i,
x_θ(j) + 1 = t_θ(j), j<n, θ(j)≥ i,
x_i = 0, j=n
=
t_θ(j), j<n,
0, j=n.
So x_ψ(1)≥…≥ x_ψ(n), i.e., (x_1, …, x_n) ∈Δ_ψ. Note also that
w_ψ (j) =
w_θ(j) = w̅_θ(j), j<n, θ(j)<i,
w_θ(j) + 1 = w̅_θ(j), j<n, θ(j)≥ i,
w_i, j=n
=
w̅_θ(j), j<n,
w_i, j=n.
We finally have that
f_τ, σ∘δ_i^0 (t_1 …, t_n-1) = f_τ, σ (t_1 …, t_i-1, 0, t_i, … t_n-1) = f_τ, σ (x_1, …, x_n)
= 1-x_ψ (1)r∑_l=1^r e_v_l + x_ψ (1) - x_ψ(2)r+1 (∑_l=1^r e_v_l + e_w_ψ(1))+ …
+ x_ψ (n-1) - x_ψ(n)r+n-1 (∑_l=1^r e_v_l + ∑_l=1^n-1 e_w_ψ(l)) + x_ψ (n)r+n (∑_l=1^r e_v_l + ∑_l=1^n e_w_ψ(l))
= 1-t_θ (1)r∑_l=1^r e_v_l + t_θ (1) - t_θ(2)r+1 (∑_l=1^r e_v_l + e_w̅_θ(1)) + …
+ t_θ (n-1) - 0r+n-1 (∑_l=1^r e_v_l + ∑_l=1^n-1 e_w̅_θ(l)) + 0r+n (∑_l=1^r e_v_l + ∑_l=1^n-1 e_w̅_θ(l) + e_w_i)
= f_τ, σ∖{ w_i } (t_1, …, t_n-1).
We can now prove the main result of this section:
The map f |P| → |K| is a homeomorphism.
In order to define a map g |K| → |P|, consider an element x∈ |K|. Let σ = {u_1 < ⋯ < u_n} be the unique simplex of K such that x = ∑_i = 1^n s_ie_u_i for (unique) numbers s_i > 0 such that ∑_i = 1^n s_i = 1. Choose a permutation α∈ S_n such that s_α(1)≥⋯≥ s_α(n).
Let
m = max{i ∈{1, …, n} | s_α(i) = s_α (1)},
and set τ = {u_α(1), …, u_α(m)}. Let ϕ be the unique order-isomorphism
{α(m+1), …, α(n)}→{1, …, n-m},
and define θ∈ S_n-m by
θ(i) = ϕ(α(m+i)). Set
t_θ(i) = (m+i)s_α(m+i) + ∑_j = m+i+1^ns_α(j).
Then
0 ≤ t_θ(n-m)≤⋯≤ t_θ(1)≤∑_j = 1^n s_α(j) = 1.
Hence (t_1, …, t_n-m) ∈Δ_θ⊆ [0,1]^n-m. We set
g(x) = [(τ,σ),(t_1, …, t_n-m)].
Using Lemma <ref>, one easily checks that g(x) does not depend on the choice of the permutation α. We have thus defined a map g |K| → |P|. Tedious but rather straightforward computations now show that f∘ g = id_|K| and g∘ f = id_|P|. Since f is a continuous map between compact Hausdorff spaces, it follows that f is a homeomorphism.
§ THE HDA
A higher-dimensional automaton (HDA) is a tuple = (P_,I_,Σ_, λ_) where P_ is a precubical set, I_∈ (P_)_0 is a vertex, called the initial state, Σ_ is a finite set of labels, and λ_ (P_)_1 →Σ_ is a map, called the labeling function, such that λ_ (d_i^0x) = λ_ (d_i^1x) for all x ∈ (P_)_2 and i ∈{1,2} <cit.>. The vertices of an HDA are also called its states.
Originally, an HDA is also equipped with a set of final states, but since we will never need final states, we omit this part of the structure. HDAs form a category in which a morphism from an HDA to an HDA is a pair (f,g) consisting of a morphism of precubical sets f P_→ P_ and a map g Σ_→Σ_ such that f(I_) = I_ and λ_(f(x)) = g(λ_(x)) for all x ∈ (P_)_1.
We turn the precubical set P defined in the previous section into an HDA by setting P_ = P, I_ = ({1},{1}), Σ_ = {1, … , N}, and λ_(τ, σ) = a
for (τ, σ) ∈ P_1 with σ∖τ = {a}. This is indeed an HDA since for (τ, σ)∈ P_2 with σ∖τ = {a < b},
λ_(d^0_1(τ, σ)) = λ_(τ, τ∪{b}) = b = λ_(τ∪{a}, σ) = λ_(d^1_1(τ, σ))
and
λ_(d^0_2(τ, σ)) = λ_(τ, τ∪{a}) = a = λ_(τ∪{b}, σ) = λ_(d^1_2(τ, σ)).
An HDA is said to be deterministic if no two edges with the same source have the same label. An HDA is said to be codeterministic if no two edges with the same target have the same label. We say that an HDA is bideterministic if it is both deterministic and codeterministic.
The HDA is bideterministic.
Let (τ, τ) be a vertex of , and let (τ, σ) and (τ, ρ) be two edges with the same label starting in (τ, τ). Suppose that σ∖τ = {a} and that ρ∖τ = {b}. Then a = λ_(τ, σ) = λ _(τ, ρ) = b. Hence σ = τ∪{a} = τ∪{b} = ρ, and so the two edges are the same. Thus, is deterministic. A similar argument shows that is codeterministic.
§.§ HDA models
An HDA is extensional if no two edges with the same endpoints have the same label. If an HDA is deterministic or codeterministic, it is extensional. A transition system is a 1-truncated extensional HDA, i.e., an extensional HDA concentrated in degrees ≤ 1.
Let be a transition system, and let R be a relation on the alphabet Σ_. The HDA model of with respect to R is the by <cit.> up to isomorphism uniquely determined HDA 𝒬 which satisfies the following conditions:
HM1 The 1-skeleton of , _≤ 1 = ((P_)_≤ 1, I_, Σ_, λ_), is .
HM2 For all x ∈ (P_𝒬)_2, λ_𝒬(d^0_2x) R λ_𝒬(d^0_1x).
HM3 For all m≥ 2 and x,y ∈ (P_𝒬)_m, if d^k_rx = d^k_ry for all r ∈{1,… ,m} and k ∈{0,1}, then x = y.
HM4 𝒬 is maximal with respect to the properties HM1–HM3, i.e., there is no HDA satisfying HM1–HM3 with P_⫋ P_.
Condition HM1 says that is built on top of by filling in empty cubes. By condition HM2, an empty square may only be filled in if the labels of its edges are related. Condition HM3 ensures that no empty cube is filled in twice in the same way. By condition HM4, all admissible empty cubes are filled in.
is the HDA model of its 1-skeleton _≤ 1 with respect to <.
Since is deterministic, _≤ 1 is indeed a transition system.
HM1 is obvious.
HM2: Let (τ, σ) be a 2-cube of . Suppose that σ∖τ = {a < b}. We have
λ_(d^0_2(τ, σ)) = λ_(τ,τ∪{a}) = a < b = λ_(τ,τ∪{b}) = λ_(d^0_1(τ, σ)).
HM3: Let m ≥ 2, and let (τ, σ), (τ', σ')∈ P_m such that d^k_r(τ, σ) = d^k_r(τ',σ') for all r ∈{1,… ,m} and k ∈{0,1}. Suppose that σ∖τ = {w_1 < … < w_m} and that σ' ∖τ' = {w_1' < … < w_m'}. Since d^0_1(τ, σ) = d^0_1(τ',σ'), we have
(τ, σ∖{w_1}) = (τ', σ' ∖{w_1'}) and therefore τ = τ'. Since d^1_1(τ, σ) = d^1_1(τ',σ'), we have
(τ∪{w_1}, σ ) = (τ' ∪{w_1'}, σ' ) and therefore σ = σ'. Thus, (τ, σ) = (τ', σ').
HM4: Let be an HDA satisfying HM1–HM3 with respect to _≤ 1 and < such that P_ = P is a precubical subset of P_. By HM1, (P_)_m = P_m for m ≤ 1. Let m ≥ 2, and suppose inductively that (P_)_m-1 = P_m-1. Let x ∈ (P_)_m. By the inductive hypothesis, d^k_ix ∈ P_m-1 for all i and k. Write d^k_ix = (τ^k_i, σ^k_i) and σ^k_i ∖τ^k_i = {w^k_i,1 < … < w^k_i,m-1}. Consider i ∈{1, …, m} and j ∈{1, …, m-1}. If m = 2, we have
w^0_i,j = w^0_i,1 = λ_(τ^0_i,τ^0_i∪{w^0_i,1}) = λ_(τ^0_i,σ^0_i) =
λ_(d^0_ix)
= λ_(d^1_ix) = λ_(τ^1_i, σ^1_i) = λ_(τ^1_i, τ^1_i ∪{w^1_i,1}) = w^1_i,1 = w^1_i,j.
If m ≥ 3, we have
λ_(d^0_1⋯ d^0_j-1d^0_j+1⋯ d^0_m-1d^0_ix)
= λ_(d^0_1⋯ d^0_j-1d^0_j+1⋯ d^0_m-1(τ^0_i,τ^0_i∪{w^0_i,1< … < w^0_i,m-1}))
= λ_(d^0_1⋯ d^0_j-1d^0_j+1⋯ d^0_m-2(τ^0_i,τ^0_i∪{w^0_i,1 < … < w^0_i,m-2}))
= ⋯
= λ_(d^0_1⋯ d^0_j-1d^0_j+1 (τ^0_i,τ^0_i∪{w^0_i,1 < … < w^0_i,j-1< w^0_i,j< w^0_i,j+1}))
= λ_(d^0_1⋯ d^0_j-1 (τ^0_i,τ^0_i∪{w^0_i,1 < … < w^0_i,j-1< w^0_i,j}))
= ⋯
= λ_(τ^0_i,τ^0_i∪{w^0_i,j})
= w^0_i,j
and
λ_(d^1_1⋯ d^1_j-1d^1_j+1⋯ d^1_m-1d^1_ix)
= λ_(d^1_1⋯ d^1_j-1d^1_j+1⋯ d^1_m-1(τ^1_i,τ^1_i∪{w^1_i,1< … < w^1_i,m-1}))
= λ_(τ^1_i∪{w^1_i,1< … < w^1_i,j-1< w^1_i,j+1< ⋯ w^1_i,m-1},
τ^1_i∪{w^1_i,1< … < w^1_i,m-1})
= w^1_i,j.
Since parallel edges in an HDA have the same label (see, e.g., <cit.>, it follows that w^0_i,j = w^1_i,j in the case m ≥ 3 as well.
Let 1 ≤ i < j ≤ m. Since
d^0_id^0_jx = d^0_i(τ^0_j, σ^0_j) = (τ^0_j, σ^0_j∖{w^0_j,i})
and
d^0_j-1d^0_ix = d^0_j-1(τ^0_i, σ^0_i) = (τ^0_i, σ^0_i∖{w^0_i,j-1}),
we have τ^0_i = τ^0_j. Set τ = τ^0_i = τ^0_j. Since
d^0_id^1_jx = d^0_i(τ^1_j, σ^1_j) = (τ^1_j, σ^1_j∖{w^1_j,i})
and
d^1_j-1d^0_ix = d^1_j-1(τ^0_i, σ^0_i) = (τ^0_i ∪{w^0_i,j-1}, σ^0_i),
we have τ^1_j = τ∪{w^0_i,j-1}. Since
d^1_id^0_jx = d^1_i(τ^0_j, σ^0_j) = (τ^0_j ∪{w^0_j,i}, σ^0_j)
and
d^0_j-1d^1_ix = d^0_j-1(τ^1_i, σ^1_i) = (τ^1_i, σ^1_i ∖{w^1_i,j-1}),
we have τ^1_i = τ∪{w^0_j,i}.
Since τ^1_j = τ∪{w^0_i,j-1} for all 1≤ i < j ≤ m, we have
w^0_1,j-1 = w^0_2,j-1 = … = w^0_j-1,j-1
for all 1 < j ≤ m. Since τ^1_i = τ∪{w^0_j,i} for all 1≤ i < j ≤ m, we have
w^0_i+1,i = w^0_i+2,i = … = w^0_m,i
for all 1≤ i < m. Since τ∪{w^0_i+1,i} = τ^1_i = τ∪{w^0_i-1,i-1} for all 1 < i < m, we have
w^0_1,i-1 = w^0_2,i-1 = … = w^0_i-1,i-1 = w^0_i+1,i = w^0_i+2,i = … = w^0_m,i
for all 1 < i < m.
Set
w_i =
w^0_i+1,i, 1 ≤ i < m,
w^0_1, m-1, i = m.
Then w_1 < … < w_m. Indeed, if m = 2, since satisfies HM1 and HM2,
w_1 = w^0_2,1 = λ_(τ^0_2, τ^0_2 ∪{w^0_2,1}) = λ_(d^0_2x) = λ_(d^0_2x)
< λ_(d^0_1x) = λ_(d^0_1x) = λ_(τ^0_1, τ^0_1 ∪{w^0_1,1}) = w^0_1,1 = w_2.
If m≥ 3, we have
w_m-1 = w^0_m,m-1 = w^0_m-2,m-2 < w^0_m-2,m-1 = w^0_1,m-1 = w_m
and
w_i = w^0_i+1, i = w^0_i+2,i < w^0_i+2,i+1 = w_i+1
for 1≤ i < m-1.
We have
d^0_m(τ, τ∪{w_1 < … < w_m})
= (τ, τ∪{w_1 < … < w_m-1})
= (τ, τ∪{w^0_2,1 < … < w^0_m, m-1})
= (τ, τ∪{w^0_m,1 < … < w^0_m, m-1})
= (τ^0_m, σ^0_m)
= d^0_mx
and
d^1_m(τ, τ∪{w_1 < … < w_m})
= (τ∪{w_m}, τ∪{w_1 < … < w_m})
= (τ∪{w_m}, τ∪{w_m}∪{w_1 < … < w_m-1})
= (τ∪{w^0_1,m-1}, τ∪{w^0_1,m-1}∪{w^0_2,1 < … < w^0_m,m-1})
= (τ^1_m, τ^1_m ∪{w^0_m,1 < … < w^0_m,m-1})
= (τ^1_m, τ^1_m ∪{w^1_m,1 < … < w^1_m,m-1})
= (τ^1_m, σ^1_m)
= d^1_mx.
For 1 ≤ i < m, we have
d^0_i(τ, τ∪{w_1 < … < w_m})
= (τ, τ∪{w_1 < … w_i-1 < w_i+1 < … < w_m-1 < w_m})
= (τ, τ∪{w^0_2,1 < … < w^0_i,i-1 < w^0_i+2,i+1 < … < w^0_m, m-1 < w^0_1,m-1})
= (τ, τ∪{w^0_i,1 < … < w^0_i,i-1 < w^0_i,i < … < w^0_i, m-2 < w^0_i,m-1})
= (τ^0_i, σ^0_i)
= d^0_ix
and
d^1_i(τ, τ∪{w_1 < … < w_m})
= (τ∪{w_i}, τ∪{w_1 < … < w_m})
= (τ∪{w^0_i+1,i}, τ∪{w^0_2,1 < … < w^0_m, m-1 < w^0_1,m-1})
= (τ∪{w^0_i+1,i}, τ∪{w^0_i+1,i}
∪{w^0_2,1 < … < w^0_i,i-1 < w^0_i+2,i+1 < … < w^0_m, m-1 < w^0_1,m-1})
= (τ∪{w^0_i+1,i}, τ∪{w^0_i+1,i}
∪{w^0_i,1 < … < w^0_i,i-1 < w^0_i,i < … < w^0_i, m-2 < w^0_i,m-1})
= (τ^1_i, τ^1_i∪{w^1_i,1 < … < w^1_i,i-1 < w^1_i,i < … < w^1_i, m-2 < w^1_i,m-1})
= (τ^1_i, σ^1_i)
= d^1_ix.
Since satisfies HM3, it follows that x = (τ, τ∪{w_1 < … < w_m}) ∈ P_m. Thus, (P_)_m = P_m.
§ ACCESSIBILITY
A state v in HDA is said to be reachable if there exists a path, i.e., a sequence of consecutive edges, from the initial state to v. An HDA in which all states are reachable is called accessible. Unreachable states are of very limited interest for the analysis of concurrent systems, since the executions of a system only pass through reachable states. Therefore, it makes sense to model only the accessible part of the state space of a system. Another important reason for doing so is the state explosion problem: the state space of a concurrent system can easily become very large, and including unreachable states in the model would dramatically aggravate this problem. Unfortunately, the HDA defined in the previous section is highly inaccessible. In this section, we show that it is possible to modify to obtain an accessible HDA of the same homotopy type. More precisely, we prove the following theorem:
Let be a bideterministic HDA which is the HDA model of its 1-skeleton with respect to a strict total order on Σ_. Suppose that is connected, i.e., |P_| is path-connected, and that has only a finite number of unreachable states (e.g., is finite). Then there exists an accessible and bideterministic HDA which is the HDA model of its 1skeleton with respect to a strict total order on Σ_ and satisfies |P_| ≃ |P_|.
For the proof, we may suppose that is not accessible. Clearly, it is enough to show that there exists a bideterministic HDA with less unreachable states than that is the HDA model of its 1-skeleton with respect to a strict total order on Σ_ and satisfies |P_| ≃ |P_|. We show first that admits an edge from an unreachable to a reachable state. Suppose that there is no such edge. Let v be an unreachable state. Since is connected, there is a sequence of vertices I_ = v_1, v_2, …, v_k = v such that for each 1 ≤ i < k there exists an edge between v_i and v_i+1. Inductively, all v_i are reachable, which is impossible.
Let e be an edge of from an unreachable state v to a reachable state w. If w = I_, we define to be the same as but with I_ = v. Suppose that w ≠ I_. Let λ_(e) = a, and let ω = (x_1,…, x_k) be a path from I_ to w with no repeated vertices, e.g., a shortest possible path. We view ω as a morphism of precubical sets 0, k → P_, where the precubical interval p, q (p,q ∈, p≤ q) is the precubical set defined by p,q _0 = {p,… , q}, p,q _1 = {[p,p+1], … , [q- 1,q]}, d_1^0[j-1,j] = j-1, d_1^1[j-1,j] = j, and p,q _n = ∅ for n > 1.
§.§ The HDA
We first extend to an HDA such that |P_| ≃ |P_|. We define the precubical set P_ by the pushout diagram
0, k⊗{2}[r, "≅"] [d, hook] 0, k[r, "ω"] P_[d, hook]
-1,0⊗{1}∪ 0, k⊗ 1,2 [rr, "ξ"']
P_.
Since the geometric realizations of the precubical sets on the left are contractible and, as is well known, the geometric realization functor preserves colimits, the inclusion |P_| ↪ |P_| is a homotopy equivalence. Let Σ_ = Σ_∪{c} for some element c ∉Σ_. We extend the labeling function of to by setting λ_(ξ(i,[1,2])) = c (i ∈{0, …, k}), λ_(ξ([i-1,i],1)) = λ_(x_i) (i ∈{1, …, k}), and λ_(ξ([-1,0],1)) = a. The initial state of is I_ = ξ(-1,1).
is bideterministic and has the same unreachable states as .
Since all edges of that are not edges of start in vertices of that are not vertices of and in no such vertex start two edges with the same label, is deterministic. Since ω has no repeated vertices, no two edges of that are not edges of end in the same vertex. Since any such edge that ends in a vertex of has label c, it follows that is codeterministic.
Since I_ is reachable in , all states of that are reachable in are also reachable in . On the other hand, since any path in from I_ to a state of intersects ω, all states of that are reachable in are also reachable in . Since all states in ξ( -1,0⊗{1}∪ 0, k⊗ 1,2 ) are reachable in , it follows that has the same unreachable states as .
Let < be the strict total order on Σ_ with respect to which is the HDA model of its 1-skeleton. We extend < to a strict total order on Σ_ by setting b< c for all b∈Σ_.
is the HDA model of _≤ 1 with respect to <.
Condition HM1 is trivially satisfied. Since satisfies HM2 and, for all i ∈{1,…, k},
λ_(d^0_2ξ([i-1,i],[1,2])) = λ_(ξ(d^0_2([i-1,i],[1,2])))
= λ_(ξ([i-1,i],1))
= λ_(x_i)
< c
= λ_(ξ(i-1,[1,2]))
= λ_(ξ(d^0_1([i-1,i],[1,2])))
= λ_(d^0_1ξ([i-1,i],[1,2])),
satisfies HM2. Let x,y ∈ (P_)_m (m≥ 2) such that d^k_rx = d^k_ry for all r ∈{1,… ,m} and k ∈{0,1}. Since satisfies HM3, x = y if x, y ∈ (P_)_m. If x ∉ (P_)_m, then m = 2 and x = ξ([i-1,i],[1,2]) for some i ∈{1, …, k}. Since x is the only 2-cube of having ξ([i-1,i],1) in its boundary, y = x. Hence satisfies HM3. Let be an HDA with P_⊆ P_ that satisfies HM1–HM3. Since is the HDA model of _≤ 1, all m-cubes of (m≥ 2) with faces in belong to . Let x ∈ (P_)_2 such that at least one edge of x does not belong to . Since every edge that starts in a vertex of belongs to , d^0_1d^0_2x is not a vertex of . Hence d^0_1d^0_2x = ξ(i,1) for some i ∈{-1,…, k}. Since λ_(d^0_2x)< λ_(d^0_1x), we have d^0_1x ≠ d^0_2x and therefore 0 ≤ i < k. Since λ_(ξ([i,i+1],1)) < λ_ (ξ(i,[1,2])) = c, we have d^0_1x = ξ(i,[1,2]) and d^0_2x = ξ([i,i+1],1). Since λ_(d^1_1x) = λ_(d^0_1x) = c, we have d^1_1x = ξ(i+1,[1,2]). Since, by Lemma <ref>, _≤ 1 = _≤ 1 is deterministic, d^1_2x = x_i+1 because d^1_2x starts in ξ(i,2) = d^0_1x_i+1 and λ_(d^1_2x) = λ_(d^0_2x) = λ_(ξ([i,i+1],1)) = λ_(x_i+1). Since satisfies HM3, x = ξ([i,i+1],[1,2]) ∈ (P_)_2. Suppose that there exist an integer m≥ 3 and an element y ∈ (P_)_m such that at least one face of y does not belong to . Then d^0_1⋯ d^0_my is not a vertex of . Hence d^0_1⋯ d^0_my = ξ(i,1) for some i. Since, by <cit.>, for all 1 ≤ i < j ≤ m,
λ_(d^0_1⋯ d^0_i-1d^0_i+1⋯ d^0_mx) < λ_(d^0_1⋯ d^0_j-1d^0_j+1⋯ d^0_mx),
m different edges start in ξ(i,1). This is not the case. Thus, = and satisfies HM4.
§.§ The HDA
We now extend to an HDA that still satisfies |P_| ≃ |P_| but in which v is reachable. Let i be the largest index in {0, …, k} such that λ_(ξ([i-1,i],1)) = a. Since is codeterministic, λ_(ξ([k-1,k],1)) = λ_(x_k) ≠ a. Hence i < k. We define the precubical set P_ by the pushout diagram
i, k+1⊗{1}∪{i,k+1}⊗ 0,1[r, "ν"] [d, hook] P_[d, hook]
i, k+1⊗ 0,1 [r,"χ"']
P_
where ν is the unique morphism of precubical sets such that ν([j-1,j],1) = ξ([j-1,j],1) (i < j ≤ k), ν([k,k+1],1) = ξ(k,[1,2]), ν(i,[0,1]) = ξ([i-1,i],1), and ν(k+1,[0,1]) = e. Note that ν is injective. Since the geometric realizations of the precubical sets on the left are contractible, the inclusion |P_| ↪ |P_| is a homotopy equivalence. Hence |P_| ≃ |P_|. We set I_ = I_ and Σ_ = Σ_ and extend the labeling function of to by setting λ_(χ(j,[0,1])) = a, λ_(χ([j-1,j],0)) = λ_(x_j) (i < j ≤ k), and λ_(χ([k,k+1],0)) = c.
is bideterministic and has less unreachable states than .
In each vertex of that is not a vertex of start exactly two edges, one with label a and the other with a different label. The only vertex of in which starts an edge of that is not an edge of is ν(i,0) = ξ(i-1,1). By definition of i, the label of this edge is different from a. Since i < k, this label is also different from c. Hence the edges starting in ν(i,0) = ξ(i-1,1) have different labels. Since, by Lemma <ref>, is deterministic, it follows that is deterministic. Since no two edges of that are not edges of end in the same vertex, no edge in with label c ends in v, and the edges ξ ([i,i+1],1), …, ξ ([k-1,k],1) have labels different from a, is codeterministic.
All states of that are reachable in are also reachable in . Since all states in χ( i, k+1⊗ 0,1 ) are reachable in and, in particular, v = χ(k+1,0) is reachable in , the number of unreachable states of is less than the number of unreachable states of and hence, by Lemma <ref>, of .
§.§ The HDA
Unfortunately, we cannot guarantee that is the HDA model of its skeleton, because the labels of the edges of the squares added to might be related in the wrong way. In the final HDA , we solve this problem. We set (P_)_m = (P_)_m for all m and define the face maps of P_ by
∂ ^k_i x = {[ d^k_3-ix, x ∈ (P_)_2 ∖ (P_)_2, λ_(d^0_1x) < λ_(d^0_2x),; d^k_ix, . ].
Then P_ is a precubical set and |P_| ≈ |P_|. Hence |P_| ≃ |P_|. We set I_ = I_ = I_, Σ_ = Σ_ = Σ_, and λ_ = λ_. Then is an HDA with _≤ 1 = _≤ 1. By Lemma <ref>, is bideterministic and has less unreachable states than . To finish the proof of Theorem <ref>, it remains to show that is the HDA model of its 1-skeleton. This is done in Proposition <ref> below.
Let be an HDA which satisfies HM2 with respect to _≤ 1 and a strict total order on Σ_, and let x be an element of P_ of degree m≥ 3. Then d^k_px ≠ d^l_qx for all 1 ≤ p < q ≤ m and k, l ∈{0,1}.
Suppose that d^k_px = d^l_qx for some 1 ≤ p < q ≤ m and k, l ∈{0,1}. Then
d^0_1 ⋯ d^0_p-1d^k_pd^0_p+2⋯ d^0_mx = d^0_1 ⋯ d^0_p-1d^0_p+1⋯ d^0_m-1d^k_px
= d^0_1 ⋯ d^0_p-1d^0_p+1⋯ d^0_m-1d^l_qx
= d^0_1 ⋯ d^0_p-1d^0_p+1⋯ d^0_q-1d^l_qd^0_q+1⋯ d^0_mx.
By the arguments given in <cit.>, it follows that
λ_(d^0_1 ⋯ d^0_pd^0_p+2⋯ d^0_mx) = λ_(d^0_1 ⋯ d^0_p-1d^0_p+1⋯ d^0_mx)
< λ_(d^0_1 ⋯ d^0_pd^0_p+2⋯ d^0_mx),
which is impossible.
is the HDA model of _≤ 1 with respect to <.
By construction, satisfies HM1 and HM2. HM3 can be shown in a similar way as for , see Lemma <ref>. Let be an HDA that contains and satisfies HM1–HM3 with respect to _≤ 1 and <. Since, by Lemma <ref>, is the HDA model of _≤ 1, all m-cubes of (m≥ 2) with faces in belong to . Let x ∈ (P_)_2 such that at least one edge of x does not belong to . Since every edge with endpoints in belongs to , x has a vertex that does not belong to . Since λ_(∂^1_2x)< λ_(∂^1_1x), we have ∂^1_1x ≠∂^1_2x. Since in no vertex that does not belong to ends more than one edge, it follows that ∂^1_1∂^1_2x ∈ (P_)_0. This implies that if ∂^0_1∂^0_2x ∈ (P_)_0, then ∂^0_1∂^0_2x = χ(i,0). Indeed, in this case, ∂^1_1∂^0_1x ∉ (P_)_0 or ∂^1_1∂^0_2x ∉ (P_)_0, and so ∂^0_1∂^0_2x is a vertex of in which starts an edge that ends in a vertex of (P_)_0∖ (P_)_0. The only such vertex is χ(i,0). Thus, there exists j ∈{i, …, k} such that ∂^0_1∂^0_2x = χ(j,0).
Suppose that ∂^0_1∂^0_2x = χ(j,0) with i < j ≤ k. Since λ_(∂^0_2x) < λ_(∂^0_1x), we have ∂^0_1x ≠∂^0_2x. Therefore there exists r ∈{1,2} such that ∂^0_rx = χ(j,[0,1]) and ∂^0_3-rx = χ([j,j+1],0). Since λ_(∂^1_rx) = λ_(∂^0_rx) = a, we have ∂^1_rx = χ(j+1,[0,1]). Since _≤ 1 = _≤ 1 is deterministic, we have ∂^1_3-rx = χ([j,j+1], 1) because ∂^1_3-rx starts in χ(j,1) = ∂^0_1χ([j,j+1], 1) and λ_(∂^1_3-rx) = λ_(∂^0_3-rx) = λ_(χ([j,j+1],0)) = λ_(χ([j,j+1],1)).
Since satisfies HM3 and < is asymmetric, it follows that x = χ([j,j+1],[0,1]) ∈ (P_)_2.
Suppose now that ∂^0_1∂^0_2x = χ(i,0). Since λ_(∂^0_2x)< λ_(∂^0_1x), we have ∂^0_1x ≠∂^0_2x. The edges starting at χ(i,0) are χ(i,[0,1]) and χ([i,i+1],0), and ξ(i-1, [1,2]) when i > 0. Since all edges starting at the endpoints of χ(i,[0,1]) and ξ(i-1, [1,2]) are edges of , there exists r ∈{1,2} such that ∂^0_rx is χ(i,[0,1]) or ξ(i-1, [1,2])}, and ∂^0_3-rx = χ([i,i+1],0). Since all edges starting at the endpoints of χ(i,[0,1]) and ξ(i-1, [1,2]) end in reachable vertices of and there exists no edge starting in a reachable vertex of and ending in χ(i+2, 0), we have ∂^1_1∂^1_2x = χ(i+1,1), ∂^1_rx= χ(i+1,[0,1]), ∂^0_rx= χ(i,[0,1]), and ∂^1_3-rx = χ([i,i+1],1). Since satisfies HM3 and < is asymmetric, it follows that x = χ([i,i+1],[0,1]) ∈ (P_)_2.
Suppose that there exists an element y ∈ (P_)_3 such that at least one face of y, say ∂^l_jy, does not belong to . Then ∂^l_jy = χ([r,r+1],[0,1]) for some i ≤ r ≤ k. Since, for some s ∈{1,2},
χ([r,r+1],0) = ∂^0_sχ([r,r+1],[0,1]) = ∂^0_s∂^l_jy = ∂^l_j-1∂^0_sy, s <j,
∂^l_j∂^0_s+1y, s ≥ j,
Lemma <ref> implies that (χ([r,r+1],0) is an edge of two distinct faces of y. Since χ([r,r+1],[0,1]) is the only 2-cube of having (χ([r,r+1],0) as an edge, this is impossible.
A simple induction now shows that (P_)_m = (P_)_m for all m ≥ 3. It follows that satisfies HM4.
§ SHARED-VARIABLE SYSTEMS
In this section, we consider shared-variable systems given by program graphs and establish our main result:
There exists a shared-variable system such that the geometric realization of its HDA model has the homotopy type of the polyhedron |K|.
§.§ Program graphs and shared-variable systems
Let V be a set of variables. The domain of a variable x, i.e., the set of its possible values, will be denoted by D_x. A program graph over V is a tuple
(L, A, T, g,
)
where L is a set of locations or local states, A is a finite set of actions, i.e., functions ∏_x ∈ V D_x →∏_x ∈ V D_x, T⊆ L × A × L is a set of transitions, g is a function that specifies a guard condition, i.e., a subset of ∏_x ∈ V D_x, for each transition, and ∈ L is an initial location (cf. <cit.>). A shared-variable system over V is a tuple (_1, …, _n, η) consisting of program graphs _i and an initial evaluation η∈∏_x ∈ V D_x.
§.§ The HDA model of a shared-variable system
Consider a shared-variable system (_1, …, _n, η) over a set of variables V, and write
_i = (L_i, A_i, T_i, g_i, _i). The state graph of (_1, …, _n, η) is the 1-truncated precubical set Q where
Q_0 = L_1 ×⋯× L_n ×∏_x ∈ V D_x,
Q_1 = ⋃_i∈{1,…,n}
t ∈ T_i L_1 ×⋯× L_i-1×{t}× L_i+1×⋯× L_n × g_i(t),
and for y = (l_1,…, l_i-1,t,l_i+1, …, l_n,γ)∈ Q_1 with t = (l^0_t, a_t, l^1_t),
d^0_1y = (l_1,…, l_i-1,l^0_t,l_i+1, …, l_n,γ)
and
d^1_1y = (l_1,…, l_i-1,l^1_t,l_i+1, …, l_n,a_t(γ)).
The initial state of the system is the state
I = (_1, … , _n, η).
The transition system model of (_1, …, _n, η) is the transition system where P_ is the largest precubical subset of Q such that all states are reachable from the initial state I, I_ = I, Σ_ = ⋃_i= 1^n {i}× A_i, and the label of an edge y = (l_1,…, l_i-1,t,l_i+1, …, l_n,γ) ∈ (P_)_1 with t = (l^0_t, a_t, l^1_t) is given by λ_(y) = (i,a_t). The HDA model of (_1, …, _n, η) is the HDA model of with respect to the relation R on Σ_ given by
(i,a) R (j,b) i < j.
In practice, the transition system model of a shared-variable system can be constructed without handling unreachable states using a procedure such as the one described in the reference manual of the Spin model checker <cit.>. HDA models of shared-variable systems written in Promela, the process description language of Spin, can be computed using the tool pg2hda <cit.>.
§.§ Proof of Theorem <ref>
Since K is connected and |K| ≈ |P| = |P_| by Theorem <ref>, the HDA is also connected. By Proposition <ref> and Theorems <ref> and <ref>, there exists an accessible and deterministic HDA which is the HDA model of its 1-skeleton with respect to a total order on Σ_ and satisfies |P_| ≃ |P_| ≈ |K|.
Suppose that Σ_ = {a_1 < … < a_n}. Consider a single variable x with domain D_x = (P_)_0, and let (_1, …, _n, η) be the shared-variable system over V = {x} where η = I_ and the program graphs _i = (L_i,A_i, T_i, g_i, _i) are defined by
* L_i = {0};
* A_i = {a̅_i} where a̅_i(v) = {[ d^1_1y, ∃ y ∈ (P_)_1 : d^0_1y = v, λ_(y) = a_i,; v, ].
* T_i = {(0,a̅_i,0)};
* g_i(0,a̅_i,0) = {d^0_1y | y ∈ (P_)_1, λ_(y) = a_i};
* _i = 0.
Since is deterministic, the action a̅_i is well defined. Let Q be the state graph of (_1, …, _n, η). We have
(P_)_0 ≅ L_1 ×…× L_n × D_x = Q_0.
Since is deterministic, the map
d^0_1{y ∈ (P_)_1 | λ_(y) = a_i} →{d^0_1y | y ∈ (P_)_1, λ_(y) = a_i} = g_i(0,a̅_i,0)
is a bijection for each i. Hence
(P_)_1 = ⋃_i∈{1,…,n}{y ∈ (P_)_1 | λ_(y) = a_i}
≅⋃_i∈{1,…,n} L_1 ×⋯× L_i-1×{(0,a̅_i,0)}× L_i+1×⋯× L_n × g_i(0,a̅_i, 0)
= Q_1.
Since for y∈ (P_)_1 with λ_(y) = a_i we have
d^0_1(0, …, 0, (0,a̅_i,0), 0, …, 0, d^0_1y) = (0, …, 0, d^0_1y)
and
d^1_1(0, …, 0, (0,a̅_i,0), 0, …, 0, d^0_1y) = (0, …, 0, a̅_i(d^0_1y)) = (0, …, 0, d^1_1y),
the precubical sets (P_)_≤ 1 and Q are isomorphic.
Let be the transition system model of (_1, …, _n, η). Since the initial state of the system, I = (0, …, 0, I_), corresponds to I_ under the isomorphism (P_)_≤ 1≅ Q and is accessible, all states of Q are reachable from I. Hence P_ = Q. We have
Σ_ = {a_1, …, a_n}≅{(1, a̅_1), … , (n,a̅_n)} = Σ_.
Since for an edge y ∈ (P_)_1 with λ_(y) = a_i we have
λ_(0,…, 0, (0, a̅_i, 0),0, …, 0, d^0_1y) = (i, a̅_i),
it follows that the transition systems _≤ 1 and are isomorphic.
Let be the HDA model of (_1, …, _n, η). Then is the HDA model of with respect to the relation
R on Σ_ given by
(i,a̅_i) R (j,a̅_j) i < j a_i < a_j.
By <cit.>, it follows that the HDAs and are isomorphic. In particular, |P_| ≈ |P_| ≃ |K|. 0pt
apalike
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